iJ/ 
 
 ^ 
 
 fe^ 
 
 i^\^ 
 ^>^%^^^ 
 
 lb. A/ 
 
 'v^i *s%^ ^^* 
 
 L 
 
 ^O0.ii\<sy 
 
 vl 
 
 <? 
 
 /^ 
 
 / 
 
 c"! 
 
 > 
 
 ■> ■> 
 
 
 ■V^ 
 
 V 
 
 /; 
 
 0^'. 
 
 A 
 
 
 Hiotpgraphic 
 
 Sciences 
 Corporation 
 
 J 
 
 
 23 WEST MAIN STREET 
 
 WEBSTER, NY. 145S0 
 
 (716) •72-4S03 
 
 ^^>*' 
 
 V 
 
 V 
 
€// 
 
 
 fe 
 
 
 ^OaiiliMi'!! 
 
 fc 
 
 I 
 
 i- 
 
 
 I 
 
i! 
 
 Technical and Bibliographic N« 
 
 The Institute has attempted to obtain the best 
 original copy available for filming. Features of ithis 
 copy which may be bibllographically unique, 
 which may alter any of the images in the 
 reproduction, or which may significantly chamge 
 the usual method of filming, are checked below. 
 
 V; 
 
 ty ♦ >^,<.i ■ , , , Ci MM ;..Ia J ."LliniU? 
 
 1 >. i/l* A 
 
 D 
 
 Coloured covers/ 
 Couverture de couieur 
 
 I I Covers damaged/ 
 
 Couverture endommagde 
 
 Covers restored and/or laminated/ 
 Couverture restaurde et/ou pellicul6e 
 
 I I Cover title missing/ 
 
 Le titre de couverture manque 
 
 □ Coloured maps/ 
 Cartes g6ographiques en couieur 
 
 □ Coloured ink (i.e. other than blue or black)/ 
 Encre de couieur (i.e. autre que bleue ou noi 
 
 I I Coloured plates and/or illustrations/ 
 
 D 
 D 
 
 D 
 
 D 
 
 Planches et/ou illustrations en couieur 
 
 Bound with other material/ 
 Reli6 avec d'autres documents 
 
 Tight binding may cause shadows or distort! 
 along interior margin/ 
 
 Lareliure serrde peut causer de I'ombre ou c 
 distortion le long de la marge intdrieure 
 
 Blank leaves added during restoration may 
 appear within the text. Whenever possible, 1 
 have been omitted from filming/ 
 II se peut que certalnes pages blanches ajou 
 lors d'une restauration apparaissent dans le 
 mais, lorsque cela 6tait possible, ces pag«)s 
 pas 6ti fllm6es. 
 
 Additional comments:/ 
 Commentaires suppl6mentaires; 
 
 />V'/ v'Jy 'fv 
 
 I / " f 
 
 r.'Onno 
 
 • 4.1 \^\^t. \t' 
 
 r,'.. J/ . 
 
 . J 
 
 va CO SOVH t n I r(:j'nrT«n 
 
 This item is filmed at the reduction ratio checked below/ 
 
 Ce document est film* au taux de reduction indiqud ci-dessous. 
 
 10X 
 
 
 
 
 14X 
 
 
 
 
 18X 
 
 
 
 
 22X 
 
 
 
 
 26X 
 
 
 
 
 30X 
 
 
 
 
 
 
 
 
 
 
 y 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 12X 
 
 
 
 
 HX 
 
 
 
 
 20X 
 
 
 
 
 24X 
 
 
 
 
 28X 
 
 
 
 
 aax 
 
I 
 
 "■s'ti 
 
 
 I / « ( 
 
 oe 
 
 rio 
 
 ar 
 nd 
 
 9P( 
 
 ve 
 oi 
 
 m 
 
 ne 
 
 illi 
 P 
 
 R 
 
 • «.i «^*^., \>' 
 
 r/.. Ji . 
 
 88 
 
 (r 
 
 la 
 10 
 
 >8l 
 
 ai 
 Hi 
 
 rai 
 
 
 
 
 4 
 
 5 
 
 6 
 
 % 
 
11 
 
 si 
 
 e 
 
 j; 
 
 t 
 
 a 
 
 ^. 
 
I 
 
I 
 
 cai 
 
 Tg 
 
 KL 
 
I 
 
 c 
 
 5 
 
 t; 
 
 ■JA 
 
 m 
 
?et 
 
 vs] 
 .t 
 go 
 ho 
 rr( 
 A 
 
 : MAC'* -i ,.'./. L-T ^\':-u ; , f, 
 
 rej 
 
 ill. 
 
 m 
 
 hi 
 
 
 
 HN 
 
I 
 
 111 
 
 P' 
 t 
 
 T 
 
 1 
 
 y 
 
 lO 
 
 )\ 
 ti 
 
 31' 
 IV 
 
 ;li 
 •e 
 d 
 
 I 
 
 S( 
 1! 
 
 t\':\. ll<',\i . 
 
 .. - . \- 
 
 bor 
 
 r. 
 
 r 
 1 
 
VI 
 
 ' . 1 1-, 
 
 power tc 
 form, A] 
 mental ( 
 
 To li( 
 "imriva 
 Public. !^ 
 line wit 
 numeric 
 " Psych 
 one of 
 teaclieri 
 than ai 
 Public 
 request 
 book o^ 
 Psycho 
 book IT 
 
 1. T 
 of num 
 makes 
 being 
 whole, 
 this s] 
 arithm 
 a true 
 num be 
 and re 
 tion; 
 of me] 
 
 ^ * 
 
 establ 
 the ui 
 Gondii 
 
 I'/ ...V 
 
 (. 11' 
 
 '. -IS n i.i\: ,\[ 
 
 ar 
 
 ) J ' .■) .yi\ I 
 
 .,<• , '.,. ■' .. * 
 
 . <H/ J. «jy v.v^v^ I, ^, -^ 
 
I 
 
 tl 
 
 '""i itr 
 
 ■a) 
 •se 
 ib( 
 
 y i 
 
 al 
 
 r»r 
 C 
 
 d 
 
 t 
 n 
 a 
 
 !a 
 
 i-a 
 
 3 
 
 a] 
 111 
 r( 
 
 g 
 hi 
 
 ct 
 
 111 
 
 » 
 
 s 
 
 s1 
 
 L'l 
 ( 
 
rf»- a^ r a r« i Mr .*i». » ». » ii g' . M ^ • 
 
 I 
 
 II 
 
 VIU 
 
 7. This 
 for algeb • 
 the syiubr 
 a higher c 
 and o\)i)Y\\ 
 his plight 
 algebra ? 
 
 He pa; 
 abstractii) 
 lay a foui 
 often thi^ 
 and all at 
 abstracti . 
 On the o; 
 ing of nv ] 
 a transit 
 building 
 power, a 
 Intelligei 
 
 8. Th: 
 work ; a 
 teachers 
 appearai i 
 
 9. Thi 
 book in 
 Meantin 
 book is 1) 
 
 We aii 
 suggesti 3 
 
 It is I ; 
 omitted 1 
 
 JuNii;, I 
 
 J J vr jn: 
 
 • I 
 
 ■ r .1 -] J I'l >. ..;i >i\ . { ,^\)(~ /"», 
 
 or . « ri;: \\\c X ji 1, 1 .- M, I-..- \ -j 
 
 r < of 
 
 iit 
 ? 
 
 mr 
 i6- 
 
 Is 
 
I 
 
 it 
 
 r 
 c 
 
CONTENTS 
 
 CIIAPTKR X 
 
 Fractions 
 
 CIiAPTER XI 
 
 Decimals . 
 
 CIIAPTER XII 
 
 Compound Quantities 
 
 157 
 
 CIIAPTER XIII 
 
 Percentage 
 
 07 
 
 ^ 
 
 CIIAPTER XIV 
 
 Interest 
 
 • ••••• 
 
 CIIAPTER XV 
 
 w 
 as 
 
 Ratio and Proportion 
 
 • • • 
 
 CIIAPTER XVI 
 
 Powers and Roots 
 
 • • 
 
 CIIAPTER XVII 
 
 ^Iensuration . 
 
 • • 
 
 CHAPTER XVIII 
 
 Metric System 
 
 • • • 
 
 CIIAPTER XIX 
 
 Miscellaneous Exercise 
 
 • • • 
 
] 
 
 57 
 
 •)7 
 
 
 CHAPTER I 
 
 
 ^^ 1 1 nT'mTTm-roiO'Q 
 
 
 i. numbers p 
 
 ft. 
 
 resfarded a^ 
 
 with 
 
 )V measure 
 
 as fl, 
 
 jill, 1 in.. 
 
 1 three-ii 
 
 e, 1 doz. er 
 
 3, Numbe. the repetit^ 
 ntity, Le. number de^ 
 V determining 1w 
 
 '^ number 
 '^e qur 
 t^ 
 
 6 
 
 i; 
 t, 
 
 n 
 1- 
 
 le 
 le 
 of 
 n- 
 te 
 
 te 
 
 re 
 en 
 
2 ARITHMETIC 
 
 repeated 8 times to measure tlie length of the room, and 
 the ratio of the lenoth of the room to the unit of measure is 
 represented by the number 8. 
 
 If a box has been found to contain 30 doz. of eggs, the 
 unit, 1 doz. eggs, lias been counted 30 times to measure the 
 quantity of eggs. 
 
 If I buy 3 qt. of milk, the milkman fills and empties the 
 unit of capacity, his one-quart measure, 3 times in order to 
 measure the quantity of milk. 
 
 5. In the above instances the wholes to be measured are : 
 the length of the ro^m, the eggs in the box, and the milk 
 sold. The units of measure are 2 ft., 1 doz., and 1 qt. The 
 numbers telling how many of these units are needed are 
 8, 30, and 3. 
 
 The unit of measure may itself be, and in exact computa- 
 tion is, measured or compared with some other unit, for con- 
 venience called the j^fi^^M^'i/ mdt ; that is, it may be a part or 
 a multiple of such unit. 
 
 si 
 
 f 
 
 •II 
 
 "ii 
 
 6. f 1, 1 in., 1 11)., are all primary units. A five-dollar 
 bill, a two-pound weight, a four-inch measure, are examples 
 of derived units, because the prlmari/ units, Ji>l, 1 lb., 1 in., 
 are repeated 5, 2, and 4 times respectively, to give the derived 
 units. 
 
 If a quantity of chestnuts be counted into groups of 4 or 
 5, the derived units, 4 or 5 chestnuts, and tlie number of 
 these groups or derived units measures the whole quantity 
 of chestnuts. Similarly, eggs counted by 4's and 6's, and 
 apples by 3's or 5's, are examples of the use of the derived 
 unit. The primary units are 1 egg and 1 apple, while 4 eggs, 
 6 eggs, 3 apples, and 5 ap[)les are derived units. 
 
DEFINITIONS 
 
 7. With reference to 1 ct., $ 1 is a derived unit, and 100 
 is the number expressing $ 1 in terms of the primary unit, 
 1 ct. 
 
 Similarly, 1 wk. is a derived unit, and 7 is the number 
 expressing 1 wk. in terms of the primary unit, 1 da. 
 
 In I ft., the primary unit of reference is 1 ft. The foot 
 is divided into 4 equal parts, one of which is the derived unit 
 of measure. The number 3 shows how many of these de- 
 rived units make up the given length. 
 
 I 
 
 •if. 
 
 Exercise 1 
 
 1. Kame three units of length used to measure short distances, 
 and state the number of times each unit must be repeated to make 
 the next larger. 
 
 2. What unit of length is used in stating the distance between 
 two cities ? 
 
 3. Name instances in which 1 sec. is used as the unit of time. 
 1 min. 1 hr. 1 da. 1 mo. 1 yr. State how often each of these 
 units must be repeated to make the next larger. 
 
 4. What is the prime standard unit for money value? For 
 weight? For area? For length? For time ? For volume ? 
 
 5. What unit of area is used to convey a definite idea of the 
 size of a farm ? Of a country ? 
 
 6. What unit is used to measure wood? 
 
 7. With what unit of capacity is milk measured ? Kerosene ? 
 
 8. What unit is used to measure a quantity of strawberries? 
 Potatoes ? Why are these convenient units for the purpose ? 
 
 9. State at least three reasons why the bushel would be an 
 inconvenient unit to measure straw berries. 
 
 10. Early in the season strawberries are sold in pint boxes; 
 later, in cpiart boxes. Explain why different units of capacity 
 are chosen. 
 
4 arithmp:tic 
 
 11. Why is the pint box chosen as the unit to measure red 
 raspberries in preference to the quart ? 
 
 12. Name different quantities which are weighed and sold by 
 the lb. By the oz. By the T. 
 
 13. Give instances in which the following units are used: 
 1 sheet, 1 quire, 1 doz. 
 
 14. In each of the following quantities name the units and 
 give the ratio of each quantity to its primary unit: 5 ft., 4 hr., 
 6 sq. in., 7 qt., 360 da., 12 oz. 
 
 15. What is the quantity which contains the unit 4 times, 
 when the unit is 6 in. ? 9 hr. ? 8 yr. ? 
 
 16. If the unit is f 4, and this unit is repeated 6 times, what 
 quantity will be produced ? 
 
 17. If the ratio of the size of a farm to the unit of area, 8 A., 
 is equal to 6, what is the size of the farm ? 
 
 18. In the following examples, what are the quantities which 
 contain their respective units the given number of times ? 
 
 Units of Measure 
 
 Numbers 
 
 $2 
 
 6 
 
 8qt. 
 
 2 
 
 7 da. 
 
 3 
 
 5hr. 
 
 4 
 
 1 doz. 
 
 15 
 
 4 in. 
 
 4 
 
 19. Name the primary units of measure, name ir two ways 
 the derived units, and state the number of derived units which 
 measure these quantities : $ j\, f ft., | yd., | of a dime, f of a 
 wk., f of a da., -| of a doz. eggs. 
 
 20. Name the coin which '^ives the derived unit of value in 
 each of the following : ^j%$ J, $ it, $ f J, $ 10, $ 5, $ 20, f of a 
 nickel, J dime, ^ of a quarter of a dollar, f of half a dollar, ^ of 
 half a dollar, |J uf half a dollar. 
 
DEFINITIONS 
 
 21, The following quantities contain their respective units 
 how often ? 
 
 Quantity 
 21 da. 
 24 hr. 
 1 gal. 
 1 min, 
 10 dimes 
 $ 18 worth of hats 
 30 ct. worth of milk 
 
 Unit 
 
 7 da. 
 8hr. 
 1 qt. 
 
 1 sec. 
 
 2 dimes 
 $3 for 1 hat 
 
 6 ct. a qt. 
 
 22. What is the unit of measure in reckoning population? 
 How many of these units give the population of the town in 
 which you live ? 
 
 23. What is the number of times each of the following units 
 must be repeated to make the next higher unit: 1 in., 1 ft., 
 1 ct., 1 dime, 1 da., 1 hr., 1 qt. ? 
 
 24. How many times must the following units of measure be 
 repeated to make 3 ft. : 2 in., 3 in., 4 in., G in., 9 in., 12 in. ? 
 
 25. State how each of the following units may be derived from 
 the next higher : 1 ft., 1 in., 50 ct., 25 ct., 1 da., 1 min., 1 qt., 
 and 1 pt. 
 
 26. A quantity of cherries is measured by using as the unit 
 as many cherries as will fill a dish holding 3 qt. ; 9 of these 
 dishes are filled. How many qt. are there in the whole 
 quantity ? 
 
 27. At 30 bu. to the A., how many bu. would there be on 
 10 A. ? What is the unit here ? What gives this particular 
 unit? If 10 bu. to the A., how many A. to produce an equal 
 quantity ? 
 
 28. What coin is equal in value to 10 of the unit 1 ct. ? 100 
 of the unit 1 ct. ? 10 of the unit 1 dime ? 
 
 29. What coin is equal to 10 of the unit 1 nickel? 5 of the 
 unit 1 nickel ? 20 of the same unit ? 
 
6 
 
 ARITHMETIC 
 
 30. How many ct. are there in 5 of the unit $1? 6 of the 
 unit 1 1 ? 10 of the unit 1 1 ? 
 
 31. What coin is equal in vahie to 500 oi the unit 1 ct. ? 
 100 of the unit 1 nickel ? 
 
 32. What coin is equal to one-tenth of a five-dollar gold piece ? 
 
 8. In the preceding diagram we have 36 dots, signifying 
 36 units of any kind, arranged in 4 rows of 9 dots each, and at 
 tlie same time 9 rows of 4 dots each. Hence we think of 36 
 as equal to 4 times 9 or 9 times 4. Arrange the dots to show 
 that 36 is equal to 3 x 12 or 12 x 3, and also to 2 x 18 or 18 x 2. 
 
 4 and 9 are called factors of 36, and 36 is called the 
 product of 4 and 9. 
 
 Ihis illustrates a law of great importance in Arithmetic. 
 
 Thus we think of 24 as equal to 2 x 12 or 12 x 2, 3 x 8 or 
 8x3, 4x6 or 6x4. 
 
 III 
 
 :i 
 
 9. If in the above arrangement we think of each dot as rei> 
 resenting |1, then the diagram shows that '|>12 ^ 12 = 6. 
 
 What other measurement is shown by the same arrange- 
 ment ? 
 
 Exercise 2 
 
 1. Arrange 30 dots in rows in as many ways as you can, and 
 express the results as in the preceding paragraph. 
 
 2. Express the following numbers as products in two ways : 
 6 {i.e. 2 X 3 or 3 X 2), 8, 10, 15, 21, and 35. 
 
DEFINITIONS 7 
 
 3. Express tlie following numbers as products in as many ways 
 as possible, and arrange the products in corresponding pairs : 12, 
 16, 20, 28, 42, and 60. 
 
 4. Give all the factors of 32, 40, and 48. 
 
 5. Place dots to show the measurement of $ 20 by a ^ 5 unit. 
 What other measurement does it show ? 
 
 6. Place dots to show the measurement of $24 by a $2 unitj 
 a $ 3 unit ; a $ 4 unit. What other measurements are shown ? 
 
 7. What is the price of 6 yd. of cheese-cloth at 8 ct. a yd.? 
 Explain each of these statements : 
 
 The whole price = 6 (8 ct.). 
 The v/hole price = 8 (6 ct.). 
 
 8. Show that 10 units of f 5 each is equal to 5 units of $ 10 
 each. 
 
 9. If a line 3 ft. long is repeated 6 times to measure the 
 length of a room, how long is the room ? How often would a 
 line 6 ft. long have to be repeated to measure the room ? 
 
 10. How many apples will be required to make 7 rows with 
 9 apples in each row ? What is the unit of measurement ? What 
 other convenient unit might be used? What would be the ratio 
 of the whole quantity to this unit ? 
 
 11. Draw a straight line 36 in. long, cut strips of paper 
 respectively 6 in., 7 in., 8 in., 9 in., 10 in., 11 in., and 12 in. 
 long. Measure along the line 3 times with each of these strips 
 of paper. 
 
 Use a yardstick divided into in. to measure your results, and 
 prove the following: 
 
 3 X 6 in. = 18 in. ; 3 X 9 in. = 27 in. ; 
 
 3 X 7 in. = 21 in. ; 3 x 10 in. = 30 in. ; 
 
 3 X 8 in. = 24 in. ; 3 x 11 in. = 33 in. ; 
 
 3 x 12 in. = 36 in. 
 
Ill 
 
 8 
 
 ARITHMETIC 
 
 !l! 
 
 i 
 
 12. Draw a line 36 in. long. Make a 3-in. measure. Measure 
 along the line respectively 6, 7, 8, 9, 10, 11, and 12 times, and 
 prove that: 
 
 6x3 in. = 18 in. 
 7x3 in. = 21 in. 
 8x3 in. = 24 in. 
 
 9 X 3 in. = 27 in. ; 
 
 10 X 3 in. = 30 in. ; 
 
 11 X 3 in. = 33 in. J 
 
 12 X 3 in. = 36 in. 
 
 13. What quantities are measured by the following: 6x4 
 in., 4x6 in., 5 x f 8, 8 x $5, 10 x 5 pears, 5 x 10 pears ? 
 
 14. What two units of length each longer than 4 in. can be 
 used to measure a line 35 in. long ? State in each case the ratio 
 of the length of the whole line to each unit. 
 
 15. What are the convenient units of money to pay a debt 
 of $35? $80? 
 
 16. What are convenient units to pay debts of 75^? 15^? 
 34^? 87^? • 
 
 17. A fruit dealer sells apples at the rate of 3 for 5^. 
 What is the unit to measure his apples ? What is the unit of 
 value? How many units are there in 24 apples? What is their 
 selling price ? 
 
 18. l)ananas are sold at the rate of 4 for 51^. What is the 
 measuring unit for the bananas ? What is the unit of value ? 
 How many units of value in 20 bananas ? 36 bananas ? What 
 is their value ? 
 
 19. Oranges are sold at 20^ a doz. What are the two meas- 
 uring units ? 
 
 20. If A can do a piece of work which is represented by 36 
 units in 9 da., how much will he do in 1 da. ? 
 
 21. If a piece of work is represented by 60 units and A can do 
 5 units in one da., in how manv da. can he do the entire work ? 
 
 B 
 
 
 
 D 
 
 E 
 
 F 
 
 22. AB is a line which represents any primary unit of measure, 
 and AQ^ AD, AE, and AF are derived units. What part is the 
 
DEFINITIONS 
 
 9 
 
 primary unit AB of each of the derived units ? What is the 
 ratio of each derived unit to the primary unit ? If AF is the pri- 
 mary unit, what would AB be ? What is the ratio of the derived 
 unit AF to the derived unit AD ? Of AD to AF? Of AC to 
 AE? OiAE to AC? Ot AF to AC? Oi AC to AF? 
 
 23. One field contains 7 units of area, and a second field con- 
 tains 9. What is the ratio of the area of the first field to the 
 second ? Of the second field to the first ? Illustrate by a drawing. 
 
 24. The distance from A to B is divided into 5 parts of 3 
 mi. erch, and that from A to C into 6 parts of 3 mi. each. What 
 is the ratio of the distance AB to AC ? Oi AC to AB? Illus- 
 trate your answer by a diagram. 
 
 25. The money in my purse is measured by the number 8 and 
 the unit f 5. I owe a debt measured by the number 6 and the 
 unit $5. What is the ratio of the debt to the money in my 
 purse ? What is the ratio of the money in my purse to the debt ? 
 How much shall I have left after paying my debt ? 
 
 26. If the amount of work required to dig a trench 800 yd. 
 long is represented by 40 units, what does 1 unit represent ? 
 
fi 
 
 CHAPTER II 
 
 NUMEEATION AND NOTATION 
 
 11 
 
 10. Numeration is co mi ting, or the expression of number 
 in words. 
 
 The ordinary system of numeration is the Decimal System^ 
 so called because it is based on the number ten. 
 
 11. The names of the first group of numbers in regular 
 succession are; one, two, three, four, five, six, seven, eight, 
 nine. 
 
 Other number-names are : ten, hundred, thousand, million, 
 billion, trillion, etc. 
 
 12. The number one applied to any unit denotes a quantity 
 which consists of a single unit of the kind named. 
 
 The number two applied to any unit denotes a quantity 
 which consists of one such unit and one unit more. 
 
 The number three applied to any unit denotes a quantity 
 which consists of two such units and one unit more. 
 
 And so on with the numbers four, five, six, seven, eight, 
 nine ; applied to any unit they denote quantities increasing 
 regularly by one such unit with each successive number. 
 
 13. The number next following nine is ten, which applied 
 to any unit denotes a quantity consisting of nine such units 
 and one unit more. 
 
 Counting now by ten units at a time, as before we counted 
 
 10 
 
NUMERATION AND NOTATION 
 
 11 
 
 by single units, we get the numbers ten, twenty, thirty, forty, 
 . . ., ninety. 
 
 The names of the numbers between ten and twenty are, in 
 order : eleven, twelve, thirteen, fourteen, . . ., nineteen. 
 
 The names of the numbers between tweuty and thirty, 
 thirty and forty, . . ., are formed by placing the names of 
 the numbers one, two, three, . . ., nine, in order after 
 twenty, thirty, . . ., ninety. 
 
 14. The number hundred applied to any unit denotes a 
 quantity which consists of ten ten-units. 
 
 Counting now by a hundred units at a time, as before we 
 counted by single units, we get the numbers one hundred, 
 two hundred, . . ., nine hundred. 
 
 The names of the numbers between one hundred and 
 two hundred, two hundred and three hundred, . . ., are 
 formed by placing the names of the numbers from one to 
 ninety-nine in regular succession after one hundred, two 
 hundred, . . ., nine hundred. 
 
 15. The number thousand applied to any unit denotes a 
 quantity which consists of ten hundred-units. 
 
 Counting now by a thousand units at a time, as before we 
 counted by single units, we get the numbers one thousand, 
 two thousand, . . ., nine thousand, ten thousan'^, eleven 
 thousand, twelve thousand, . . ., twenty thousand, . . ., 
 one hundred thousand, . . ., two hundred thousand, . . ., 
 nine hundred and ninety-nine thousand. 
 
 The names of the numbers between one thousand and two 
 thousand, two thousand and three thousand, . . ., are formed 
 by placing in order the names of the numbers from one to 
 nine hundred and ninety-nine, — the numbers preceding a 
 
12 
 
 ARITHMETIC 
 
 thousand, — after one thousand, two thousand, . . ., nine 
 hundred and nmety-nine thousand. 
 
 16. The number million applied to any unit denotes a 
 quantity whit3h consists of a thousand thousand-units. 
 
 The number billion applied to any unit denotes a quantity 
 which consists of a thousand million-units. 
 
 The number trillion applied to any unit denotes a quantity 
 which consists of a thousand billion-units. 
 
 m 
 
 • i 
 
 17. The number tenth applied to any unit denotes that 
 quantity of which ten make up the unit. 
 
 The number hundredth applied to any unit denotes that 
 quantity of which ten make up one tenth of the unit. 
 
 Consequently, one hundred of the hundredths of any unit 
 make up that unit. 
 
 The number thousandth applied to any unit denotes that 
 quantity of which ten make up the one hundredth of the 
 unit. 
 
 Consequently, one thousand of the thousandths of any unit 
 make up that unit ; and so on. 
 
 18. We count by the tenth of a unit at a time, as before 
 we counted from one to nine by a single unit each time ; 
 thus ; one tenth, two tenths, . . ., nine tenths. 
 
 We count by a hundredth of a unit at a time, as before we 
 counted from one to ninety-nine by a single unit each time ; 
 thus: one hundredth, two hundredths, . . ., ninety-nine 
 hundredths. 
 
 We count the thousandths of a unit, from one to nine 
 hundred and ninety-nine of the same in like manner as we 
 count thousands of the unit, and so on. 
 
 .r:,.vJ: 
 
NUMERATION AND NOTATION 
 
 13 
 
 19. Notation is the art of expressing numbers by means of 
 certain number symbols called numerals or figures. 
 
 20. The Arabic NnmeraU^ styled also Figures^ are 
 
 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 
 denoting naught, one, two, three, four,five, six, seven, eight,nine 
 respectively. The first of these is called naughty cipher, or 
 zero; the remaining nine are called digits. By means of 
 these numerals and a dot called the decimal pointy we can 
 write down any number expressed decimally. Tlie method 
 of doing so may be described as follows : 
 
 A figure immediately to the left of the decimal point de- 
 notes so many single units. 
 
 A figure immediately to the left of the eingle-units figure 
 denotes so many tens of the units, while a figure immediately 
 to the right of the single-units figure denotes so many tenths 
 of the unit.. 
 
 Figures to the left of the tens-figure, taking them in order 
 from right to left, denote so many hundreds of the unit, so 
 many thousands of the unit, so many ten-thousands of the 
 unit, so many hundred-thousands of the unit, so many mil- 
 lions of the unit, etc. 
 
 Figures to the right of the tenths-figure, taking them in 
 order from left to right, denote so many hundredths of the 
 unit, so many thousandths of the unit, so many ten-thousandths 
 of the unit, so many hundred-thousandths of the unit, so many 
 millionths of the unit, etc. 
 
 The function of the decimal point is to mark the place of 
 the standard unit when the quantity is measured. 
 
 21. Instead of speaking of " the number denoted by " 5, 
 75, or 375, we may for brevity speak of the number 5, 75, or 
 375. 
 
14 
 
 ARITHMETIC 
 
 I 
 
 11 : 
 In 
 
 22. Consider the following quantities : 
 
 
 
 
 9 
 
 4 
 
 yd. 
 
 
 
 
 59 
 
 a 
 
 
 
 
 259 
 
 it 
 
 
 
 
 3259 
 
 (( 
 
 
 
 / 
 
 43259 
 
 (( 
 
 
 
 843259 
 
 (( 
 
 
 
 
 59.7 
 
 (( 
 
 
 
 
 59.76 
 
 (( 
 
 
 
 
 59.761 
 
 u 
 
 
 
 
 59.7613 
 
 a 
 
 
 
 
 59.76132 
 
 it 
 
 9 denotes 9 of the unit one yd. 
 
 
 5 " 
 
 5 " 
 
 
 ten yd. 
 
 
 2 " 
 
 2 " 
 
 
 one hundred 
 
 yd- 
 
 3 " 
 
 3 " 
 
 
 one thousand yd. 
 
 4 » 
 
 4 " 
 
 
 ten thousanc 
 
 I yd. 
 
 8 " 
 
 8 " 
 
 
 one hundred thousand yd. 
 
 7 " 
 
 7 " 
 
 
 one tenth of 
 
 a yd. 
 
 6 " 
 
 6 " 
 
 
 one one-hundredth of a yd. 
 
 1 " 
 
 1 « 
 
 
 one one-thousandth of a yd. 
 
 3 " 
 
 3 " 
 
 
 one ten-thousandth of a yd. 
 
 2 " 
 
 2 " 
 
 
 one one-hundred thousandth of a yd. 
 
 23. The number 8 always denotes 8 of the unit ; 86 denotes 
 8 of the ten-unit or 80 of the unit and 6 of the unit, and is 
 read eighty-dx of the unit. 
 
 865 denotes 8 of the hundred-\\w\t and 6 of the ten-wmi and 
 5 of the unit ; i.e. 800, 60, and 5 of the unit, and is read eight 
 hundred mid sixty-jive of the unit. 
 
 Thus, the numbers 8, 80, 865 always denote eight, eighty- 
 
 .JL1.MUU 
 
 taorac 
 
NUMERATION AND NOTATION 
 
 15 
 
 six, eight hundred and sixty-five respectively, the position of 
 the figures in each case giving the unit. For example : 
 
 In the number 865,865,865 each 865 is read eight hundred 
 and sixty-five, the difference being in the unit only ; 865 of 
 the million-unit, 865 of the thousand-unit, and 865 of the 
 one-unit, which is the primary unit of reference. 
 
 This number is read eight hundred and sixty-five million 
 eight hundred and sixty-five thousand eight hundred and 
 sixty-five. 
 
 Exercise 3 
 
 1. Express in words the numbers given in Exercises 11, 13, 
 19, 20, and 21. 
 
 Express in words : 
 
 2. 3,409,035; 42,590,709; 6,003,040. 
 
 3. 20,394,678; 4,007,890; 8,000,006; 70,001,002. 
 
 Exercise 4 
 
 Express in words : 
 
 1. .7 ton ; .64 ton ; .643 ton. 3. 9.403 hr. ; 29.04 min. ; .09 sec. 
 
 2. 4.92 1b.; 8.09 1b.; 2.734 1b. 4. 7.456 A. ; 6.7985 A. 
 5. 8452.69 sq. mi. ; 21.4394 A. 
 
 ft. Express in words the nmnbers given in Exeroiises 14 and 24. 
 
 Exercise 5 
 
 Write in figures : 
 
 1. Three hundred and forty-nine ; eight thousand four hundred 
 and sixty -nine : nine thousand five hundred and seventy. 
 
 2. Twenty-nine thousand one hundred and thirty -four; fifty 
 thousand eight hundred and seventy-six ; seventy-eight thousand 
 three hundred. 
 
 3. Nine hundred and fifty-two thousand seven hundred and 
 forty ; six hundred and forty -nine thousand nine hundred and 
 five ; nine hundred thousand eight hundred and sixty-four. 
 
16 
 
 ARITHMETIC 
 
 4. One hundred and sixty-eight thousand six hundred and 
 eigliteen; throe liundred and twelve thousand seven hundred and 
 forty-two; four hundred and sixty-one thousand eight hundred 
 and twenty-one. 
 
 5. Seven liundred and four thousand and thirty; three hun- 
 (IhmI thousand two hundred and four; one hundred thousand and 
 fifty. 
 
 6. Five million two hundred and ninety-five thousand three 
 lumdred and three. 
 
 7. Sixty-four million seven hundred and eight thousand one 
 hundred and thirty-four. 
 
 8. Seventy-eight million four thousand and eighty-five. 
 
 9. Six million six thousand and six. 
 
 10. What is the value of six thousand units of $4 each ? Of 
 six million units of ^d each ? 
 
 Exercise 6 
 
 Kx|)ress in figures 
 
 1. Five tenths; two, and sixty-seven hundredths; nine hiin- 
 driHlMis; four thousandths. 
 
 2. Nine humlred ami twelve thousandths; seven hundred and 
 four thousandths; tive thousand four hundred and sixteen ten 
 thousandths. 
 
 3. Four hundred ami iifty-three, and four hundredths; four 
 hundred anil iifty, and one hundred and twenty-six thousandths. 
 
 4. Nine hundred and six thousandths; nine hundred and six 
 ten thousandths; twenty, and forty-tive thousandths. 
 
 5. Seventeen, and seven ten thousandths; three liundred and 
 (luee, and nine hundred and nine ten thousandths. 
 
 ?*BP 
 
NUMERATION AND NOTATION 
 
 Exercise 7 
 
 17 
 
 1. Name each unit in 34G yd. and state tl^e number of units of 
 each kind. 
 
 2. In question 1, what is the ratio of each unit to the next 
 unit to the right of it ? 
 
 3. What part is each unit of the next unit to the left? 
 
 4. What quantity expressed in single units is indicated by 3 
 in question 1 ? By 4 ? By G ? 
 
 5. Express in words 3.4G feet. Name each unit. 
 
 6. Fill in the blanks : % 679 is equal to G units of , 
 
 7 units of , and 9 units of — — . 
 
 7. Write as one number, 8 of the hundred-mntj 5 of the ten- 
 unit, and G of the one-unit. 
 
 8. Name the two cliiel: units in 843,294 and state the number 
 of units of each kind. Express the number in words. 
 
 9. Write the name of each of the six units in $294,785 and 
 the number of each unit. 
 
 10. In the quantity $ GGGjGGG, which G represents the largest 
 sum of money ? Which tlie smallest ? 
 
 11. Write as one number G7 of the thousand-unit and 413 of 
 the one-unit. 
 
 12. Write as one number 5 of the million-unit, 4G3 of the 
 thousand-unit, and 7G8 of the one-unit. 
 
 13. Write as one number .'U9 of the thousand-unit and 258 of 
 the one-unit. 
 
 14. Write as one number .''G5 of the million-unit, 829 of the 
 thousand-unit, and G()4 of the one-unit. 
 
 15. Which is the largest unit used in giving the population of 
 the largest city in the United States ? The length of the largest 
 river ? Tlie height of the high ?st mountain ? The area of the 
 largest state ? The area of tlie smallest state? 
 
 
 
 
 
18 
 
 ARITHMETIC 
 
 ^\'l 
 
 The Roman Notation 
 
 24. The Arabic Notation is the one in general use. It 
 was introduced into Europe by the Arabs. The system of 
 notation which was used among the Romans is now used only 
 to denote the chapters and sections of books, etc. 
 
 25. The following letters are used to denote numbers, and 
 their values are written below: 
 
 I. 
 
 V. 
 
 X. 
 
 L. 
 
 C. 
 
 D. 
 
 M. 
 
 1. 
 
 5. 
 
 10. 
 
 50. 
 
 100. 
 
 500. 
 
 1000. 
 
 26. The numbers 6, 8, 15, 20 are represented thus : 
 
 VI. VIII. XV. XX. 
 
 Hence if a character in the Roman Notation be followed by 
 another of equal or less value, the number denoted by the ex- 
 pression is equal to the sum of the simple values. 
 
 27. The numbers 4, 9, 40, and 90 are represented by IV., 
 1.2v., A.Ij., A-O. 
 
 Hence if a character in the Roman Notation is followed by 
 one of greater value than itself, the number denoted by the 
 expression is the difference of their simple values. 
 
 28. Express 1896 in Roman numerals. 
 
 1896 = 1000, 800, 90, and 6. 
 1000 = M. 
 800 = DCCC. 
 90=XC. 
 6= VI. 
 .-. 1896 = MDCCCXCVI. 
 
NUMERATION AND NOTATION 
 
 Hence to write any number in Roman numerals, separate 
 the number into its different parts, and write down the parts 
 in order, beginning at the left. 
 
 Exercise 8 
 
 Write in Roman numerals : 
 
 1. 14, 25, 54, 89, 99. 
 
 2. 178, 304, 871, 982, 999. 
 
 3. 1204, 1590, 1756, 1876, 1895. 
 
 Write in figures : 
 
 4. XLVI., LXXIX., XCIV., LXXXIII. 
 
 5. XCIX., CXXXIX., CLX. 
 
 6. DLIV., MDCIL, MDCCCXIX., MXC. 
 
 !■ V ' 
 
 lV.1 
 
 it 
 
m I 
 
 CHAPTER III 
 
 ADDITION 
 
 1:1 
 
 
 29. Let the length of a room be measured by the parts, 
 2 ft., 3 ft., 4 ft., and 5 ft. Here the common unit of meas- 
 ure, 1 ft., has been repeated 2, 3, 4, and 5 times to measure 
 the parts. 
 
 The number of units in all is the sum found primarily by 
 counting 2, 3, 4, and 5, or 14 units of 1 ft. Hence the 
 length of the room which is now definitely measured is 
 14 ft. 
 
 Addition may, therefore, be considered as the operation of 
 finding the quantity, which, as a whole, is made up of two or 
 more given quantities as its parts. Each of these quantities 
 must have the same measuring unit. Not only is it im- 
 possible to add 5 ft. to 4 min., it is impossible to add 5 ft. 
 to 4 in. ; i.e, to express without change of unit the whole 
 quantity by a number of either ft. or in. 
 
 The parts added are called Addends. 
 
 The Sum is the quantity obtained by adding the quantities 
 expressed in terms of a common unit. 
 
 30. The Sign of Addition is +, and is read plus; thus 
 6 + 8 is read 6 plus 8. 
 
 The Sign of Equality is =, and is read equals or equal; 
 thus 4 + 5 = 9 is read 4 plus 5 equals 9. 
 
 31. I bought 8 farms of 50 A. each, 6 farms of 50 A., 
 and 4 farms of 50 A. How much did I buy altogether? 
 
 20 
 
ADDITION 
 
 21 
 
 Here we are required to find the whole quantity measured by the sum of 
 3, 6, and 4 farms of 50 A. 
 
 .-. the whole quantity = 13 farms of 50 A. 
 
 t 
 
 ♦I 
 
 Exercise 9 
 
 1. What quantity is measured by the parts 2 yd., 6 yd., 
 
 and 7 yd. ? 
 
 2. What sum of money is equal to 4 five-cent pieces, 9 five- 
 cent pieces, and 5 five-cent pieces ? 
 
 3. How much IS 8 fifty -dollar bills, 4 fifty-dollar bills, and 
 9 fifty-dollar bills ? 
 
 4. I paid out in one day 6 ten-dollar bills, 8 ten-dollar bills, 
 and 5 ten-dollar bills. How much did I spend altogether ? 
 
 5. If I sell two lots, one for 8 units of value, and the other 
 for 6 units, what do I get for both, the unit of value being f 100 ? 
 
 6. A fruit dealer who arranges his apples in piles of 6 for 5 
 ct. sells 1 pile to each of a company of 4 persons, and 3 piles to 
 another customer. How much does he sell altogether? 
 
 7. A speculator buys 5 farms of 100 A. for $5000, 6 farms 
 of 100 A. for 17000, and 3 farms of 100 A. for $4000. How 
 much land did he buy ? If $ 1000 is the unit of value, how many 
 units did he pay out for all the farms ? 
 
 8. What is the quantity denoted by the sum 6, 7, and 5 times 
 the measuring unit ? 
 
 9. 2 in. -f 5 in. -f 4 in. = ? 2 f t. -f- 5 ft. + 4 ft. = ? 
 
 10. f3 + f4 + $6 = ? 3 ten-dollar bills + 4 ten-dollar bills 
 + 6 ten-dollar bills = ? 
 
 11. A horse was bought for 10 ten-dollar bills, and a carriage 
 for 12 ten-dollar bills. How much was paid for both ? 
 
 12. A man paid out at one time 4 five-dollar bills, at another 
 6 five-dollar bills, and again 3 five-dollar bills. How much did 
 he pay out altogether ? 
 
 
in 
 
 1 ' < 
 
 22 
 
 ARITHMETIC 
 
 13. A fruit dealer arranges his fruit in piles of 3 each. He 
 has 20 piles of apples, 10 of oranges, and 30 of plums. How 
 much has he altogether ? If there were 30 of each kind in a pile, 
 what number would express the aggregate ? 
 
 14. A fruit dealer sells his apples at the rate of 3 for 5 cents. 
 He sells five cents' worth to each of 8 customers. How much did 
 he sell ? 
 
 15. What is the sum of two quantities, one denoted by 9 times 
 the measuring unit, and the other by 6 times the measuring unit? 
 
 16. If I paid 30 units of f 100 each for a lot, and built a house 
 upon it which cost me 40 units of $ 100 each, what was the total 
 cost ? Iff 1000 is the unit, what is the number expressing the 
 total cost ? 
 
 17. A horse which travels at the rate of 8 mi. an hr. goes 
 from A to B in 3 hr., from B to C in 2 hr., and from C to D 
 in 4 hr. If 8 mi. is the unit of length, what is the number 
 expressing the distance from A to D ? 
 
 32. Drill on the following addition combinations to secure 
 accuracy and rapidity : 
 
 3 1 
 3. 6 
 
 1112 12 12 
 
 2 3 
 
 12 3 4 
 
 12 3 2. 
 
 -} -J —J - > 
 
 12 3 4 
 
 8 
 
 4 3. 5 . 
 
 -> - ) —) 
 
 12 3 4 5 
 
 5 4. 7 
 
 , _, _ , _, _, _, 
 
 6 
 
 5 4. 
 
 2 3 4 5 
 
 3 4 5 6 
 
 7 6 5. 9 8 7 6 
 
 _, _, _, _, 
 
 -> -> -) — > 
 
 4 5 G 5 
 
 9 8 7. 9 
 
 _, _, ._ , 
 
 5. 9 8 7 6. 9 8 7 6. 
 
 — J — ) -,' -> - > -) — J -J — ? 
 
 67677889 
 
 8 7. 9 8. 9 8. 9. 9 
 
 ■J — , - , _, _ , _, _ , _ , _. 
 
 Enlarge each combination thus : 
 
 8 8 8 8 18 28 18 38 68 
 ? 19 29^ etc.; ? _9 _9^ etc. ; 15 29 39 etc. ; 
 80 80 80 180 280 
 
 90 190 p.p . 90 90 90 
 — — , etc. , — — , 
 
 etc. 
 
 
I'Vn 
 
 ADDITION 
 
 28 
 
 Give such problems as the following requiring instantane- 
 ous answers : 
 
 How many sq. ft. in 1 sq. yd. 8 sq. ft.? 1 sq. yd. 6 sq. 
 ft.? etc. 
 
 How many qt. in 1 pk. 4 qt. ? etc. 
 
 When 7 is the number added, base the problem on days ; 
 thus: 
 
 How many da. in 1 wk. 4 da. ? 
 
 When the number is 6, on minutes ; thus : 
 
 How many sec. in 1 min. 30 sec, and so on. 
 
 
 Exercise 10 
 
 Using any unit, count to the number next larger than 100. 
 
 from ; from 1. 
 
 from ; from 2. 
 
 from ; from 1, 2, and 3 separately. 
 
 from ; from 1, 2, 3, and 4. 
 
 from 0, 1, 2, 3, 4, and 5. 
 
 from 0, 1, 2, 3, 4, 5, and 6. 
 
 from 0, 1, 2, 3, 4, 5, 6, and 7. 
 
 from 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. 
 
 33. A person paid $38 for a cow, $146 for a horse, and 
 • 255 for a carriage. Find the cost of all. 
 
 ^y 
 
 • 
 
 1. 
 
 2's 
 
 2. 
 
 3's 
 
 3. 
 
 4's 
 
 4. 
 
 5's 
 
 5. 
 
 6's 
 
 6. 
 
 7's 
 
 7. 
 
 8's 
 
 8. 
 
 9's 
 
 I 38 
 146 
 255 
 
 $439 
 
 In this problem we are required to find the cost which is the 
 whole measured by the parts $ 38, $ 146, and ^ 255. 
 
 This may, for convenience, be broken up into the sum of 5, 6, 
 and 8 units of $1, 5, 4, and 3 units of i|10, and 2 and 1 units 
 of $ 100. The sum of 5, C, and 8 units of -^ 1 = 19 units of $ 1 = 1 
 unit of $ 10 and 9 of the $ 1 unit. 
 
 Add the 1 unit of $ 10 in with the tens' column. 
 
 
 Si' 
 
 hi- 
 
rll 
 
 24 
 
 ARITHMETIC 
 
 Si 
 
 iHi 
 
 
 983 
 
 t 
 
 ,! 5G7 
 
 
 ;i 842 
 ,i 2626 
 
 J'! 
 
 1 
 
 
 The sum of 1,5, 4, and 3 units of $ 10 = 13 units of $ 10, or 1 unit of 
 $ 100, and 3 units of $ 10. 
 
 Add the 1 unit of $ 100 in with the hundreds' column. 
 
 The sum of 1,2, and 1 unit of ,«$ 100 = 4 units of $ 100. 
 
 Hence the cost = 4 units of $ 100, 3 units of $ 10, and 9 units of $ 1 = $ 439. 
 
 34. If the primary unit is 1 da., and the derived unit 
 1 yr., how many primary units of time are there in one 
 derived and 243 primary units? 
 
 Here we are required to find the sum of 365 and 243 primary units. The 
 result = 008 primary units. 
 
 Find the sum of tlie following numbers, using the unit 
 employed in stating your age : 
 
 234 •*• the sum = 2G26 yr., since 1 yr. is the unit of age. 
 
 Add thus, beginning at the units' column : 9, 12, 16 ; write down 
 6 and carry 1 to the tens' column; 5, 11, 19, 22; write down 2 
 under the tens' column and carry 2 to the hundreds' column; 10, 15, 
 24, 26 ; write down 26, putting the 6 under the hundreds' column. 
 
 To prove the answer correct, add downward. If the same 
 answer is obtained, the result is likely to be correct. 
 
 Exercise 11 
 
 Find the sum of the following quantities and explain your 
 work clearly. Name all the primary units and also the quanti- 
 ties which measure the parts. Prove each answer correct by 
 beginning at the top and adding down. 
 
 •i- 
 
 1. 
 
 $441 
 234 
 518 
 
 
 4. 
 
 543 
 
 min. 
 
 
 GGG 
 
 (( 
 
 
 752 
 
 u 
 
 
 231 
 
 u 
 
 2. 
 
 341 ct. 
 
 
 225 " 
 
 
 343 " 
 
 5. 
 
 G35hr 
 
 
 87 " 
 
 
 256 " 
 
 
 742 " 
 
 3. 
 
 532 da 
 
 
 233 " 
 
 
 154 " 
 
 6. 
 
 247 in. 
 
 
 859 " 
 
 
 23 " 
 
 
 271 " 
 
 .1 I 
 
 m 
 
 1 
 

 
 ADDITION 
 
 
 
 7. 
 
 2576 ft. 
 
 8. 2598 yd. 
 
 9. 
 
 4397 mi. 
 
 
 3491 " 
 
 6776 " 
 
 
 8999 " 
 
 
 7743 " 
 
 4259 " 
 
 
 5637 " 
 
 
 8988 " 
 64251 oz. 
 
 7362 " 
 
 12. 
 
 8249 " 
 
 10. 
 
 11. 89435 1b. 
 
 850439 T 
 
 
 3789 " 
 
 62789 " 
 
 
 973642 " 
 
 
 45278 " 
 
 576 " 
 
 
 845867 '' 
 
 
 99 " 
 6472 " 
 
 43 " 
 
 
 939894 '' 
 
 
 13. $453798 
 
 768795 '' 
 
 
 
 649879 " 
 
 
 
 
 
 667788 
 
 
 
 
 
 549763 
 
 
 
 
 
 438925 
 
 
 
 
 
 648888 
 
 
 
 
 
 999999 
 
 
 
 25 
 
 
 n'.N 
 
 f > 
 
 ! r 
 
 14. Find the number of days in 1 leap yr. and 257 da. 
 
 15. If the unit of length 1 ft. is repeated 5280 times to 
 measure a mi., how many such units are there in 1 mi. and 
 4754 ft. ? In 1 mi. 1 yd. 2 ft. ? 
 
 16. If 1 mi. is represented by the number 63,360 when the 
 primary unit is 1 in., how many times will the primary unit 
 measure the distance 1 mi. 5769 in. ? The distance 1 mi. 1 yd. 
 1 ft. 8 in. ? 
 
 Find the sum of the following quantities, the unit of measure 
 being that in selling retail the following: In (17) cherries, 
 (18) kerosene, (19) raspberries, (20) sugar, (21) coal, (22) cloth, 
 (23) potatoes, (24) eggs. 
 
 

 26 
 
 
 ms 
 
 W I 
 
 
 PI 
 
 m ■: 
 
 h ^- 
 
 17. 84 
 93 
 89 
 75 
 91 
 27 
 30 
 45 
 
 21. 
 
 542 
 
 879 
 666 
 257 
 389 
 983 
 365 
 874 
 
 22 
 
 ARTTIIMETIC 
 
 18. 99 
 
 19. 893 
 
 77 
 
 254 
 
 86 
 
 767 
 
 25 
 
 899 
 
 43 
 
 654 
 
 88 
 
 473 
 
 76 
 
 129 
 
 52 
 
 895 
 
 . 9834 
 
 23. 7594 
 
 729 
 
 821 
 
 8345 
 
 2357 
 
 728 
 
 8463 
 
 3403 
 
 1525 
 
 17 
 
 7469 
 
 295 
 
 2856 
 
 8943 
 
 8888 
 
 20. 733 
 
 842 
 951 
 258 
 365 
 874 
 935 
 273 
 
 24. 5846 
 7593 
 3819 
 5578 
 2904 
 8392 
 9576 
 2882 
 
 35. If tlie unit 1 mi. contains 5280 ft., how many ft. are 
 there in 4 such units ? 
 
 Here we are required to find the sum of four equal addends, each of which 
 
 is 5280 ft. ; thus : 
 
 5280 ft. 
 
 5280 " 
 
 5280 " 
 
 5280 " 
 
 21120 ft. 
 
 Exercise 12 
 
 1. If 1 mi. contains 1760 yd., find, by adding, the number of 
 yd. in 2 mi. In 3 mi. In 4 mi. 
 
 2. If the unit 1 mi. contains 320 rd., how many rd. in 9 such 
 units ? 
 
 3. One sq. ft. contains 144 sq. in. How many sq. in. are 
 there in two oblongs, one containing 3 sq. ft. and the other 
 4 sq. ft.? 
 
^m 
 
 ADDITION 
 
 27 
 
 4. One cu. ft. contains 1728 cu. in. Show by adding that 12 
 volumes, each containing 144 cu. in., will be equal to 1 cu. ft. 
 
 5. One gal. contains 231 cu. in. Show that a gallon measure 
 can be filled Avith water and emptied 7 times into a measure 
 containing 1 cu. ft. without causing it to overflow, but not 8 
 times. 
 
 6. If the unit of area, 1 A., contains 4840 sq. yd., how many 
 sq. yd. are there in 6 such units ? 
 
 7. If there are 197 school days in 1 yr., find, by adding, the 
 number of school days in 8 yr. 
 
 8. One sq. mi. contains 640 A. How many A. are there in 
 8 sq. mi. ? 
 
 9. Find, by adding, the number of da. in 6 ordinary yr. of 
 365 da. each and 2 leap yr. of 366 da. each. 
 
 1*1 
 * 
 
 m 
 
 
 Exercise 13 
 
 In each of the following questions, state in each case which is 
 the whole quantity to be measured and what are the parts meas- 
 uring it. 
 
 1. A spent the following sums of money: |425, $342, $673, 
 and $ 897. How much did he spend altogether ? 
 
 2. An encyclopaedia consists of three volumes. In the first 
 there are 693 pages, in the second 745, and the third 892. Find 
 the number of pages in the encyclopaedia. 
 
 3. Using the table in § 171, find the number of days in 
 the first six months of the year. In the last six months. In a 
 year. 
 
 4. Find the number of days in the three spring months. In 
 the three summer months. In the three fall months. In the 
 three winter months. 
 
 
rr 
 
 
 28 
 
 ARITHMETIC 
 
 II ft 
 
 I 'r 
 
 5. If the average number of persons to the sq. mi. is taken 
 as the unit of population, and the unit for California is 8 persons, 
 what is the unit of population for the following states ? 
 
 A, which has twice, and B, which has three times the average 
 population of California ; C, whose unit is the sum of the units of 
 A and B. 
 
 6. Find the total area of these lakes : 
 
 Lake Erie, area 7,750 sq. mi. ; 
 
 Lake Ontario, area G,950 sq. mi. ; 
 Lake Michigan, area 22,000 sq. mi. ; 
 Lake Superior, area 31,500 sq. mi. 
 
 7. A merchant bought 150 yd. of cloth for $ 232, 254 yd. for 
 f 175, 1875 yd. for $ 2395, and 640 yd. for 1 1966. Find the 
 number of yd. bought and the total cost. 
 
 8. How many years are there between the establishment of 
 the Republic of Rome in 509 u.c, and the Declaration of Inde- 
 pendence in 1776 A.D. ? 
 
 9. What is the sum of 4, 6, 9, 7, 5, 8, 3, 6, and 7, when the 
 unit of value is |1 ? {$ 10 ? $ 100 ? 
 
 10. What is the value of the quantity denoted by the sum of 3, 
 9, 12, 6, and the unit ^ 1000 ? $ 10,000 ? $ 100,000 ? $ 1,000,000 ? 
 
 11. Find the sum of three hundred and seventy-six thousand 
 and fifty-four ; one hundred and ninety-seven thousand two hun- 
 dred and fifty-one; four hundred and fifty-seven thousand six 
 hundred and forty-nine. 
 
 12. The population of Maine is 661,086; New Hampshire, 
 376,530; Vermont, 332,422; Massachusetts, 2,238,t43; Rhode 
 Island, 345,506, and Connecticut, 746,258. Find the population 
 of the New England States. 
 
 13. According to the census of 1890, the population of the 
 seven largest cities in the United States is : New York, 1,515,301 ; 
 Chicago, 1,099,850; Philadelphia, 1,046,964; Brooklyn, 806,343; 
 
'■m 
 
 ADDITION 
 
 29 
 
 St. Louis, 451,770; Boston, 448,477, and Baltimore, 434,439. 
 Find the total population. 
 
 14. Find the sum of the following measures: one 1-inch unit, 
 one 12-inch unit, one 3C-inch unit, one 198-inch unit, and one 
 63360-inch unit. 
 
 15. Find the sum of the following : 2 of the one-unit, 4 of the 
 ten-unit, 6 of the hundred-unit, 3 of the thousand-unit, and 9 of 
 the ten-thousand unit quantities. 
 
 16. Find the number of times a clock strikes from a quarter of 
 nine a.m. until a quarter of nine p.m. 
 
 17. A, B, and C engaged in trade ; A put in 1 3475, B f 4593, 
 and C as much as the other two together. How much money was 
 put into the business ? 
 
 18. A man, dying, willed to his widow f 6875; to his son, 
 $ 4294, and to his daughter, ^ 3875. What was the value of his 
 estate ? 
 
 19. The census of 1890 gave the negro population of Georgia 
 858,996; Mississippi, 744,749; South Carolina, 689,141; Ala- 
 bama, 679,299; Virginia, 635,858; North Carolina, 562,565; 
 Louisiana, 560,192 ; Texas, 489,588 ; Tennessee, 430,881 ; Arkan- 
 sas, 309,427. Find the entire negro population of these states. 
 
 20. The area of the basin of the Colorado River is 250,000 
 sq. mi.; Columbia, 250,000; Mackenzie River, 440,000; Mis- 
 souri-Mississippi, 1,250,000; Nelson, 355,000; Rio Grande, 
 180,000; St. Lawrence, 350,000. Find the total area of these 
 river basins. 
 
 21. What is the area of the New England States; that of 
 Maine being in sq. mi. 33,040, of New Hampshire 9305, of 
 Vermont 9565, of Massachusetts 8315, of Rhode Island 1256, 
 of Connecticut 4990 ? 
 
 22. A father left his eldest son f 24,000 more than he left his 
 second son, and the second son $ 7560 more than the third ; to 
 the third he left $ 60,480. What was the eldest son's portion, 
 and what sum did the father leave to his three sons ? 
 
 *f 
 
 
 lit 
 
I 
 
 ri 
 
 30 
 
 ARITHMETIC 
 
 ^ 
 
 i:' 1 
 
 "' i 
 
 hi 
 
 hr 
 
 tvrl ► 
 
 23. If in question 6, 100 sq. mi. = unit of measure, and in 
 question 20, 1000 sq. mi. = the unit, what numbers express the 
 respective aggregates ? 
 
 24. Add vertically and horizontally the following statement of 
 eight weeks' cash receipts : 
 
 Ai 
 
 
 MON. 
 
 TUES. 
 
 Wed. 
 
 Thur. 
 
 Fiji. 
 
 Sat. 
 
 Total. 
 
 1st 
 
 $3862.93 
 
 $1391.76 
 
 $6760.68 
 
 $1098.91 
 
 $1696.65 
 
 $ 43.68 
 
 
 2d 
 
 396.74 
 
 6108.37 
 
 864.39 
 
 964.26 
 
 167.69 
 
 1864.86 
 
 
 3d 
 
 1768.63 
 
 467.89 
 
 2035.68 
 
 3165.03 
 
 691.83 
 
 786.97 
 
 
 4th 
 
 3976.98 
 
 76.05 
 
 364.76 
 
 93.68 
 
 1948.39 
 
 1759.46 
 
 
 6th 
 
 263.76 
 
 1036.84 
 
 36.10 
 
 386.41 
 
 3.45 
 
 1396.71 
 
 
 6th 
 
 1569.83 
 
 1932.57 
 
 1268.16 
 
 8.37 
 
 279.72 
 
 67.85 
 
 
 7th 
 
 62.24 
 
 318.62 
 
 134.36 
 
 1763.29 
 
 1468.29 
 
 643.66 
 
 
 8th 
 
 194.87 
 
 3.85 
 
 7643.82 
 
 685.38 
 
 765.42 
 
 39.67 
 
 
 Total 
 
 
 
 
 
 
 
 
 64 
 
 6e 
 
 25. Add vertically and horizontally the following statement : 
 
 Total 
 
 ! 1169.84 
 
 909.58 
 
 675.72 
 
 2078.28 
 
 312.83 
 
 1062.47 
 
 339.11 
 
 1732.60 
 
 1237.50 
 
 113.50 
 
 3661.00 
 
 1139.67 
 
 $3650.12 
 
 866.78 
 
 742.49 
 
 1180.66 
 
 1638.24 
 
 342.66 
 
 687.23 
 
 614.02 
 
 3839.26 
 
 1291.98 
 
 973.03 
 
 070.22 
 
 1 189.10 
 914.19 
 
 1064.70 
 119.25 
 
 2010.72 
 108.00 
 216.17 
 667.00 
 777.00 
 112.50 
 311.20 
 
 1201.64 
 
 1 97.22 
 239.49 
 196.17 
 
 8418.00 
 
 1642.24 
 349.96 
 
 1020.00 
 000.00 
 130.70 
 
 1850.14 
 630.99 
 
 7357.61 
 
 $ 20.55 
 297.02 
 869.09 
 
 2223.42 
 
 6300.20 
 136.97 
 
 1124.60 
 476.00 
 
 4666.05 
 738.76 
 243.44 
 262.47 
 
 $ 861.02 
 
 312.00 
 
 1477.42 
 
 508.36 
 
 110.02 
 
 1214.03 
 
 1732.26 
 
 138.60 
 
 1097.47 
 
 1204.74 
 
 142.91 
 
 694.02 
 
 Total. 
 
 tl 
 fi 
 
 e 
 
 t( 
 I 
 
ADDITION 
 
 31 
 
 ^1 
 
 If. 
 
 36. Find the sum of: 2.46, 23.973, 15.025, 643.319, and 
 .468. 
 
 2.46 
 23.973 
 15.025 
 643.319 
 
 .468 
 
 685.245 
 
 Since we can add numbers of the same unit, we write the 
 addends so tliat units will be under units, tenths under tenths, 
 and so on. This is easily done by placing the decimal points 
 directly below each other. Then, beginning at the right, we 
 add the figures as if they were integers, and place the decimal 
 point in the sum between the units' and tenths' column. 
 
 Add: 
 
 1. 3.456 
 4.593 
 
 Exercise 14 
 
 2. 
 
 7 245 
 9.864 
 
 27.43 
 18.314 
 5.687 
 34.986 
 
 3. 
 
 76.425 
 39.639 
 28.764 
 21.385 
 
 Write in columns and add : 
 
 4. 4.396 + 7.295 + 6.478 + 5.765. 
 
 5. .432 + .987 + .593 + M6, 
 
 6. 84.63 4- 46.892 + 24.7 -f 95.657. 
 
 7. f 24.375 + 1 95.875 -f- $ 16.125 -|- $ 19.50. 
 
 8. Find the capacity of four bins, the first of which will con- 
 tain 66.384 bu., the second 89.645 bu., the third 27.437 bu., and 
 the fourth 75.938 bu. 
 
 9. What is the area of a farm which is divided into three 
 fields containing, respectively, 25.936 A., 14.56 A., and 24.504 A. ? 
 
 10. Four towns, A, B, C, D, lie on a road running directly 
 east and west. The distance from A to B is 5.693 mi., from B 
 to C 8.421 mi., from C to D 12.768 mi. Find the distance from 
 AtoD. 
 
 
 IM 
 
 ll 
 
 fe^ 
 
 i l! 
 
 li 
 
|h ! 
 
 CHAPTER IV 
 
 ul ' I 
 
 I- ■ 
 
 'vi 
 
 
 SUBTEAOTION 
 
 37. A man who earned $14 a week, spends $5 a week for 
 his board. How much has he left ? 
 
 We are here given the whole quantity, or $14, and one 
 part, or $ 5, and we are required to find the other part. 
 
 The question may be viewed in two ways: How much 
 must be added to -$5 to make $14 ? Or how much must be 
 taken from $ 14 to leave $ 5 ? The answer in both cases is 
 known from addition. $5 and $9 are two quantities making 
 $ 14. Therefore, if one of them, $ 5, is given, the other must 
 be $9. Or, in other words, $9 is the difference between 
 f 14 and $^. It is this view of difference that gives the 
 name Subtraction. 
 
 V r 
 
 t 
 I 
 
 Hi 
 
 38. Subtraction may therefore be defined as the operation 
 of finding the part of a given quantity that remains when a 
 given part has been taken from the quantity. 
 
 The given quantity is called the Minuend, and the given 
 part tlie Subtrahend, while the part that remains is called 
 the Difference or Remainder. 
 
 39. The Sign of Subtraction, — , is called minus. Thus 
 8 — 6 is read 8 minus 6, and signities that 6 is to be sub- 
 tracted from 8. 
 
 83 
 
 
■mm 
 
 ^ 
 
 SUBTRACTION 
 
 Exercise 15 
 
 Read the following questions, filling in the blanks 
 
 33 
 
 1. 6 and 7 are 
 
 8 and 9 are 
 
 4 and 8 are — . 
 
 2. 4 and 6 are — , 4 and — are 10, 9 and — are 15. 
 
 3. 2 and — are 11, 3 and — are 8, 6 and — are 14. 
 
 4. 22 and — are 25, 4 and — are 36, 9 and — are 27. 
 
 5. 8 and — are 29, 5 and — are 16, 6 and — are 48. 
 
 Subtract (Note : Let the process be 7iot 6 from 9 leaves 3, but 
 6 and 3 are 9) : 
 
 9 19 29 39 49 69 99 
 
 ^' 6' "6' 
 
 6' 6' 6' 6' 6 
 
 90 190. 8. 28. 80. 380 
 60' 60' 5' 5' 50' 50* 
 12 120 32 320 14 54 
 
 8. 
 
 7 ' 70 ' 7 ' 
 
 70' 8' 
 
 8 
 
 What numbers added respectively to 9, 7, 6, 8, 5, and 4, make 
 9. 12? 10. 15? 11. 17? 12. 14? 13. 18? 14. 16? 
 
 40. Drill, as in § 32, in addition, on the fundamental sub- 
 tractions, connecting with corresponding additions, until 
 accuracy and rapidity are secured ; thus : 
 
 9 19. 29. 39. 49. 59 
 
 8' 8' 8' 8' 8' 8 
 
 90 190 290 390 490 
 
 80 
 
 L7 
 8 
 
 80 
 8 
 
 80 ' 80 ' 80 
 
 t7. 55 
 8' 8' 8 
 
 ; and so on. 
 and so on. 
 
 17. .27. 37. 47. 57. ^^^^^^^^ 
 
 41. A man who owned 18 farms of 50 A., sold 7 of them. 
 How much had he left ? 
 
 Because 7 -f 11 = 18, it is evident that he had left 11 farms of 50 A. 
 each. 
 
 D 
 
 'm 
 
 
 ei 
 
 
34 
 
 ARTTHMETTC 
 
 Exercise 16 
 
 III •: 
 
 "fi: 
 
 ■ r « 
 
 If 
 
 1. 1) ft. -f ? = '0 ft. 1) yd. + ? = 16 yd. 
 
 2. How many dimes must l)e added to 6 dimes to get 14 
 dimes ? What is the difference between 14 dimes and 6 dimes ? 
 How much less is G live-dollar bills than 14 iive-dollar bills ? 
 
 3. A fruit dealer sold 10 piles of 3 apples each. How many 
 had he left if he had at first 15 piles of 3 apples ? 
 
 4. A fruit dealer arranges his oranges into 12 piles of 4 
 oranges each. He sells 8 piles. How many has he left ? 
 
 5. A fruit dealer who sells apples at the rate of 3 for 5 
 ct., arranges his apples into 19 groups of 3 each. He sells 
 5 ct. worth to each of 12 customers. How much has he still 
 remaining ? 
 
 6. I owe a debt of 12 ten-dollar bills and have 5 ten-dollar 
 bills in my pocket. If I pay this toward the debt, how much 
 do I still owe ? 
 
 7. What must be added to 15 units to get 20 units? Taken 
 from 20 units to get 15 units ? To get 5 units ? If I sell my 
 house, which cost 20 units of value of $ 100 each, for 25 units, 
 how much did I gain on the transaction ? 
 
 8. What is the difference between a quantity denoted by 14 
 times the measuring unit and one denoted by 8 times the meas- 
 uring unit ? 
 
 9. A person who has f 50 pays a debt of 330. How much 
 money has he left ? What number expresses this remainder if 
 f 10 is the unit of measure? If |»5 is the unit of measure? 
 If f 4 is the unit ? If $ 2 is the unit ? 
 
 10. A speculator bought 12 farms of 100 A. each for $6000. 
 He sold 4 of those farms for f 3000. How much land had he 
 left ? If $ 1000 is the unit of money, express in terms of this 
 unit the difference between the buying and the selling price of 
 the 4 farms. 
 
 'i 
 
 4 
 
^^ 
 
 SUBTRACTION 
 
 85 
 
 42. The following method of subtraction, which is nearly 
 always adopted in making change, is almost universally 
 employed by professional computers and is considered by 
 many teachers the best way to perform subtraction. 
 
 It is superior in accuracy and rapidity to the method of 
 the next paragraph. 
 
 It is based on the principle that the sum of the subtrahend 
 and remainder is equal to the minuend. 
 
 From 875 take 451. 
 
 875 
 
 451 
 424 
 
 Thus : 1 and 4 are 5 ; 5 and 2 arc 7 ; 4 and 
 4 are 8. In this operation let the pupil fancy 
 that he is doing addition with the sum at the 
 top, and as he works set down the figures, 4, 2, 
 and 4. 
 
 'li 
 
 
 43. A merchant bought 965 yd. of silk and sold 723 yd. 
 How much had he left ? 
 
 Here we are required to find the undefined part. This is the difference 
 between the measured whole, or 905 yd. , and the given part, 723 yd. 
 
 965 yd. = the measured whole. 
 
 723 yd. = the measured part. 
 
 242 yd. = the difference, which is now definitely known. 
 
 Explanation. — As in addition, we write units under units, tens under 
 tens, and hundreds under hundreds. Beginning with the units, we say 3 units 
 from 5 units leaves 2 units, which we write below the line in the units' column. 
 Then 2 tens from 6 tens leaves 4 tens. Place this in the tens' column. 
 Lastly, 7 hundreds from 9 hundreds leaves 2 hundreds, which we write in the 
 hundreds' column. 
 
 This difference, 242 yd,, is the other part, which is now definitely 
 measured. 
 
 Exercise 17 
 
 Subtract, and prove your answer correct in each case : 
 
 1. 
 
 ^946 
 324 
 
 2. 
 
 785 lb. 
 323 " 
 
 3. 
 
 659 T 
 236 " 
 
 4. 
 
 897 da. 
 683 " 
 
 i 
 
 : i [J 
 
36 
 
 ARITHMETIC 
 
 ■1 1 
 
 1^' i 
 
 lif^f. 
 
 m 
 
 m 
 
 m^ 
 
 ';li' 
 
 it 
 
 A 
 
 ■ ' I r 
 
 m 
 
 5. 
 
 8. 
 
 8498 hr. 
 2361 " 
 
 $7684 
 6450 
 
 6. 
 9. 
 
 9999 min. 
 7265 " 
 
 $8697 
 1082 
 
 7. 
 
 10. 
 
 8395 sec. 
 4073 " 
 
 $ 2578 
 1506 
 
 Exercise 18 
 
 In the following questions, name (1) the undefined part, (2) the 
 whole quantity, (3) the given part. 
 
 1. A merchant sold 246 yd. from a piece of cloth 258 yd. in 
 length. How many yd. had he remaining ? 
 
 2. A person deposited in a bank $8495, but shortly after drew 
 out $1035. How much had he left in the bank ? If $10 is the 
 unit of value instead of $ 1, what number expresses the amount 
 left in the bank ? 
 
 3. On Tuesday a merchant deposited in a bank $3475, on 
 Wednesday $ 4690. If he withdrew $ 1010 on Thursday, how 
 much did he still have on deposit ? 
 
 4. What is the difference between 1 yr. and 213 da.? 
 
 5. A bankrupt has debts amounting to $ 8496 ; his assets are 
 $3015. How much more does he owe than he can pay ? 
 
 6. A man left property to the value of $36,875 to his two 
 children. The son received $ 14,250 ; what was the daughter's 
 share ? 
 
 7. At an election the successful candidate received 953 votes, 
 and the unsuccessful candidate 613 votes. Find the majority of 
 the former. 
 
 44. Computers' Method. 
 From 94,275 take 67,492. 
 94275 
 
 Thus : 2 and 3 are 5 ; 9 and 8, 17 ; 
 carry 1 to 4 as in addition, making it 5 ; 
 5 and 7 are 12 ; carry 1 to 7, making it 
 8 ; 8 and 6 are 14; carry 1 to 6, making 
 it 7 ; 7 and 2 are 9. 
 
 67492 
 
 26783 
 
 The numbers 3, 8, 7, G, and 2 are written down in order to give the 
 remainder. 
 
 
 26' 
 
 il; 
 
SUBTRACTION 
 
 37 
 
 642 
 875 
 
 267 
 
 45. Find the difference between 642 and 375. 
 
 As we cannot take 5 units from 2 units, we take 1 ten from the 4 
 tens, and adding this 1 ten, which equals 10 units, to the 2 units, we 
 have 12 units. Then 5 units from 12 units leaves 7 units, which we 
 write under the units' column. Now as we took 1 ten from 4 tens, 
 we have left only 3 tens ; we borrow 1 hundred from the 6 hundreds, 
 and considering the 1 hundred as 10 tens, we add it to the 3 tens, making 13 
 tens ; then 7 tens from 13 tens leaves C tens, which we write under the tens' 
 column. 
 
 Now as we took 1 hundred from Jmndreds, we have left only 5 hundreds ; 
 hence we subtract 3 hundreds from 5 hundreds, leaving only 2 hundreds, 
 which we write in the hundreds' column. 
 
 The remainder, or difference, is thus 2 hundreds, 6 tens, and 7 units, 
 or 267. 
 
 
 i:Ji^i^|! 
 
 ';■ I": 
 
 ^^l: 
 
 Exercise 19 
 
 In the following questions prove the correctness of your results 
 by adding the two parts and finding the sum equal to the whole 
 quantity. 
 
 4. 
 
 7. 
 
 10. 
 
 13. 
 
 16. 
 
 $653 
 269 
 
 921 min. 
 
 87 
 
 (( 
 
 3849 yr. 
 2567 " 
 
 8000 yd. 
 5348 " 
 
 43970 pt. 
 26784 " 
 
 3406 bu. 
 29143 " 
 
 2. 
 
 5. 
 
 8. 
 
 11. 
 
 14. 
 
 17. 
 
 19. 
 
 850439 T. 
 473642 " 
 
 307 dimes 
 
 (( 
 
 268 
 
 255 hr. 
 99 " 
 
 9345 in. 
 8367 " 
 
 9041 rd. 
 
 7385 " 
 
 50062 qt. 
 37891 " 
 
 986403 oz. 
 
 728547 " 
 
 20. 
 
 3. 
 
 6. 
 
 9. 
 
 12. 
 
 15. 
 
 18. 
 
 642 sec. 
 375 " 
 
 907 da. 
 
 859 " 
 
 7007 ft. 
 6609 " 
 
 7968 mi. 
 2693 " 
 
 12009 gal. 
 11376 " 
 
 620703 lb. 
 
 444444 " 
 
 759826 A. 
 378934 " 
 
 (if '; 
 
 
 1'^' 
 
 |-H: 
 
 

 'I 
 
 
 J'; 
 
 If 
 
 II 
 
 
 ( I 
 
 
 38 
 
 * Subtract: 
 
 1. 57261 
 
 38877 
 
 4. 
 
 7. 
 
 89437 
 15790 
 
 654375 
 412884 
 
 10. 
 13. 
 
 233826 
 204739 
 
 164326 
 48476 
 
 * Subtract : 
 
 1. 755903 
 699004 
 
 4. 100794 
 81685 
 
 7. 
 
 4731246 
 4342760 
 
 10. 
 
 3801572 
 2003789 
 
 13. 
 
 1217191 
 1038182 
 
 16. 
 
 5468305 
 1490673 
 
 19. 
 
 8235460 
 3530089 
 
 ARITHMETIC 
 Exercise 20 
 
 2. 40359 
 9998 
 
 5. 
 
 67182 
 30293 
 
 8. 
 
 986392 
 826957 
 
 11. 
 
 310865 
 270326 
 
 14. 
 
 982623 
 897674 
 
 Exercise 21 
 
 2. 
 
 640021 
 400569 
 
 5. 
 
 143812 
 109758 
 
 8. 
 
 9487352 
 5999999 
 
 11. 
 
 5745861 
 2837154 
 
 14. 
 
 4100293 
 1925867 
 
 17. 
 
 7086543 
 2889454 
 
 20. 
 
 2679953 
 1346397 
 
 3. 
 
 10000 
 1021 
 
 6. 
 
 81349 
 47538 
 
 9. 
 
 303233 
 192001 
 
 12. 
 
 605487 
 584598 
 
 15. 
 
 1000101 
 
 707707 
 
 3. 716287 
 662763 
 
 6. 
 
 948735 
 473596 
 
 9. 
 
 1737682 
 739908 
 
 12. 
 
 5048650 
 4243091 
 
 15. 
 
 2047000 
 1054888 
 
 18. 
 
 1671498 
 536819 
 
 21. 
 
 1521815 
 1432568 
 
 * Use computers' method of subtraction. 
 
SUBTRACTION 
 
 89 
 
 Exercise 22 
 
 1. Subtract 231 cu. in. from 2772 cu. in., and from the remain- 
 der, and so on, until no remainder is left. If 1 gal. contains 
 231 cu. in., how many gal. are there in 2772 cu. in. ? 
 
 2. Subtract 320 rd. from 2880 rd., and from each remainder 
 until none is left. If the unit of length, 1 mi., is equal to 320 rd., 
 how many such units are there in 2880 rd. ? 
 
 3. Subtract 1760 yd. from 15,840 yd., and from each remain- 
 der until none is left. If 1 mi. contains 17G0 yd., how many mi. 
 are there in 15,840 yd. ? 
 
 4. Subtract, as in question 3, 5280 ft. from 68,640 ft. If 1 
 mi. is equal to 5280 ft., how many mi. are equal to 68,640 ft. ? 
 
 5. Subtract, as in question 3, 144 sq. in. from 864 sq. in. If 
 1 sq. ft. contains 144 sq. in., how many sq. ft. are there in 864 
 sq. in. ? 
 
 6. Subtract, as in question 3, 4840 sq. yd. from 53,240 sq. yd. 
 If 1 A. contains 4840 sq. yd., what is the number of A. in 
 53,240 sq. yd. ? 
 
 7. Subtract, as in question 3, 1728 cu. in. from 15,552 cu. in. 
 If 1 cu. ft. contains 1728 cu. in., how many cu. ft. are there in 
 15,552 cu. in. ? 
 
 8. Subtract, as in question 3, 365 da. 3 times, and 366 da. once, 
 from 2922 da. How many yr. are there in 2922 da. ? 
 
 1; 
 
 il 
 
 ■1 ?! 
 
 Exercise 23 
 
 Solve the following questions and verify your answers. 
 
 1 . Subtract $ 819 from f 918, explaining the process. 
 
 2. A speculator sold cattle at a loss of $3145 and some horses 
 at a gain of f 2578. How much did he lose on both transactions ? 
 
 3. A merchant exchanges a stock of goods worth $6725, and 
 a house worth $3120, with a farmer for a farm valued at $5900, 
 
 
 5 
 
 ^•1 
 
 I: 
 
 m 
 
 t; 
 
 i > 
 
 V 
 
 m 
 
40 
 
 ARITHMETIC 
 
 
 
 il 
 
 
 P Hi 
 
 (IHfl 
 
 p 111. 
 
 the farmer paying the balance in money. What sum must t' 
 merchant receive ? 
 
 4. If the measured quantity be 0748 bbl. of sugar, and one 
 of the parts is 1987 bbl., find the other part. 
 
 5. Make and solve a question in which the whole quantity 
 and one part are given. 
 
 6. A lends B f) 1)780; B repays A by giving him bank stock 
 to the amount of J$ 194(), a farm worth $0385, and the balance in 
 cash. How much cash did 13 pay A ? 
 
 7. A is worth |0215, B is worth $870 less than A, and C is 
 worth as much as A and B together, lacking $ 2o43. How much 
 are B and C worth, respectively ? How much are all three worth ? 
 
 8. How much larger is Lake Erie than Lake Ontario ? Lake 
 Superior than Lake Michigan ? The total areas of the three 
 smaller lakes than Lake Superior ? (For the areas of these lakes 
 see Ex. 13, question 0, in Addition.) 
 
 If 100 sq. mi. is the unit, what number expresses these 
 ditferences ? 
 
 9. What is tlie difference between 043 and 579 'v ^ the unit 
 of value is f 1? $10? $100? $1000? $10,000? ifi) 100,000? 
 $1,000,000? 
 
 10. How much greater are 253 units of $ 1000 than 1804 units 
 of $ 100 ? What number expresses the difference when $ 100 is 
 the unit of measure ? When $ 10 is the unit of measure ? 
 
 11. A man bought a house and lot for $8450. He spent 
 $ 1379 in improvements and $ 212 for insurance. He then sold 
 the house and lot for $12,000; did he gain or lose, and how 
 much ? 
 
 12. A collector received $1300 from five men; from the first 
 he received $307, from the second $194 less than from the first, 
 from the third $ 30 more than from the second, from the fourth 
 as much as from the second and third together. How much did 
 he collect from the fifth ? 
 
 i 1 
 
 
SUBTRACTION 
 
 41 
 
 13. From the difference between 784 and 8395, take the 
 difference between 17,012 and 21,410. 
 
 14. Two men start from the same poini} and travel in the 
 same direction. The first travels 84 mi. in one day and the 
 second G9 mi. How far were they apart at the end of the first 
 day ? If they had travelled in opposite directions, how far would 
 they have been apart ? 
 
 15. The population of Texas in 1890 was 2,235,523, and of 
 Illinois, 3,82G,.'>51. How much greater was the population of 
 Illinois in 1890 than that of Texas ? What is the unit in this 
 question ? 
 
 16. The length of the St. Lawrence River is 2000 mi., and 
 the area drained by it is 350,000 sq. mi. The length of the 
 Amazon is 4000 mi., and the area drained by it is 2,500,000 
 sq. mi. What is the ratio of the length of the Amazon to that 
 of the St. Lawrence ? Show by subtracting 350,000 sq. mi. 
 successively from 2,500,000 sq. )ni., how many times greater is 
 the area drained by the Amazon than, that drained by the St. 
 LawrencC;, and find the lemainder. 
 
 17. The area of Texas is 205,780 sq. mi., of England, 50,800 
 sq. mi., and of Germany, 208,700 sq. mi. How much larger 
 is Texas than the united area of England and Germany? 
 
 18. The population of Kew York State in 1870 was 4,387,404, 
 in 1880 it was 5,082,871, and in 1890 it was 5,997,853. What 
 was the increase in population from 1870 to 1880 ? From 1880 
 to 1890 ? How much greater ^vas the latter increase than the 
 former ? 
 
 19. If the colored population of South Carolina in 1890 was 
 089,141, and the total population 1,151,149, how much larger 
 was the colored population in 1890 than the white popuhation ? 
 
 20. The area of Texas is 265,780 sq. mi., of Illinois 56,650 
 sq. mi., of Oklahoma 39,030 sq. mi. and of the District of Co- 
 lumbia 70 sq. mi. If 10 sq. mi. be cut off of Texas for an Indian 
 
 ^1 
 
 Ml 
 
 ^ 
 
 ! m 
 
 li!' 
 
42 
 
 ARITHMETIC 
 
 M 
 
 
 IN ti 
 
 m 
 
 m 
 
 Reservation, into how many states can the remainder be divided, 
 making as many as possible of the size of Illinois, and then of 
 Oklahoma, and of the Disiricc of Columbia? 
 
 46. From 25.3846 take 18.6397. 
 
 We write units under units, tenths under tenths, and so 
 on. lieginninfi; at the right, we subtract as if the figures were 
 integers, and place the decimal point in the difference between 
 the units' and the tenths' column. 
 
 Do this problem by the computers^ method. 
 
 25.3846 
 18.6397 
 
 "6.7449 
 
 Find the difference : 
 
 1. 26.437 
 15.254 
 
 Exercise 24 
 
 2. 94.568 
 29.783 
 
 3. 
 
 102.4951 
 
 58.2876 
 
 From : 
 
 4. 75.093 take 34.267. 
 
 5. 6.4297 take 3.5824. 
 
 6. 41.7453 take 27.937. 
 
 7. 3.1111 take 1.4682. 
 
 8. 3.1416 take .9885. 
 
 9. A car containing 24.875 T of coal was divided between 
 A and B. A received 11.375 T. ; what did B get ? 
 
 10. Show by successive subtractions that a field containing 
 52.584 A. can be divided into 6 fields, each containing 8.764 A. 
 
 
 m 
 
 
 
 fe>' 
 
CHAPTER V 
 
 MULTIPLICATION 
 
 47. 1. Beginning with $2, add by $2, till you reach J!^ 26. 
 What are the $ 2 called ? Addends. What the result ? Sum. 
 
 2. In getting this swm have you definitely thought of how 
 many $2 there are? No. Do you know from the sum how 
 many there are ? No. 
 
 3. If you add f 2 to $ 2, etc., till you reach the sum, f 182, do 
 you know how many twos there are ? No. 
 
 4. How do you look upon the sum $ 2G (say) and the $ 2 ? 
 The f 26 is simply the sum of an unknown number of % 2. 
 
 5. Now count the number of ^2. There are 13. Did you 
 think of this 13 in the addition process ? No. 
 
 6. Now consider this 13 in relation to the addend $ 2, and the 
 sum $ 26, what neiv idea is introduced ? The idea of hoio many 
 times f 2 is repeated to make $ 26? 
 
 7. Then what is the number \\\\\c\). measures $26? 13. What 
 is the unit of measure? $2. From what you know of number 
 say what ratio 13 is ? The ratio of $ 26 to $ 2. W^e say at OTvce 
 (without adding) that 13 times $ 2 is $ 26. 
 
 8. In this do we depend at all on addition ? Yes. We first 
 find the sum, and connect this in memory with the number of 
 times the addend is repeated. 
 
 9. But is it then correct to say that the processes f 2 4- $2 -f 
 $ 2 ... = $ 26, is identical with the process 13 x f 2 = f 26 ? No; 
 
 43 
 
 1 « 
 » 
 
 .t • 
 
 ■it 
 
 
 I 
 
•I' k 
 
 44 
 
 ARITHMETIC 
 
 for 13 represents the " new idea " referred to, and f 2 has 
 become a definite unit of measure, which witli 13 denotes the 
 quantity $2G. The addend has become a factor, and the sum 
 a product. 
 
 I 
 
 I 
 
 I 
 
 i 
 
 
 y 
 
 48. Find the cost of 9 yd. of cloth at $5 a yd. 
 
 (1) Here we think of $5 as a derived unit measuring the value of 1 yd. 
 Hence the cost of 9 yd. is equal to 9 x $ 5, or to f 45. 
 
 (2) Thus 45, the number of primary units in the total cost, is called the 
 product of the nujnber of primary units in the derived unit $5, vsrhich is 5, 
 by the number of units, viz. 9, in the given quantity of cloth. 
 
 (3) In the above example the total cost was given by 9 units of $5 each, 
 and after mnltiplication by 45 units of $ 1 each. Thus multiplication does 
 not change the total cost {i.e. the measured quantity); it changes only the 
 number which measures it (in this case from 9 to 45) by changing the unit 
 of measure, $5, to the primary unit, $1. 
 
 The numbers to be nmltiplied together, viz. 9 and 5, are called factors of 
 the product, i.e. of the number that measures the quantity. 
 
 49. Multiplication is the operation of finding the number 
 of primary units in a quantity expressed by a given number 
 of derived units, or, more briefly, 
 
 Multiplication is the operation of finding the product of 
 two numbers. 
 
 The Multiplicand is the derived unit of measure. 
 
 The Multiplier denotes how many times this unit of meas- 
 ure is to be repeated, i.e. it denotes the ratio of the measured 
 quantity to the unit of measure. 
 
 I 
 
 50. 8 X 16 is read 8 times |G. 
 
 16 X 8 is read 16 nmltiplied by 8. 
 X is the Sign of Multiplication. 
 
 Hir 
 
MULTIPLICATION 
 
 45 
 
 Multiplication Table 
 
 Twice 
 
 Three times 
 
 Four times 
 
 Five times 
 
 Six 
 
 times 
 
 Seven times 
 
 1 is 2 
 
 1 is 3 
 
 1 is 4 
 
 1 
 
 is 5 
 
 1 
 
 is 6 
 
 1 is 7 
 
 2 " 4 
 
 2 " 6 
 
 2 " 8 
 
 2 
 
 " 10 
 
 2 
 
 " 12 
 
 2 " 14 
 
 3 " 6 
 
 3 " 
 
 3 " 12 
 
 3 
 
 " 15 
 
 3 
 
 " 18 
 
 3 " 21 
 
 4 " 8 
 
 4 " 12 
 
 4 " 16 
 
 4 
 
 " 20 
 
 4 
 
 " 24 
 
 4 " 28 
 
 5 " 10 
 
 5 " 15 
 
 5 " 20 
 
 5 
 
 " 25 
 
 5 
 
 " 30 
 
 5 " 35 
 
 6 " 12 
 
 6 " 18 
 
 6 " 24 
 
 6 
 
 " 30 
 
 6 
 
 " 36 
 
 6 " 42 
 
 7 " 14 
 
 7 " 21 
 
 7 " 28 
 
 7 
 
 " 35 
 
 i 
 
 " 42 
 
 7 " 49 
 
 8 " 16 
 
 8 " 24 
 
 8 " 32 
 
 8 
 
 '' 40 
 
 8 
 
 " 48 
 
 8 " 56 
 
 9 " 18 
 
 9 " 27 
 
 9 '■ 36 
 
 9 
 
 " 45 
 
 9 
 
 " 54 
 
 9 " 63 
 
 10 " 20 
 
 10 " 30 
 
 10 " 40 
 
 10 
 
 " 50 
 
 10 
 
 " 60 
 
 10 " 70 
 
 11 " 22 
 
 11 " 33 
 
 11 " 44 
 
 11 
 
 " 55 
 
 11 
 
 " 66 
 
 11 " 77 
 
 12 " 24 
 
 12 '' 36 
 
 12 " 48 
 
 12 
 
 " 00 
 
 12 
 
 " 72 
 
 12 " 84 
 
 
 ■I ' 
 
 Eight times 
 
 Nine times 
 
 Ten times 
 
 Eleven times 
 
 Twelve times 
 
 1 is 8 
 
 1 is 9 
 
 1 is 10 
 
 1 is 11 
 
 1 is 12 
 
 2 " 16 
 
 2 " 18 
 
 2 " 20 
 
 2 ' 
 
 ' 22 
 
 2 " 24 
 
 3 " 24 
 
 3 '' 27 
 
 3 " 30 
 
 3 ' 
 
 ' 33 
 
 3 " 36 
 
 4 " 32 
 
 4 " 36 
 
 4 " 40 
 
 4 ' 
 
 i 44 
 
 4 " 48 
 
 5 " 40 
 
 5 " 45 
 
 5 " 50 
 
 5 ' 
 
 ' 55 
 
 5 " 60 
 
 6 " 48 
 
 6 " 54 
 
 6 " 60 
 
 6 ' 
 
 ' 66 
 
 6 " 72 
 
 7 " 56 
 
 7 " 63 
 
 7 " 70 
 
 7 ' 
 
 . 77 
 
 7 " 84 
 
 8 " 64 
 
 8 " 72 
 
 8 " 80 
 
 8 ' 
 
 ' 88 
 
 8 " 96 
 
 9 " 72 
 
 9 " 81 
 
 9 " 90 
 
 9 ' 
 
 ' 99 
 
 9 '^ 108 
 
 10 " 80 
 
 10 " 90 
 
 10 " 100 
 
 10 ' 
 
 ' 110 
 
 10 " 120 
 
 11 " 88 
 
 11 '' 99 
 
 11 " 110 
 
 11 ' 
 
 ' 121 
 
 11 " 132 
 
 12 '< 96 
 
 12 " 108 
 
 12 " 120 
 
 12 ' 
 
 ' 132 
 
 12 " 144 
 
 I 
 
 (N 
 
 Hi 
 
r 
 
 46 
 
 ARITHMETIC 
 
 I'. 
 
 
 •fe 
 
 m 
 w 
 
 h U 
 
 lli 
 
 t 
 
 ^p 
 
 -srs 
 
 id 
 
 51. Develop at least a portion of the multiplication table by measuring. 
 Let a line be drawn on the board at least 86 in. long, and let 12 strips of 
 paper be cut, respectively, 1 in., 2 in., 3 in., •••, 12 in. long, representing 
 different units of measure. 
 
 To develop, for instance, the table of 3's, measure along the line 3 
 times with each of these measures. Then with a yard ruler divided into 
 inches measure each result in turn. 
 
 Hence 3x1 in. = 3 in. 
 
 3x2 in. = 6 in. 
 3x3 in. = 9 in. 
 etc. = etc. 
 3 X 12 in. = 36 in. 
 
 52. From the above diagram it is evident that 32 is equal 
 to 4 X 8, or 8x4. 
 
 Arrange the dots to show that 32 is equal to 2 x 16, or 
 16 x2. 
 
 How often is 32 measured by 4 ? 8 ? 2 ? 16 ? 
 
 Exercise 25 
 
 1. Arrange dots, representing any nnits, to show that 28=4 x 7, 
 or 7x4. 
 
 2. Give the factors of 45 (9 x 5 or 5 x 9), GCy, 56, 72, 96, 63, 
 90, 54, 99, 84, 132, 108. 
 
 3. Give the factors of 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. 
 
 4. Arrange 30 dots to show how often 30 is measured by 10. 
 
 5. How often is 72 measured by 9? 8 ? ? 12? 4? 18? 3? 
 24? 2? 36? 
 
 6. If one factor of 96 is 12, wliat is the other ? If one factor 
 is 8, what is the other ? 
 
 ta| 
 h( 
 
 ;n 
 
MULTIPLICATION 
 
 47 
 
 7. What will 5 yd. of cloth cost at |4 a yd.? What will 
 4 yd. cost at f 5 a yd. ? 
 
 8. If 9 men can do a piece of work in 6 da., how long will it 
 take 1 man to do it ? If 6 men can do a piece of work in 9 da., 
 how long will it take 1 man to do it ? 
 
 53. Suggestions regarding the Multiplication Table. 
 
 1. The tables of 10 and 11 present no difficulty, the product being 
 directly associated with the digit. 
 
 2. The table of 9 can be similarly remembered. The product is made 
 up of tens and units. In the table (up to 10 x 9) the tens' digit is always 
 one less than tlie number multiplied. The sum of the digits is 9. 
 
 3. In the table of 5, the unit digit is 5 for odd multiplicands, and for 
 even ones. 
 
 4. By association ; thus if 6 x 9 = 54, then 9 x G = 54. 
 
 5. Require the table to be memorized in regular order ; also, so that it 
 can be given by the pupil in irregular order, thus : 9 x 4 = 30, 9 x C = 54, 
 9 X 10 = 90, etc. 
 
 G. Drill, requiring instantaneous oral and written answers to such ques- 
 tions as : What is 6x7? 9x8? 8x9? 
 
 7. Drill, requiring instantaneous answers : What isGx7+4? 9x8 + 3? 
 8x9 + 7? 
 
 8. Extend the table thus : 5 x 70 = ? 7 x 800 = ? 4 x 120 = ? 
 
 9. Drill thus : 42 - 7 = ? 84 - 12 = ? 54 - 8 = ? 
 
 10. Give two factors, 
 each less than 12, of 3G, 
 54, etc. 
 
 11. What part of 28 
 is 12, IG ? Of 36 is 20, 
 28 ? (using the table of 
 
 4 as a basis). 
 
 12. What is the quan- 
 tity whose ratio to the 
 unit $9 is ecjual to 6? 
 
 13. How many in. in 
 
 5 ft. 4 in. ? How many 
 da. in 4 wk. 6 da. ? 
 
 Scale II m. s I fn, 
 
 1 
 
 A ^ 
 
 
 1 1'. H 
 
 ^H 
 
 ' 91 
 
 i 
 
 \i 
 
48 
 
 ARITHMETIC 
 
 I 
 
 54. (1) Find the area of an oblong 5 in. long and 3 in. 
 wide. 
 
 Let the oblong be divided into 3 strips by lines 1 in. apart, as in the 
 figure. 
 
 The area of 1 strip = 5 sq. in. 
 .*. the area of the oblong = 3 x 5 sq. in. 
 
 = 15 sq. in. 
 
 In this figure there are two units of measurement. The smaller or pri- 
 mary unit of area is 1 sq. in., and is repeated 5 times to make the larger 
 unit, or the strip ; the larger or derived unit of one strip is 5 sq. in. and is 
 repeated 3 times to make the oblong which is now measured. Make a 
 rectangle 5 in. long and 3 in. wide. Divide it as in the figure and make a 
 mental picture of the resulting figure. 
 
 (2) Reduce 9 pk. 6 qt. to qt. 
 
 9 pk. = 9 X 8 qt. = 72 qt. 
 .-. 9 pk. qt. = 78 qt. 
 Here the problem is to add 6 qt. to 9 units of 8 qt. each. 
 
 
 '•n 
 
 
 fV'h- 
 
 m 
 
 I 
 
 m 
 
 rf 
 
 ft- 
 
 Scale : J in = I in. 
 
 (3) Find the volume of a rectangular solid 5 in. long, 
 
 4 in. wide, and 3 in. tliick. 
 
 Let the solid be divided into 3 slices by horizontal planes 1 in. apart. 
 Let the upper slice be divided into 6 rows by vertical planes 1 in. apart. 
 
MULTIPLICATION 
 
 49 
 
 Let the right-hand row be divided into 4 cu. in, by vertical planes 1 in. 
 apart. 
 
 Tlie volume of 1 row = 4 cu. in. 
 The volume of 5 rows or 1 slice = 5 x 4 cu. in. 
 The volume of 3 slices or the solid = 3 x 5 x 4 cu. in. 
 
 = 60 cu. in. 
 
 In this solid, the three units of volume, in order of size, are the primary 
 unit or 1 cu. in., and the derived units, i.e. the rows or 4 cu. in., and the 
 slice or 6x4, i.e. 20 cu. in. 
 
 The solid is made up of how many units of each kind ? Each unit is 
 made up of how many of the next smaller? 
 
 
 •:« 
 
 Exercise 26 
 
 In the following exercise, where possible, make drawings and 
 mental pictures : 
 
 1. Find the area of tlie following oblongs, draw the figure, 
 and name the primary and derived units of area: 4 in. by 6 in.; 
 7 in. by 9 in. ; 8 in. by 11 in.; 9 in. by 12 in. What is the ratio 
 of the area of the oblong to the primary unit ? To the derived 
 unit ? What is the ratio of the derived to the primary unit ? 
 
 2. Find the number of sq. ft. in a sq. yd. ; of sq. in. in a 
 sq. ft. 
 
 3. Find the area of the floors of the following rooms whose 
 dimensions are: 6 yd., 7 yd.; 6 yd., 8 yd.; 8 yd., 9 yd.; 11 yd., 
 12 yd. 
 
 4. Find the area of squares whose sides are: 1, 2, 3, 4, 5, 6, 
 7, 8, 9, 10, 11, and 12 in., respectively. 
 
 5. Find the area of a room made up of 8 units of 9 sq. yd. 
 
 6. If the unit of length is 4 ft., what is the unit of area ? 
 
 7. What is the volume whose smallest unit, 1 cu. in., is re- 
 peated 3 times to make the next larger, this (> times to make the 
 next larger, and this 8 times to make the volume ? 
 
 8. Find the volumes of rectangular solids whose dimensions 
 are : 2, 3, and 4 in. ; 2, 4, and 9 in. ; 3, 4, and 7 in. ; 2, 5, and 9 
 
 B 
 
 III 
 
50 
 
 ARITHMETIC 
 
 in. What are the volumes of the primary and of the derived 
 units of volume ? 
 
 9. Find the number of eu. ft. in a eu. yd. 
 
 10. If 4 in. is the unit of length, what is the unit of volume ? 
 
 11. Find the volumes of the cubes whose sides are respec- 
 tively 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 in. 
 
 12. Reduce to lower denominations: 
 
 (1) 5 yd. 2 ft. ; 7 yd. 1 ft. ; 8 yd. 2 ft. ; 9 yd. 1 ft. ; 11 yd. 1 ft. ; 
 12 yd. 2 ft. 
 
 (2) 5 ft. 4 in. ; 8 ft. 10 in. ; 7 ft. 8 in. ; 11 ft. 3 in. ; 9 ft. 7 in. ; 
 12 ft. 6 in. 
 
 13. 3 sq. yd. 5 sq. ft. ; 5 sq. yd. 7 sq. ft. ; 12 sq. yd. 6 sq. ft. ; 
 
 8 sq. yd. 8 sq. ft. ; 9 sq. yd. 2 sq. ft. ; 11 sq. yd. 10 sq. ft. 
 
 14. 6 qt. 1 pt. ; 8 qt. 1 pt. ; 11 qt. 1 pt. ; 7 pk. 6 qt. ; 9 pk. 4 qt. ; 
 
 9 bu. 3 pk. ; 8 bu. 2 pk. ; 11 bu. 3 pk. ; 7 gal. 2 qt. ; 9 gal. 3 qt. 
 
 15. 7 wk. 4 da. ; 9 wk. 2 da. ; 11 wk. 6 da. ; 8 wk. 1 da. ; 12 wk. 
 
 5 da. ; 5 hr. 40 min. ; 8 hr. 9 min. ; 9 hr. 22 min. ; 12 hr. 45 min. ; 
 
 6 da. 4 hr. ; 8 da. 2 hr. 
 
 16. How many hr. are there in 1 wk. ? 
 
 17. The perimeter of a room is found by adding twice the 
 width to twice the length. Find the perimeter of rooms whose 
 dimensions are : yd., 8 yd. ; 7 yd., 9 yd. ; 8 yd., 11 yd. ; 11 ft., 
 12 ft; 13 ft., 14 ft. 
 
 t 
 
 \>> 
 
 55. What is the cost of 6 town lots at 1894 a lot? 
 
 Here we think of the whole cost as 6 units of $894 each. 
 Explanation. — The unit $804 may be considered as made up of 4 units 
 of one dollar, 9 units of ten dollars, and 8 units of one hundred dollars. 
 
 6x4 units of one dollar = 24 units of one dollar = 2 units of 
 ten dollars + 4 units of one dollar. 
 
 6x9 units of ten dollars = 54 units of ten dollars. 
 54 units of ten dollars + 2 units of ten dollars =: 56 units of ten 
 dollars = 5 units of one hundred dollars + 6 units of ten dollars. 
 
 '1894 
 6 
 
 16364 
 
 6x8 units of one hundred dollare = 48 units of one hundred dollars. 
 
MULTIPLICATION 
 
 51 
 
 48 units of one hundred dollars + 5 units of one hundred dollars = 63 
 units of one hundred dollars = 5 units of one thousand dollars + 3 units of 
 one hundred dollars. 
 
 5 units of one thousand dollars + 3 units of one hundred dollars + 6 units 
 of ten dollars + 4 units of one dollar = $ 5304. 
 
 56. The method generally followed by the pupil is : 
 
 6 times 4 = 24 ; 6 times 9 -. 54 ; 64 and 2 = 56 ; 6 times 8 = 48 ; 48 and 
 5 = 63. 
 
 This process is too slow. If the pupils have been well drilled in § 53, 
 No. 7, they will be able to add in the digit to be carried instantaneously, and 
 should be trained thus : 
 
 6 X 4 = 24 ; 6 X = 56 (adding in the 2 in one process). 
 6 X 8 = 53 (adding in the 5) . 
 
 Exercise 27 
 
 Multiply separately : 
 
 1. 231 by 2, 4, 6, 8, 10, and 12. 
 
 2. 690 by 3, 5, 1, 9, and 11. 
 
 3. 897 by 4, 6, 8, 10, and 12. 
 
 4. 2463 by 3, 5, 7, 9, and 11. 
 
 5. 5781 by 2, 4, 8, 10, and 12. 
 
 6. 9654 by 3, 5, 7, 9, and 11. 
 
 7. 8267 by 2, 4, 6, 8, 10, and 12. 
 
 8. 5280 by 3, 5, 7, 9, and 12. 
 
 9. 1728 by 2, 4, 6, 8, 10, and 12. 
 
 10. 4840 by 3, 5, 7, 9, and 11. 
 
 11. 63,360 by 2, 4, 6, 8, 10, and 12. 
 
 12. 24,793 by 3, 5, 7, 9, and 11. 
 
 13. 98,654 by 2, 4, 6, 8, 10, and 12. 
 
 14. 89,743 by 3, 5, 7, 9, and 11. 
 
 15. 64,789 by 2, 4, 6, 8, 10, and 12. 
 
 16. Solve the questions in Exercise 12 by multiplication. 
 
I 
 
 52 
 
 ARITHMETIC 
 
 Exercise 28 
 
 1. Find the peri 
 
 meters, 
 
 and also the 
 
 areas 
 
 of the four walls 
 
 of rooms whose dimensions 
 
 are: 
 
 
 
 Length 
 
 
 Width 
 
 
 Height 
 
 16 ft. 
 
 
 14 ft. 
 
 
 8 ft. 
 
 17 " 
 
 
 15 " 
 
 
 8 " 
 
 20 " 
 
 
 18 " 
 
 
 9 " 
 
 22 
 
 a 
 
 20 
 
 a 
 
 12 
 
 u 
 
 How 
 
 2. A cd. of wood is 8 ft. long, 4 ft. wide, 4 ft. high, 
 many cu. ft. does it contain ? 
 
 3. Find the number of cu. in. in a cu. ft. 
 
 4. A gal. of water will exactly fill a rectangular box 11 in. 
 long, 7 in. wide, and 3 in. high. Fnid the number of cu. in. in 
 a gal. 
 
 5. If there are 1760 yd. in 1 mi., find the number of yd. in 
 
 2 mi. ; 5 mi. ; 8 mi. ; 9 mi. ; 12 mi. 
 
 6. If there are 5280 ft. in 1 mi., find the number of ft. in 
 
 3 mi. ; 6 mi. ; 7 mi. ; 9 mi. ; 11 mi. 
 
 7. Reduce to sq. in. : 5 sq. ft. ; 8 sq. ft. ; 10 sq. ft. ; 12 sq. ft. 
 
 8. Reduce to sq. ft. : 3547 sq. yd.; 8426 sq. yd.; 9819 sq. yd. 
 
 9. If 1 sq. mi. contains 640 A., find how many A. there are 
 in 6 sq. mi. ; 8 sq. mi. ; 10 sq. mi. ; 12 sq. mi. 
 
 10. Reduce to cu. in. : 4 cu. ft. ; 5 cu. ft. ; 7 cu. ft. ; 9 cu. ft. ; 
 
 11 cu. ft. 
 
 11. Reduce to da. : 453 wk. ; 769 wk. ; 827 wk. ; 852 wk. 
 
 12. Reduce to qt. : 765 gal. ; 917 gal. ; 763 gal. ; 789 gal. 
 
 13. Reduce to qt. : 735 pk. ; 892 pk. ; 679 pk. ; 728 pk. 
 
 14. Reduce to da. : 3 yr. ; 5 yr. ; 6 yr. ; 8 yr. ; 11 yr. ; 
 
 12 yr. (1 yr. = 365 da.) 
 
 15. If there are 640 A. in 1 sq. mi., find the number of A. 
 in 6 sq. mi. ; in 9 sq. mi. ; in 12 sq. mi. 
 
Mill 
 
 MULTIPLICATION 
 
 53 
 
 16. Show that 4 units of one dollar multiplied by 7 tens is 
 equal to the product found by multiplying 4 units of ten dollars 
 by 7. 
 
 17. Show that 9 units of ten dollars multiplied by 7 tens is 
 equal to G3 units of one hundred dollars. 
 
 57. If the number expressing the ratio of the measured 
 quantity to the unit of measure ^894 is 76, what is the 
 quantity ? 
 
 <$ 894 '^ '^^ explanation is similar to that given in § 55. Since, when 
 
 _- 7 is used as a multiplier, the 4 units of one dollar are multiplied 
 
 ____ by 7 tens, the product is the same as that found by multiplying 
 
 «5364 4 units of ten dollars by 7. This is 28 units of ten dollars, and is 
 
 6258 6qual to 2 units of one hundred dollars and 8 units of ten dollars. 
 
 Hence the 8 is written under the G in the tens' column, and the 
 
 ^ 0< 944 2 is carried to be added in the hundreds' column, and so on. 
 To prove the answer correct, multiply 70 by 894 ; thus : 
 
 76 
 
 894 
 
 304 
 
 684 
 
 608 
 
 .-. the answer is correct. 67944 
 
 Exercise 29 
 
 Multiply and prove your answers correct : 
 
 1. 423 by 36. 
 
 2. 479 by 32. 
 
 3. 295 by 16. 
 
 4. 798 by 24. 
 
 5. 581 by 52. 
 
 6. 649 by 27. 
 
 11. 
 
 8647 by 
 
 365 
 
 12. 
 
 7245 by 
 
 168 
 
 13. 
 
 8939 by 
 
 224 
 
 14. 
 
 655S by 
 
 144 
 
 15. 
 
 9275 by 
 
 231 
 
 7. 959 by 24. 
 
 8. 764 by 31. 
 
 9. 953 by 56. 
 10. 825 by 48. 
 
 16. 9475 by 1760. 
 
 17. 8213 by 5280. 
 
 18. 4781 by 1728. 
 
 19. 5893 by 2240. 
 
 20. 6439 by 1728. 
 
 ii 
 
 ii 
 
 ''l4ij 
 
 1^ 
 
 t'l: 
 
I y 
 
 h 
 
 ! 
 
 54 
 
 ARITHMETIC 
 
 Multiply : 
 
 1. 8245 by 684. 
 
 2. 7()39 by 797. 
 
 3. 5927 by 395. 
 
 4. 4399 by 927. 
 6. 8999 by 868. 
 
 Exercise 30 
 
 6. 8746 by 675. 
 
 7. 9687 by 897. 
 
 8. 4786 by 478. 
 
 9. 9467 by 769. 
 10. 8769 by 567. 
 
 I < 
 
 la i 
 
 i 
 
 Exercise 31 
 
 In the following questions state which is the unit of measure 
 and which is the number : 
 
 1. If 1 mi. contains 320 ril., find the number of rd. in 2897 mi. 
 
 2. If 1 mi. contains 1760 yd., find the number of yd. in 1679 mi. 
 
 3. If 1 mi. contains 5280 ft., find the number of ft. in 834 mi. 
 
 4. If 1 sq. ft. contains 144 sq. in., find the number of sq. in. in 
 a rectangle 27 ft. long and 18 ft. wide. 
 
 5. Find the number of sq. ft. in a garden, the shape of an 
 oblong, which is 16 yd. long and 14 yd. wide. 
 
 6. If 1 sq. mi. contains 640 A., how many A. are there in a 
 township containing 36 sq. mi. ? 
 
 7. Find the value of 6 units of land at $ 85 a unit. 
 
 8. If 1 in. be taken as the unit of length, how many units of 
 area are there in the surface of a box whose dimensions are 
 respectively 2, 3, and 4 of the next higher unit of length ? 
 
 9. If 1 ft. be taken as the primary unit of length, what is 
 the number of the primary units of volume in a rectangular solid 
 whose dimensions contain 4, 5, and 6 of the next higher unit of 
 length? 
 
 \i i 
 
^iH 
 
 MULTIPLICATION 
 
 56 
 
 10. If the unit of length is 2 ft., what is the unit of volume ? 
 How many cu. ft. are there in the volume of a rectangular solid 
 whose dimensions contain respectively »^, 4, and 5 such units 
 of length ? 
 
 11. What is the area of the surface of this solid in square 
 feet? 
 
 12. If 1 cu. ft. contains 1728 cu. in., find the number of cu. 
 in. in a rectangular solid 8 ft. long, 4 ft. wide, and 4 ft. high. 
 
 13. If 1 cu. yd. contains 27 cu. ft., find the number of cu. ft. 
 in a rectangular solid Avhose dimensions are 0, 4, and 3 yd. 
 
 14. If 1 cd. of wood contains 128 cu. ft., how many cu. ft. are 
 there in 936 cd. ? 
 
 15. If 1 gal. contains 231 cu. in., how many cu. in. are there in 
 a vessel which contains 888 gal. ? 
 
 16. A quantity contains the unit 365 da. 896 times. Find the 
 quantity. 
 
 58. (1) A drover bought 36 horses at $145 a head, and 96 
 cows at $28 a head. Which cost the more, and how much ? 
 
 Here the problem is to find the difference between 3G units of $ 145 each 
 and 96 units of $28 each. 
 
 (2) A's barn cost $175, his house 16 times as much, and his 
 farm cost as much as both. What was the cost of all ? 
 
 Here the problem is to find the cost of 1 + 16 -f 17, or 34 units of $ 175 
 each. 
 
 Exercise 32 
 
 1. Exemplify the truth that two or more factors will give the 
 same product in whatever order they are multiplied. 
 
 2. A speculator bought 150 head of cattle and 47 mules. He 
 made a profit of $ 13 a head on the former and f 17 each on the 
 latter. What was gained by the speculation ? 
 
 
 iiij 
 
m 
 
 56 
 
 ARITHMETIC 
 
 
 h 
 
 K*-! 
 
 
 rr'" 
 
 3. A sliip sailed HO hr. at the rate of 11 mi. per hr., when she 
 encountered a storm of 10 liours' duration, which drove her back 
 at the rate of 14 mi. per lir. How far from port was she at the 
 expiration of the 72 hr. ? 
 
 4. A is worth $ 12G5, B is worth 4 times as much as A and 
 $ 183, and C is worth 8 times as much as A and B lacking f 2343. 
 How much are B and C worth, respectively ? How much are 
 they all worth ? 
 
 5. Make out a bill for the following goods: 
 
 23 yd. cotton @ 11^ ; 13 yd. gingham @ 23^ ; 
 25 yd. flannel @ 37^ ; 18 yd. tweed @ $ 1.50 ; 
 12 yd. serge @ f 1.75; 6 yd. broadcloth @ f 4.50. 
 
 6. A produce merchant exchanged 48 bu. of oats at 39^ per 
 bu. and 13 bbl. of apples at f 3.85 a bbl. for 200 lb. of butter at 
 37^ a lb. How much should he pay to settle the account ? 
 
 7. A grain dealer buys 4795 bu. of wheat in Chicago at 63^ a 
 bu., and ships it to New York at a cost of 3^ a bu. Find his 
 gain if he sells it in New York for 71^ a bu. 
 
 8. A man bought 51 horses at $ 97 each, and sold them at 
 $ 136 each. How much did he gain? 
 
 9. Find the amount of the following bill : 
 
 63 brooms, at 16^ each ; 
 13 yd. print, at 11^ per yd. ; 
 17 lb. tea, at 35^ per lb. ; 
 4 doz. oranges, at 4^ each ; 
 287 lb. sugar, at 5^ per lb. ; 
 84 eggs, at 13^ per doz, 
 
 10. Bought oranges at the rate ot 18^ a doz., and sold them 
 at the rate of 6 oranges for 15^. How much did I gain on 
 11 boxes, each containing 20 doz.? 
 
 11 
 
 
T-''^ 
 
 MULTTPLTCATTON 
 
 57 
 
 11. If in question 10 two boxes were spoiled, what would have 
 been the gain ? 
 
 12. A grocer buys 150 lb. of coffee at 23^ a lb., and 39 lb. of 
 chiccory at (J^ a lb. He pays a duty of 2^ a lb. on each, and 
 mixes and sells the mixture at 33^ a lb. Find his profit. 
 
 13. A book contains 457 pages, each page containing 39 lines, 
 averaging 13 words to a line. Find the number of words in the 
 book. 
 
 14. How far will a bicyclist travel in 27 da., if he travels 9 hr. 
 a da. at 12 mi. an hr. ? 
 
 15. Two vessels start from the same point and travel, the one 
 down a river at the rate of 15 mi. an hr., the other up the river at 
 the rate of 9 mi. an hr. How far will they be apart in 8 hr. ? 
 
 16. If the first vessel travelled up the river at the rate of 12 mi. 
 an hr., how far apart would they be in 8 hr, ? 
 
 17. A speculator bought 45 A. of land at $ 05 an A., and 63 A. 
 at $ 78 an A. If he sold the whole at $ 75 an A., how much did 
 he gain or lose ? 
 
 59. Multiply .048 by 6. 
 
 :948 
 6 
 
 5.688 
 
 948 thoiLsaiuUliR inuUipUed by 6 equals 5088 thousandths or 
 5.088. 
 
 Exercise 33 
 
 Multiply: 
 
 1. .5 by 9. 
 
 2. .8 by 3. 
 
 3. .26 by 3. 
 
 4. .39 by 8. 
 
 5. .624 by 3. 
 
 6. .842 by 9. 
 
 7. .1416 by 25. 
 
 8. .988 by 76. 
 
 •I 
 
 J Hi a 
 
 
58 
 
 ARITHMETIC 
 
 
 f 
 
 i 
 
 9. 3.54 by 12. 12. 3.295 by 16. 
 
 10. .543 by 36. 13. .7568 by 144. 
 
 11. 4.79 by 32. 14. 8.754 by 172. 
 
 15. The circumference of a circle is 3.1416 times the diameter. 
 What is the circumference of a circle whose diameter is 27 yd. ? 
 
 16. Find the number of sq. yd. in 1 sq. ch., there being 30.25 
 sq. yd. in 1 sq. rd., and 16 sq. rd. in 1 sq. ch. 
 
 17. A. cu. ft. contains 7.48 gal. of water. Find how many gal. 
 can be poured into a tin-lined box, whose interior dimensions are 
 3 by 4 by 6 ft. 
 
 18. Find the weight of a rectangular solid of oak, 3 ft. by 
 2 ft. by 1 ft., weighing 47.375 lb. per cu. ft. 
 
 19. A drover bought 12 sheep at $ 5.375 per head, 36 at 
 $4,625, and 212 at Jj^ 4.125. Find the total cost. 
 
 20. Multiply each of the following numbers by 10 : .43; .576; 
 4.23; .017; 89.4263. 
 
 21. Write down the product of each of the following numbers 
 multiplied by 10 : 7.4 ; 8.94() ; 5.32 ; .008 ; 62.9347. 
 
 22. State how to write the product obtained by multiplying a 
 decimal by 10. 
 
 23. Multiply by 100 each of the following: .435; 8.027; 9.12; 
 46.5928. 
 
 24. State how to write the product obtained by multiplying a 
 decimal by 100. I^y 1000. By 10,000. 
 
 l:^ 
 
 I 
 
 a ft 
 
 hi 
 
■J 
 
 ■I* 
 
 t 
 
 CHAPTER VI 
 
 DIVISION 
 
 60. What will 4 oranges cost at 5^ apiece ? If 4 oranges 
 cost 20^, what is the cost of each? What must 50 be multi- 
 plied by to get 20^ ? 
 
 At 15 a yd., how much will 9 yd. of cloth cost? What 
 must $S he multiplied by to get 127? At $S a yd., how 
 many yd. can be bought for $ 27 ? 
 
 61. Of what product are 5 and 6 the factors? (30.) If 5 
 is one factor of 30, what is the other? Of what product are 
 12 and 4 the factors? If 4 is one factor of 48, what is the 
 other? K 6 is one factor of 42, what is the other? If 9 is 
 one factor of 63, what is the other? 4 is one factor of each 
 of the following numbers; what are the other factors? 24, 
 3(3, 20, 28, and 16. 
 
 62. In Multiplication we are given two factors and we are 
 required to find their product. 
 
 In Division, on the other hand, we are given the product, 
 and also one of the factors, and we are required to tind the 
 other factor. 
 
 Thus: Find how many yd. of cloth at $5 a yd. can be 
 bought for 145? 
 
 In this problem, 45, the number measuring the cost of the 
 cloth, is -the product of two factors. One of these is 5. the 
 ffiven number, which measures the value of the unit, and 
 the other is 9, which is the number of yards. 
 
 50 
 
 [f. 
 
 lip 
 
 % 
 
■ 
 
 60 
 
 ARITHMETIC 
 
 El 
 
 If 9 yd. of cloth cost -145, what will 1 yd. cost? 
 As before, 45 is the product of two factors. The given fac- 
 tor is 9, and the required factor, 5. Therefore 1 yd. costs 1 5. 
 
 63. Division is the operation of finding either of two fac- 
 tors, when their product and the other factor are given. 
 
 The factor found is called the Quotient. It shows how 
 often the Divisor is contained in the Dividend. 
 
 The given factor is called the Divisor. 
 
 The given product of the Quotient and Divisor is called 
 the Dividend. 
 
 When the Divisor is not contained an exact number of 
 times, the excess is called the Remainder. See § 71. 
 
 64. The Sign of Division is -^ ; thus |8-4-|2 = 4 is read 
 $ 8 divided by 1 2 is equal to 4. 
 
 1 8 ^ 2 = !| 4, is read $ 8 divided by 2 is equal to $ 4. 
 9-5-3 may also be written |, where 9 is the dividend and 3 
 the divisor. 
 
 65. When the divisor does not exceed 12, the operation 
 can be performed mentally, and the process is called Short 
 Division. 
 
 When all the different steps of the division are written, the 
 process is called Long Division. 
 
 66. Suggestions to the Teacher. — Give questions 
 similar to the following, in order to secure facility in in- 
 terpreting results and accuracy and rapidity in using the 
 nrati plication table. 
 
 Thus : (1) A jn-oduct is 72 ; one factor is 8. Find the other. 
 
 (2) What is 136 ^4? 172-^19? 172-5-9? 132-f-ll? 
 Associate simple practical questions with these numbers. 
 
 (3) Extend the table thus: Divide 210 by 7 ; 3500 by 5; 
 450 by 90 ; 840 by 12. 
 
M i 
 
 DIVISION 
 
 61 
 
 
 (4) Give the quotient and remainder when 86 is divided 
 by 7 ; 93 by 12 ; 43 by 6. 
 
 (5) Reduce to the next higher denomination : 32 qt. ; 40^ ; 
 96 in. ; 45 da. ; 450 min. 
 
 (6) The unit of area is 9 sq. rd. What number expresses 
 the ratio of the area of a lield containing 270 sq. rd. to the 
 unit of area ? 
 
 67. If 1 T. of coal costs 1 6, how many T. will 14764 buy? 
 
 Here the product is ^4704 ; one factor is $6, and we are required to find 
 the other factor, which is the number of T. 
 
 6 )4764 
 794 
 
 6 divides 47 of the hundreds' unit 7 times in the hundreds' place, with a 
 remainder 5 of the hundreds' unit ; 5 of tlie hundreds' unit and 6 of the 
 tens' unit equal 56 of the tens' unit. 
 
 6 divides 56 of the tens' unit times in the tens' place, with a remainder 
 2 of the tens' unit. 2 of the tens' unit and 4 of the one-unit, equal 24, which 
 divided by 6 equals 4. 
 
 .•. the number of T. = 794. 
 
 68. (1) Find the length of an oblong which contains 96 
 sq. in. and is 8 in. wide. 
 
 5 
 
 c 
 
 U 
 
 ''1 
 
 
 w. 
 
62 
 
 ARITHMETIC 
 
 i ih 
 
 Cut off from the oblong a strip 1 ft. wide. This strip contains 8 sq. in. 
 Here we are given the whole quantity, or 90 sq. in., and the measuring unit, 
 8 sq. in. The number 12 gives the immber of primary units of 1 in. con- 
 tained in the length. Therefore the length is 12 in. 
 The area of 1 strip 1 in. wide = 8 sq. in. 
 The number of strips 1 in. wide = 90 sq. in. -^ 8 sq. in. = 12. 
 
 .-. the length = 12 in. 
 
 (2) Find the number of yd. of carpet required to carpet 
 a room 32 ft. long and 26 ft. wide, the carpet running 
 lengthwise, if each strip is 2 ft. wide. 
 
 The number of strips of carpet = 20 ft. -f- 2 ft. = 13. 
 The length of the carpet = 13 x 32 ft, =410 ft. 
 
 = 138 yd. 2 ft. 
 .*. 138 yd. 2 ft. of carpet are needed. 
 
 Make a diagram for this question on the scale of ^ in. to 
 1ft. 
 
 (3) Reduce 6498 da. to wk. 
 
 In this question the time is expressed in terms of the primary unit, 1 da., 
 and we are reciuired to express it in terms of the derived unit, 1 wk. or 7 da. 
 Dividing 0498 da. by 7 da., the result is 928 wk. 2 da. 
 
 Exercise 34 
 
 In the following exercise prove the correctness of your answers : 
 
 1 . Reduce to qt, : 36 pt, ; 78 pt. ; 96 pt, ; 65 pt, ; and 
 257 pts. 
 
 2. Find the number of strips of carpet 2 ft, wide required to 
 carpet rooms respectively 24,, 28, and 32 ft. wide. 
 
 3. Reduce to yd.: 384 ft; 456 ft.; 723 ft.; 807 ft.; 
 5280 ft. 
 
DIVISION 
 
 63 
 
 4. Find the number of strips of carpet 3 ft. wide required to 
 carpet rooms respectively 27, 33, and 39 ft. wide. 
 
 5. Find the number of yards of carpet required to carpet a 
 room 24 ft. long and 18 ft. wide, the carpet being 2 ft. wide and 
 running lengthwise. Draw a diagram. 
 
 6. In question 5, find the number of yards required in case 
 the carpet runs across the room, and draw the diagram. 
 
 7. Reduce to gal. : 576 qt. ; 893 qt. ; 798 (it. ; 962 qt. 
 
 8. Divide by 5 : 475 ; 827 ; 593 ; 890 ; 646. 
 
 9. Divide by 6: 252; 435; 728; 846; 777. 
 
 10. Reduce to wk. : 245 da.; 365 da.; 678 da.; 899 da.; 
 987 da. 
 
 11. Reduce to pk : 32 qt. ; 892 qt. ; 958 qt. ; 2456 qt. ; 
 9472 qt. 
 
 12. Reduce to sq. yd.: 756 sq. ft.; 894 sq. ft.; 3478 sq. ft.; 
 9864 s(i. ft. 
 
 13. Reduce to dimes: 620^; 840^; 729^; 843^; 5246^; 
 8795^. 
 
 14. Divide by 11: 451; 628; 847; 956; 8297; 7887. 
 
 15. How many dozen are there in 842 units ? 957 units ? 1459 
 units ? 4596 units ? 
 
 16. Reduce to ft. : 459 in. ; 897 in. ; 2641 in. ; 63,360 in. 
 
 17. Find the length of the sides of stiuares whose areas are 
 respectively 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, and 144 sq. in. 
 
 18. Find the length of an oblong 6 ft. wide which contains 
 258 S(i. ft. 
 
 19. What is the width of an oblong which contains 15 units of 
 area, the length containing 5 units of length, and the unit of 
 length being 4 f t. ? 
 
 if: 
 
 
■ J 
 
 ] 
 
 64 
 
 arithmp:tic 
 
 Exercise 35 
 
 Find the quotient and remainder and prove your results correct : 
 
 1. 36 -h 2; 48 --3; 72-4; 65-5; 84-6. 
 
 2. 56-7; 96-8; 81-9; 80-10; 77-11; 48 
 
 3. 249-3; 842-6; 941-8; 654-9. 
 
 4. 137 - 2; 439 - 5 ; 849 - 7 ; 999 - 12. 
 
 12. 
 
 5. 
 
 6)743 
 
 11)682 
 
 9)847 
 
 10)895 
 
 6. 
 
 5)679 
 
 4)673 
 
 8)976 
 
 12)899 
 
 7. 
 
 7)2457 
 
 10)6430 
 
 9)1978 
 
 12)4994 
 
 8. 
 
 6)3975 
 
 2)2557 
 
 11)7381 
 
 3)4565 
 
 9. 
 
 6)4544 
 
 12)8256 
 
 5)1935 
 
 4)3191 
 
 10. 
 
 11)8149 
 
 8)7919 
 
 9)4676 
 
 7)6769 
 
 11. 
 
 4)2319 
 
 5)9847 
 
 6)1764 
 
 12)9543 
 
 12. 
 
 11)8682 
 
 8)7798 
 
 9)7992 
 
 12)63360 
 
 Divide by short division : 
 
 13. 200 )14800 200 )14876 300 )14976 
 
 14. 300 )29654 400 )98743 500 )25931 
 
 69. (1) A father dying left an estate valued at $48,832 to 
 
 be divided equally among his wife, his two sons, and his four 
 
 daughters. What was the share of each ? 
 
 Ill this problem we are required to find the share of each, which is the 
 unit of measure. In order to find this, we are given the value of the estate, 
 which is the whole quantity, and one factor, which is the number of shares ; 
 viz. 1 + 2 + 4, or 7. Therefore, dividing the whole quantity by 7, we find 
 the share of each to be $ ()970. 
 
 (2) The area of the four walls of a room whose dimensions 
 are 8 yd. and 6 yd. is 112 sq. yd. Find the height of the 
 room. 
 
 
m 
 
 DIVISION 
 
 65 
 
 We are here required to find the number of yd. in the height of the 
 room. In order to find it, we are given the area, which is the measured 
 whole. 
 
 We are also given the dimensions with which we can find the perimeter of 
 the rooms, and thus find the measuring unit as in §54. 
 
 The perimeter = twice the sum of G -|- 8, or 14 yd. = 28 yd. 
 
 The area of a strip 1 yd. wide running around the room = 28 sq. yd. 
 
 The number of yd. in the height =112 sq. yd. -4- 28 sq. yd. = 4. 
 
 .-. the heiglit = 4 yd. 
 
 
 ' if'. 
 
 ■''*■, 
 
 -(■ 
 
 I 
 
 70. If we multiply 59 by 724, we do so by the following 
 process : 
 
 59 
 
 724 
 
 236 
 118 
 413 
 
 42716 
 
 or, rearranging the work, 
 we have this : 
 
 59 
 724 
 
 413~ 
 118 
 236 
 
 42716 
 
 We now wish to arrive at a method of recovering 724 
 from the dividend 42,716 and the divisor 59. It is evident 
 that if from 42,716 we subtract 413 of the hundreds' digit, 
 which is the product of 59 and 7 times the hundreds' digit, 
 and from the remainder, 118 of the tens' digit, which is the 
 product of 59 and twice the tens' digit, and from this 
 remainder, 236, which is the product of 59 and 4, we shall 
 liave no remainder. 
 
 Hence to divide 42,716 by 59, we have the following 
 method : 
 
 First divide 59 into 427 to get the quotient 7 of the 
 hundreds' digit, multiply 57 by 7, and subtract the prod- 
 uct 413 from 427, leaving 14 of the hundreds' digit, which 
 with the 1 of the tens' digit makes 141 of the tens' digit. 
 
 ': "11 
 
 t i 
 
ee 
 
 ARITHMETIC 
 
 Divide this 141 by 59, and we have the quotient 2 of the 
 tens' digit. Multiply 59 by 2, and subtract the product from 
 141, leaving 23 of the tens' digit, which with the 6 makes 
 236. Divide 236 by 59, and we have the quotient 4. Multi- 
 ply 59 by 4, and subtract the product from 236, and there 
 is no remainder. 
 
 59)42710(724 
 413 
 
 141 
 118 
 
 71. (1) 
 
 236 
 236 
 
 Divisor Dividend Quotient 
 
 31 ) 7598 ( 245 
 62 
 
 139 
 124 
 
 158 
 
 IKK 
 
 3 Remainder 
 
 81 divides 76 of the hundreds' unit two hundred times. Put 2 as the first 
 term in the quotient. 
 
 Multiply 31 by 2, and subtract the product 62 from 75. The remainder is 
 13 of the hundreds' unit. Annex tlie 9 tens of the dividend, making 139 
 tens. 31 divides 139 tens 4 tens times. Put 4 as the second term in the 
 quotient. Multiply 31 by 4, and subtract tlie product 124 from 139. The 
 remainder is 15 of the tens' unit. Annex the 8 units, making 158 units. 31 
 divides 158 units 5 times. Put 5 as the third term in the quotient. Multiply 
 31 by 5, and subtract the product 155 from 158. Tlie remainder is 3. 
 
 What is the quotient on dividing 7598 by 31 ? What does it show ? What 
 is the remainder ? What does it sliow ? See § 63. 
 
DIVISION 
 
 67 
 
 
 I 
 
 
 (2) To prove that the answer in the last example is 
 
 correct : 
 
 245 Quotient 
 
 31 Divisor 
 
 245 
 
 735 
 
 7595 Product 
 3 Remainder 
 
 7598 Dividend 
 
 .'. the answer is correct. Or thus, by division, 
 
 245)7598(31 
 735 
 
 248 
 245 
 
 3 
 
 .-. 245 quotient and 3 remainder is the correct answer. 
 
 72. Divide 39,726 by 87. 
 87)39726(456 
 348 
 
 492 
 435 
 
 576 
 522 
 
 In this division name the unit to which each remain- 
 der and each partial dividend belongs. 
 
 54 
 
 
 i 
 
 73. Trial divisor and trial dividend. 
 
 The work of finding the quotients can be much simplified by using the 
 trial divisor and trial dividend. 
 
 Thus in § 71, as 31 is nearer 30 than 40, the trial divisor is 3. Dividing 3 
 into the trial dividends 7, 13, and 15, the quotients are 2, 4, and 5. 
 
 In § 72, as 87 is nearer 00 than 80, the trial divisor is 9. Dividing 9 into 
 the trial dividends 39, 49, and 57, the quotients are 4, 5, and 6. 
 
 ili^ 
 
68 
 
 ARITHMETIC 
 
 In general, if the divisor is Gl, 02, m, 015, 027, or 034, the trial divisor 
 isO. 
 
 If the divisor is 07, 08, 09, 075, 080, or 007, the trial divisor is 7. 
 
 If the divisor is 04, 05, or 00, the use of tlie trial divisor is less certain ; 
 but the rule is to use as the trial divisor for 04, 7 for 00, and either 6 or 7 
 for 05. 
 
 Name the trial divisors in the next exercise. 
 
 Exercise 36 
 
 l\ 
 
 Find the quotients and remainders of the following, and 
 prove the answers to the odd num'ueis correct by niultii^lying, 
 and the even numbers correct by dividing. 
 
 1. 712^31. 
 
 2. 2341 -^ 51. 
 
 3. 6287 --71. 
 
 4. 2195-80. 
 6. 5894-91. 
 
 6. 2068 --22. 
 
 7. 3572-42. 
 
 8. 1576-5-62. 
 
 9. 8189-4-82. 
 
 10. 6285 H- 23. 
 
 11. 7549 --53. 
 
 12. 8476-63. 
 
 13. 9989-93. 
 
 14. 7948-29. 
 
 15. 8543-49. 
 
 16. 9765 --69. 
 
 17. 8720-89. 
 
 18. 8888-38. 
 
 19. 9894-18. 
 
 20. 9320 --58. 
 
 21. 16,.324--78. 
 
 22. 30,086-98. 
 
 23. 18,874 --27. 
 
 25. 36,989 --67. 
 
 26. 52,298 --87. 
 
 27. 75,643 --97. 
 
 28. 23,877-24. 
 
 29. 38,753-34. 
 
 30. 63,056-64. 
 
 31. 74,111-25. 
 
 32. 96,433 --75. 
 
 33. 56,159-95. 
 
 34. 27,766-36. 
 
 35. 56,139-56. 
 
 36. 78,045-76. 
 
 24. 21,803-37. 
 
 37. Reduce 9.872 qt. to bu. (1 bu. = 32 qt.). 
 
 38. Make simple practical problems based on questions 26-36. 
 
 State how to find the quotient and remainder in any ques- 
 tion in long division. In the following exercise, before divid- 
 ing, make a careful guess as to what the quotient will be. 
 
 I 
 
or 
 
 7 
 
 d 
 
 3) 
 
 1. 10,377^13. 
 
 2. 29,452-14. 
 
 3. 99,024-15. 
 
 4. 87,043-- 16. 
 
 5. 03,277 -r- 17. 
 
 6. 04,935 H- 18. 
 
 7. 99,058-19. 
 
 8. 29,943-99. 
 
 DIVISION 
 
 Exercise 37 
 
 9. 37,847-80. 
 
 10. 84,374-45. 
 
 11. 22,158-23. 
 
 12. 84,999-09. 
 
 13. 15,273-34. 
 
 14. 42,905-88. 
 
 15. 335,290-1-47. 
 
 16. 582,934-50. 
 
 69 
 
 25. 004,820 -J- 29. 
 
 26. 253,789^90. 
 
 17. 854,300-49. 
 
 18. 537,047-30. 
 
 19. 024,839-75. 
 
 20. 802,000-33. 
 
 21. 203,204-54. 
 
 22. 407,989-^08. 
 
 23. 407,989-07. 
 
 24. 033,000-70. 
 
 27. 494,358 -=- 05. 
 
 28. 832,010-^79. 
 
 In the following exercise, before dividing, make a careful 
 guess as to what the quotient will be. 
 
 Exercise 38 
 
 12. $307,989-470. 
 
 13. 578,243 oil. in. — 231 cii. in. 
 
 14. 578,243 rd. - 320 rd. 
 
 15. 987,055 cii. ft. -^ 128 cu. ft. 
 
 16. f 128,821 - 300. 
 
 17. 599,047 -T- 170. 
 
 18. 313,947 da. -^ 305 da. 
 
 19. 444,555-300." 
 
 20. 574,381 A. - 040 A. 
 
 21. 987,432 sq. in. — 144 sq. in. 
 
 1. 395,207-7-105. 
 
 2. 300,498-^207. 
 
 3. 227,870-121. 
 
 4. 407,253 -^ 309. 
 
 5. 839,428 -- 224. 
 
 6. 719,888-4-421. 
 
 7. 584,287-593. 
 
 8. 495,038 -- 784. 
 
 9. 597,445 -r- 050. 
 
 10. 380,777-4-921. 
 
 11. 811,394-4-075. 22. 358,049-4-528. 
 
 23. Solve Exercise 22 by division. 
 
 24. Make simple practical problems based on questions 12-22. 
 
 
 
 I n 
 
 m 
 
 •nh' 
 
 ji 
 
 isH. 
 
 !n 
 
70 
 
 ARITHMETIC 
 
 Exercise 39 
 
 1. Divide $324 among A, B, and C, giving 15 twice as miicli 
 as A, and C tliree times as much as B. 
 
 2. A rod 540 in. long lias a piece of 8 in. cut off from it, 
 then another piece of the same length, then another, and so on. 
 How often may this be done, and what is the length of the piece 
 remaining at last ? 
 
 3. The sum of f 15,108 was paid for a number of sheep at 
 $ 6 apiece. Find the number of sheep. 
 
 4. What is the object of division when the measuring unit 
 and the measured quantity are given ? When the number and 
 the measured quantity are given ? 
 
 5. A merchant sold a (piantity of silk at | 3 a yd., and an 
 equal quantity at $5 a yd. How much did he sell of each 
 kind if he received J$ 3816 for the goods? 
 
 6. A merchant sold a quantity of cloth at ^ 3 a yd., and 
 twice as much at | 2 a yd., the whole amounting to 1 2005. 
 How much did he sell altogether? 
 
 7. What number must be added to 91 to make it exactly 
 divisible by 8 ? 
 
 8. A farmer mixed 15 bu. of oats, worth 40^ per bu., with 
 5 bu. of corn, worth 80^ j)er bu. What is the mixture worth 
 per bu. ? 
 
 9. The expense of carpeting a room was $45; but if tlie 
 breadth had been 3 ft. less than it was, the expense would have 
 been $ 36. Find the breadth of the room. 
 
 10. How much water must be mixed with 600 gal. of wine, at 
 $2.50 per gal., in order to make the mixture worth $2 per 
 gal? 
 
 11. Divide $448 among 2 men, 3 women, and 4 children, giv- 
 ing each man three times and each woman twice as much as each 
 child. 
 
'W' 
 
 DIVISION 
 
 Exercise 40 
 
 1. 49;^,287 yd. ^ ITGO yd. 
 
 2. 298,456 ft. h- 5280 ft. 
 
 3. 140,008 cii. in.^1728 cu. in. 
 
 4. 080,442 cu. iii.-j-2150 cu. in. 
 
 5. 998,209 11).- 2240 lb. 
 
 6. 857,804 gr. -- 5700 gr. 
 
 7. .398,125 gr. -- 7000 gr. 
 
 71 
 
 8. <S19,0;U -:- 4972 
 
 9. 819,034 -^ 3264 
 
 10. 205,039 -H 7459 
 
 11. 720,998-9543 
 
 12. 337,877-9961 
 
 13. 098,206^8456 
 
 14. 729,453 -- 5879 
 
 Exercise 41 
 
 1. Find the -raniber of mi. in 56,978 rd. 
 
 2. Find the nund)ei' of mi. in 86,300 yd. 
 
 3. Find the number of mi. in 34,720 ft. 
 
 4. Find the number of mi. in 740,360 in. 
 
 5. Find the number of Si{. ft. in a room 263 in. long and 
 248 in. wide. 
 
 6. Find the number of to\vnshi}).s, each containing 36 s(i[. mi., 
 which can be made out of a section of land in the form of an 
 oblong 27 mi. long and 12 mi. wide. 
 
 7. If 1 cu. ft. contains 1728 cu. in., find tlie nund)er of cu. ft. 
 in 632,194 cu. in. 
 
 8. If 1 gal. contains 231 cu. in., find the luunber of cu. ft. 
 in 576 gal. 
 
 9. Find the largest number of gal. of water wliich can be 
 emptied into a vessel containing 1 cu. ft., without overflowing. 
 
 10. How many gal. of water will fill a vessel containing 
 77 cu. ft. ? 
 
 11. Find the number of yr., of 365 da. each, in 84,678 da. 
 
 12. Find the wages diu> a workman who has worked 423 hr. 
 at $ 1.50 a da., of 9 hr. each. 
 
 4 5 
 
 1,: i: 
 
 t 
 
 I ^ 
 
 li 
 
 u 
 
 1. 
 I,. 
 
 ! f I 
 
lit: 
 
 [Ml 
 
 72 
 
 ARITHMETIC 
 
 I 
 
 13. Bought ()4() 1)11. of barley at the rate of 32 bu. for f 20.08, 
 and sold it at the rate of 10 bu. for f 8.75. Find my profit on 
 the transaction. 
 
 14. The earth's polar diameter contains 41,707,796 ft., and 
 the differei^ce between the ecpiatorial and the polar is one-292nd 
 part of the latter. Find the difference between the two in miles. 
 
 15. A farmer bought land from B at f 00 per A., and the 
 same quantity from C at $ 85 per A. The whole amounted to 
 $ 53,215. How many A. did he buy from each ? 
 
 16. What number must be added to 7,809,456 to render it ex- 
 actly divisible by 8975 ? 
 
 17. $ 90.90 are shared among 4 men, 5 women, and 6 children, 
 so as to give to each man twice as much as to each woman, and to 
 each woman three times as much as to a child. What do the 
 women get ? 
 
 18. A farmer laid out .1f> 71,778 in purchasing an e(pial number 
 of sheep, hogs, and cows. Fach sheep cost $ 6, each hog twice as 
 much as a sheep, and each cow twice as niuch as a hog. How 
 many of each did he buy ? 
 
 19. A speculator gave $ 18,810 for horses, and sold a certain 
 number of them for $ TWM), at $ 85 each, losing thereby $ 10 each. 
 For how much each must he sell the remainder so as to gain 
 $ 2180 on the whole ? 
 
 20. In 161,384 in., how maiiy i\:\. ? 
 
 21. If ()H bales of linen contain f)7,048 yd., and each bale 
 contains 34 ])ieces, and v.uh \)']rvo llie same number of yd., 
 how many yd. are there in ';;('! i pi.'ce? 
 
 22. If the quotient is 50<w\ when ilic divisor is 2001 and the 
 remainder 100, what is the divuuiid '.' 
 
 23. Divide 10,149 by 7, and the (inotient by 5; thence deduce 
 the true remainder, and show that it is the same as after the 
 division of 10,149 by 35. 
 
DIVISION 
 
 73 
 
 24. What iiuuiber is that to which if oS and H tinie.s 38 be 
 added, and the sum so found be increased by 7 times itself, the 
 total sum is 2400 ? 
 
 25. A dealer in horses gave {$ 9900 for a certain rnnnber. and 
 sold a part of them for $ ,')825 at $ (S5 each, and by so doing lost 
 f 5 a head. For how much per head must he sell the remainder 
 so as to gain $ 1140 on the whole ? 
 
 26. A receives on 225 shares of mining stoc^k an annual 
 dividend of $ 96 a share ; and B receives the same total annual 
 dividend on 270 shares of oil stock. Find tlie annual dividend on 
 one share of B's stock. 
 
 27. A drover bought a number of cattle for Ij^ 17,100 and sold a 
 certain number of them for .f 12,474 at f 12() a head, gaining on 
 those he sold $2574. How many did he buy at lirst, and how 
 much did he gain on each sold ? 
 
 28. The fore wheel of a carriage is 8 ft. in circumference, and 
 in a distance of V4 mi. makes 2IU0 revvolutions more than the 
 hind wheel. Find the circumference of the hind wlieel. 
 
 29. Divide $2()40.75 among 4 men, women, and 8 children, 
 giving to each child double a woman's sliare, ;ind to each woman 
 triple a man's share. 
 
 30. A grain merchant bought 40,(>40 lb. of wheat at .|)1.2() 
 per bu., and shipped it to New Vork at an expense of o^ per 
 bu. Before he sold it there was a loss in hniidling, etc., of -^\- 
 of the original weight; his ])rofit on the transaction was li?()9.(S5. 
 At what price did he sell the wheat? (A bu. of wheat weighs 
 GO lb.) 
 
 31. How many sq. ft. of glass are recpiired to glaze 5 windows, 
 each containing 14 panes of glass, the panes measuring 17 in. by 
 15 in. ? 
 
 32. If the average number of people to the scp mi. is called 
 the unit of population, lind to the nearest integer this unit for 
 your own state and also for the following states: 
 
 
 
 
 •I 
 
 > !l 
 
 :'m 
 
 i1 
 
 m 
 
u 
 
 ARITHMETIC 
 
 State 
 
 Ml LBS 
 
 Population (185 
 
 Massachusetts, 
 
 8,315 
 
 2,238,943 
 
 New York, 
 
 49,170 
 
 5,997,853 
 
 Illinois, 
 
 56,()50 
 
 3,826,351 
 
 Texas, 
 
 265J80 
 
 2,235,523 
 
 California, 
 
 158,3G0 
 
 1,208,130 
 
 South Dakota, 
 
 77,650 
 
 328,808 
 
 -.'li 
 
 fv 
 
 Account for the difference in the units of population. 
 
 33. If the unit of population for one state is 16 times as great 
 as that of another, but the area of the second is 4 times as great 
 as the first, compare the population of the two states. 
 
 34. The length of the Missouri-Mississippi River is 4200 mi., 
 and the area drained by it is 1,250,000 sq. mi. If this area is 
 (ionceived of as a rectangle whose length is 4200 mi., find its 
 width to the nearest integer. 
 
 35. Find, as in question 34, the width of the rectangle for 
 these rivers: 
 
 UlVER 
 
 LkN<JTII IV 
 M 1 LK8 
 
 AkKA DKAINKI) 
 
 IN Sqitakk Miles 
 
 St. Lawrence, 
 
 2000 
 
 350,000 
 
 Amazon, 
 
 4000 
 
 2,500,000 
 
 La Platca, 
 
 2300 
 
 1,250,000 
 
 fli' 
 
 
 74. Divide 77.908 by 8. 
 
 8) 77.968 8 Is contained in 77 units 9 times with remainder 5 units. 8 is 
 
 9 746 contained in 50 tenths 7 tentlis times with remainder 3 tenths. 
 8 is contained in JJG hundredths 4 hundredtlis times with remain- 
 der 4 hundredtlis. 8 is contained in 48 thousandths C thousandtlis times witli 
 no remainder. 
 
 Hence the operation of dividing a dividend containing a decimal is similar 
 to that of dividing when the dividend does not contain a decimal. Care 
 must be taken to insert the decimal point in the quotient as in the example, 
 immediately after the units' figure is used in the dividend. 
 

 DIVISION 
 
 75 
 
 Ezerc7.se 42 
 
 Divide : 
 
 1. 12.Gby6. 
 
 2. 7.56 by 7. 
 
 3. 16.38 by 13. 
 
 4. 239.76 by 37. 
 
 5. 596.36 by 17. 
 
 6. 889.92 by 72. 
 
 7. 195.2544 by 473. 
 
 8. 192.4947 by 171. 
 
 9. Divide 389.904 A. of land equally among 16 persons. 
 
 10. If 43 bu. of wheat cost $37,625, find the cost of 1 bu. 
 
 11. Divide the following numbers by 10 : 
 
 47.39 ; 543.21 ; 62 ; 7.64. 
 
 12. Write down the quotients obtained on dividing the follow- 
 ing numbers by 10 : 
 
 64.52 ; 3.9 ; 742.63 ; 95.614, 
 
 13. State how to find the quotient without actual division, 
 when a decimal is divided by 10. 
 
 14. Divide by 100 : 
 
 792.6 ; 8943.62 ; 54.15 ; 89467.1. 
 
 15. State how to find the quotient without actual division, 
 when a decimal is divided by 100. By 1000. 
 
 
 1 
 
k 
 
 CHAPTER VII 
 
 OOMPARISON OF NUMBEES 
 
 75. If 14 be multiplied in turn by the numbers 1, 2, 3, 4, 
 
 5, 6, 7, 8, 9, 10, 11, 12, the products will be 14, |8, 1 12, $1G, 
 820, 124, 128, 132, i36, 140, |44, 148. 
 
 Hence, taking $4: as the unit of measure, and noting how 
 often the unit is repeated to produce 14, 112, $8, i20, $36, 
 we find that these quantities are represented by 1, 3, 2, 5, 
 and 9 units, respectively. 
 
 Exercise 43 
 
 What is the largest unit which will measure each of the 
 following quantities, and what is the number of units in each 
 of them ? 
 
 1. I 27, $ 24, f 12, $ ao, $ a% $ 36 (unit $ 3). 
 
 2. 1 45, f 10, $ 20, $ 55, $ .35, $ 00 (unit f 5). 
 
 3. 42, 18, 30, 0, 54, and 00 sljeep. 
 
 4. 03, 84, 14, 35, 49, and 77 bbl. of flour. 
 
 5. 32, 10, 48, 50, 80. p.nd 9() iiiin. 
 8. 54, 27, 9, 81, ;J0, and 99 men. 
 
 7. 80, 20, 90, 110, 70, ami 120 bu. of oats. 
 
 8. 99, 06, 33, 55, 121, and 132 T. of coal. 
 
 9. 108, 24, 72, 132, 48, and 60 lb. of tea, 
 
 7i^ 
 
COMPARISON OF NUMBEUS 
 
 77 
 
 V, 
 
 76. If 25 slieep cost 1120, what will 10 sheep cost? 
 
 Taking 5 sheep as the unit of value, 25 sheep contain 5 units, and 10 sheep 
 
 2 units. 
 
 5 units cost .$120, 
 
 1 unit costs $ 24. 
 
 .-. 2 units cost S^ 48. 
 
 Exercise 44 
 
 1. If 3G yd. of cloth cost J$42, what will 24 yd. cost at the 
 same rate? 
 
 2. A iniller sold 35 bbl. of flour for f 12G. How much will 
 he receive for 15 bbl. at the same rate? 
 
 3. A train runs 32 mi. in 48min. At the same rate, what 
 distance will it run in 54 niin. ? 
 
 4. If 84 men can dig a trench in 3G da., how long will it 
 take 108 men to dig a trench of the same size ? 
 
 5. If 88 horses eat 33 bu. of oats in 1 da., how many bu. 
 will 48 horses eat in the same time? 
 
 6. If 45 men can reap a field of 30 A. in a certain time, how 
 many A. woidd 25 men reap in the same time ? 
 
 7. A bankrupt pays $35 out of every .1i> (53 owed. How much 
 shall I receive if I am his debtor to tlie extent of .1^ 81 ? 
 
 8. If 32 T. of coal cost .1^184, what will 88 T. cost at the 
 same rate ? 
 
 9. If 50 men can do a piece of work in 21 da., how long 
 will it ta-vC 24 men to do it ? 
 
 10. How many lb. of tea can be bought for $50, at the rate 
 of f 10 for 34 lb. ? 
 
 11. Tea is bought at 72^ a lb., and sold for 84^ a lb. The 
 buying price is what pare of the selling price ? 
 
 12. The cost of fencing 132 rd. of railway is $ 117. What is 
 the eoh'.t of fencing 83 rd. ? 
 
 *: 
 i] 
 
 II 
 
 I 
 
 ! 1 
 
 t 1 
 
 li 
 
 >*»... '. ■'. 
 
78 
 
 ARITHMETIC 
 
 li 
 
 13. If a 12-(it.. pail is just filled by (5 units of milk, how many 
 qt. are there in a pail which will hold 5 units ? AVhat is the 
 unit ? 
 
 14. Fifty-four niin. are represented by 9 units of time. How 
 many min. are there in 7 units ? 
 
 15. If 4 ft. is the unit of length, what is the unit of area? 
 
 16. If 4 ft. is the unit of length, how many units of area are 
 there in an oblong whose dimensions are 24 and 30 ft. ? 
 
 17. If 5 ft. is the unit of length, what is the unit of volume ? 
 
 18. If 5 ft. is the unit of length, how many units of volume 
 are there in a rectangular solia 30 ft. long, 25 ft. wide, and 20 ft. 
 high ? 
 
 19. Divide $ 84 among two persons, giving the first $ 3 for 
 every $4 the second will get. 
 
 20. Divide $SS between A, B, and C, ",o that A will get $2 
 and B $ 4 for every $ 5 C gets. 
 
 21. A sum of money is divided between A and B. If A, who 
 gets ^ 2 for every $ 5 B gets, receives f 24, how much does B 
 receive? AVhat sum was divided? 
 
 22. A sum of money is divided between A and B. A receives 
 $ 2 for every JfiJ 5 B receives. If B gets $ 24 more than A, how 
 much did each receive ? What sum was divided ? 
 
 23. INIake and solve problems similar to 19, 20, 21, and 22. 
 
 24. A man Avhose weekly income is $ 15 spends $ 2 out of 
 every .f5 5 he earns for board. AVhat does his board cost him 
 per wk. ? 
 
 77. (1) What is the ratio of 120 to ^45? 
 
 Let $[■> be taken as the unit of measure. 
 Then $20 is measured by 4 times the unit. 
 And $45 is measured by times the unit. 
 .-. $ 20 is I of $ 4&. 
 
 th 
 
lii 
 
 III 
 
 COMPARISON OF NUMBERS 
 
 79 
 
 (2) If 22 yd. of cloth cost |1G, what will 33 yd. cost at 
 the same rate ? 
 
 Take 11 yd. as tlic unit of length. 
 
 Then 33 yd. = :] of 22 yd. 
 
 .-. 33 yd. cost j| of $ 10 or $24. 
 
 Ksercise 45 
 
 1. What is the ratio of f 18 to $24'/ $35 to f 55? $28 
 to $ G3 ? 
 
 2. What is the ratio of IG hr. to 5G hr. ? 72 hr. to 45 hr. ? 
 
 3. What is the ratio of GO lui. to 25 mi. ? 99 A. to 55 A. ? 
 
 4. If 45 cd. of wood cost 1 1G2, what will 20 cd. cost ? 
 
 5. If 21 T. of hay cost f 174, what will 70 T. cost ? 
 
 6. If G3 men can dig a trench in IG da., how long will it take 
 18 men to dig it ? 
 
 7. If a piece of cloth 15 ft. long and 3 ft. wide costs $ 18, 
 what will a similar piece 20 ft. long and 4 ft. wide cost ? 
 
 8. Divide f 9G between A and B so that A will get $5 for 
 every $ 7 B will get. 
 
 9. Divide $240 between A and B so that the two parts will 
 be in the ratio of their ages, which are 8 and 12 yr. 
 
 10. Divide $ 460 among three persons, A, B, and C, so that 
 the three portions will be to each other as the numbers 5, 8, and 
 7, respectively. 
 
 11. A bankrupt has three creditors, to whom the sums due are 
 as the numbers 2, 3, and 4. If his assets are valne«l at $ 540, 
 find the sum each will receive. 
 
 12. A tract of land is divided into two farms in the ratio of 
 2 to 3. If the whole tract contains 480 A., what is the size of 
 each farm ? 
 
 I 
 
 « 
 
 1!: 
 
 I 
 
CIIAMER VIII 
 
 iih 
 
 i-.vi| 
 
 u'y 
 
 
 SQUARE ROOT 
 
 78. The product of 3 and 3 is 9 ; of 5 and 5 is 25. The 
 squares whose sides measure 3 and 5 units of length contain 
 9 and 25 units of square measure. We say that 9 is the 
 square of 3 and that 25 is the square of 5; tliat 3 is the 
 square root of 9 and 5 the square root of 25. 
 
 The square of 3 is written 3^ and the square root of 9 is 
 indicated thus: V9. 
 
 2 is called tlie Exponent, and ^ the Radical Sign. 3^ is also 
 called the second Power of 3. 
 
 79. The Square of a number is the product found by mul- 
 tiplying the number by itself. 
 
 Til us the squares of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 
 are 1, 4, 9, 10, 25, 36, 49, 64, 81, 100. 
 
 80. The square root of a luunber is that number which 
 multi[)lied by itself is e(][ual to the given number. 
 
 Thus the square roots of 1, 4, 9, 16, 25, 3t), 49, 64, 81, 100, 
 are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. 
 
 81. Tupils should inoniovizo the tables in the two preceding paragraphs 
 and be able to answer instantly such (piestions as the following : 
 
 What is the tirst figure in the s(iuare root of 27 ? G8 ? 70? 4.']? 80? 
 
 
 pi' 
 If 
 
 Exercise 46 
 
 AVrite the following products as powers : 
 1. 5 X 5. 2. 7 X 7. 
 
 80 
 
 3. 24 X 24. 
 
SQUARE ROOT 
 
 81 
 
 Write the following powers as products and find their valnes 
 
 8. (Id'- 
 
 9- my- 
 
 4. 
 
 S\ 
 
 6. 46*'^. 
 
 5. 
 
 132. 
 
 7. (if. 
 
 Prove the following 
 
 statements : 
 
 10. 
 
 V41) = 7. 
 
 13. V324 = 18. 
 
 11. 
 
 V169 = 13. 
 
 14. V9G1=31. 
 
 12. 
 
 V72y = 27. 
 
 15. V2025 = 45. 
 Exercise 47 
 
 16. V|H_=f5. 
 
 17. VHe=ii. 
 
 1. Find the squares of the numbers from 10 to 20 and commit 
 the results to memory. 
 
 2. Find the squares of 25, 28, 54, 75, and 99. 
 
 3. From the results in § 79 state how many digits are found 
 in the square of a number of 1 digit. 
 
 4. From the results obtained in questions 1 and 2, state how 
 many digits are found in the square of a number of 2 digits. 
 
 5. Find the squares of 175, 199, 24G, 402, 814, 999. 
 
 6. From the results in question 5 state liow many digits are 
 found in the s<piare of a number of 3 digits. 
 
 7. Judging from the results obtained in questions 1, 2, and 5, 
 state the number of digits in the scpiare root of a square number 
 that contains 3 digits ; 4 digits ; G digits ; 7 digits ; 8 digits. 
 
 8. How many digits in the square correspond to 1 digit in 
 the square root ? 
 
 9. What is the square root of 400 ? Of 900 ? 
 
 10. The square root of G25 lies between what two numbers? 
 
 11. Find the square of 10, 20, 30, 40, 50, GO, 70, 80. and 90. 
 
 12. Between what numbers does the square root of 1225 lie ? 
 Of 4225 ? Of 2304 ? Of 8281 ? Of 8704 ? 
 
 13. What is the square of 200 ? Of 300 ? 
 
 I 
 
 I 
 
 ■:1- 
 
 ■J 
 
 u:i! 
 
 
 'i\ 
 
I 
 
 1 
 
IMAGE EVALUATION 
 TEST TARGET (MT-3) 
 
 1.0 
 
 I.I 
 
 12.8 
 
 li^m 
 
 |Z5 
 
 ■ 2.2 
 
 m m 
 
 US 
 
 m 
 
 u 
 
 14.0 
 
 I 
 
 2.0 
 
 1.8 
 
 
 11.25 lU 1.6 
 
 III ^=SBS 11^^^ ^^S 
 
 
 < 
 
 6" 
 
 ► 
 
 V] 
 
 vl 
 
 
 *;; 
 
 #// « 
 
 y 
 
 ^ 
 
 Hiotographic 
 
 Sciences 
 
 Corporation 
 
 23 WEST MAIN STREET 
 
 WEBSTER, N.Y. UStO 
 
 (716)172-4503 
 

 ;V4, 
 
82 
 
 ARITHMETIC 
 
 14. The square root of 71,289 lies between what two numbers ? 
 
 15. Find the squares of 100, 200, 800, etc., up to 900. 
 
 18. Between what two numbers does the square root of 271,441 
 lie ? Of 795,664 ? 
 
 82. The following explanation will make clear the method 
 of finding the square root of a number of 3 or 4 digits. 
 
 24 
 24 
 16 
 
 80 
 
 80 
 
 400 
 
 576 
 
 20 
 
 40 
 
 Thus 24, which is made up of two parts, 20 and 4, has for its 
 square 576, which is seen to be made up of 400, the square of 20 ; 
 16, the square of 4 ; and twice the product of 20 and 4. 
 
 Now to recover 24 from 676, we know that its hundreds' digit 5, 
 showing that the number is between 400 and 900, gives the tens' digit of 
 the root, so that we know one of the parts of the root, viz. 20. The 
 square of 20 is 400, and the rest of the given number, 176, must be 
 2 times 20, multiplied by the other part, together with the square of 
 the other part. Multiplying 20 by 2, and using the product 40 as a 
 576^20-4-4 divisor with 176 as dividend, we get the quotient 4. 
 
 176 
 160 
 
 16 
 16 
 
 Multiplying 40 by 4 and subtracting the product 160 
 from 176, we get the remainder 16, 
 which is the square of 4. Therefore 
 20 + 4 or 24 is the square root of 676. 
 The work of extracting the square 
 root may be simpliiied by leaving out 
 the unnecessary zeros, thus : 
 
 44 
 
 5'76(24 
 4 
 
 176 
 176 
 
 83. (1) The method of discovering the square root of a 
 number of 5 or 6 digits is similar to that for finding the 
 square root of numbers of 3 or 4 digits. 
 
 246 
 
 246 
 
 36 
 
 1440 
 
 1440 
 
 57600 
 
 60516 
 
 Thus 246, which is made up of two parts, 240 and 6, has for 
 its square 60,616, which is seen to be made up of 67,600, the square 
 of 240 ; 36, the square of 6 ; and twice the product of 240 and 6. 
 
SQUARE ROOT 
 
 83 
 
 44 
 
 480 
 
 6 
 
 6'0516(240 + 6 
 4 
 
 205 
 176 
 
 2916 
 
 2880 
 
 (2) Hence proceeding with 605 as in § 82 
 with 576, we get in the square root 24 tens or 
 240. Multiplying 240 by 2 and using the product 
 480 as a divisor with 2916 as a dividend, the 
 quotient is found to ])e 6. 
 
 Multiplying the 480 | 
 
 36 
 36 
 
 by 6, and subtracting 
 the product 2880 from 
 2916 we have the re- 
 mainder 36, which is 
 the square of 6. There- 
 
 44 
 
 486 
 
 6'05'16(246 
 4 
 
 205 
 176 
 
 2916 
 2916 
 
 fore 240 + 6 or 246 is the square root of 60516. 
 
 Leaving out the unnecessary zeros, the work may 
 be simplified as in the contracted form. 
 
 The number whose square root is to be subtracted should be pointed off 
 into groups of two figures, as in the preceding examples, beginning with the 
 units' figure. 
 
 84. To find the square root of |||. 
 
 The square roots of 289 and 625 are found to be 17 and 25. 
 Hence Vf|| = H- 
 
 Exercise 48 
 
 Find the square root and prove your answer correct : 
 
 1. 324. 5. 3025. 9. 71,824. 13. ^e^. 
 
 2. 529. 6. 6889. 10. 101,761. 14. J^Vi- 
 
 3. 841. 7. 4096. 11. 465,124. 15. 30^. 
 
 4. 1156. 8. 9409. 12. 998,001. 16. 22^%. 
 17. Compare the process of extracting the square root of a 
 
 number with long division, stating in what respect the process is 
 similar, and where it is different. 
 
 85. (1) Find the length of the side of a square containing 
 4225 sq. in. 
 
 The square is measured by 4225 units of 1 sq. in. ; therefore the side 
 is measured by \/4225, or 65 units of 1 in,, i.e. the length of the square 
 is 65 in. 
 
 ,Ji|! 
 
 iii 
 
 ■!li 
 
 W 
 
 ".*« 
 
84 
 
 ARITHMETIC 
 
 (2) The sides of a rectangular field containing 735 sq. rd. 
 
 are as 3 to 5. Find their length. 
 
 The field contains 3 x 5, or 15 units of area. 
 
 Tiie area of one unit = 735 sq. rd. ^ 15, or 49 sq. rd. 
 
 The side of a square containing 49 sq. rd. = 7 rd. 
 
 .-. the sides of the field are 3 x 7 rd., or 21 rd., and 5x7 rd., or 35 rd. 
 
 Statement of Solution 
 
 First find the number of units of area in the field. Divide this number 
 into the area, and find the unit of area. Then find the unit of length, which 
 is the length of the side of the unit of area. 
 
 Multiply the unit of length by 3 and 5 respectively, to find the sides of the 
 field. 
 
 (3) To find the area of a triangle, the lengths of whose sides are given, 
 find one-half the sum of the number of units of length in the sides. Sub- 
 tract from this the number of units of length in each side separately. Find 
 the product of these four results. The square root of this product is the 
 number of units of area in the given triangle. 
 
 Find the area of a triangle whose sides are 5 in., 12 in., 
 
 and 13 in. respectively. 
 
 The sum = 5 + 12 + 13 = 30. 
 One-half this sum = 15. 
 
 15- 5 = 10. 15 X 10 X 3 X 2 = 900. 
 
 15-12= 3. \/900 = 30. 
 
 15 — 13 = 2. ••. the area of the triangle = 30 sq. in. 
 
 Exercise 49 
 
 1. Find the length of the side of an enclosure in the form of 
 a square containing 386 sq. yd. 7 sq. ft. 
 
 2. A park contains 9408 sq. yd., and it is 3 times as long as 
 it is wide. Find its length and width. 
 
 3. A merchant bought a number of yards of cloth, paying as 
 many cents for each yard as there were yards. The whole cost 
 $ 56.25. How many yards did he buy, and at what price per 
 yard ? 
 
SQUARE ROOT 
 
 85 
 
 ■H 
 
 4. What is one of the two equal factors of 24,336 ? 
 
 5. A rectangular field, the sides of which are in the ratio of 
 4 to 7, contains 4032 sq. rd. Find the length of each side, and 
 the cost of fencing it at f 4 per rd. 
 
 6. A body of soldiers in column form 567 ranks, 7 abreast. If 
 they were drawn up in solid square, how many would there be on 
 each side ? 
 
 7. Find the side of a square which is equal in area to the sum 
 of the area of two squares, the sides of which are 6 and 8 in. 
 long. 
 
 8. Draw two lines respectively 6 and 8 in. long, at right 
 angles. Join their extremities by a straight line. Measure this 
 line and show that it is equal to the side of the square found in 
 question 7. 
 
 9. Work problems similar to 7 and 8, using the following as 
 the lengths of the sides of the smaller squares : 3 in., 4 in. ; 5 in., 
 12 in. ; 8 in., 15 in. 
 
 10. From the preceding three questions make a rule showing 
 how to find the length of the hypotenuse of a right triangle when 
 the lengths of the other two sides are known. 
 
 11. What is the hypotenuse of a right triangle whose sides are 
 21 ft. and 28 ft. ? 15 ft., 36 ft. ? 56 ft., 105 ft. ? 
 
 12. Find the side of a square equal in area to the difference of 
 the area of the two squares whose sides are 41 ft. and 9 ft. 
 
 13. What is the altitude of a right triangle whose hypotenuse 
 and base are 34 ft., 16 ft. ? 205 ft., 45 ft. ? 136 ft., 64 ft. ? 
 
 14. The top of a ladder rests against the side of a building 84 
 ft. from the ground, and its foot is 35 ft. from the wall. Find 
 the length of the ladder. 
 
 15. A ladder 51 ft. long stands close against a building. How 
 far must the foot be drawn out that the top may be lowered 6 ft. ? 
 
 16. Find the diagonal of a rectangular field whose sides are 
 144 yd. and 60 yd. 
 
 \4 
 
 
86 
 
 ARITHMETIC 
 
 17. Find the side of a square equal in area to a rectangle 
 whose sides are 148 yd. and 333 yd. Find the difference between 
 the perimeters of the rectangle and square. 
 
 18. Find the area of the largest rectangle which can be en- 
 closed by a line 36 in. long. 
 
 19. A field in the form of a rectangle whose sides are as 3 to 4 
 contains 432 sq. rd. How much do I save by crossing along its 
 diagonal instead of going along its two sides ? 
 
 20. State in as few words as possible how you have solved each 
 of the preceding questions. 
 
 21. The sides of a triangle are 8 in., 15 in., and 17 in. Find its 
 area. 
 
 22. Find the area of an oblong whose bides are 8 in. and 15 in. 
 What is the ratio of its area to that of the triangle given in ques- 
 tion 21 ? Why is this so ? 
 
 23. Find the areas of the triangles whose sides are : 
 
 21 in., 28 in., 35 in. 
 
 24 in., 45 in., 51 in. 
 
 9 in., 40 in., 41 in. 
 
 24. The sides of a rectangle containing 34,992 sq. ft. are as 
 4 to 3. Find them. 
 
 25. One side of an oblong is | as long as the other, and its area 
 is 67,335 sq. yd. Find the length of each side. 
 
i!;t 
 
 CHAPTER IX 
 
 GREATEST COMMON MEASURE AND LEAST COMMON 
 
 MULTIPLE 
 
 86. Name all the units of length which will exactly meas- 
 ure 15 in. 
 
 They are 1 in., 3 in., 5 in., and 15 in. 
 
 87. Find all the different units of length that will exactly 
 measure 12 ft. and 18 ft. 
 
 The measures of 12 ft. are 1, 2, 3, 4, 6, and 12 ft. 
 
 The measures of 18 ft. are 1, 2, 3, 6, 9, and 18 ft. 
 
 It is evident that all the common measures of 12 ft. and 
 18 ft. are 1, 2, 3, and 6 ft., and that the greater common 
 measure is 6 ft. 
 
 A Common Measure of two or more quantities is a unit that 
 will exactly divide each of them. 
 
 The Greatest Common Measure (G. C. M.) of two or more 
 quantities is the largest unit which will exactly divide each 
 of them. 
 
 For convenience we speak of the common measure or the 
 greatest common measure of two or more numbers. 
 
 Exercise 50 
 
 Find all the common measures and the greatest common meas- 
 ure of : 
 
 1. 16 ft., 28 ft. 3. 54 yd., 72 yd. 5. 32 qt., 56 qt. 
 
 2. f60, $90. 4. 42 mi., 105 mi. 6. 27 oz., 47 oz. 
 
 7. 21pt., 91pt. 8. 84bu., 91bu. 
 
 87 
 
 i 
 
88 
 
 ARITHMETIC 
 
 9. Find all the measures that can be used to measure the 
 capacity of each of two baskets containing 20 qt. and 32 qt. 
 
 10. Find the lengths of the two longest boards that can be 
 used to build a fence around a garden 30 ft. long and 24 ft. wide. 
 
 11. Make questions similar to 9 and 10, using the quantities 
 in problems 1 to 8. 
 
 Prime Numbers 
 
 88. A Prime Number is one that can be divided only by 
 unity and itself, as 5, 11, and 13. 
 
 Select the prime numbers : 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. 
 
 The prime factors of a number are the prime numbers 
 which when multiplied together give it; thus, 3, 3, and 5 
 are the prime factors of 45. 
 
 89. Find the prime factors of 168. 
 
 2 
 2 
 2 
 3 
 
 168 
 84 
 42 
 21 
 
 7 
 
 That is, 168 = 2 X 84 ; = 2 x 2 x 42 ; = 2 x 2 x 2 x 21 ; 
 
 = 2x2x2x3x7; 
 = 23 X 3 X 7. 
 
 .*. The prime factors of 168 are 2, 3, and 7. 
 23 is a short way of writing 2x2x2. 
 
 3 is called the exponent of 2, and denotes that 2 has been taken as a factor 
 three times. 
 
 Exercise 51 
 
 1. Name the even numbers from 1 to 100. 
 
 2. Name the odd numbers from 1 to 100. 
 
 3. Name the prime numbers from 12 to 100. 
 Find the prime factors of: 
 
 4. 30; 36; m-, 48; 84; m-, 196; 195; 231. 
 
GREATEST COMMON MEASURE 
 
 89 
 
 5. 86; 147; 104; 132; 78; 135; 342; 255. 
 
 6. 336; 408; 372; 565; 342; 484; 375; 861. 
 
 7. What prime factors are common to 30 and 36 ? 66 and 
 132? 147 and 336? 135 and 255? 
 
 90. (1) A man owns a rectangular lot 210 ft. long and 
 144 ft. wide. Find the length of the longest board that 
 can be used to fence it. 
 
 We are required to find the length of the longest board, i.e. the G. C. M. 
 of 144 ft. and 210 ft. 
 
 144 = 2 X 2 X 2 X 2 X 3 X 3. 
 210 = 2 X 3 X 5 X 7. 
 
 Thus, the G. C. M. of 144 and 210 = 2 x 3 = 6. 
 .-. the length of the longest board is ft. 
 To prove the answer correct : 
 
 The number of boards required for the length = 210 -4- 6 = 35. 
 
 The nuHiber of boards required for the width = 144 -r- 6 = 24. 
 35 and 24 have no common measure except unity, .-. 6 ft. is the correct 
 answer. 
 
 (2) A certain school consists of 132 pupils in the high 
 
 school, 154 in the grammar, and 198 in the primary grades. 
 
 If each group is divided into sections of the same number 
 
 containing as many pupils as possible, how many pupils will 
 
 there be in each section ? 
 
 We are required to find the number of pupils in each section, i.e. the 
 G. C. M. of 132, 154, and 198 pupils. 
 
 2 
 
 132 
 
 154 
 
 198 
 
 11 
 
 66 
 
 77 
 
 99 
 
 
 6 
 
 7 
 
 9 
 
 Since 2 and 11 are the only common factors, the G. C. M. of 132, 154, and 
 198 is 2 X 11, or 22. 
 
 .-. each section will contain 22 pupils. 
 
 Exercise 52 
 
 1. Draw two lines, one 15 in. and the other 21 in. long. What 
 is the longest line that can be used to measure both lines ? 
 
"■«piP«!5WW!!W» 
 
 11 
 
 90 
 
 ARITHMETIC 
 
 2. What is the longest line that will exactly measure two 
 lines 28 and 32 in. long ? 
 
 3. What is the longest line that will exactly measure three 
 lines respectively 20, 30, and 45 in. long ? 
 
 4. What is the largest unit of capacity that can be used to 
 measure the quantity of oil in each of two vessels, one containing 
 16 qt. and the other 36 qt. ? 
 
 5. What is the largest unit of money that can be used to pay 
 each of two debts, one of $ 45 and the other of ^ 80 ? 
 
 Exercise 53 
 
 1. A certain school consists of 132 junior and 99 senior stu- 
 dents. How might each of the two classes be divided so that the 
 whole school should be distributed into equal sections ? 
 
 2. A gentleman has a piece of ground, the sides of which 
 measure 225 ft., 297 ft., and 369 ft. He wishes to enclose it 
 with a fence having panels of uniform length. What is the 
 longest panel that can be used for that purpose ? 
 
 3. A teacher having a school of 144 boys and 128 girls divided 
 it into the largest possible equal classes, so that each class of 
 girls should number the same as each class of boys. What was 
 the number of classes ? 
 
 4. There is a street 354 rd. long, and the land on one side of 
 this street is owned by three persons, A, B, and C. A has 102 
 rd. fronting the street, B 114 rd., and C 138 rd. They agree to 
 divide their land into village lots in such a manner that the 
 lots shall be of the greatest width that will allow each person to 
 form an exact number of lots out of his land. What is this 
 width ? 
 
 5. A farmer has 240 bu. of wheat and 920 bu. of oats, which 
 he desires to put into the least number of boxes of the same 
 capacity, without mixing the two kinds of grain. Find how 
 many bu. each box must hold. 
 
GREATEST COMMON MEASURE 
 
 91 
 
 6. If 1 lb. Avoirdupois contains 7000 gr., and 1 lb. Troy 5760 
 gr., iind the greatest weight that will measure both 1 lb. Troy and 
 1 lb. Avoirdupois. 
 
 7. A farmer has 66 bu. of corn and 90 bu. of wheat, which he 
 wishes to put into sacks of equal size, and without mixing the 
 two kinds of grain. How many bu. must each sack contain in 
 order to be as large as possible ? 
 
 Exercise 54 
 
 Find the G. C. M. of : 
 
 1. 40,56. 
 
 2. 42,54. 
 
 3. 81,105. 
 
 4. 108,162. 
 
 5. 63, 91. 
 
 6. 90,105. 
 
 7. 102,114. 
 
 8. 75,175. 
 
 9. 210,455. 
 
 10. 287,369. 
 
 11. 230,506. 
 
 12. 42, 72, 180. 
 
 13. 60,135,165. 
 
 14. 210,462,546. 
 
 15. 395,474,632. 
 
 16. 666,738,954. 
 
 17. Prove that your answer is a common factor by dividing it 
 into each of the numbers. Prove that it is the greatest common 
 measure by examining your quotients and finding that they have 
 no common measure except unity. 
 
 18. State how to find the G. C. M. of two numbers. Of three 
 numbers. 
 
 19. Make questions similar to those in the previous exercise. 
 
 91. Seven divides 126 and 35. It also divides their sum, 161, 23 times, and 
 their difference, 91, 13 times. Seven also divides the sum of 35 and 4 x 126, or 
 539, 77 times, and the difference between 126 and 3 x 35, or 21, 3 times. 
 
 Thus any number, as 7, that divides two other numbers, as 126 and 35, 
 will divide their sum or difference. It will also divide the sum or difference 
 of any multiples of these numbers. 
 
92 
 
 ARITHMETIC 
 
 Exercise 55 
 Prove the principle stated above : 
 
 1. Divisor C, numbers 84, 30. 
 
 2. Divisor 8, numbers 88, 24. 
 
 3. Divisor 13, numbers 65, 26. 
 
 4. Divisor 19, numbers 133, 38. 
 
 92. When the factors of the number cannot be easily found, the follow- 
 ing method is used : 
 
 741)898(1 
 
 741 
 
 162)741(4 
 608 
 
 133)152(1 
 133 
 
 19)133(7 
 133 
 .-. the G. C. M. of 741 and 893 is 19. 
 To prove that 19 is the G. C. M. of 741 and 893 : 
 
 First. 19 divides 133, and therefore it divides 19 + 133, or 162. 19 divides 
 133 and 152, and therefore it divides 133 and 4 times 152, or 741. 19 divides 
 152 and 741, and therefore it divides 152 + 741, or 893. Therefore 19 is a 
 common measure of 741 and 893. 
 
 Again. It is also the G. C. M. Any number which divides 893 and 741 
 will divide their difference, or 152. Any number which divides 741 and 162 
 will divide the difference between 741 and 4 times 152, or 133. Any num- 
 ber which divides 152 and 133 will divide their difference, or 19. But 19 is 
 the largest number that divides 19 ; therefore 19 is the G. C. M. of 741 and 
 893. 
 
 93. The following more compact form may be used after the pupils under- 
 stand the method given above. It will be observed that the quotients are 
 omitted and that the divisors 741, 152, 133, and 19 are alternately on the 
 left and right side of the vertical line. 
 
 741 
 608 
 
 133 
 133 
 
 893 
 741 
 
 152 
 133 
 19 G. C.JVI. 
 
GREATEST COiMMON MEASURE 
 
 93 
 
 94. To find the G. C. M. of three numbers, find first the 
 G. C. M. of two of them, and then of the result and the third 
 number. 
 
 Exercise 56 
 
 Find the G. C. M. of: 
 
 1. 145,203. 
 
 2. 344,559. 
 
 3. 465,682. 
 
 4. 1781,4384. 
 
 5. 3423,3248. 
 
 6. 4807,9545. 
 
 7. 11,682,19,626. 
 
 8. 31,416,54,593. 
 
 9. 2487,8413. 
 
 10. 495,891,1155. 
 
 11. 3066,4818,8541. 
 
 12. 15,561, 11,115, 13,585. 
 
 ml 
 
 13. State how to find the G. C. M. of two numbers. 
 
 Exercise 57 
 
 1. What is meant by saying that one number is a common 
 measure of two or more numbers? Also, the greatest common 
 measure'^ 
 
 2. Show by means of the examples in the preceding exercise, 
 that the greatest common measure of two numbers can never 
 exceed the difference of the numbers. • 
 
 3. In the following pairs of numbers, select those that are 
 prime to each other, and those that are not prime to each other : 
 12, 18 ; 8, 15 ; 12, 17 ; 20, 21 ; 28, 35 ; 13, 29 ; 36, 48 ; 31, 47. 
 
 4. Explain what is meant by one number being prime to 
 another. When two numbers are prime to each other are they 
 necessarily prime ? Give examples. 
 
 5. What is the product of the three consecutive numbers, 11, 
 12, 13 ? 
 
 6. The product of three consecutive numbers is 120. Find 
 the numbers. 
 
94 
 
 ARITHMETIC 
 
 7. The product of four consecutive numbers is 1680. Find 
 them. 
 
 8. Find the G. C. M. of 90, 150, and 168 by resolving the 
 numbers into their prime factors. When several numbers have 
 been resolved into their prime factors, which of these factors 
 must be taken to form by their product the greatest common 
 measure of the numbers? 
 
 9. Find all the common measures of 210 and 462. Prove 
 that every common measure of these two numbers is a measure 
 of their difference. 
 
 10. Find the G. C. M. of 13,515 and 13,787. 
 
 11. How many rails will enclose a field 3143 ft. long by 2471 
 ft. wide, provided the fence is straight and 8 rails high, and the 
 longest that can be used ? 
 
 12. Find the G. C. M. of 169,037 and 66,429. 
 
 Least Common Multiple 
 
 95. The quantity 15 in. is measured by the unit 5 in., 
 3 times, and is therefore called a multiple of 5 in. 
 
 The quantity 18 lb. is exactly divisible by the units 1 lb., 
 2 lb., 3 lb., 6 lb., and 9 lb., and is a multiple of each one of 
 them. Thus 18 lb. is equal to 18(1 lb.), 9(2 lb.), 6(3 lb.), 
 or 3(6 lb.). 
 
 Select from the following quantities the multiples of the 
 unit $3: 112, $16, 118, 1 25, and $27. Name all the units 
 that will exactly measure the quantity 24 hr. 
 
 Any quantity is a multiple of a unit of measure when it is 
 exactly divisible by the unit. 
 
 96. Thirty days is exactly divisible by the units 3 da. and 
 5 da., and is, therefore, a common multiple of 3 da. and 5 da. 
 
^ 
 
 LEAST COMMON MULTIPLE 
 
 95 
 
 One quantity is a common multiple of two or more units 
 when the former is exactly divisible by each of the latter. 
 
 Thirty days is the least quantity that is exactly divisible 
 by the units 6 da. and 10 da., and is, therefore, the least com- 
 mon multiple of 6 da. and 10 da. 
 
 The Least Common Multiple (L. C. M.) of two or more units 
 is the least quantity that is exactly divisible by each of them. 
 
 For convenience we speak of one number being a multiple 
 of another, or a common multiple, or the least common mul- 
 tiple of two or more numbers. 
 
 97. In problems in Greatest Common Measure, we are 
 given two or more quantities and are required to find the 
 largest unit that will measure each of them. In problems 
 in Least Common Multiple, we are given two or more units 
 of measure and are required to find the least quantity that 
 can be measured by each of the units. 
 
 98. 3, 6, 9, 12, 15, 18, 21, 24, are multiples of 3. 
 
 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, are multiples of 2. 
 
 .*. 6, 12, 18, 24, are common multiples of 2 and 3, and it is 
 evident that 6 is the L. C. M. of 2 and 3. 
 
 The second common multiple, 12, is 2 x 6. 
 
 The third common multiple, 18, is 3 x 6. 
 
 The fourth common multiple, 24, is 4 x 6. 
 
 What are the fifth and sixth common multiples of 2 and 3? 
 In the same way find common multiples and the L. C. M. of 
 2 and 5, 3 and 4, 4 and 6, 8 and 12. 
 
 99. The preceding paragraph shows that the L. C. M. of two prime num- 
 bers, as 2 and 8, 2 and 5, 3 and 4, are their products 6, 10, 12, while the 
 L. C. M. of two numbers not prime, as 4 and 6, and 8 and 12, which are 12 
 and 24, are less than their products, and are in both cases equal to the prod- 
 uct of the numbers divided by their G. C. M. 
 
 
 111 
 III 
 
 •Ik 
 
 : Hi 
 
96 
 
 ARITHMETIC 
 
 100. (1) Find the shortest distance which can be exactly 
 measured by two lines respectively 36 ft. and 48 ft. long. 
 
 We are here required to find the shortest distance, i.e. the L. C. M., of the 
 units 36 ft. and 48 ft. 
 
 36 = 2x2x3x3. 
 48 = 2x2x2x2x3. 
 
 Thus the L. C. M. of 36 and 48 = 2 x 2 x 2 x 2 x 3 x 3 = 144. 
 .'. the shortest distance is 144 ft. 
 
 (2) Find the L. C. M. of 24, 30, 36. 
 
 6 
 
 24 
 
 30 
 
 36 
 
 2 
 
 4 
 
 5 
 
 6 
 
 
 2 
 
 5 
 
 3 
 
 Here 6 and 2 are the factors common to two or more of the numbers, and 
 2, 5, and 3 are the factors not common. 
 
 .-. the L. C. M. = 6 X 2 X 2 X 5 X 3 = 360. 
 
 State how to find the L. C. M. of two or more numbers. 
 
 Show by division that 24, 30, and 36 are all factors of their L. C. M. 360. 
 
 (3) What is the least number of bu. of wheat that will 
 
 make an exact number of full loads for three drays, hauling 
 
 respectively 24, 30, and 36 bu. a load ? 
 
 We are required to find the least number of bu., i.e. the L. C. M. of the 
 units 24, 30, and 36 bu., which is 300 bu. 
 
 To prove the answer correct : 
 
 Dividing 360 bu. by 24, 30, and 36 bu. respectively, we find the number 
 of loads to be 15, 12, and 10. 15, 12, and 10 have no common factor. 
 
 .'. 360 bu. is the least number of bu. 
 
 101. (1) Find the L. C. M. of 14, 21, 54, ^^, 84. 
 
 3 
 
 U n 54 
 
 56 
 
 84 
 
 2 
 
 18 
 
 66 
 
 t^ 
 
 
 9 
 
 28 
 
 
 .: the L. C. M. = 3 X 2 X 9 X 28 = 1512. 
 
 14 is erased since it is a factor of 56, and 21 since it is a factor of 84. In 
 the second line, 28 is erased since it is a factor of 56. 
 
LEAST COMMON MULTIPLE 
 
 97 
 
 (2) Find the L. C. M. of 481 and 1665. 
 
 481)1665(3 
 1443 
 
 222)481(2 
 444 
 
 37)222(6 
 222 
 
 481 
 
 37 = 13 ; 1665 -h 37 = 45. 
 
 .-. the L. C. M. of 481 and 1665 = 37 X 13 x 45 = 481 x 45 = 21,645. 
 State how to find the L. C. M. of two numbers. 
 Show by division that 481 and 1665 are both factors of 21,645. 
 Show that the L. C. M. 21,645 is equal to the product of 481 and 1665 
 divided by their G. C. M. 37. 
 
 Exercise 58 
 
 Solve the following problems and name the units of measure- 
 ment : 
 
 i. How long is the L. C. M. of two lines, one 6 in. long, and 
 the other 10 in. long ? How many times is it measured by the 
 6-in. line ? By the 10-in. line ? 
 
 2. Find the L. C. M. of three lines respectively 12 in., 15 in., 
 and 18 in. 
 
 3. Four bells toll at intervals of 3, 7, 12, and 14 sec. respec- 
 tively, and begin to toll at the same instant. When will they 
 next toll together ? 
 
 4. If in two days A can build 28 rd. of fencing, B 50 rd.," 
 C 16 rd, and D 40 rd., find the least number of rd. that will 
 furnish an exact number of days' work. 
 
 5. When is a number a common multiple of two or more 
 numbers, and when the least common multiple ? 
 
 6. What is the least number of A. that will admit of being 
 divided into a number of farms containing 150, 200, or 250 A. 
 each? 
 
 7. In each question, prove your answer correct. 
 
 H 
 
 
 
" 
 
 98 
 
 ARITHMETIC 
 
 Find the L. C. M. of : 
 
 1. 4,8,16,32. 4. 
 
 2. 3,6,9,12. 5. 
 
 3. 24,30,36,45. 6. 
 10. 30,21,40,28,24, 
 
 Find the L. C. M. of the 
 12. i,i,i. 13 
 
 IK 2 5 1 1 _8 
 1A 4 2 1 14 
 
 Exercise 59 
 
 15, 18, 28, 36. 
 12, 20, 21, 45. 
 22, 33, 30, 44. 
 56. 11. 
 
 7. 65,26,56,52. 
 
 8. 36,48,60,54. 
 
 9. 33, 27, 55, 135. 
 56, 36, 63, 28, 72. 
 
 denominators of these fractions : 
 
 6 .7. 1 9 
 TSf t^} 6TF* 
 
 ^** 7> h TI> Tff* 
 
 17. 
 18. 
 
 18 5 1 
 
 12 7 11 
 
 3? zy 12? Ti"* 
 
 Find the L. C. M. of: 
 
 1. 266, 703. 3. 
 
 2. 1173,1702. 4. 
 
 Exercise 60 
 
 3045, 4515. 
 3589, 2257. 
 
 Exercise 61 
 
 5. 8159,14,227. 
 
 6. 10,959, 12,753. 
 
 1. Show by examples that the L. C. M. of two numbers can 
 never exceed their product. 
 
 2. Find the L.C. M. of 90, 150, and 168 by resolving the num- 
 bers into their prime factors. When several numbers have been 
 resolved into their prime factors, which of these factors must be 
 taken to form by their product the L. C. M. of the numbers ? 
 
 3. A, B, C, and D start together, and travel the same way 
 around an island which is 600 mi. in circuit. A goes 20 mi. 
 per da., B 30, C 25, and I) 40. How long must their journeying 
 continue, in order that they may all come together again ? 
 
 4. A shepherd, on telling his sheep, found that when he told 
 them out by twos, threes, fours, and fives, he had none left, and 
 he knew his flock was above 300, but less than 400. What num- 
 ber had he ? 
 
" 
 
 LEAST COMMON MULTIPLE 
 
 99 
 
 5. Three bodies move uniformly in similar orbits round the 
 same centre in 87, 224, 365 da. respectively. Supposing all three 
 in conjunction at a given time, find after how many days they 
 will be in conjunction again. 
 
 6. An island is 48 mi. in circumference. A, B, and C have 
 to walk round till they all arrive together at the starting-point. 
 A walks 2, B 3, and C 4 mi. an hour. How many times must 
 each go round before the task is accomplished ? 
 
 7. Find the L. C. M. of 11, 14, 28, 22, 7, 56, 27, 81, 54, and 36. 
 
 8. If 1 lb. Avoirdupois contains 7000 gr., and 1 lb. Troy 
 5760 gr., find the least weight which can be expressed without 
 fractions in both lb. Troy and lb. Avoirdupois. 
 
 9. Prove by example that the product of the H. C. F. and the 
 L. C. M. of two numbers is equal to the product of the numbers, 
 and state why this is so. 
 
 10. The L. C. M. of 391 and another number is 12,121, and the 
 G. C. M. is 23. Find the other number. 
 
 11. Along a certain path 1600 yd. long, there is a house every 
 50 yd., and a tree every 20 yd. How many houses will have 
 a tree in front? 
 
 12. The periods of three planets which move uniformly in circu- 
 lar orbits round the sun, are respectively 200, 250, and 300 da. 
 Supposing their positions relatively to each other and the sun to 
 be given at any moment, determine how many da. must elapse 
 before they again have exactly the same relative positions. 
 
 I' 
 
 m 
 
 
 
CHAPTER X 
 
 FEAOTIONS 
 
 102. The expression 5 ft. denotes a quantity measured by 
 the number 5 and the unit 1 ft. The expression 5x2 in. 
 denotes a quantity measured by the number 5 and the unit 
 2 in. The expression 5 x ^ ft. denotes a quantity measured 
 by the number 5 and the unit one-sixth of a ft. 
 
 103. The expression | ft. denotes that the quantity 1 ft. 
 is conceived as made up of 6 equal parts or units, and that 
 5 of these parts or units have been taken to measure the 
 quantity denoted by | ft. 
 
 The primary unit, 1 ft., has been divided into 6 equal 
 parts to give the direct measuring unit, which is ^ ft., or 
 2 in. The number of these units in the given quantity is 5. 
 The ratio of the given quantity to the direct measuring unit 
 is 5. 
 
 104. The quantity represented by | ft. contains 5 direct 
 measuring units, and the primary unit, 1 ft., contains 6 of 
 these units. Hence the fraction |^ expresses the ratio of the 
 quantity denoted hy | ft. to the primary unit, 1 ft. 
 
 AD 
 
 ab 
 
 105. Draw a line, AB^ 1 ft. long. Divide it into 6 equal parts, or units, 
 each ^ of a ft. long. Draw a second line, (7/>, above the first, containing 5 of 
 these units, and use these two lines to illustrate the preceding paragraph. 
 
 100 
 
FRACTIONS 
 
 101 
 
 Exercise 62 
 
 In the following questions give the number of direct measuring 
 units that makes up the primary unit. Name the direct measur- 
 ing unit in two ways. Give the ratio of the quantity to the direct 
 measuring unit. Give the ratio of th3 quantity to the primary 
 unit. 
 
 1. 
 
 f ft. 
 
 5. 
 
 2. 
 
 fyd. 
 
 6. 
 
 3. 
 
 j)b. 
 
 7. 
 
 4. 
 
 i sq. yd. 
 
 8. 
 
 3 
 2 
 
 9. 
 10. 
 11. 
 12. 
 
 U sq. ft. 
 f cu. yd. 
 ^ wk. 
 
 2 4 vr 
 
 13. 
 
 f da. 
 
 14. 
 
 f da. 
 
 15. 
 
 U hr- 
 
 16. 
 
 1 min. 
 
 106. A Fraction is a number in which the unit of measure 
 is a definite part of some primary unit of the same kind. 
 
 The denominator shows into how many parts the primary 
 unit is divided to give the direct unit of measure ; it also 
 names this unit. The numerator shows the number of them 
 that measures the quantity. 
 
 A proper fraction, as an expression of measured quantity, 
 is one in which the numerator is less than the denominator. 
 Select the proper fractions : |, |, |, |, |. 
 
 
 
 B 
 
 D 
 
 K 
 
 ICn. Let AB represent some quantity measured by 5 
 units, each equal to AC^ and DE as measured by 6 units, 
 each equal to A C. Then if we think of AB in relation to 
 DU^ we think of 5 units in relation to 6 units, and this rela- 
 tion or ratio is expressed by the fraction |. 
 
 Similarly, the fraction, or number |, expresses the ratio of 
 85 to $6, 5 hr. to 6 hr,, 5 mi. to 6 mi., 5 (8 ft.) to 6 (8 ft.), 
 5 (12 lb.) to 6 (12 lb.), or, generally, 5 of any unit to of 
 the same unit. 
 
 !■;, 
 
 13 
 Hi 
 
 ^me- 
 
 5MJr. 
 
 -I 
 
HKH^^^W^ 
 
 102 
 
 ARITHMETIC 
 
 Similarly, it expresses the ratio of 20 lb. (i.e. 5x4 lb.) 
 to 24 lb. (i.e. 6x4 lb.), and so on. 
 
 Exercise 63 
 
 Write the fraction which expresses the ratio of the following 
 quantities : 
 
 1. 2 ft. to 5 ft.; 2 ft, to 1yd. 
 
 2. $4 to $9; $8 to $12; 1 dime to $1. 
 
 3. 3 qt. to 4 qt. ; 2 qt. to 1 gal. 
 
 4. 12 yd. to 32 yd. ; 4 in. to 2 ft. 
 
 5. If 32 yd. of cloth cost $48, what part of $48 will 12 yd. 
 cost ? How many dollars will 12 ydv cost ? 
 
 6. 24 men to 36 men ; 18 men to 24 men. 
 
 7. If 18 men can do a piece of work in 32 da., in what part 
 of 32 da. can 24 men do the same work ? How many da. ? 
 
 108. Let AB represent some definitely measured quantity, 
 as 4 ft. or 16 ft., and let it be divided as shown in the 
 diagram. 
 
FRACTIONS 
 
 103 
 
 It is evident from this diagram that |, |, |, |, |, |, |^, ^|, 
 ^|, of a quantity measure it, and are all equal. It is also evi- 
 dent that J, |, |, |, j^Q, j^g' 1^6' ^^ ^ quantity measure one-half 
 of it, and are all equal. 
 
 Similarly, J, |, ^gi ^^ ^ quantity measure one-third of it, 
 and are equal. Similarly, ^ = ^q. 
 
 ::) 
 
 Exercise 64 
 
 1. Find|of$8; }of$8; foffS. 
 
 2. Find ^ of f 30; y2^of$30; f^offSO. 
 
 3. Find ^ of 36 ft. ; find also respectively |, |, y\, r^-g, and J| 
 of 36 ft. 
 
 4. Find | of 24 dimes ; | of 24 dimes ; -fj of 24 dimes. 
 
 5. Find f of 28 lb. ; jf of 28 lb. ; |§ of 28 lb. 
 
 6. Find f of 32 da. ; f of 32 da. ; if of 32 da. ; |f of 32 da. 
 
 7. Name three fractions equal to |; three equal to J; three 
 equal to |. 
 
 8. Name two fractions each equal to J, and prove your results 
 correct by taking each fraction of $ 24. 
 
 i| 
 
 Exercise 65 
 
 1. If 1 da. is made up of 24 measuring units, find the number 
 .of such units respectively in i, ^, i, i, J^, ■^^, of a da. 
 
 2. If 1 da. is made-up of 24 measuring units, find the number 
 of such units respectively in i, J, |, |, ■^, and if of a da. 
 
 3. If 1 hr. is made up of 60 measuring units, find the number 
 of units in ^, -f-^, ^, of an hr. 
 
 4. If 1 wk. contains 168 units of time, how many units of 
 time are there respectively in |, y\, ^, and /^ of a wk. ? 
 
 
104 
 
 ARITHMETIC 
 
 5. If 1 sq. ft. (contains 144 units of area, find the number of 
 units of area in ], H, ^\^, }§, and Jf of a sq. ft. respectively. 
 
 6. If $ 1 contains 100 measuring units, find the number of units 
 that measure $ J, f -/o, ^ ih '"^"^^ ^ to" I'espectively. 
 
 7. If 1 lb. is rei)resented by a 16-unit quantity, what part of 
 a lb. is a 1-unit quantity ? a 2-unit quantity ? a 4-unit quantity ? 
 an 8-unit quantity? What part of a pound is eight 1-unit 
 quantities ? four 2-unit quantities ? two 4-unit quantities ? one 
 8-unit quantity ? 
 
 8. What actual coins and how many of each kind are equal 
 respectively to |, |, -fjj, and i § of f 1 ? 
 
 9. From the above examples state what changes can be made 
 in the numerator and denominator of a fraction without altering 
 the value of the fraction. 
 
 109. Ill tliis exercise, the quantities 1 da., 1 hr., 1 wk., 1 sq. ft, $ 1, and 
 1 lb. have been said to contain a certain number of units. Thus, 1 da. con- 
 tains 24 units of 1 hr. ; 1 hr., 60 units of 1 min. ; 1 wk. , 168 units of 1 hr. 
 Any measured quantity may, however, be represented by any convenient 
 number of units. Tims 1 da. may be measured by 75 units, ^1 by 24 units, 
 a piece of work by 40 units, and so on. 
 
 110. (1) Express 9 yd. as eighths of a yd. 
 
 A L_ I I I \ ! \ B 
 
 
 Let the line AB be drawn to represent 1 yd. Think of 1 yd. as contain- 
 ing 8 units of length, as shown in the diagram. 
 
 Then 9 yd. will contain 9 x 8, or 72 of these units. 
 Therefore 9 yd. is equal to \2_ of 1 yd. 
 
 (2) Express $ | as a fraction with 20 as a denominator. 
 
 Think of $ 1 as containing 20 units of 5^ each. Then $ | contains | of 
 20, or 15 of these units. Therefore, $ | = $ |^. 
 
FllACTIONS 
 
 105 
 
 Exercise 66 
 
 Express as fra(;tioiis : 
 
 1. $8asl()ths. 
 
 2. $4as20ths. 
 
 3. 9 yd. as 3ds. 
 
 4. 6 ft. as 12ths. 
 
 5. 14 gal. as 4ths. 
 
 6. 11 wk. as 7ths. 
 
 7. 5 yr. as 12ths. 
 
 8. o da. as 24ths. 
 
 9. 4 mill, as COths. 
 
 10. 6 hr. as COths. 
 
 11. 8 pk. as Sths. 
 
 12. 12 bu. as 4ths. 
 
 13. Ill each of these questions name the direct measuring unit 
 in your fraction as an actual unit of measure in common use. 
 
 14. Express as fractions with 100 as denominator: L 4, I, JL, 
 
 1L3329 
 '^Jif T5> i> Zf Zf T'TT- 
 
 Keduce, illustrating your work by diagrams : 
 
 15. I ft. to 12ths. 19. I da. to 24ths. 
 
 16. I yd. to 33ds. 20. j% to 78ths. 
 
 17. I lb. to 16ths. 21. j\ to 45ths. 
 
 18. f gal. to 28ths. 22. li to 54ths. 
 
 23. What are the new units of measure in your results? 
 Where possible, identify them with actual units in common use. 
 
 111. (1) Express 8 in. as a fraction of 1 ft. 
 
 8 in. = /a or | of a ft. = | ft. 
 
 (2) A man's capital is represented by 20 units of value. 
 He invests \ of it in land and J of the remainder in bank 
 stock. How many units did he invest in bank stock, and 
 what part is it of his entire capital? 
 
 The amount invested in land = | of 20 units, or 5 units. 
 
 The remainder = 20 — 5, or 15 units. 
 
 The amount invested in stock = J of 15, or 5 units. 
 
 .*. the amount invested in stock =2^5, or \ of his entire capital. 
 
 
 m 
 
 : •Ik -'I 
 
 11 
 
106 
 
 ARITHMETIC 
 
 
 M 
 
 Bzercise 67 
 
 1. Express as a fraction of a ft. : 
 
 3 in., 4 in., 5 in., 6 in., 9 in., 10 in. 
 
 2. Express as a fraction of a lb. Avoir. : 
 
 14 oz., C oz., 9 oz., 12 oz., 15 oz. 
 
 3. Express as a fraction of a mo. : 
 
 12 da., 15 da., 18 da., 20 da., 25 da. 
 
 4. If 1 mi. is the measuring unit, what number measures 
 each of the following : 
 
 20 rd.? 35 rd.? 80 rd. ? 120 rd. ? 150 rd. ? 
 6. Express as a fraction of a da. : 
 
 6 hr., 9 hr., 12 hr., 15 hr., 18 hr. 
 
 6. Express as fractions of a dollar : 
 
 25, 50, 75, CO, 80, and 100 cents. 
 
 7. If $4 is used as a measure of ^4, what is the number ex- 
 pressing the measurement ? If f 4 is used as a measure of $ 3, 
 what is the number ? 
 
 8. If cloth is 40^ a yd., how much can I buy for 30 ^ ? 
 
 9. If oranges cost 30 ^ a doz., what part of a doz. can I buy 
 for 25^ ? How many oranges ? 
 
 10. A man walks a certain distance in 4 hr. What part of 
 it does he walk in 1 hr. ? In 2 hr. ? In i an hr. ? 
 
 11. A can do a piece of work in 3 hr., 13 the same amount in 
 4 hr. If the work be measured by 12 units, how many units 
 will A do in an hr. ? How many B ? How many both working 
 together ? What part will their joint work for an hr. be of the 
 whole work ? 
 
 12. A received a certain sum of money, B twice as much, and 
 C as much as A and B together. How many units measure the 
 entire amount ? B receives what part of the whole sum ? 
 
 13. Given that pure water contains 15 parts by weight of oxy- 
 gen, and 2 parts of hydrogen, what part of the weight of a gallon 
 of water is hydrogen ? 
 
FRACTIONS 
 
 107 
 
 14. Six brothers join in paying a debt of $ 700. The eldest 
 pays ^ of it, and each of the others \ of the remainder. How 
 much does the eldest pay ? How much does each of the five pay ? 
 This is what part of the whole debt ? 
 
 16. If a pipe empties a tank at the rate of 12 gal. in 1 min., 
 what is the rate per sec. ? 
 
 16. $ 40 is divided among A, B, and C, caving A J of it, B |, 
 and C the remainder. What is the sum of A and B's shares ? 
 What is C's share ? C's share is what part of the whole sum ? 
 
 17. The value of a mine is represented by 10 units of money. 
 A man who owns | of it sells } of his share. How many units 
 did he own ? How many did he sell ? What part of the whole 
 mine did he sell ? 
 
 18. The length of an oblong is 8 ft., and the width 4 ft. 
 What part of the perimeter is the length ? 
 
 19. An oblong 6 ft. wide and 8 ft. long is divided into strips, 
 each 1 ft. wide, made' by drawing lines parallel to the length. 
 What part of the area of the oblong is the area of one strip ? 
 
 20. The area of an oblong is 10 sq. ft. What part of its area 
 is that of a square whose side is 2 ft. ? 
 
 21. If 20 units measure the cost of a lb. of tea sold at a 
 gain of /^ of the cost, how many units are gained by selling ? 
 What is the selling price ? The cost price is what part of the 
 selling price ? What is the ratio of the selling price to the cost ? 
 The gain is what part of the selling price ? 
 
 22. A has 12 marbles, and B has 3. They play together, and 
 A loses J of his marbles. How many has B now ? What part 
 are they of what A now has ? 
 
 23. The value of a house is measured by 5 units. The lot on 
 which it stands is worth ^ as much as the house. What is the 
 measure of the value of the house and lot ? The value of the lot 
 is worth what part of both together ? 
 
 
 
 .i*'U 
 
 -AM 
 ill 
 
108 
 
 ARITHMETIC 
 
 24. A and B set out at the same time from places 42 mi. 
 apart, and meet at the end of G hr. A travels at the rate of 
 3 mi. an hr. How far does B travel ? What is B's rate ? A's 
 rate is what part of B's ? 
 
 25. Apples are sold at the rate of 12 for a dime, and bananas 
 at the rate of 8 for a dime. Compare the value of an apple with 
 that of a banana. 
 
 Note. — Let the dime be measured by 24 units. 
 
 26. A crew can row 6 mi. an hour in still water. What is the 
 rate of rowing up stream in a current running at the rate of 
 2 mi. an hr. ? What is the rate down stream ? What is the 
 ratio of the rate down stream to that up stream ? 
 
 Reduction of Fractions 
 
 112. A fraction is in its lowest terms when its numerator 
 and denominator have no common factor. 
 (1) Reduce ||J| to its lowest terms. 
 
 Tlie object of reduction is to give a more definite idea of the value of the 
 quantity by expressing the ratio in the smallest numbers. 
 
 315 
 
 10 5 
 1 tlu 
 
 135=|_7_. 
 
 The common factors are 3, 3, and 5. 
 
 The effect of dividing each term of the first fraction by 3 is to make each 
 measuring unit 3 times as large, and to reduce the number of these units 
 to one-third as many. Similarly, with the second division by 3, and the 
 third by 6. 
 
 (2) Reduce |^| to its lowest terms. 
 
 713 
 558 
 155 
 124 
 31 
 
 992 
 713 
 279 
 155 
 124 
 124 
 
 713 ^ 713 -^ 3 1 ^ 23 
 992 992-31 32* 
 
FRACTIONS 
 
 109 
 
 In this instance the direct measuring unit of the fraction || is 31 times 
 as great as that of the fraction |^|, but only ^^j- as many units are needed to 
 measure the quantity denoted by |^| or f| of the primary unit. 
 
 (3) A real estate dealer bought a house for 12545 and 
 sold it for $1679. Find the ratio of the selling price to the 
 cost price. 
 
 The selling price = if |f of the cost price 
 
 = If of the cost price. 
 The G. C. M. of 1679 and 2545 is 73. 
 
 Dividing both numerator and denominator by 73, we have the required 
 ratio equal to |?. 
 
 r>.ii 
 
 Exercise 68 
 
 Reduce to its lowest terms 
 
 1. 
 
 $if. 
 
 6. 
 
 2. 
 
 ^iJ- 
 
 7. 
 
 3. 
 
 $M. 
 
 8. 
 
 4. 
 
 $}f- 
 
 9. 
 
 5. 
 
 ^if 
 
 10. 
 
 48 
 
 14 
 6^- 
 
 3 2 
 
 13. 
 99* 
 
 5 4 
 
 11. 
 12. 
 13. 
 14. 
 
 35 
 ¥T- 
 
 84 
 WW' 
 
 24 
 
 8¥* 
 
 _4 8 
 
 15. If I pay f 60 for a bicycle and afterward sell it for f 45, 
 what part of the cost do I sell it for ? 
 
 16. A merchant sells 55 yd. of cloth from a piece containing 
 66 yd. What part of the whole piece does he sell ? 
 
 17. A man buys a horse for f 128 and sells it for f 112. Find 
 the selling price as a fraction of the cost. 
 
 18. On an investment of ^ 88 a merchant gains $ 33. Compare 
 the gain with the cost. 
 
 19. Sixty days is what fraction of a year ? 
 
 20. Out of a farm containing 135 A., 81 A. were sold. Find 
 what part of the farm was sold. 
 
 •'■"3 
 i4i 
 
no 
 
 ARITHMETIC 
 Exercise 69 
 
 Keduce to the lowest terms : 
 
 1 2 34 
 ^' SI'S- 
 
 2. 
 
 559 
 
 3. 
 
 4. 
 
 45 9_ 
 1 1^9* 
 
 2124 
 319^' 
 
 K 39 9 6 
 O. f 
 
 6. 
 
 ^B^T- 
 
 1161 
 T¥T7- 
 
 7. 
 8. 
 
 1218 
 
 161 
 
 9. A person made $ 282 on an investment of f 329. What 
 part of the cost did he gain ? 
 
 10. A speculator paid $ 1071 for a lot and shortly after sold it 
 at an advance of ^ 259. Find his gain as a fraction (in its lowest 
 terms) of the cost price. 
 
 11. A man insured his house which cost $ 2552 for f 1798. 
 For what part of the cost did he insure it ? 
 
 12. I paid $ 1449 for a house and insured it so that in case of 
 fire I should lose only f 184. What fraction of the cost would I 
 lose in case of fire ? 
 
 13. Property to the value of $ 3341 was assessed for $ 1542. 
 Find the ratio of the assessed valuation to the real value of the 
 property. 
 
 14. Having $ 5943, I invested $ 5094 in business. What part 
 of my money did I invest in business ? 
 
 113. Such an expression (as $ 21, e.g.) denotes a quantity 
 in which two units of measure of different values have been 
 used : a primary unit and parts of this, a derived unit. 
 
 ^2Jis the number 7 in disguise. To make it number in 
 the strict sense, we must express the quantity in the smaller 
 unit of measure, i.e. as $^; this as a quantity can be counted; 
 $ 2 J cannot be counted. 
 
 An improper fraction as an expression of measured quantity 
 is one whose numerator is equal to or greater than its de- 
 nominator ; as J, |, |, ^^. 
 
FRACTIONS 
 
 111 
 
 (1) Reduce to an improper fraction 75f yd. 
 
 Let 1 yd. =3 units of length of | yd. each. 
 Then 75 yd. = 225 units of length of ^ yd. each, 
 and f yd. = 2 units of length of i yd. each. 
 .-. 75f yd. = 227 units of length of I yd. each, or ^^ yd. 
 Hence if we count the smaller unit i yd. 227 times, we measure the whole 
 quantity 75| yd. 
 
 Exercise 70 
 
 Reduce to improper fractions : 
 
 1. 
 2. 
 3. 
 4. 
 5. 
 6. 
 7. 
 
 4 J. 
 16f 
 
 8. 20fgal. 
 
 9. 8fpk. 
 5A lb. 
 
 3iyd. 
 
 5,^ ft. 
 
 10. 
 
 11. 
 
 12. 
 13. 
 14. 
 
 12f. 
 
 Q5 
 
 6tV 
 
 15. 
 16. 
 17. 
 18. 
 19. 
 20. 
 
 3^. 
 20f. 
 21t. 
 19f 
 
 7-6 
 
 40J. 
 
 21. What are the two units of measurement in each question ? 
 What is the unit of measurement in the result ? How many of 
 these units must be counted to measure the quantities ? 
 
 22. What is the quantity which is measured by the unit J ft. 
 and the number 9 ? If the same quantity be measured by the 
 number 3, what is the measuring unit ? 
 
 23. If 9 boys receive $^ each, what sum was divided? If 
 the same sum had been divided among 4 boys, what would each 
 have received ? 
 
 Exercise 71 
 
 Reduce to improper fractions : 
 1. 
 2. 
 3. 
 4. 
 
 ISA- 
 
 6. 
 
 34^. 
 
 9. 
 
 C4if 
 
 13. 
 
 29ii% 
 
 21A- 
 
 6. 
 
 481f 
 
 10. 
 
 152f 
 
 14. 
 
 87|tf 
 
 29H- 
 
 7. 
 
 51M- 
 
 11. 
 
 98A- 
 
 15. 
 
 341if 
 
 41^- 
 
 8. 
 
 36tf. 
 
 12. 
 
 168f. 
 
 16. 
 
 453ff^ 
 
 l«l 
 
112 
 
 ARITHMETIC 
 
 114. (1) Which is greater, | ft. or f ft.? 
 
 Consider the ft. as having 42 measuring units, then | of this is 35, and 
 f is 36 units. 
 
 .-. f ft. is greater than f ft. by 1 unit, or ^^ ft. 
 
 (2) Compare the quantities $|, f^V' ^1* 
 
 Let $ 1 be represented by 24 units of value. Then $ |, $ j\, f |, are respec- 
 tively equal to 18, 14, and 15 units. Hence the first fraction is the greatest 
 and the second the least. 
 
 Eaiercise 72 
 
 Find the greatest and least of the following : 
 
 1. I yd., f yd. 4. I ft., J ft., H ft. 
 
 2. # "I", # f . 5. 
 
 3. $ YU, ^ -g-, $ 2V ■ ^• 
 
 7. State how to find the greatest and least of a number of 
 fractions. 
 
 1 _8_ 11 
 
 4> 11) 22* 
 
 2 A 11 
 
 ¥' 8? 1 «• 
 
 115. Express ^^- yd. in terms of the primary and derived 
 
 units of measure. 
 
 The primary unit 1 yd. contains 5 derived units of I of a yd. each. 
 Hence 2^^- yd., i.e. 23 derived units = 4 primary and 3 derived units ; 
 
 = 4| yd. 
 Or, more simply, 5)23 
 
 I 
 
 Exercise 73 
 
 Express in terms of the primary and direct measuring units : 
 1. $-^/. 5. -VV-yr. 
 
 ft 293 ff- 
 ^- iia 7 28 4 vfl 11 4849 IK 
 
 4. ^^wk. 8. V/rd. 12. %¥-lb. 
 
 13. What are the two units of measurement in the answers to 
 the above problems ? 
 
 2. $-V-. 
 
 3. H^da. 
 
 9. \%^-m[. 
 
 10. 3 7 8 oz. 
 
FRACTIONS 
 
 113 
 
 Exercise 74 
 
 Reduce to integers and proper fractions : 
 
 1 8048 O 208 3 5 o 894 3 A 
 
 9_9 41 
 " 6T • 
 
 5 79 3 
 '3 8 9"' 
 
 6- %W- 
 
 Addition of Fractions 
 
 116. The sum of 3 dimes and 4 dimes = 7 dimes. 
 The sum of I {^ and % {^ = $ /^ . 
 The sum of 5 oz. and 8 oz. = 13 oz. 
 The sum of ^^ lb. and -^q lb. = ^| lb. 
 
 Here the direct measuring unit of value is $ ^^ or 1 dime, 
 that of weight ^^g lb. or 1 oz. 
 
 (1) Add J ft., I ft., I ft. 
 
 The L.C.M. of the denominators is seen to be 12 ; the ft. is considered as 
 measured by 12 equal parts or units. 
 
 1 ft. = 6 units ; f ft. = 8 units ; f ft. = 9 units. 
 The sum = 6 + 8 + 9 or 23 units 
 = f I ft. or 111 ft. 
 .*. the sum = ljl ft. 
 
 Prove the correctness of this result by drawing a line and measuring off 
 ^ ft., I ft., and I ft., and also lii ft. 
 
 (2) Find the sum of |21|, $15|, $13|, $S-^\. 
 
 The L. C. D. = 72, . •. the dollar is considered as measured by 72 units. 
 
 I f + •$ I + $ f + $ r^5 = 27 + 60 + 32 + 42 or 161 units, i.e. $ -W" ^^ $ 2||. 
 $ 21 + $ 15 + .^ 13 + $ 8 = ^ 57. 
 
 . •. the sum = $ 57 + $ 2} | = $ 59J J. 
 
 Find the sum of ; 
 
 1. 
 
 f , f i. 
 
 2. jY It'-j T2 ^^' 
 I 
 
 Exercise 75 
 
 3. 
 
 T^ 
 
 oz., j\ oz. 
 
 4. 1^1 da., ^J da. 
 
 5. ^ wk., ^ wk. 
 
 6. 1^ mill., J^ min. 
 
114 
 
 ARITHMETIC 
 
 Exercise 76 
 
 In the following exercise what ia a convenient number of direct 
 measuring units by which to represent the primary units ^1, 
 1 ft., 1 lb., 1 gal., and so on ? 
 
 Find the sum of : 
 1. fi, $i. 2. $i, $i,ff. 3. fft., fft. 4. J ft., f ft., I ft. 
 
 Prove your answers to 3 and 4 correct by measuring with 
 a ruler. 
 
 5. J lb., I lb., If lb. 6. } gal., -J gal. 7. f bu., J bu., H bu. 
 8- fyr., Ifyr. 9. f da., f da., J da. 
 
 Prove your results correct by reducing each fractional quantity 
 to a lower denomination (as f |^ to ct., f ft. to in., J lb. to oz., 
 and so on) and also your result. Then add the integers. 
 
 Exercise 77 
 
 Find the sum of the following fractional parts of any unit : 
 
 3 4 2 
 
 •?? Zy ^• 
 
 5 8 IS 
 
 12? TTJ ^TT' 
 
 JL 19. 11 3 7 
 
 4. 
 
 ft 3 7 13 14 15 
 **• "8' tf TSf TS' ??• 
 
 O 7 16 9 
 '*'• 12> "JT> IS' 
 
 q 1 1 11 1 2 
 ^' 15' 2 1> ^Z' 
 
 7. 3f, 8iV 23^, 18H, 46/^. 
 
 8. 2f, 3f, 5f, 9j\, 28if- 
 
 9. 23^, 32i|, 43if, 62if 
 
 10 1.S-17 47_9 4 ,S4_8_1_ IP, 27 
 
 11. State how to find the sum of two or more (1) proper frac- 
 tions, (2) mixed numbers. 
 
 12. Explain clearly the principles involved in finding the sum 
 of two fractions. 
 
FRACTIONS 
 
 115 
 
 Subtraction of Fractions 
 117. (1) Find the difference between | hr. and ^^g hr. 
 
 Let 1 hr. = 30 units of time. 
 
 Then f hr. = 26 units and j\ hr. = 16 units. 
 
 .*. tlie difference = 26 — 16 or 9 units, i.e. -^q or ^^ hr. 
 
 Prove this result by expressing f, j\, and ^^ hr. in min., and taking the 
 difference of the first two. 
 
 (2) Find the value of : 
 
 = $2. 
 The L. C. D. = 144. 
 
 Let $ 1 =144 units. 
 
 Then $ f § = 116 units ; $ ^| = 61 units. 
 
 116-61=66. 
 
 (3) Subtract 6^ da. from 9| da. 
 
 Let 1 da. = 24 units of time. 
 
 Then 9| da. - 6^2 ^a. = 3 da. + 9 units - 14 units 
 
 = 2 da. -f 33 units — 14 units 
 = 2 da. + 19 units 
 
 Find the value of : 
 
 1. 
 
 TIT 
 
 -^ 
 
 TS- 
 
 = 2^1 da. 
 
 
 
 Exercise 78 
 
 
 
 4. 
 
 1 gal. - 
 
 i gal. 
 
 5. 
 
 H ^^' - 
 
 - 1 fi lir 
 6 "^* 
 
 6. 
 
 if da. - 
 
 - A da. 
 
 2. ^7^ ft. -^ ft. 
 
 3. f pk.-f pk. 
 
 7. What is the direct unit of measure in each question ? Ex- 
 press your answer in two ways. 
 
 8. Prove results by reducing each fraction to the next lower 
 denomination and then subtracting. 
 
 
 «H! 
 
 m 
 
 "m 
 
116 
 
 AKITHMETIC 
 
 Exercise 79 
 
 Find the difference as a fraction of any unit of measure : 
 2- i 
 
 2 
 ^• 
 
 5 
 6' 
 
 Q 6 
 
 2 
 
 R 4-1—8 
 
 4 _8 ? 
 
 6. 
 
 13 
 
 5 
 
 7. Give the three steps required in subtracting one proper 
 fraction from another. 
 
 8. 
 
 12 -|. 
 
 9. 
 
 5-A- 
 
 10. 
 
 8-A- 
 
 11. 
 
 7-4A- 
 
 12. 
 
 9 - 6tf . 
 
 13. 12-8Jf. 
 
 18. 
 
 ^H 
 
 -m- 
 
 14. 8j9^-5tV 
 
 19. 
 
 4r\ 
 
 -Iff 
 
 15. 9|J-4i|. 
 
 20 
 
 46*- 
 
 -39f 
 
 16. 18H-12/^. 
 
 21. 
 
 95f 
 
 -84f. 
 
 17. 4iJ-2fi. 
 
 
 
 
 Exercise 80 
 
 
 
 
 1. I of the value of a horse = $ 60. 
 ^ of the value of a horse = 
 
 I of the value of a horse = 
 .-. the horse is worth $ . 
 
 Fill out the blanks. 
 
 2. If I of the value of a farm is $ 9000, what is the value of J 
 of the farm ? What is the farm worth ? 
 
 3. A person sold a cow, gaining* f of the cost price. If he 
 gained $ 12, what did the cow cost him ? 
 
 4. A man lost in business f of his property. His loss was 
 $4500; what was his property worth ? 
 
 5. A boy lost f of his marbles, and then had 60 marbles left. 
 What fraction of his marbles did he have left? How many 
 marbles had he at first? 
 
 6. If I of the cost of a gal. of a wine is $ 3, what was the 
 cost? 
 
FRACTIONS 
 
 117 
 
 7. A inercluuit sold potatoes at 75^ a bu., gaining J of the cost. 
 The selling price was what fraction of the cost ? What was the 
 cost price per bu. of the potatoes ? 
 
 8. A merchant sold cloth for 80^ a yd., thereby losing J of the 
 cost. Find the cost. 
 
 Multiplication of Fuaotions 
 
 118. (1) Find the cost of 12 yd. of cloth at || per yd. 
 
 We are required to find the quantity measured by the number 12 and the 
 measuring unit $ |. 
 
 12 yd. cost V x$J = $9. 
 
 (2) Find the cost of | yd. at $12 a yd. 
 
 The cost is measured by the number f and the unit $ 12. 
 
 f yd. costs I of $12 = $10. 
 Explain the solution ^ yd. costs 12 x $ f = $ 10. 
 
 (3) Find the area of a floor 12 ft. long and 9| ft. wide. 
 
 The area is measured by the unit 9^ sq. ft., which is the area of 1 
 strip, and the number 12. 
 
 The area of 1 strip = ()| or '^£ sq. ft. 
 
 .'. the total area = \^ x ^'^ sq. ft. = 117 sq. ft. 
 
 (4) Reduce | ft. to in. 
 
 8 
 
 8 X'/ • 32 ; 
 
 - ft. = - X ^in. = —in. = 10| in. 
 9 Jl 1 S 
 
 3 
 
 Exercise 81 
 
 Find the cost of : 
 
 1. 10 yd. of cloth at f f per yd. 
 
 2. 12 yd. at f If per yd. 
 
 3. 9 yd. at^2|ayd. 
 
 4. I yd. at f 6 a yd. 
 
 5. If the cost price is measured by the number | and the unit 
 $ o, find the cost. 
 
 I'l: 
 
 I 
 
118 
 
 ARITHMETIC 
 
 Reduce to in. : 
 
 6. f ft. 
 
 Reduce to yd. : 
 
 9. 8 rd. 
 
 Reduce to qt. : 
 
 12. 8| gal. 
 
 7. 23^ ft. 
 
 10. 12 rd. 
 
 8. 5fft. 
 
 11. 17 id. 
 
 13. 12iigal. 
 
 Find the areas of the foUowing rooms : 
 
 14. Length 24 ft., width 14 ft. 10 in. 
 
 15. Length 14 ft., width 12 ft. 6 in. 
 
 16. Length 18 ft., width 14 ft. 10 in. 
 
 17. Length 19 ft., width 16 ft. 4 in. 
 
 18. Length 22 ft., width IG ft. 9 in. 
 
 19. Length 28 ft., width 20 ft. 11 in. 
 
 Find the area of the walls of a room whose: 
 
 20. Perimeter is 80 ft., height 9 ft. 6 in. 
 
 21. Perimeter is 63 ft., height 10 ft. 8 in. 
 
 22. Perimeter is 67 ft., height 8 ft. 9 in. 
 
 119. (1) Find the cost of 12f yd. of cloth at $1| a yd. 
 
 12| yd. cost 12| x $ If = -V" X ^ V = $ W = $ IHI- 
 
 (2) Find the area of the four walls of a room whose 
 
 perimeter is 62 ft. 8 in., and height 8 ft. 9 in. 
 
 The perimeter = 62 ft. 8 in. = 62| ft. = if^ ft. 
 The height = 8 ft. 9 in. = 8| ft. = ^V" ft. 
 47 
 
 TOO on 
 
 . '. the area = ^ x -j- sq. ft. = 6481^ sq. ft. 
 
 (3) Reduce | rd. to yd. 
 
 4 
 
 ? rd. = § X 51 yd. = ? X ^ = — or 4 1 yd. 
 9 9 '^ 9 ^ 9 ■' 
 
FRACTIONS 
 
 119 
 
 ii 
 
 Exercise 82 
 Reduce to yd. : 
 
 1. J rd., 2i rd., 6j\ rd., 3^% rd. 
 
 Find the cost of the following : 
 
 2. 8| yd. at $ 3t a yd. ; 4^ yd. at $ 2| a yd. ; 
 20| yd. at ^ 7^ a yd. ; 5^ yd. at $ 2 J a yd. 
 
 3. 21 J lb. of sugar at 5J^ a lb. ; 
 13 J lb. of sugar at 5^^ a lb. 
 
 4. 17f yd. of cotton at 11^^ a yd. 
 
 5. 15 J- doz. of eggs at 14|^ a doz. 
 
 6. 81- T. of hay at $ 11| a T. 
 
 7. Find the area of the floor of a room whose dimensions are : 
 (1) 12 ft. 6 in., 10 ft. 8 in. ; (2) 16 ft. 4 in., 11 ft. 3 in. ; (3) 18 ft. 
 8 in., 15 ft. 3 in. 
 
 8. Find the area of the four walls of a room whose perimeter 
 and height are respectively : (1) 52 ft. 6 in., 9 ft. 4 in. ; (2) 63 ft. 
 4 in., 10 ft. 6 in. 
 
 9. On J of a field I planted potatoes ; on | of the remainder I 
 sowed wheat. What part of the field did I sow with wheat ? 
 
 10. I withdrew from the bank f of my deposit and then ^ of 
 the remainder. What part of the original deposit did I take out 
 the second time ? 
 
 11. A man who owns ^^ of a ship sells J of his share. What 
 fraction of his former £;hare does he still own ? What fraction 
 of the ship ? If he had sold J of his share, what part of the 
 ship would he have still owned ? 
 
 12. A grain dealer invested f of his money in wheat, and f of 
 the remainder in oats. What part of his money did he invest in 
 oats ? If he invested ^ 3000 in oats, how much did he have at 
 first? 
 
 13. The owner of a farm valued at f 12,000 sells f of it to 
 one man, and J of the remainder to another. What part of the 
 
 !' 
 
 
 I 
 
120 
 
 ARITHMETIC 
 
 farm does he sell to the second man, and what should he get for 
 it? 
 
 14. l^'our brothers enter into partnership ; the eldest puts in J 
 of the capital and the others the remainder in equal shares. 
 What part of the entire capital does each of the younger 
 brothers put in ? If they each put in f 2000, what is the entire 
 (\apital ? 
 
 15. If I own I of J of a business, what part do I own? If I 
 sell J of my share, what part of the business do I still own ? 
 
 16. A man left his farm to be divided among his three sons ; 
 the oldest got 80 A., the second ^ of the farm, and the youngest 
 I as much as the other two. Trove that the farm contained 
 210 A. 
 
 17. If the loss is measured by the number | and the unit 
 $ 17^ tons, what is the loss ? 
 
 If the gain is measured by the number ^ and the nnit f 11|, 
 what is the gain ? 
 
 120. A owns a farm containing 81 1 A., B owns ^Q-^2 -^•'> 
 and C 64^ J A. How many A. do they own altogether? 
 
 Here we are required to liud the whole quantity measured by the parts, 
 813 A.,96yV A., 04}-!L A. 
 Let 1 A. = 120 units. 
 
 Then f A. = 45 units; /^ A. = 70 units ; }^ A. = 88 units. 
 .-. f A.-\-^.2 A.+ \\ A. = 45 4- 70 + 88, or 208 units = f §^ A. = 1/^% A. 
 81 A. + 90 A. -f 64 A. = 241 A. 
 .-. the sum = 241 A. + 1t¥(J A. = 242 //(j A. 
 
 121. A sum of money is divided among 4 persons. The 
 first receives |^, the second ^, the third ^, and the fourth the 
 remainder. It is found that the first received $700 more 
 than the fourth. Find the sum received by each. 
 
 Consider the sum of money as made up of 60 units. 
 The first receives \ of 60 or 20 units ; the second 15, and the third 12 
 units. 
 
FRACTIONS 
 
 121 
 
 The three receive 20 4 15 -\- 12 or 47 units. 
 
 The fourth receives (JO — 47 or l^J units. 
 
 The first receives 20 — V] or 7 units more than the fourth. 
 
 7 units = j$ 700. 
 
 1 unit = #100. 
 
 .-. the first received 20 units or .^2000, tlie second .1^1500, the third $1200, 
 and the fourtli $ ir>00. 
 
 Exercise 83 
 
 1. What is the combined weight of three men, the first of 
 whom weighs 125 J^ lb., the second 147| lb., and T.i^ third 175| lb.? 
 
 2. A grocer has three bbl. of oil; the first ccmtains 18| gal., 
 the second 24|, and the third lOJ. Find how much there was in 
 the three bbl. 
 
 3. If three crocks contain respectively 8|, 12|, and 14|§- lb. 
 of butter, how much do the first two contain more than the third ? 
 
 4. Four farms join each other; the first contains 125f A., the 
 second 78| A., the third 9GJ- A., and the fourth llOf A. Find 
 the total area. 
 
 5. A person paid $ 165| for a horse, and $ 23J- more than that 
 for a carriage, and shortly after sold them at a loss of f 46 1. 
 What was the selling price ? 
 
 6. What fraction subtracted from the sum of J and ^ will have 
 unity for remainder ? 
 
 7. If f of a school term exceed J of it by 18 L da., how many 
 da. are there in the whole term ? 
 
 8. I am the owner of | of a ship worth $ 30,000, and sell J of 
 the ship. What part of it will then belong to me, and what 
 will it be worth ? 
 
 9. Add together the greatest and least of the fractions f , J, -J-|-, 
 J^, and subtract this sum from the sum of the other two frac- 
 tions. 
 
 i 
 I 
 
 
122 
 
 ARITHMETIC 
 
 ■i, 
 ■I 
 
 10. A man invested ^ his fortune in land, ^ of it in bank stock, 
 ^ in provincial debentures, and lost the remainder, $8000, in 
 speculation. What was his fortune at first ? 
 
 11. Explain why fractions having different denominators must 
 be altered in form before their sum or difference can be expressed 
 by one fraction. 
 
 12. A gentleman sold ^\ of an estate to one person, and ^ to 
 a second. What part of the estate did he still retain ? If this 
 is worth f 8346, what is the value of the estate ? 
 
 13. If a man fills J of a cask with brandy, \ with wine, and J 
 with water, and it lacks 21J gal. of being full, how many gal. 
 will it contain ? 
 
 14. Show that the fraction ^ ^ "*" ^ , is greater than 4 and less 
 
 12 + 14 ® 
 
 than ^. 
 
 
 15. One person expends $ 5 for coal at $ 7 per T. ; and another 
 $ 6 at $9 per T. Which of them obtains the greater quantity 
 of coal? 
 
 16. An estate worth $ 10,000 is left to A, B, and C ; f to A, f 
 to B, and the remainder to C. Find C's portion and its value. 
 
 17. If during the day I pay out J, then J, next y^^, and lastly 
 Y^5^ of the money I had in the morning, what fraction of it have I 
 left? If the sum left amounts to $ 1.54, what sum had I at first? 
 
 18. (a) How much must be added to the denominator of ^, that 
 the resulting fraction may be equal to ^^ ? 
 
 (6) How much must be subtracted from the denominator of -^ 
 that the resulting fraction may be equal to |- ? 
 
 19. In a certain subscription list, | of the number of subscrip- 
 tions are for f 5 each, J are for $ 4 each, ^ are for f 2 each, J are 
 for $ 1 each, and the remaining subscriptions, amounting to 
 $10.50, are for 50.^ each. Find the whole number of sub- 
 scribers, and the total amount of their subscriptions. 
 
' i'V:^.,.jy^\ , -r, '■':'^:: 
 
 FRACTIONS 
 
 123 
 
 20. A man lost \ of his property in speculation ; he afterwards 
 purchased a partnership in business for f 16,000, and had still 
 f 6000 left. What was he worth at first ? 
 
 21. The sum paid for 491 gal. of oil, including a dutj"" on 
 each gal. which amounts to ^ of the cost price of a gal., is 
 $ 1719.12. Find the duty on each gal. 
 
 22. A house and lot cost f 3600 ; the value of the lot is ^ that 
 of the house. Find the value of each. 
 
 23. What must be the length of a plot of ground, if the breadth 
 is 15| ft., that its area may contain 46 sq. yd. ? 
 
 24. A boy gives ^ of his marbles to A, J to B, and the rest to 
 C. C loses 20, and has then 70 less than A. How many had 
 each at first ? 
 
 25. A has $ 3 more than J of the whole of a sum of money ; B 
 has $ 4 more than J of the whole ; and C has $ 5 less than ^ of 
 the whole. Find the sum divided. 
 
 26. After spending $ 10 less than f of my money, I had $ 15 
 more than ^-^ of it left. How much had I at first ? 
 
 i.ji. 
 
 122. (1) Find the product of Jf x 6| x 7^ |. 
 Reducing to improper fractions and cancelling, the product 
 
 8 
 
 19 
 
 1% ^^ rr- 
 
 v^ 9 n 
 
 = 1M=16|. 
 9 ^ 
 
 (2) Find the volume of a solid whose dimensions are 2| 
 
 in., 2§ in., and 4J in. 
 
 The vohime = 2§ x 2f x 4^ cu. in. 
 = V^ X f X f cu. in. 
 = If 5 or 31^ cu. in. 
 
 The smallest unit of volume is 1 cu. in. 
 
 The next larger unit of volume is \\ cu. in. 
 
 The largest unit of volume is 2| x 4| cu. in. = 12 cu. in. 
 
 How many units of each magnitude measure the solid ? 
 
 I 
 
 !■;, 
 
 
124 
 
 ARITHMETIC 
 
 Exercise 84 
 
 Find the volume of the solids whose dimensions are : 
 1. 2 in., 4 in., GJ in. 4. j\ ft., 2f ft., 4| ft. 
 
 5. I ft., f ft., Y% ft. 
 
 6. A yd. 236T yd, 8i yd. 
 
 2. H in., 21 in., 2| in. 
 
 3. 2J^ in., 232^ in., 4| in. 
 
 7. What are the units of volume in the first and second 
 questions ? 
 
 8. How many units of each magnitude measure the solids in 
 the first and second questions ? 
 
 Fin 
 
 d the product of : 
 
 
 
 9. 
 
 1 X y X j\. 
 
 12. 
 
 U X 2^3_ X 13f 
 
 10. 
 
 A X f X If. 
 
 13. 
 
 A X 2| X 3if 
 
 11. 
 
 10 V 1 V 28 
 
 14. 
 
 16i X 4y\ X 2^\ 
 
 123. (1) A owns | of a ship and B the remainder, and | of 
 
 the difference between their shares is $1500. Wliat is the 
 
 vessel worth ? 
 
 Represent the value of the ship by 20 units of money. 
 A owns f of the sliip or 8 units, B | of the ship or 12 units. 
 The difference between their shares = 12 — 8 or 4 units. | of tlie differ- 
 ence between their shares = 3 units. 
 
 3 units = $ 1500. 
 1 unit = 500. 
 20 units =10,000. 
 .'. the vessel is wortli $ 10,000. 
 
 (2) A man lost | of the value of his horse by selling it for 
 $ 60. For what should he have sold it to gain | of its value ? 
 
 Let the value of the horse = 5 units of money. 
 
 Then the loss on selling = 2 units of money. 
 The first selling price = 3 units of money. 
 
 The second selling price = 7 units of money. 
 .*. the second selling price = ^ of $ GO = $ 140. 
 
FRACTIONS 
 
 125 
 
 (3) A person who has | of a mine sells j of his share for 
 $6000. What is the value of the whole mine? 
 
 Let the value of the mine be measured by 20 units of money. 
 
 I of the mine = 8 units. 
 
 The amount sold = | of 8 units = units. 
 
 6 units = $6000. 
 
 1 unit = 1000. 
 
 20 units = 20,000. 
 
 .•• the value of the mine = $ 20,000. 
 
 
 ■IT 
 
 i 
 
 Eacercise 85 
 
 1. A grocer huys tea at 64^ a lb., and sells it so as to gain 
 -J of the cost price. Find his receipts on 0043 lb. 
 
 2. A man bought a horse for $ 120, which was $ 30 less than 
 ^ of one and a half times what he sold him for. Hosv much did 
 he make on the sale of the horse ? 
 
 3. If I own } of f of I of a ship worth f 20,000, and sell J of 
 the ship, what will the part I have left be worth ? 
 
 4. The owner of a ship which was valued at f 10,000 sells f 
 of it for ,113800, and then ^ of the remainder for .f 1800. What 
 did he gain or lose by the transaction ? 
 
 $. A man has $ 4000 in the bank. He drew out 2^^^ of it, aiul 
 then ^ of the remainder, and afterwards deposited i of what he 
 had drawn out. How much had he then in the bank? 
 
 6. A man divided a farm among three sons; to the first he 
 gave 80 A., to the second 4. of the whole, and to the third J as 
 much as to both the others. How many A. did the farm con- 
 tain? 
 
 7. Divide $05.80 between two persons, so that one shall re- 
 ceive one-third as much again as the other. 
 
 8. Five brothers join in paying a sum of money ; the eldest 
 pays ^ of it, and the others pay the remainder in equal shares. 
 
 ' 
 
 
 
126 
 
 ARITHMETIC 
 
 It is found that the eldest brother pays $300y^^ more than a 
 younger brother's share. Find the sum of money. 
 
 9. Show clearly that both terms of a fraction can be multi- 
 plied by the same number, without changing the value of the 
 fraction. 
 
 10. A man commenced business with a capital of ^8000; the 
 first, year he gained f40 for every flOO invested, adding his 
 gain to his capital; the second year he gained $25 for every 
 $ 100 invested, adding his gain as before ; the third year he lost 
 ^ of his accumulated capital. How much did he make in the 
 three years ? 
 
 11. If I of f of an A. produces 43 bu. of potatoes, how many 
 bu. will an A. produce ? 
 
 12. A paid $ GO per A. for his farm, which was | as much as 
 B paid per A. for his farm of 150 A. Find the entire cost of 
 B's farm. 
 
 13. If the sum paid for 247 gal. of spirits amounts, together 
 with the duty, to $ 859.50, and the duty on 1 gal. be -J- part of its 
 original cost, what is the duty i)er gal. ? 
 
 14. A piece of cloth, when measured with a yd. measure 
 which is I of an in. too short, appears to be lOi yd. long. 
 What is its true length ? 
 
 15. I had a sum of money, of which I paid away ^, then i of 
 the remainder, then | of what was still left, and found that I had 
 still left half a dollar less than J of J of the whole. What sum 
 had I at first ? 
 
 16. A man who owns ^^ of a mill sells | of his share. What 
 fraction of the mill does he still own ? Had he sold | of the mill, 
 what fraction of the mill would he still have owned? 
 
 17. Find the sum of the greatest and least of the fractions f , 
 y\, f, and /^; the sum of the other two; and the difference of 
 these sums. 
 
FRACTIONS 
 
 127 
 
 
 18. If when -^t^ of a certain time has elapsed, and then 1 hr., 
 and then f of the remainder of the time, it is found that 16 miu. 
 of the time still i^mains, what was the whole time ? 
 
 19. A man sold ^ of his wheat and then ^ of the remainder, 
 and next \^ oi what then remained, and had 18 bu. more than ^ 
 of his wheat left. How many bu. had he at first ? 
 
 20. A man sold \ of his fg-rm, then \ of the remainder, then \ 
 of what remained, then \ of what still remained, and he then 
 found that he had sold altogether 72 A. more than he had 
 remaining. How many A. had he at first? 
 
 i 
 
 I 
 
 Division of Fractions 
 
 124. Divide ^ ft. by | ft. 
 
 Here we are given the whole measured quantity 3| ft. and the unit of 
 measure f ft., and we are required to find the number which expresses the 
 ratio of the whole quantity to the unit. Consider the ft. as measured by C 
 units of length. Then 3^ ft. contains 20 units, and | ft. contains 5 units. 
 
 .*. the number or ratio = 4. 
 
 Prove that f ft. divides 3^ ft. 4 times by drawing a line 3i ft. or 40 in. 
 long, and measuring it with a line | ft. or 10 in. long. Prove that the 
 quotient 4 is correct by multiplying ^ ft. by 4 and finding the product to 
 be 3^ ft. 
 
 125. If we represent any quantity by | of itself, we think 
 of the quantity as made up of 5 units, and the unit as equal 
 to \ of the quantity. Hence we may think of any quan- 
 tity as equal to 5 x J (i.e. five times one-fifth) of itself. 
 
 What, then, is the meaning of the terms 5 and \ considered 
 separately ? 5 shows the relation or the ratio of the quantity 
 to the unit of measure^ and \ shows the ratio of the unit of 
 measure to the quantity. Numbers thus mutually related 
 are said to be reciprocal. 
 
 B 
 
 ^ 1 — . 1 1 \ \ 
 
 4ill 
 
 
128 
 
 ARITHMETIC 
 
 V 
 
 Again, let the line AB, which is divided into 5 equal parts, represent 
 the primary unit, $1. Then AC represents the quantity denoted by $|. 
 Hence it is evident from the diagram that *i is the ratio of AG to AB, i.e. of 
 the quantity denoted by $ § to the primary unit, $ 1 ; also, that its recip- 
 rocal I is the ratio of AB to AC, i.e. of the primary unit, $1, to the 
 quantity denoted by $ |. Similarly, the ratio of $ 1 (which contains 2 
 units) to the quantity $| (which contains 3 units) is equp' to |, i.e. to the 
 reciprocal of |. 
 
 Exercise 86 
 
 ani 
 
 ! • 
 
 1. How many units of money are there in f 1 and also in $ |, 
 the unit being one-half dollar ? 
 
 How many in f 1 and also in $ |, the unit being one-third of 
 a dollar ? 
 
 How many in f 1 and also in $ |, the unit being one-fifth of 
 a dollar ? 
 
 2. What is the ratio of 1 1 to $ f ? ||? ff? $21? |i? 
 
 i? 
 
 4'^ 
 
 T • 
 
 3. What part is $ 1 of each of these quantities : $ 2 ? | -U ? 
 
 $ V- ? 
 
 3i? 
 
 4_2_ 9 
 * 1 1 • 
 
 4. What is the ratio of 1 lb. to I lb. ? fib.? 2f lb. ? 22 1b.? 
 
 5. What is the ratio of 2 lb. to each quantity given in ques- 
 tion 4 ? Of 3 lb. ? Of 4 lb. ? 
 
 6. What is the ratio of li lb. to each quantity given in ques- 
 tion 4? 
 
 7. If J of a yd. cost 1 1, what part of f 1 will 1 yd. cost ? 
 
 8. If j of a yd. cost $ 1, what part of f 1 will 1 yd. cost ? 
 
 9. If f of a yd. cost $2^, what part of f 21 will 1 yd. cost? 
 How much will 1 yd. cost ? 
 
 10. If 31 yd. of cloth cost fl If, what will 1 yd. cost ? 
 
 11. If 4^ lb. of sugar cost 2(\^, what will 1 lb. cost ? 
 
FRACTIONS 
 
 129 
 
 I, 
 
 12. If f of a yd. of cloth costs 32^, what will 1 yd. cost? 
 What will 3 yd. cost ? 41 yd. ? ^\ yd. ? 
 
 13. If 7 J lb. of raisins cost 85^, what will 4^ lb. cost? 
 
 126. (1) At f 5 a yd., how much lace can be bought for 
 
 Here we are given the whole quantity or $ ^'g , and the measuring unit $ 5, 
 and we are required to find the number. 
 
 Consider $ 1 as measured by 10 units. 
 
 Then $i\j contains 7 units and $6 contains 50 units. Therefore the 
 number of yd. = $ j^^ -t- $ 5 = 7 units -;- 60 units = /g^. 
 
 (2) Find the value of If ^ $|. 
 
 Consider $ 1 as made up of 28 units. Then $ ^ is equal to 24 of these 
 units and $ | to 35 of these units. Therefore, $ J^ -r- $ | = 24 units -h 35 units 
 = f |. Hence, to divide one fractional quantity by another, we first reduce 
 them to the same unit. 
 
 127. It may b^ observed that || is equal to | multiplied 
 by |, the reciprocal of J. Hence^ to divide one fraction hy 
 another^ multiply the first fraction hy the reciprocal of the 
 second. 
 
 Exercise 87 
 
 In the following exercise, solve (1) by reducing the quantities 
 to the same unit, (2) by multiplying by the reciprocal : 
 
 1. ff^ff. 3. fpk. -Jpk. 
 
 2. fyd.-iyd. 4. 2^ - f . 6. /^ - 
 
 5 I -i- -5- 
 
 7'.' 
 
 4 
 
 128. (1) Reduce 21 ft. to a fraction of 1 yd. 
 
 1 ft. = 1 yd. 
 
 2J ft. = 2J X 1 yd. = I X i yd. = I yd. 
 
 Give the solution which follows from the fact that 2| ft. is the whole 
 measured quantity, 3 ft, the unit of measurement, and tliat we are required 
 to find the number. 
 
 K 
 
130 
 
 ARITHMETIC 
 
 ! 
 
 I 
 
 (2) Find the number of strips required to carpet a room 
 
 12 ft. wide witii carpet 2J ft. wide. 
 
 The number of strips = 12 ft. -=- 2J ft. = V- X I = 4|, i.e. 5. 
 . •. 6 strips are required. 
 
 Exercise 88 
 
 1. Reduce to the fraction of a ft. : IJ in., 2 J in., 4|^ in., 2J in. 
 
 2. Reduce to the fraction of a yd. : 1^ ft., 2§ ft., 1^ ft., 2^ ft. 
 
 3. Reduce to fractions of $100: $12^, f 37|^, $62|, $87 J, 
 $6i, $6|. 
 
 4. If I paid f y\ for 6 yd. of cloth, what will 1 yd. cost ? 
 
 5. If I paid $4i for 14 yd. of cloth, what will 1 yd. cost? 
 
 6. If f f is considered as measured by 5 units, what is the 
 value of 1 unit ? 
 
 7. If $ 8| is measured by the unit $ 5, what is the number 
 expressing the measurement ? 
 
 8. At f 2^ a yd., how much lace can be bought for $ ^ ? 
 
 9. At f 3| a yd., how much lace can be bought for $ ^^ ? 
 
 10. Find the number of strips of carpet required to carpet a 
 room 23 ft. 4 in. long, with carpet 2 ft. wide. With carpet 3 ft. 
 wide. 
 
 11. Find the lengths of the following rooms : 
 
 Area 466| sq. ft., width 20 ft. 
 Area 2421 sq. ft., width 15 ft. 
 Area 681^ sq. ft., width 25 ft. 
 Area 877| sq. ft., width 27 ft. 
 
 129. (1) Divide If ft- by | ft. 
 
 lJft.-^fft. = J-f = Jxf = 4f. 
 
 Prove this by drawing a line 1| ft. long, and dividing it into parts f ft. 
 long. 
 
FRACTIONS 
 
 131 
 
 (2) If I paid I J for | of a yd. of cloth, what is the price 
 per yd.? Here we are given the whole measured quantity, 
 that is, $ |, and the number, or |, and we are required to find 
 the measuring unit. 
 
 f of a yd. costs $ |, 
 . •. 1 yd. costs $ I -4- 
 
 3 _ 
 
 5 — 
 
 Ix | = $U, 
 
 Exercise 89 
 
 1. Eeduce to the fraction of a rd. : 2J yd., If yd., 4| yd., 6J yd., 
 
 4tV yd. 
 
 2. Pind and prove by actual measurement the following: 
 2Jft.^Jft.; lJft.H-Jft.; 2Jft.^ift.; 1 J f t. -r-i- f t. ; 4 J f t. ^ ^A^ f t. ; 
 6|ft.-5-/^ft. 
 
 3. At $3^1^ a bbl., how much flour can I buy for $ 17^? At 
 ^ 9^ a yd., how much lace can be bought for $ 2^ ? 
 
 4. Find the number of strips of carpet required to carpet each 
 of the following rooms : 
 
 Width, 22 ft. 6 in. ; width of carpet, 2^ ft. (22 1- ft. -^ 2^ ft.) 
 Width, 24 ft. 3 in. ; width of carpet, 2| ft. 
 Width, 8 yds. 2 ft. ; width of carpet, f yd. 
 Width, 9 yds. 1 ft. ; width of carpet, f yd. 
 
 5. Find the length of each of the following rooms : 
 
 Area of floor, 126f sq. yd. ; width, 8 yd. 2 ft. 
 Area of floor, 133^ sq. yd. ; width, 9J yd. 
 Area of floor, 3341^ sq. ft. ; width, 15 ft. 9 in. 
 Area of floor, 253J sq. ft. ; width, 10 ft. 8 in. 
 
 6. Find the perimeter of each of the following rooms ; 
 
 Area of the Walls Height 
 
 533i sq. ft. 8 ft. 4 in. 
 
 7041 sq. ft. 9 ft. 4 in. 
 
 877^ sq. ft. 10 ft. 9 in. 
 
 65^ sq. yd. 2 yd. 2 ft. 
 
 It! 
 
 ^ 
 
 
■VMMHia 
 
 132 
 
 ARITHMETIC 
 
 7. If f of the sum a man paid for a horse is f 83J, what did 
 the horse cost ? 
 
 8. If I of a yd. of cloth cost ^ f , what will 1 yd. cost ? 
 
 ar€ 
 mc 
 
 130. What is the ratio of 8^ da. to 1-}^ da. ? 
 The ratio of 8f da. to 1^ da. = 8f -- If^ = ^ -^ f I = r X U = 4 
 
 
 Exercise 90 
 
 1. What is the ratio of 1 5| to f 3| ? 
 
 2. What is the ratio of IJf T. to Gf T. ? 
 
 3. If 6f T. of hay cost $42, what part of f 42 will IJf T. 
 
 cost ? How much will l^f T. cost ? 
 
 4. If $ ^^ is represented by unity, what number will represent 
 
 H? 
 
 5. Find the value of (1) 5} h- 3f ; (2) 7 J h- ^^^ ; (3) 2 j-\ -=- 5|^ ; 
 (4) If I - 2if 
 
 42 
 
 131. Simplify^. 
 
 42 
 
 Zl — 42 _:_ Kl 
 
 14 V 4—8 
 T — "3" -^ 2T — 9* 
 
 Or thus, multiplying the numerator and denominator by 12, the L. C. M. 
 of 3 and 4, we have 
 
 6^ 63 ~ 9' 
 
 th{ 
 soi 
 
 Fii 
 
 Exercise 91 
 
 Simplify, using both methods : 
 
 1 * 
 
 A* — • 
 
 2. X. 
 
 If 
 
 3. 
 
 2J 
 5f 
 
 4. 
 
 
 6. 
 
 HI' 
 
 7. 6^ 
 
 8. 
 
 17i 
 
 it 
 25J- 
 
 fro 
 as 
 
 W] 
 < 
 
 of 
 
 « 
 
 He 
 
FRACTIONS 
 
 133 
 
 
 132. (1 ) A man earns i4J a cla., and his daily expenses 
 
 are $1J. How many da. will it take him to save enough 
 
 money to juy a bicycle costing #68J? 
 
 The sum saved each day = |!4J - $ IJ = $2\^. 
 .-. the number of days = 58^ ^ 2];] = 25. 
 
 (2) A farm of 340 A. was divided between two sons, so 
 that I of tlie youngest son's share was equal to | of the eldest 
 son's share. Find the size of each farm. 
 
 I of the youngest son's sluire = 5 of the eldest son's share. 
 
 The youngest son's share = | -=- 5, or § of the eldest son's share. 
 •*• ff + y» o^ "V 0^ *^^^ oldest son's share = 840 A. 
 
 . •. the oldest son's share = 340 A. -.- \} = 180 A. 
 
 .-. the youngest son's share = ^ of 180 A. = 100 A. 
 
 Complete the following solution of question (2) : 
 
 Let I of the yoniigcst sou's share = G units. 
 
 (3) Sold tea at 90^ per lb., having gained 2% of the cost. 
 Find the selling price per lb. if he had lost ^q. 
 
 Let the cost of 1 lb. be measured by 20 units. 
 Then ^^ of the cost price = 3 units. 
 The first selling price = 23 units. 
 The second selling price = 17 units. 
 .'. the second selling price = H of the first 
 
 = iJof 90^ = 66if^ 
 
 fl 
 
 ii 
 
 
 m 
 
 •| I,! 
 
 Exercise 92 
 
 1. A vessel holds 2^^ qt. How many times can it be filled 
 from a barrel containing 31^ gal. of oil ? After filling the vessel 
 as often as possible, how much oil will remain in the barrel ? 
 What fraction of a vesselful will this remaining quantity be ? 
 
 2. Divide the sum of the fractions ^ and y*^ by the product 
 of y^Y and if, and reduce the result to its lowest terms. 
 
 3. The bottom of a cistern measures 7 ft. 6 in. by 3 ft. 2 in. 
 How deep must it be to contain 76 cu. ft. of water ? 
 
 'I 
 
 i (, 
 
 (Ill Wj 
 f+ .3 
 
 ■^ ' 
 
 'u 
 
134 
 
 ARITHMETIC 
 
 
 V- 
 
 4. A man sold 24 horses for $ 150 each ; on ...vit of them he 
 gained y\y of what they cost ; and on the remainder he lost ^ of 
 what they cost. Find his whole gain or loss. 
 
 5. l^y selling cigars at the rate of $2.00 for 4 doz., it was 
 fonnd that J of the cost was gained. Find the price at which 
 each cigar ought to have been sold in order to gain -^j^ of the 
 original cost. 
 
 6. James received a present of some money. He gave J of 
 it to his sister, and ^ of the remainder to his brother, and kept 
 the rest, $ 4, for himself. How much did he receive, and how 
 much did his brother get ? 
 
 7. A tree of 140 ft. in length was broken into two pieces by 
 falling, and -^j of the longer piece was equal to | of the shorter. 
 Find the length of each piece. 
 
 8. A owns I of a ship and B the remainder, and } of the 
 difference between their shares is $1500. What is the vessel 
 worth ? 
 
 9. A person who has f of a min^ sells J of his share for 
 f 6000. What is the value of the wh mine ? 
 
 10. Divide $ 2880 between A and B so that | of A's share will 
 be equal to | of B's. 
 
 11. If 13 were added to a certain number, | of | of the sum 
 would be 40. Find the number. 
 
 12. Three partners. A, B, and C, gain $ 17,100 ; A's gain and 
 C's are together $ 11;000, and | of A's is equal to ^\ of C's. Find 
 each man's gain. 
 
 13. A grocer in selling goods sells 15| oz. for 1 lb. How 
 much does he cheat a customer who buys to the amount of $ 40 ? 
 
 14. The numerator of a certain fraction is \ as much again 
 as its denominator, and the sum of the numerator and denomi- 
 nator is 352. Find the fraction. 
 
 15 
 How far will it have gone in |J of a min. ? 
 
 A cannon ball travels at the rate of 1500 ft. in 1^ sec. 
 
FRACTIONS 
 
 135 
 
 16. A man divides the value of his estate e(iually among his 
 three sons. The first son gains an amount ec^ual to \ of what he 
 has ; the second loses ^ of what he has^ and the third gains | of 
 what he has, and then loses ^ of what he has after his gain ; and 
 now the sons together have $ 300 less than the value of the estate. 
 What was its value? 
 
 17. Given, that pure water is composed of oxygen and hydro- 
 gen in the proportion by weight of 15 to 2, find the weight 
 of each in a cu. ft. of water. (A cu. ft. of water weighs 
 1000 oz.) 
 
 18. A line A is half as long again as B, and B is one-quarter 
 as long again as C. What fraction of the length of A is equal to 
 J of the length of C? 
 
 19. A and B go into business with equal sums of money. A 
 gains a sum equal to ^ of what he had at first, and B loses $ 60. 
 A then has 1| times as much as B. What sums had they at first ? 
 
 20. A and B sit down to play. A has 5J- times as much money 
 as B. At the first game B wins | of A's money. What fraction 
 of B's money must A win back so that they may have equal 
 shares ? 
 
 21. A young man's salary increased ^ every year; his expenses 
 each year were ^ of his salary, and at the end of 4 years he had 
 saved $ 1050. Find his last year's salary. 
 
 22. Find a fraction equivalent to ^\, and having its numerator 
 44 less than its denominator. 
 
 23. A trader spent the first year $ 40, and added to his capital 
 J of what he had left; the second yea^r he spent $50, and added 
 to his capital ^ of what he had left ; the third year he spent $ 60, 
 and added to his capital ^ of Avhat he had left ; he finds that his 
 capital is now 1^^ of what it was at first. Find the original 
 capital. 
 
 24. In a field in which cows and sheep were grazing, ^ of the 
 total number were cows ; but when 3 cows more were driven in, 
 
 id 
 
 ii 
 
 s 
 
^ 
 
 136 
 
 ARITHMETIC 
 
 the latter numbered ^^ of the whole. How many sheep were 
 there ? 
 
 25. Two persons, A and B, finish a work in 20 da., which B 
 by himself could do in 50 da. In what time could A finish it by 
 himself? How much more of the work is done by A than 
 byB? 
 
 Let the work be represented by 100 units of work. 
 
 A and B do 100 units h- 20 or 5 units in 1 da. 
 B does 100 units -;- 50 or 2 units in 1 da. 
 A does 5 — 2 or 3 units in 1 da. 
 .-. A does the work in 100 ^ 3 or 33|^ da. 
 Again, 
 
 A does 3 — 2 or 1 unit more than B in 1 day. 
 
 A does 20 units more than B in 20 days. 
 .*. A does ^§j^ or ^ of the work more than B. 
 
 26. If A can do a piece of work in 3 da., and B in 4 da., in 
 what time can both working together do the work ? 
 
 27. A alone can do a piece of work in 11 da., and B alone 
 can do it in 17 da. How long would they take to do it to- 
 gether ? 
 
 28. A and B can do a piece of work together in 6 da., and B 
 can do \ of the same in 1\ da. How long would each be in 
 doing it alone ? 
 
 29. A can do a piece of work in 5, B in 6, and C in 8 da. 
 If A and B work at it 2 da. each, how long will it take B and 
 C to finish it ? 
 
 30. A and B can do a piece of work in 3 da., B and C in 6 
 da., and A and C in 4 da. If $ 16 be paid for the woik, what is 
 each man worth per da. ? 
 
 31. A and B can do a piece of work alone in 15 and 18 da. 
 respectively ; they work together at it for 3 da., when B leaves, 
 but A continues, and after 3 da. is joined by C, and they finish 
 it together in 4 da. In what time would C do the work by 
 himself ? 
 
FRACTIONS 
 
 137 
 
 i 
 
 t 
 
 G. C. M. AND L. C. M. OF Fractions 
 
 133. To find the G.C. M. of ^ and 2| of any unit. 
 
 Consider the unit as divided into 15 units, then 5| is equal to 80, and 2f 
 to 36 of these units. Here the G. C. M. is equal to the G. C. M. of 80 and 36, 
 or to 4 of these units, i.e. to j\ of the primary unit. 
 
 Hence we have the following rule : 
 
 To find the Gr. C. M. of a number of fractions., reduce the 
 fractions to a common unit., and then find the Gr. C. M. of the 
 resulting numbers. Express the result as a fraction of the pri- 
 mary/ unit. 
 
 134. To find the L. C. M. of 51 and 2|. 
 
 Reduce the fractions, as in the preceding paragraph, to a common unit. 
 The L. C. M. of 80 and 36 is 720, and the L. C. M. is equal to ^^V', or 48 of 
 the primary unit. 
 
 Hence., to find the L. 0. M. of a number of fractions., reduce 
 them to a common unit., and then find the L. C. M. of the re- 
 sulting numbers. Express this result as a fraction of the pri- 
 mary unit. 
 
 s 
 
 Find the G. C. M. of : 
 1. -i/and \0-. 
 Find the L. C. M. of: 
 
 Exercise 93 
 
 2. I and If. 
 
 3. 2^1 and 7^V- 
 
 6. 4|,if, and4f. 
 
 5. 1^, 21, and 3|. 
 
 7. A man has a triangular field, of which the sides are 115|- 
 ft., 128i ft., and 134f ft. Find the length of the longest boards 
 of equal length that can be used in fencing it without cutting 
 a board. 
 
 8. The driving wheels of a locomotive are 17^ ft. in circum- 
 ference, and the trucks 10|. What distance must the train move 
 to bring wheel and truck into same relative positions as at 
 starting ? 
 
 1. ! 
 
 -♦1 ' .1 
 
 il 
 
 <!( .ii 
 
 L 
 
 
 
.> 
 
 138 
 
 ARITHMETIC 
 
 I: 
 
 1) 
 
 9. The side of a field is 19^ rd. in length, and the end 
 IQys ^'^- What is the longest board that can be used in fenc- 
 ing both side and end of the field, so that no fraction (of a board) 
 may be left ? 
 
 10. The distance between the post-office and schoolhouse is 
 half a mile. A carriage stands on the pavement in front of the 
 schoolhouse. The fore wheels are lOy*^ ft. in circumference, the 
 hind wheels 15f ft. A chalk mark is made upon the upper 
 side of each wheel. How often will the 4 chalk marks be all up 
 together while the carriage drives to the post-office ? 
 
 11. Show how to find the least common multiple of two or 
 more fractions. A, B, and C start at a given place to travel 
 round an island 120 mi. in circumference; A's rate is 5| mi. 
 a da., B's 8J, C's 9f . In what time will they all be together 
 again ? 
 
CHAPTER XI 
 
 liS 
 
 t » i 
 
 DECIMALS 
 
 135. The standard unit of money is f 1. One dime is 
 one tenth of $ 1, and 1 f^ is one tenth of 1 dime. Hence 
 we write |1, 1 dime, and 1^ thus, using the dollar sign: 
 fill. 
 
 Write in terms of |1, the sum of: 
 
 (1) $1, 2 dimes, and 4^. 
 
 (2) 1 ten-dollar bill, 11, 1 dime, and 1^. 
 
 (3) 1 one-hundred dollar bill, 1 ten-dollar bill, $1, 1 dime, 
 and 1^. 
 
 136. In the metric system of measures, the standard unit 
 is 1 meter.* 
 
 One decimeter is one tenth of 1 meter, 1 centimeter is 
 one tenth of 1 decimeter, and 1 millimeter is one tenth of 
 1 centimeter. Since the same relation holds between these 
 units as between 1 dime and H, 1^ and 1 dime, we can 
 express the results of measurements in terms of 1 meter, 
 as we express money in terms of $1. 
 
 Hence if we measure with the metric stick a distance equal 
 to 1 meter, 1 decimeter, and 1 centimeter, we can express the 
 distance thus: 1.11 meters. 
 
 Measure the following distances and express them in 
 meters : 
 
 * See Chapter XVIII. If the teacher has not a metric stick, the work in 
 decimals may be based on dollars, cents, and mills. 
 
 139 
 
 »1 
 
 ''i ''! 
 
 
 
 i 
 
 I - »' 
 
 Vyii 
 
" 
 
 ill 
 
 
 
 140 
 
 ARITHMETIC 
 
 (1) 2 meters, 3 decimeters, and 4 centimeters. 
 
 (2) 1 meter, 2 decimeters, 3 centimeters, and 5 milli- 
 meters. 
 
 (3) 1 meter, 1 decimeter, 1 centimeter, and 1 millimeter. 
 
 137. Again, 1 decameter is equal to 10 meters, and 1 
 hectometer to 10 decameters. Hence we can write 1 hec- 
 tometer, 1 decameter, 1 meter, 1 decimeter, 1 centimeter, 
 and 1 millimeter, thus : 111.111 meters. 
 
 Write in terms of the meter, the sum of: 
 
 (1) 1 hectometer, 2 decameters, 9 meters, 7 decimeters, 
 6 centimeters, and 8 millimeters. 
 
 (2) 3 hectometers, 1 decameter, 5 meters, 2 decimeters, 4 
 centimeters, and 6 millimeters. 
 
 138. Notation and Numeration. — Consider the num- 
 ber 111 : the firet 1, beginning at the right, denotes one unit ; 
 the second, one ten or ten units ; the third, 07ie hundred or 
 one hundred units. 
 
 The third 1 is equivalent to one hundred times the first 1, 
 and to ten times the second 1 ; the second is equivalent to 
 ten times the first 1, and to one tenth of the third 1 ; the first 
 1 is equivalent to one tenth of the second 1, and to one hun- 
 dredth of the third 1. 
 
 Now rewrite the number 111, place a point after the first 
 1 to indicate that this 1 is to be regarded as representing the 
 standard unit, and then place after the point three I's, so 
 that we have 
 
 111.111. 
 
 We may ask what each of these I's should mean, if the 
 same relation is to hold among successive digits that we 
 
DECIMALS 
 
 141 
 
 have supposed hitherto to hold. The 1 after the point would 
 naturally mean one tenth. The next 1 to the right would 
 naturally mean one hundredth. It is one tenth of the pre- 
 ceding one — that is, one tenth of one tenth. 
 
 Similarly, the next 1 would signify one thousandth, and 
 would equal one hundredth of the one tenth or one tenth 
 of the one hundredth. Thus, the number 111.111 may be 
 written as follows: One hundred, one ten, one unit, one 
 tenth, one hundredth, and one thousandth. 
 
 Again, the 1 to the extreme right is 1 thousandth; the 
 next 1 is, from its position, equivalent to 10 thousandths; 
 and the next 1 is 100 thousandths. So that to the right 
 of the point we have 111 thousandths. The whole number 
 may now be read, one hundred and eleven, and one hundred 
 and eleven thousandths. 
 
 139. A Decimal Fraction or a decimal is one which has for 
 its denominator 10, 100, 1000, or some power of 10. 
 
 The Power of a number is the product found by multiply- 
 ing the number by itself one or more times ; thus, 100 or 10^ 
 is the second power of 10; 1000 or 10^, the third power 
 of 10. 
 
 The denominator of a decimal fraction is never expressed ; 
 thus -^Q and -f^ are written as decimals, .5 and .67. 
 
 The point placed to the right of the one-unit and between 
 it and the tenth-unit is called the decimal point. 
 
 140. To change a decimal to a vulgar fraction in its lowest 
 
 terms, 
 
 .425 = ,%%% = #0 = H- 
 
 6.0364 = Q^UU = ^'lU^' 
 
 Conversely, 6^%% = 6.293,' 
 
 92r§b = 92.026. 
 
 ill;! 
 
 n, 
 
 :\ 1 
 
 ■ \4 
 
 ^ 
 
 i^& 
 
 Ii4 
 
 ' ii 
 
 ifnii 
 
 k.: -..i; 
 
 ^m 
 
142 
 
 ARITHMETIC 
 
 I 
 
 Exercise 94 
 
 1. Read the numbers in Exercises 95 and 96, expressing them 
 in terms of different units, as 1 meter, 1 mi., 1 lb. 
 
 2. Read .5, .05, .005, 0005. 
 
 3. What is the ratio of .5 to .05 ? To .005 ? To .0005 ? 
 
 4. .0005 is what part of .005 ? Of .05 ? Of .5 ? 
 
 5. Explain how it is that the insertion of a zero between the 
 point and the 5 in the decimal .5 changes the value of the deci- 
 mal. 
 
 6. Read .5, .50, .500, .5000. 
 
 7. What is the ratio of .5 to .50 ? To .500 ? To .5000 ? 
 
 8. .5000 is what part of .500 ? Of .50 ? Of .5 ? 
 
 9. Explain why the addition of zeros to the right does not 
 change the value. 
 
 10. Name the decimal consisting of one digit which lies near- 
 est to .54, .6Q, .92, .78, and .85. 
 
 11. Reduce the following decimals to vulgar fractions in their 
 lowest terms : 
 
 .8, .45, .06, .0004, .375, .0625, .006, .00600. 
 
 12. Express the following fractions as decimals : 
 
 J 4 3. 6 _2 4.S 2 9 Q 8 OfiO 5 4 17 9 
 
 T1T> T5(5> TIFT' lDU¥? TTFTFITJ ^TTTF' '^"♦^T17TFirT» *^ ' TTyTHFTnT' 
 
 Express in figures : 
 
 13. Four tenths ; two, and five hundredths ; six thousandths. 
 
 14. Six hundred and five, and twenty-eight-thousandths ; three 
 thousand and twenty-nine, and sixty-five ten-thousandths; two, 
 and four hundred and nine millionths. 
 
 15. Eighteen, and two ten-thousandths; nine hundred, and 
 twenty-nine ten-millionths ; one hundred, and one ten-thou- 
 sandth. 
 
DECIMALS 
 
 143 
 
 Addition op Decimals 
 
 141. What is the sum of 4.9, 6.084, 24.32, and .8976 ? 
 
 In arranging the numbers, be careful to put the deci- 
 mal points directly under each othev, thus bringing units 
 under units, tenths under tenths, ( tc. Then begin at 
 the lowest order and add as if the figures were integers, 
 putting the decimal between tlie unit and the tenths' 
 place. 
 
 4.9 
 
 6.084 
 24.32 
 
 .8976 
 
 36.2016 
 
 Exercise 95 
 
 Find the sum and prove your answers correct by adding down 
 the columns : 
 
 1. 53.67 2. 9.7 3. .8592 
 
 4.009 20.492 913.74 
 
 821.64 .0487 21.0106 
 
 2.182 918.0006 47.9 
 
 4. Explain step by step the process of addition in the first 
 three examples. 
 
 Write in columns and add : 
 
 5. 6.5 + 32.47 + 2.048 + 59. 
 
 6. .452 + 4.08 + .646 + .06 + 49.027. 
 
 7. 4.0406 + 213.939 + 2.91 + 3.04. 
 
 8. 89432.1 + 7.65439 + .0084 + 8400. 
 
 9. 27.064 + .0012 + 394.2001 + .819. 
 
 10. How many yd. are there in five pieces of cloth, the first 
 of which contains 37.5 yd., the second 26.75, the third 14.375, 
 the fourth 36.5, and the fifth 63.125 ? 
 
 11. Four sections of land contain the following areas : 24.729 
 sq. mi., 92.04 sq. mi., 8.007 sq. mi., and 36.429 sq. mi. Find the 
 total area. 
 
 12. Find the sum of sixty-one ten-thousandths; eight, and 
 seven thousand six hundred and ninety-five ten-thousandths; 
 nine thousand seven hundred and eighty-six. 
 
 "t,''' 
 
 « 
 
 'at 
 
 *■ f. 
 
 'i',i 
 
 i «l 
 
•*' 
 
 144 
 
 ARITHMETIC 
 
 13. Find the sum of the following: twenty-four ten-thou- 
 sandths; nine, and three thousand and four ten-thousandths; 
 two hundred and seventy-seven, and six hundred thousandths ; 
 nine, and nine thousandths. 
 
 14. Find the sum of 4 of the tenths' unit, 6 of the thousandths' 
 unit, and 8 of the millionths' unit. 
 
 15. What is the sum of 4, 6, 9, 7, 5, 8, 3, 6, and 7, when the 
 unit of value is fl? ^.01? ^.001? $.0001? 
 
 in 
 
 il 
 
 Subtraction of Decimals 
 142. From 29.364 take 3.87049. 
 
 29.36400 
 3.87049 
 
 29.364 
 3.87049 
 
 As in addition of decimals, place the decimal points under each other, thus 
 placing units under units, tenths under tenths, etc. 
 
 As the value of the decimal is not changed by annexing zeros to the right 
 of the decimal, annex in this case two zeros. Subtract as in whole numbers, 
 and place the decimal point in the remainder between the unit and the tenths' 
 place. 
 
 
 
 Exercise 96 
 
 
 From 
 
 1. 8.43 
 
 2. 13.47016 3. .503 
 
 4. .52 
 
 Take 
 
 2.95 
 
 2.0984 .28914 
 
 .13064 
 
 5. Explain step by step the process of subtraction in the first 
 three examples. 
 
 Find the difference and prove your answers correct : 
 
 6. .62 -.47. 10. .07 -.059. 13. .7304 - .67. 
 
 7. .73 -.35. 11. 8.9-3.4265. 14. 4.8295-3.9998. 
 
 8. .894 -.406. 12. 39.42-15.9879. 15. 2.03 - .00428. 
 
 9. .74 -.365. 
 
DECIMALS 
 
 145 
 
 16. Explain whether .067 or .068 is nearer .06748, and express 
 in words the difference in each case. 
 
 17. The length of a seconds pendulum is 39.1392 in., and 
 that of a meter is 39.371 in. Find the difference in their 
 Icagths. 
 
 18. Find the difference between the length of 1 meter and 
 1yd. 
 
 19. From a piece of cloth containing 35.5 yd., a merchant 
 sold 12.75 yd. How much was left ? 
 
 20. Take 1 millionth from 1 thousandth. 
 
 21. Find the difference between sixty-four ten-thousandths 
 and seventy -five tenths. Add this difference to their sum. 
 
 22. Find the difference between $ HfW ^^^ ^^^• 
 
 23. The mercury in a barometer rose .121 in., .073 in., and 
 .019 in. in three successive days; it fell .054 in. and .065 in. 
 during the two following days, rose .053 in. on the sixth day, and 
 fell .028 in. on the seventh day. If its height at the beginning 
 of the first day was 30.078 in., what was its height at the close 
 of the seventh day ? 
 
 Multiplication of Decimals 
 
 143. (1) ju multiplied by 10 = 7, and therefore .7 multiplied by 10 = 7. 
 f§J multiplied by 10 = ^j%^-, or 62.7, and therefore 6.27 multiplied by 
 10 = 62.7. 
 
 (2) ^5% multiplied by 100 = 75, and therefore .75 multiplied by 100 = 75. 
 m^ multiplied by 100 = 4^5 = 627.5, and therefore 6.275 multiplied by 
 
 100 = 627.5. 
 
 (3) 62^58 multiplied by 1000 = ^Vo- = 6275.8, and therefore 6.2758 
 multiplied by 1000 = 6275.8. 
 
 Exercise 97 
 
 1. Multiply each of the following numbers by 10 : 
 
 .6, .8, .84, .95, .842, .763. 
 
 2. Multiply by 10 : .06, .04, .005, .0123. 
 
 4.** 
 
 hi 
 
 
 ;i 
 
 11 
 
 11 
 it 
 
 !te! 
 
 If 
 
 fi 
 
 II 
 
146 
 
 ARITHMETIC 
 
 
 3. Multiply by 10 : 42.3, 5.69, .478. 
 
 4. State how to multiply a decimal by 10, without actually 
 doing the work of multiplication. 
 
 5. Multiply by 100: .84, 9.65, .763, .003, .04, .246. 
 
 6. State how to multiply a decimal by 100, without actually 
 doing the work of multiplication. 
 
 7. Multiply by 1000: .982, .0642, .0009, .008, .0123. 
 
 8. State how to multiply a decimal by 1000. By 10,000. By 
 100,000. 
 
 9. Multiply by 100 : .86, 8.6, .9, .060, 9.8, .4. 
 
 10. Multiply by 1000 : .594, 5.94, 59.4, .007, .07, .7, 3.14, 2.5. 
 
 11. State how to divide a decimal by 10. By 100. By 1000. 
 By 10,000. 
 
 12. Divide by 10: 27, 82.19, 4.8, 52.93, .4, .06, .009. 
 
 13. Divide by 100: 482, 76, 415.62, 8.1, .78, .4, .09, 789.46. 
 
 14. Divide by 1000 : 643, 2459.7, .69, 2.31, .03, .009. 
 
 144. (1) Multiply 6.24 by 46. 
 
 6.24 
 46 
 
 37.44 
 249.6 
 
 287.04 
 
 Hero we multiply 4 hundredths by 6, and the product is 24 hundredths, 
 or 2 tenths and 4 hundredths. Again, we multiply 2 tenths by 6, and, add- 
 ing in the 2 tenths, the result is 14 tenths, or 1 unit and 4 tenths, and so on. 
 Next, multiplying by 4, we must write the results one place to the left, as in 
 the multiplication of integers. In multiplying, it is as well to omit the deci- 
 mal points from the partial products 37.44 and 249.6. 
 
DECIMALS 
 
 147 
 
 (2) Multiply 6.24 by 4.6. 
 
 6.24 
 
 4.6 
 
 3744 
 
 2496 
 
 28.704 
 
 This differs from the former question only in that the multiplier 4.6 is 
 one-tenth of 46, and therefore the product is one-tenth as large, or 28.704. 
 
 145. From this and other similar problems the rule can be 
 deduced : 
 
 To multiply two decimals^ proceed as if they were integers^ 
 and mark off in the product as many places as there are in 
 both multiplier and multiplicand. 
 
 
 
 146. (1) Multiply 2.66 by .94. 
 
 2.66 
 .94 
 1024 
 2304 
 
 2.4064 
 
 (2) Multiply .249 by .035. 
 
 .249 
 ■035 
 1245 
 
 747 
 
 .008716 
 
 Explanation. — 
 
 .249 X .035 = ^%%% X jUn = TuVoVa^ = -008715. 
 
 Exercise 98 
 
 1. Find .05 of f 26 ; .06 of $ 248 ; .04 of $ 16. 
 
 2. Find .08 of $46.50; .03 of $894.75; .07 of $2389.20. 
 
 3. Find .3 of $ 250 ; .9 of $ 64 ; .8 of $ 76. 
 
 4. Find .025 of f 324 ; .035 of $ 704.60; .045 of $ 852.94. 
 
 ■'f 
 
 
 
 
 II 
 

 ■it 
 
 148 
 
 ARITHMETIC 
 
 5. Find .375 of 45 mi. ; .0625 of 640 A. ; .0875 of $415.60. 
 
 Multiply : 
 
 6. 4.8 X5.12; .21x4.67. 
 
 7. 3.1416 X.02; 1.46 x .39. 
 
 8. .004 X.99; .004 x .005. 
 
 9. f 249 X 1.04. 
 
 10. .84 X .251 ; 2.04 x .0037. 
 
 11. .8 X .8; .09 X .09. 
 
 12. .1 X .1 ; .01 X .01. 
 
 13. .2 X .2; .7 X .7. 
 
 14. .375 X2.15; .0375 x 2.15. 
 
 15. .051 X .042 ; .014 x .0038. 
 
 16. Find the area of a rectangle 6.2 ft. long by 4.7 ft. wide. 
 
 17. Find the area of a rectangle 7.5 ft. wide by 3.4 ft. long. 
 
 18. Find the value of a rectangular solid whose dimensions 
 are 2.5, 4.6, and 5.8 in. 
 
 19. Find the volume of a solid whose dimensions are 8.8, 6.6, 
 and 4.4 in. 
 
 20. If my gain on selling an article is measured by the num- 
 ber .23 and the cost $ 265, find the gain. 
 
 21. If my loss on selling an article that cost $ 226.50 is meas- 
 ured by the number .34 and the cost, find the loss. 
 
 22. Multiply $ 240 by 1.05 twice in succession. 
 
 23. Multiply f 325 by 1.06 three times in succession. 
 
 24. Multiply $415.80 by 1.04 twice and the result by 1.02. 
 
 25. Multiply $ 75 by 1.045 three times in succession. 
 
 26. 
 
 Multiply $975 by 1.065 three 
 
 times in 
 
 succession. ] 
 
 27. 
 
 Multiply 3.1416 by 8 
 
 ; by 7.5; 
 
 by .04. 
 
 
 28. 
 
 Multiply 7.48 by 9.1 ; 
 
 by .04 ; 
 
 by .006. 
 
 
 29. 
 
 Multiply the square of 5 by 3.1416. 
 
 

 DECIMALS 
 
 149 
 
 30. Measure in inches and decimals of an inch the diameters 
 of several circles and multiply the result in each case by 8.141(). 
 Then measure the circumference and note which are greater, the 
 products or the lengths of the circumferences. 
 
 .41, 
 
 ■ »r 
 
 ni 
 
 Exercise 99 
 
 1. The length of a wall, according to the French metric 
 system, is 9.48 meters. Find its length in in., the length of 1 
 meter being 39.371 in. 
 
 2. Multiply the sum of 2.616, .00132, and 1.0448 by .62639. 
 
 3. A piece of land is 63.5 rd. long, and 27.75 rd. wide. What 
 will it cost to fence it at f .875 per rd. ? 
 
 4. Multiply 10.5 by 1.05 and reduce the result to a fraction 
 in its lowest terms. 
 
 5. The specific gravity of atmospheric air compared with 
 water is .0012. I ask for the specific gravity of common gas, and 
 am told it is .45 compared with air. Find its specific gravity 
 compared with water. 
 
 6. A lumber merchant bought 106,250 ft. of lumber at 
 $ 14.375 per M., and retailed it at $ 1.75 per C. Find his gain. 
 
 7. Water is composed of two gases, oxygen and hydrogen, in 
 the proportion of 88.9 to 11.1. What weight is there of each in 
 a cubic yard of water ? (1 cu. ft. of water weighs 1000 oz.) 
 
 8. The chain for measuring land is GG ft. long. What is 
 the length in yards of a fence that measures 2456 ch., and how 
 much would it cost at $ 8.86 per yd. ? 
 
 9. Find the area of a rectangular field whose breadth is 
 78.23 ch., and length 85.40 ch., there being 10 sq. ch. in 1 A. 
 
 10. A farmer bought 48.125 T. of hay ; for 20.25 T. of it he 
 paid $16 per T., and for the rest f 18.2625 per T. ; he sold 
 the whole at the average price of $ .945 per cwt. How much 
 did he gain or lose ? 
 
 ■] ■ 
 
 
 f. t 
 
 ■'Hi 
 
 '■• HI S 
 
 ■* ■•' H 
 
150 
 
 ARITHMETIC 
 
 11. A person sold .15 of an estate to one person, and then ^ 
 of the remainder to another person. What part of the estate did 
 he still retain ? 
 
 12. If a business produces an annual return of $ 12,000, and of 
 three partners one has .465 and another .28 share of the profits, 
 how much money falls to the share of the third partner ? 
 
 13. A merchant sells 2S.5 yd. of cloth which cost him 25^ 
 a yd., for 37.5^ a yd. What was his gain ? 
 
 Division of Decimals 
 
 
 8)24 8)2. 
 3 
 
 4 8}^ 
 3 
 
 24 
 .03 
 
 That is, 
 
 24 divided by 8 = 3. 
 
 
 
 I 
 
 24 tenths divided by 
 
 24 hundredths divided by 
 
 8 = 3 tenths = .3. 
 8 = 3 hundredths 
 
 = .03 
 
 
 6).0018 
 .0003 
 
 46)251.62(5.47 
 230 
 216 
 
 
 
 
 184 
 322 
 
 
 
 
 322 
 
 
 That is, 18 ten-thousandths -4-6 = 3 ten-thousandtlis = .0003 ; and 26,162 
 hundredths -r- 46 = 547 hundredths = 5.47. 
 
 147. 5)15 
 
 50 )150 
 3 
 
 50 0):^ 500 
 3 
 
 500 0)15000 
 3 
 
 Therefore, if we multiply the divisor by 10, 100, 1000, and 
 so on, and at the same time multiply the dividend by 10, 
 100, or 1 000, and so on, the quotient remains unchanged. 
 
 (1) Find the value of 4.1262 -5- .69. 
 
 The quotient of 4.1262 h- .69 is the same as that of 412.62 4- 69. Here 
 we multiplied each number by 100. 
 
DECIMALS 
 
 69)412.62(5.98 
 345 
 676 
 621 
 552 
 552 
 
 .-. 4.1202 -- .09 = 5.98. 
 
 151 
 
 W 
 
 In this division the 412 is 412 units, and the quotient 5 is therefore 5 units ; 
 the first remainder, 676, is 676 tentlis, and the quotient 9 is therefore 9 tenths ; 
 the second remainder, 552, is 652 liundredths, and the quotient 8 is therefore 
 8 hundredths. 
 
 (2) Find correct to the third decimal place the quotient 
 of 8.94 -t- 3.1416. 
 
 3.1416)8.94( 
 31416)89400(2.845 
 
 62832 
 
 265680 
 
 251328 
 
 143520 
 
 125664 
 
 17856 
 
 15708 
 
 2148 
 
 " Pi 
 
 » (I 
 
 fU 
 
 .'. 8.94 -H 3.1416 = 2.845, correct to three decimal places. 
 
 Exercise 100 
 
 Divide, proving your answer correct to every third question : 
 
 1. 25.68 -J- 3.21. 3. 8.54 --.07. 
 
 2. 10.836 -f- 5.16. 4. 1 49.92 H- .065. 
 
 5. f 54.75 -i- .98, correct to three decimal places. 
 
 6. f 75.60 -f- .99, correct to three decimal places. 
 
 7. $64.26^-1.02. 
 
 8. f 84.3648 -h 1.04 twice in succession and the result by 1.02. 
 
152 
 
 ARITHMETIC 
 
 9. f 16989.7728 -r- 1.04 twice in succession and the result by 
 1.02. 
 
 10. 2450.90 -r- .998, correct to three decimal places. 
 
 11. f 23881 -7- 1.06 three times in succession, correct to three 
 decimal places. 
 
 12. 1 11.19195 --$4.8665. 
 
 13. .00081 H- 27, and 1.77089 by 4.735. 
 
 14. 1 -H .1, by .01, and 1 -f- .0001. 
 
 15. 31.5 -H .126; 5.2 -.32. 
 
 16. 12.6 H- .0012, and .065341 -- .000475. 
 
 17. 3.012 -4- .0006. 
 
 18. 130.4 -- .0004 and .004, and 46.634205 ^ 4807.65. 
 
 19. 1.69 H- 1.3, by .13, by 13, and by .013. 
 
 20. 816 -I- .0004. 
 
 21. .00005 -5- 2.5, by 25, and by .0000025. 
 
 22. 32.5 H- 8.7; .02 -h 1.7, correct to four decimal places. 
 
 23. .009384 -h .0063, correct to four decimal places. 
 
 24. 37.24 -^ 2.9 ; .0719 ^ 27.53, correct to four decimal places. 
 
 25. Measure in inches and decimals of an inch the diameter of 
 any circle, and divide the result into the length of the circum- 
 ference. Do this until you find the quotient to be nearly 3.1416. 
 
 Exercise 101 
 
 1. If a gal. contains 231 cu. in. and a cu. ft. 1728 cu. in., 
 find correct to two decimal places the number of gal. in a cu. ft. 
 of water. 
 
 2. Using the result obtained in question 1, find the number 
 of gal. of water that a rectangular tin 2 ft.,- by 3 ft., by 4 ft. 
 will hold. 
 
 3. If a bu. contain 2150.42 cu. in., find the number of cu. in. 
 in a dry qt. 
 
DECIMALS 
 
 163 
 
 4. Find correct to three decimal places the number of cu. ft. 
 in a bu. 
 
 5. By what decimal part of 1 in. does .0009 of 1 ft. exceed 
 .00003 of 1 yd. ? 
 
 6. Find cost of 7225 lb. coal at $ 7.25 per ton of 2000 lb. 
 
 7. Find the value of 
 
 3.0005 X .006 
 .0009 
 
 8. A man paid $2,896,863.50 for land and sold 56.25 A. at 
 $ 31 an A. ; the remainder then stood him at $ 20.05 an A. How 
 many A. did he buy ? 
 
 9. Water expands when freezing so that a cu. ft. of water 
 becomes 1.089 cu. ft. of ice. Find how many cu. ft. of water 
 there are in an iceberg which is 900 ft. long, 88 ft. broad, and 
 220 ft. high. 
 
 10. Show by examples that a decimal is divided by 10,000 by 
 removing the decimal point in the dividend four places towards 
 the left. 
 
 11. A creditor receives $ 1.50 for every $4 of what was due to 
 him, and thereby loses f 301.05. What was the sum due ? 
 
 12. Divide .0075 by 6.4, and explain the reason for fixing the 
 position of the decimal point in the quotient. 
 
 13. A person expended .*$ 55.92 in tea at $.875 per lb., coffee 
 at $.1875, and sugar at $.1025, buying an equal quantity of each. 
 How many lb. of each did he buy ? 
 
 14. A gentleman whose real property is .834 of his personal 
 property leaves the former, amounting to $ 10,008, to his eldest 
 son ; and the latter to be equally shared by him and two others. 
 Find the amount received by each. 
 
 IB. A merchant expended $ 280.60 in purchasing cloth at 95^ 
 a yd., at $ 1.37 a yd., and at 73^ a yd., buying the same quantity 
 of each. Find the entire number of yd. purchased. 
 
 i 
 
 M 'ii 
 "J 
 
 J: 
 
 m 
 
 M 
 
 
 m 
 
 
154 
 
 ARITHMETIC 
 
 16. Suppose unity to represent .0012, what number represents 
 .0001 ? 
 
 17. Find the earth's equatorial diameter in miles, supposing 
 the sun's diameter, which is 111.454 times as great as the equa- 
 torial diameter of the earth, to be 883,345 mi. 
 
 18. The French meter is 39.371 in. in length. Express the 
 length of 25 meters as a decimal of an English mi., there being 
 5280 ft. in 1 mi. and 12 in. in 1 ft. 
 
 19. The total Indian population on the reservations in 1880 
 was 255,327, while the area of the Indian reservations, in the 
 United States, was 241,800 sq. mi. What average quantity of 
 land was occupied by 100 Indians ? 
 
 20. The total Indian population on the reservations in 1893 
 was 249,366, while the area of the Indian reservations was 
 134,176 sq. mi. What average quantity of land was occupied 
 by 100 Indians ? 
 
 21. The area of the reservations in 1893 was how many thou- 
 sandths of that in 1880 ? 
 
 22. The Indian population on the reservations in 1893 was 
 how many thousandths of that in 1880 ? 
 
 23. Find the quantity of coal consumed by a steamer for a 
 voyage of 4043 mi., supposing her rate per hr. to be 16.172 mi. 
 and her consumption of coal 87 T. per da. 
 
 Reduction of Decimals 
 
 148. (1) Reduce .275 to a common fraction. 
 
 07.r_ 2 75 _ 5 5 —11 
 
 (2) Reduce to a common fraction .08J. 
 
 ^ 100 300 12 
 
DECIMALS 
 
 Exercise 102 
 
 Reduce to common fractions in their lowest terms 
 
 155 
 
 1. 
 
 .5. 
 
 6. 
 
 .625. 
 
 11. 
 
 .33J. 
 
 16. 
 
 8.9375. 
 
 2. 
 
 .25. 
 
 7. 
 
 .125. 
 
 12. 
 
 .03J. 
 
 17. 
 
 29.975. 
 
 3. 
 
 .75. 
 
 8. 
 
 .0625. 
 
 13. 
 
 •06i. 
 
 18. 
 
 18.06|. 
 
 4. 
 
 .60. 
 
 9. 
 
 .875. 
 
 14. 
 
 .66|. 
 
 19. 
 
 6.00f 
 
 5. 
 
 .375. 
 
 10. 
 
 .16|. 
 
 15. 
 
 .14f 
 
 20. 
 
 249.00075. 
 
 21. If my gain on selling a lb. of tea is measured by the 
 number .125, and the cost 72^, what was my gain on 1 lb.? 
 
 22. If a crate of berries which cost $ 1.35 was sold at a gain 
 of .22| of the cost, find the gain. 
 
 23. I bought a farm for f 4800, and sold it at a loss of .375 of 
 the cost price. Find the selling price. 
 
 24. A merchant sold coffee at a gain of .33^ of the cost. If 
 his gain on a quantity of coffee was $ 12, what did it cost him ? 
 
 25. A grain merchant sold wheat for $3400, gaining .06^ of 
 the cost. Find the cost price. 
 
 149. (1) Divide 7 by 8, expressing the result as a decimal. 
 
 8)7.000 
 .875 
 
 Now 7^8 = |. .-. I = .875. 
 
 Proof 
 
 (2) Reduce ^| to a decimal. 
 
 16)150(.9375 
 144 
 
 60 
 48 
 
 120 
 112 
 
 .-. II =.9376. 
 
 80 
 80 
 
 '1 
 
 ?« 
 
) 
 
 1 I 
 
 I: 
 
 :■ 
 
 ii 
 
 156 
 
 ARITHMETIC 
 
 (3) Reduce 9|| to a decimal correct to four decimal places. 
 
 23) 180 (.7820 
 161 
 190 
 184 
 
 60 
 46 
 140 
 138 
 2 
 
 /. 9^1 = 9.7820, correct to four decimal places. 
 
 Exercise 103 
 
 Reduce to decimals and prove results : 
 
 l4i^ 213 57 o 
 
 5. t¥8, t'^. ih- 6. 6^1 3if. 7. 
 
 8. Find correct to four decimal places the value of y^^, |^, 
 
 15 9 
 TB-J TF> T¥- 
 
 4. 
 
 TSf YH' 
 
 5 9 3 2 31 
 
 4? 1^> 1T> TQ' 
 
 14 16 
 TffJ 2T* 
 
 9. Butter bought for 25^ a lb. was sold for 28^ a lb. 
 Express the gain as a decimal of the cost. 
 
 10. I bought a sto]-e for f 6912, and sold it for $ 5184. Ex- 
 press the selling price as a decimal of the cost. 
 
 11. A real estate agent sold land which cost him $6240 at 
 a loss of $ 2340. What was his loss on each $ 1000 invested ? 
 
 12. A ch. contains 66 ft., and a mi. 5280 ft. What decimal 
 part of a mi. is a ch. ? 
 
CHAPTER XII 
 
 •« 
 
 COMPOUND QUANTITIES 
 
 150. Quantities like 4 yd., 3| lb., and 6J gal. are called 
 simple quantities^ because they are expressed in terms of a 
 single unit of measurement. 
 
 Quantities like 3 lb. 8 oz., 6 gal. 1 qt., are called compound 
 quantities, because they are expressed in terms of two or more 
 units of measurement. 
 
 
 151. The units of money are the units which are used to 
 measure the value^ of things. The one-dollar gold piece is at 
 present (July, 1896) the prime unit or standard of value in 
 the United States and Canada. 
 
 152. Units op Value 
 
 United States Monev 
 
 10 mills (ni.)= 1 cent (ct. or f) 
 10 cents = 1 dime (d.) 
 
 10 dimes = 1 dollar ($) 
 10 dollars = 1 eagle (E.) 
 
 The coins of the United States are : 
 
 Bronze : the cent. 
 
 Nickel : the five-cent piece. 
 
 Silver : the dime, quarter-dollar, half-dollar, and dollar. 
 
 Gold : the quarter-eagle, half-eagle, eagle, and double-eagle. 
 
 153. Sterling Money is the money of Great Britain and 
 Ireland. 
 
 167 
 
 
 m 
 
 I 
 
158 
 
 ARITHMETIC 
 
 The prime unit is 1 pound, whose value is $ 4.8665. 
 The pound, when coined, is called the sovereign. 
 
 British ou Sterling Money 
 
 4 farthings (far.) = 1 penny (d) 
 12 pence = 1 shilling (s.) 
 
 20 shillings = 1 pound (£) 
 
 5 shillings = 1 crown 
 
 21 shillings = 1 guinea 
 
 154. The unit of French Money is 1 franc, which is worth 19. 3j^. 
 The unit of German Money is 1 mark, which is worth 23.85^. 
 
 Exercise 104 
 
 1. How many mills are there in 2^ ? 3^ ? J^ ? 1 J^ ? 2\^ ? 
 
 2. How many cents are there in 40 mills ? GO mills ? 15 
 mills ? 5 mills ? 25 mills ? 
 
 3. State orally the table of English Money. 
 
 4. Reduce to farthings : od.\ Crf. 2 far. ; 9d. 3 far. 
 
 5. Reduce to pence : 4 s.; 8 s. 5 c?. ; 12 s. 6 cZ. 
 
 6. Reduce to shillings : £ 7 ; £ 2 12s. ; £9 7s. 
 
 7. How many pence are there in Js. ? |s. ? |s. ? |s. ? |s. ? 
 
 8. How many shillings and pence are there in £ | ? £ J ? 
 £f? £J? 
 
 9. How many shillings are there in £ .3 ? £ .7 ? £ .25 ? 
 £ .33 J ? 
 
 10. What fraction of a shilling is 3(?. ? 4d ? 8d. ? 9d? 
 10 d? 
 
 11. What is the value of £ 1 in American money ? Of £ 10 ? 
 Of £100? 
 
 12. How many shillings and pence are there in 60 d. ? 84 c?. ? 
 39 d? 58 d.? 112 d? 
 
COMPOUND QUANTITIES 
 
 159 
 
 13. How many pounds and shillings are there in 80s. ? 65s. ? 
 120 s.? 48 s.? 
 
 14. How many pounds and shillings in 1 guinea ? 4 guineas ? 
 6 guineas ? 9 guineas ? 
 
 15. What decimal of a pound is 10 .s. ? 12 s. ? 17 s. ? 24 s. ? 
 
 16. What part of a crown is 1 s. ? How many crowns in 10 s. ? 
 20 s.? 
 
 17. Express 2 guineas in sovereigns and shillings. 
 
 18. What is the value of 10 francs in U. S. money ? 
 
 19. What is the value of 100 marks in U. S. money ? 
 
 20. What is the cost in cents of 3 books at 1 franc each ? 
 
 21. What is the difference in value between 100 marks and 
 100 francs ? 
 
 Units of Weight 
 
 155. Avoirdupois Weight is used for weighing everything ex- 
 cept jewels, precious metals, and medicines when dispensed. 
 The prime unit of weight is 1 pound Avoirdupois. 
 
 AvoiKDUPOis Weight 
 
 16 ounces (oz.) = 1 pound (lb.) 
 100 pounds = 1 hundredweight (cwt.) 
 
 20 hundredweight = 1 ton (T.) 
 
 In the United States Custom House, and in weighing iron and coal at the 
 mines, the long hundredweiglit and the long ton are used. 
 
 112 pounds — 1 long hundredweight 
 2240 pounds = 1 long ton 
 
 One pound Avoirdupois = 7000 grains 
 One ounce Avoirdupois = 437 J grains 
 
 i 
 
 Exercise 105 
 
 1. State orally the table of Avoirdupois Weight. 
 
 2. Reduce to oz. : 1 lb. 8 oz. ; 2 lb. 4 oz. 
 
160 
 
 ARITHMETIC 
 
 r 
 
 1 
 
 3. Express 1 lb. 8 oz. as <a fraction of 2 lb. 4 oz. 
 
 4. Express 1 lb. 8 oz. as a decimal of 2 lb. 4 oz. 
 
 5. What part of 1 lb. is 4 oz. ? 12 oz. ? 2 oz. ? 8 oz. ? 
 Express your results also as decimals. 
 
 6. What part of 1 T. is 400 lb. ? 800 lb. ? 1500 lb. ? 
 
 7. A coal dealer buys coal by the car-load at the mines. How 
 many more lb. of coal does he get" for 1 T. than he gives ? 
 
 8. Show by dividing 7000 gr. by 1(5, that 1 oz. Avoir, is equal 
 to 437^ gr. 
 
 9. One oz. is what part of 1 lb. ? ^ of an oz. is what part of 
 lib.? 
 
 10. A farmer sells 3 cows whose united weight is 1 T. 5 cwt. 
 What is the average weight of the cows ? 
 
 156. Troy Weight is chiefly used for weighing gold, silver, 
 and jewels. 
 
 TuoY Weight 
 
 24 grains (gr.) = 1 pennyweight (pwt.) 
 20 pennyweights = 1 ounce (oz.) 
 12 omices = 1 pound (lb.) 
 
 One pound Troy = 6760 grains 
 One ounce Troy = 480 grains 
 
 Exercise 106 
 
 1. State orally the table of Troy Weight. 
 
 2. Reduce to gr. : 1 pwt. 16 gr. ; 2 pwt. 12 gr. 
 
 3. What is the ratio of 2 pwt. 12 gr. to 1 pwt. 16 gr. ? 
 
 4. Express 1 pwt. 16 gr. as a decimal of 2 pwt. 12 gr. 
 
 5. Reduce 1 oz. to gr. 
 
 6. Divide 5760 gr. by 12, and verify your result in question 6. 
 
COMPOUND QUANTITIES 
 
 161 
 
 7. By how many gr. is 1 lb. Avoir, heavier than 1 lb. Troy ? 
 
 8. By how many gr. is 1 oz. Troy heavier than 1 oz. Avoir. ? 
 
 9. How many oz. and pwt. are there in J lb. ? | lb. ? f lb. ? 
 f lb. ? 
 
 10. How many pwt. are there in .25 oz. ? .4 oz. ? .35 oz. ? 
 
 11. What part of a lb. is 8 oz. ? 1 oz. ? | oz. ? | oz. ? 
 
 12. If coal is worth $G a T., how many lb. can be bought 
 forf3? f2? 
 
 13. A cu. ft. of water contains 1000 oz. How many lb. does a 
 cu. ft. of water weigh ? 
 
 157. Druggists buy their medicines by Avoirdupois Weight, 
 but use Apothecaries' Weight in mixing and in selling medi- 
 cines. 
 
 Apotiiecauies' Weight 
 
 20 grains (gr.)=: 1 scruple (3) 
 
 3 scruples = 1 dram (3) 
 
 8 drams = 1 ounce ( 5 ) 
 
 12 ounces = 1 pound (lb.) 
 
 One pound Apothecaries' weight = 67G0 grains 
 One ounce Apothecaries' weight = 480 grains 
 
 4 
 ^ 
 
 '!.': 
 
 
 Exercise 107 
 
 1. State orally the table of Apothecaries' Weight. 
 
 2. How many ounces in 16 3 ? 40 3 ? T2 3 ? 
 
 3. How many drams in 9 S ? 18 3 ? 54 ^ ? 
 
 4. What part of a pound is 4 5 ? 95? 10 5? 
 
 5. How many scruples in 3 3 and 2 3? In 40 gr. ? In 120 
 
 gr. 
 
162 
 
 AUITIIMETIC 
 
 Units ok Lkncjth 
 
 158. The prime or standard unit of length is 1 yard. 
 
 Long Mkahi'ui!: 
 
 12 inclies (in.) = 1 foot (ft.) 
 
 3 feet = 1 yard (yd.) 
 
 GJ yards or 10 J feet = 1 rod (rd.) 
 320 rods = 1 mile (mi.) 
 
 1 mi. = 320 rd. = 1700 yd. = 5280 ft. 
 
 A hand, used in measuring the height of horses, = 4 in. ; a knot, used in 
 navigation, = 0080 ft. or 1.15 mi. 
 
 A fathom, used in measuring depth at sea, = ft. 
 
 159. Surveyor's Linear Measure is used by surveyors in 
 
 measuring land. The prime unit is 1 chain, called Gunter^a 
 
 Chain. 
 
 SuKVKYou's Linear Measukb 
 
 100 links (1.)= 1 chain (ch.) 
 80 chains = 1 mile (mi.) 
 
 1 ch. = 4 rd. = 22 yd. = 00 ft. = 702 in. 
 1 link = 7.02 in. 
 
 160. Mark off in the schoolroom 1 ft., 1 yd., and 1 rd. 
 Locate two points exactly 1 mi. apart. 
 
 Exercise 108 
 
 1. How many in. are there in 1 yd. ? | yd. ? J yd. ? J yd. ? 
 
 i yd. '^ 
 
 2. Reduce to yd. : 4 rd. ; 8 rd. ; 32 rd. ; 320 rd. ; 1 mi. 
 
 3. Reduce to ft. : 18 yd. ; 7G yd. ; 17G yd. ; 17G0 yd. ; 1 mi. 
 
 4. Show that 1 mi. = 320 rd. = 17()() yd. = 5280 ft. = G3,3G0 in. 
 
 5. How many yd. are there in .3 mi. ? .8 mi. ? .25 mi. ? 
 .06^ mi. ? 
 
 6. What part of a ft. is 3 in. ? 6 in. ? 8 in. ? 10 in. ? 
 
 7. What part of a yd. is 12 in. ? 18 in. ? 24 in. ? 27 in. ? 
 
COMPOUND QUAXTITTES 
 
 163 
 
 8. What ])art of a nl. is 1 yd. ? :\ yd. ? 5 yd. ? n\ yd. ? 
 
 9. What part of a mi. is 40 rd. ? 200 yd. ? 280 yd. ? 
 
 10. How many yd. aro there in 1 rd. ? How many ft. ? How 
 many in. ? 
 
 11. How many ch. are there in IVJO rd. ? How many rd. are 
 there in 1 eh. ? 
 
 12. Show that 1 ch. = 4 rd. = 22 yd. = 0() ft. = 792 in. 
 
 13. How many in. are there in 100 links ? In 1 link ? 
 
 1^ 
 
 Units of SuufA('K ok Squaiik Mkasuke 
 161. Surface has two dimensions, — length and breadth. 
 
 162. The prime unit of area is 1 square yard, which, like 
 1 square inch, 1 square foot, 1 square rod, and 1 square mile, 
 is derived from the corresponding unit of linear measure. 
 
 The measure of 1 ft. is 12, the unit beiug 1 in. 
 
 The measure of 1 sq. ft. is 144, the unit being 1 sq. in. 
 
 .•. 1 sq. ft. = 144 sq. in. 
 
 The measure of 1 yd. is 8, the unit being 1 ft. 
 
 The measure of 1 sq. yd. is 9, the unit being 1 sq. ft. 
 
 • •. 1 sq. yd. = 9 sq. ft. 
 
 The measure of 1 rd. is 5 J, the unit being 1 yd. 
 
 The measure of 1 sq. rd. is (5^ x 6^), or 30], the unit being 1 sq. yd. 
 
 • •. 1 sq. rd. = 30^ sq. yd. 
 
 Illustrate the above by drawing 1 sq. ft., 1 sq. yd., and 1 sq rd., and divid- 
 hig each into tlie next lower units of area. 
 
 In the case of 1 sq. rd. draw according to the scale of 4 in. to 1 rd. 
 
 Slrface or Square Measure 
 
 144 square inches (sq. in.)= 1 square foot (sq. ft.) 
 9 square feet = 1 square yard (sq. yd.) 
 
 30| square yards = 1 square rod (sq. rd. ) 
 
 160 square rods = 1 acre (A.) 
 
 640 acres = 1 square mile (sq. mi.) 
 
 10 square chains = 1 acre ; 1 acre = 4840 square yards 
 
164 
 
 ARITHMETIC 
 
 163. A township is 6 mi. square, and is divided, as 
 in the accompanying figure, into 36 sections, each 1 mi. 
 square. 
 
 6 
 
 5 
 
 4 
 
 3 
 
 2 
 
 1 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 18 
 
 17 
 
 16 
 
 15 
 
 14 
 
 13 
 
 19 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 30 
 
 29 
 
 28 
 
 27 
 
 26 
 
 25 
 
 31 
 
 32 
 
 33 
 
 34 
 
 35 
 
 36 
 
 W. H 
 320 ACRES 
 
 N E.Ji 
 160 ACRES 
 
 N. Ji^OFS.E.M 
 aO ACRES 
 
 8.W.X0F 
 
 8.E. H 
 
 40 ACRES 
 
 8.E.)iOF 
 
 8.E. M 
 40 ACRES 
 
 TOWNSHIP. 
 
 SECTION. 
 
 Locate sections 8, 22, and 36 in the drawing. Draw a 
 township on a scale of 1 in. to 1 mi., and divide it into 
 36 sections, numbering each section. Divide one of these 
 sections into 4 square farms of 160 A. each, and name 
 each farm according to its position in the section. Divide 
 a second section into 8 i*ectangular farms of 80 A., and a 
 third into 16 squjire farms of 40 A., and locate each farm as 
 before. 
 
 Exercise 109 
 
 1. Howinimy sq. in. are there in2 sq. ft. ? 4 sq. ft. ? 9 sq. ft. ? 
 
 2. Ifow many sq. ft. in 5 sq. yd. ? i sq. yd. ? \- sq. yd.? 
 
 3. One sq. rd. is equal to how many sq. yd. ? 4 sq. rd. ? 16 
 S(i. rd. Y 160 sq. rd. ? 1 A. ? 
 
 4. What part of a sq. rd. is 1 sq. yd. ? 
 
 5. llediice to sq. rd. : 4(S4 sq. yd.; 151 J^ sq. yd. 
 
 6. Wliat part of an A. is 80 sq. rd. ? 120 sq. rd. ? 
 
 7. How many sq. oh. are there in 160 sq. rd. ? One sq. ch. 
 equals how many sq. rd. ? 
 
1 
 
 COMPOUND QUANTITIES 
 
 165 
 
 8. Reduce to A. : 5 sq. mi. ; 8 sq. mi. 
 
 9. If 1 sq. mi. is the unit of area, find the number which 
 expresses the measure of 2500 A. Of 4 townships. 
 
 10. Into how many townships can a county be divided which 
 contains 324 sq. mi. ? 
 
 11. What is the area of a square 6 ft. in length ? 
 
 12. What is the difference in area between two figures, one 6 
 in. sq. and the other containing G sq. in. ? 
 
 Illustrate by a drawing. 
 
 *4. 
 
 •1 •. 
 
 Units of Volume 
 
 164. A volume has three dimensions, — length, breadth, 
 and thickness. 
 
 165. The prime unit of volume is 1 cubic yard, which, 
 like 1 cubic inch and 1 cubic foot, is derived from the cor- 
 responding unit of linear measure. 
 
 The measure of the volume of 1 cu. ft. = 12 x 12 x 12 = 1728, the unit 
 of volume being 1 cu. in. 
 
 The measure of the volume of 1 cu. yd. = 3 x 3 x 3 = 27, the unit of 
 volume being 1 cu. ft. 
 
 Cunic oit VoLUMK Mkasurb 
 
 1728 cubic inches (cu. in.)= 1 cubic foot (cu. ft.) 
 27 cubic feet = 1 cubic yard (cu. yd.) 
 
 166. Five wood and rough stono are measured by the cord (cd.). The 
 cord is a pile 8 ft. long, 4 ft. wide, and 4 ft. high. It contains 128 cu. ft. 
 One cord foot (cd. ft.) is 1 ft. in length of the cord. Its volume is 16 cu. ft. 
 
 A cu. yd. of earth is called a load. 
 
 IIow many loads of dirt are there in a pile 15 ft. long, 12 ft. wide, and 
 ft. deep ? 
 
 167. Mark off in one corner 1 cu. ft., 1 cu. yd., and 1 cd. 
 Divide the cd. into cd. ft. 
 
 
166 
 
 ARITPIMETIC 
 
 3 
 
 Units of Capacity 
 
 168. The prime unit of capacity is 1 gallon. 
 
 Liquid Measure 
 
 4 gills (gi.)^ 1 pint (pt.) 
 2 pints = 1 quart (qt. ) 
 4 quarts = 1 gallon (gal.) 
 
 169. The capacity of cisterns, reservoirs, and the like is often expressed 
 in barrels (bbl.) of 31. ^ gal. each, or in hogsheads (hhd.) of 63 gal. each, A 
 gal. contains 231 cu. in. Have a tin box made 11 in. long, 7 in. wide, and 
 3 in. deep, and note that 1 gal. of water will just fill it. 
 
 Dry Measure 
 
 2 pints (pt.)= 1 quart (qt.) 
 8 quarts = 1 peck (pk.) 
 4 pecks = 1 bushel (bu. ) 
 
 One bushel contains 2150.42 cubic inches 
 
 170. Apothecaries' Fluid Measure is used by druggists in 
 mixing medicines. 
 
 Apothecaries' Fluid Measure 
 
 00 minims {%)= 1 fluid dram (f 3) 
 
 8 fluid drams = 1 fluid ounce (f 5) 
 10 fluid ounces = 1 pint (O.) 
 
 8 i)ints = 1 gallon (Cong.) 
 
 One minim is about equal to 1 drop 
 
 Exercise 110 
 
 1. What part of 1 gal. is 1 qt. ? 1 pt.? 
 
 2. What is the number of cu. in. in 1 gal. ? In 1 qt., liquid 
 measure ? In 1 pt., liquid measure ? 
 
 3. What part of 1 bu. is 1 qt. ? 1 ])t. ? 
 
 4. What is l:ie number of cu. in. in 1 bu. ? In 1 qt., dry 
 measure ? In 1 pt., dry measure ? 
 
 5. How many more cu. in. are contained in 1 qt., dry measure, 
 than in 1 qt., liquid measure ? 
 
COMPOUND QUANTITIES 
 
 167 
 
 6. Show that 1 bu. is nearly equal to 9.31 gal. 
 
 7. Find the number of cu. ft. in 1 cd. 
 
 8. Find the number of cd. of wood in a pile 30 ft. long, 6 ft. 
 high, and 4 ft. wide. 
 
 9. Find the cost of a pile of wood 24 ft. long, 5 J ft. high, and 
 4 ft. wide, at ^ 6 a cd. 
 
 Units of Time 
 
 171. The prime unit of time is 1 day. 
 
 Measure of Time 
 60 seconds (sec.)= 1 minute (min.) 
 
 60 minutes 
 24 hours 
 7 days 
 
 365 days 
 
 366 days 
 100 years 
 
 = 1 hour (hr.) 
 
 = 1 day (da. ) 
 
 — 1 week (wk.) 
 
 = 1 common year (yr.) 
 
 = 1 leap year (1. yr.) 
 
 = 1 century (C.) 
 
 The year is divided into 12 calendar months : 
 
 January (Jan.) 31 da. 
 
 February (Feb.) . . 28 or 29 " 
 
 March 31 " 
 
 April 30 " 
 
 May 31 " 
 
 June 30 " 
 
 July 31 da. 
 
 August (Aug.) 31 
 
 September (Sept.) .... 30 
 
 October (Oct.) 31 
 
 November (Nov.) .... 30 
 
 December (Dec.) .... 31 
 
 In business transactions, 1 mo. is generally taken as equal to 30 da., and 
 
 1 yr. as equal to 360 da. 
 
 The following lines are useful in enabling one to remember the number 
 
 of days in a month : 
 
 " Thirty days hath September, 
 
 April, June, and November." 
 
 A year is the period of the earth's revolution about the sun. It consists of 
 365 da. 5 hr. 48 min. 50 sec. 
 
 A common year lacks 11 min. 10 sec. of being 365 da. hr., or 365 J da. 
 Hence when we take 365 da. to a common year, and 360 da. to a leap 
 year, we increase each year by 11 min. 10 sec. In 400 years this amounts 
 to a little over 3 da. For that reason three out of four centennial years are 
 
 1! 
 
 
 *1 
 
 
 :l!f 
 
168 
 
 ARITHMETIC 
 
 counted as common years, i.e. the centennial years that do not divide 
 equally by 400 have only 305 da. 
 
 Exercise 111 
 
 1. Express 9 lir. as a fraction of a \vk. 
 
 2. Express 12 sec. as a deciiiml of a miii. 
 
 3. Express 146 da. as a fraction of a yr. 
 
 4. Express as a fraction of a nio. : 10 da.; 15 da.; 18 da. 
 
 5. Express as a decimal of a nio. : 21 da. ; 18 da. ; 27 da. 
 
 6. Find the number of da. between Jan. o and Feb. 4; March 
 27 and April .'50; Oct. 24 and Dec. 11. 
 
 ?. How many da. are there in Nov.? Jan.? Dec? April? 
 Feb. ? 
 
 8. State the number of dii. in each of the following, and also 
 what part of a yr. each is, there being iJO da. to the mo. and 
 360 da. to the yr. : 1 mo. 10 da. ; 2 mo. 12 da.; 7 mo. 6 da. 
 
 ■!■ i 
 
 Exercise 112 
 
 In the questions in the following exercise add 3 days to the 
 given time (called days of gnice) to find the day on which the 
 note is due. 
 
 Find the date on which a note falls due, which I promise to pay : 
 
 1. Three months after March 3, 1894. 
 
 2. Four months after June 13, 1896. 
 
 3. Ninety days after May 13, 1890. 
 
 4. Hixty days after Sept. 16, 1895. 
 
 5. Ninety days after June 4, 1895. 
 
 Find the ex.'ict nund)er of days between the day on which each 
 of the following notes is discounted and the day on which it is 
 due : 
 
 6. Day of discount, May 7, 1894; due June 6, 1894. 
 
 7. Day of discount, June 27, 1894; due Oct. 16, 1894. 
 
COMPOUND QUANTITIES 169 
 
 8. Day of discount, Sept. 4, 1894 ; due Oct. .30, 1894. 
 
 9. Day of discount, Dec. 2.'], 1894 ; due Feb. 20, 1894. 
 10. Day of discount, Jan. 15, 1892; due May 1, 1892. 
 
 Circular or Angular Measure 
 
 172. Angular Measure is used to measure arcs, angles, and 
 in determining latitude, longitude, direction, the position of 
 vessels at seu, and the like. 
 
 173. A Circle is a plane figure contained by one line called 
 the circumference, all points of which are equally distant 
 from a point witliin it called the centre. 
 
 One-half of the circumference is called the semicircum- 
 ference, and one-fourth a quadrant. 
 
 An arc is any portion of the circumference. 
 
 A line drawn through the centre and terminated at both 
 extremities by the circumference is called the diameter. 
 
 The line drawn from the 
 centre and terminated by 
 the circumference is called 
 the radius. 
 
 In the figure the line OB lias 
 revolved from OA through one- 
 fourth of a revolution. The an- 
 gle AOB is called a right angle 
 and contains 00°. 
 
 OA and OB are said to be per- 
 pendicular to each other. 
 
 Angular Measure 
 
 60 stconds (") = 1 rainute (') 
 60 minutes = 1 degree (°) 
 300 degrees = 1 circumference (C.) 
 
 The circumference of the earth at the equator = 21,002 mi. 
 
 The length of a degree at the equator = 24,002 mi. ~ 360 = 69.17 mi. 
 
 t^ 
 
 m 
 
 
 i-'^. 
 
 I 
 
 ^g 
 
170 
 
 ARITHMETIC 
 
 Exercise 113 
 
 1. What part of a revolution is 1 right angle ? 2 right 
 angles ? 4 right angles ? 
 
 2. What part of a revolution is G0° ? 45° ? 225° ? 
 
 3. What is the length of the arc of 1° in a circle whose cir- 
 cumference is 30)0 yd. ? 
 
 4. If an arc of 3° is G ft. long, what is the length of the cir- 
 cumference of the circle ? 
 
 MiSCKLLANKOUS UnITS 
 
 174, NUMBEHS 
 
 12 units = 1 dozen (doz.) 
 12 dozen = 1 gross 
 12 gross = 1 great gross 
 20 units = 1 score 
 
 Paper 
 
 24 sheets = 1 quire 
 20 quires = 1 ream 
 2 reams = 1 bundle 
 5 bundles = 1 bale 
 
 Miscellaneous Wekjuts 
 
 175. A bushel of wheat := 60 lb. 
 
 A bushel of beans = 00 lb. 
 
 A bushel of clover seed = 00 lb. 
 A bushel of shelled corn = 50 lb. 
 A bushel of rye ~ 50 lb. 
 
 A bushel of barley = 48 lb. 
 
 A bushel of oats = o2 lb. 
 
 A bushel of potatoes = (50 lb. 
 A bushel of coarse salt (domestic) = 56 lb. 
 
 These are the legal luunber of pounds per bushel in Michigan, Indiana, 
 Illinois, Wisconsin, Iowa, Missouri, and New York. 
 
 On the Chicago Board of Trade seeds arc sold by the cental. 
 
 A barrel of flour = 196 lb. 
 
 A barrel of pork or beef = 200 lb. 
 A cental of grain = 100 lb. 
 
 Exercise 114 
 
 1. Pens are sold in boxes containing 1 gross. 
 are there in a box ? 
 
 How many pens 
 
 m 
 
COMPOUND QUANTITIES 
 
 171 
 
 2. Lead pencils are sold in boxes containing | gross. How 
 many pencils are there in a box ? 
 
 3. How many packages of lead pencils of 1 doz. each are 
 there in a box ? 
 
 4. Eggs are packed in crates holding 30 doz. How many eggs 
 in a crate ? 
 
 5. How many sheets of paper in 20 quires ? 
 
 6. What articles of food weigh GO lb. to the bu. ? 
 
 7. How many bu. of wheat weigh as much as 10 bu. of bar- 
 ley? 
 
 8. What is the ratio of the weight of 1 bu. of barley to that 
 of 1 bu. of oats ? 
 
 176. A certain room is 8 yd. long. Here the unit of 
 measurement is 1 yd., and the measure of the length of 
 the room is the number 8. 
 
 The area of the floor of a room is 192 sq. yd. 6 sq. ft. 
 Here the measure of the area is the sum of 192 units of 
 1 sq. yd. and 6 units of 1 sq. ft. 
 
 A pitcher holds | of a gal. of water. 
 
 Here the measure of the capacity of the pitcher is | and 
 the unit of measurement is 1 gal. 
 
 
 Exercise 115 
 
 Name the measures of the following quantities, and the units 
 of measurement : 
 
 1. The volume of a cistern which holds 450 cu. ft. 
 
 2. The volume of a cistern which holds 1200 gal. 
 
 3. The area of a field which contains 8| A. 
 
 4. The value of a house worth $ 2800. What are the meas- 
 ures of the value of the house with the following units : f 5, $ 10, 
 $ 50, f 100 ? 
 
 
 frfii . 
 
172 
 
 ARITHMETIC 
 
 5. The weight of 200 lb. of sugar. What are the measures of 
 the weight with 1 oz., 1 cwt., and 1 T. as units ? 
 
 6. The weight of a quantity of tea weighing 336 lb. If 1 
 long cwt. is used as the unit, what is the measure ? 
 
 7. The weight of 8 oz. of gold. If 1 pwt. is the unit of meas- 
 urement, what number expresses the measure? What number 
 measures the weight when 1 lb. is the unit ? 
 
 8. The weight of 40 gr. of quinine. What is the measure 
 when 1 3 is the unit? 
 
 9. What are the measures of 1 lb. of gold, of lead, and of 
 quinine, when the unit of measurement is 1 oz.? When 1 gr. 
 is the unit ? 
 
 10. The length of the circumference of a circle found to be 22 
 yd. long. What are the measures of the circumference when 1 ft. 
 and 1 rd. are the measures ? 
 
 11. The length between two points which measures 4 eh. 
 What are the measures of the length when the units are 1 mi. 
 and 1 link ? 
 
 12. The area of a field whi^h contains 80 sq. rd. What are 
 the measures of the field when the units are 1 A. and 1 sq. yd. ? 
 
 13. The capacity of a pitcher which contains | of a gal. of 
 water. What is the measure when the unit is 1 qt. ? 
 
 14. The capacity of a basket which holds 6^ qt. What are 
 the measures, the units being 1 pt. and 1 pk. ? 
 
 15. The time of a rainstorm, which lasted 2 hr. What are 
 the measures, the units being 1 min. and 1 da. ? 
 
 16. The weight of a silver cup which is 15 oz. 12 pwt. 12 gr. 
 
 177. The fundamental units used in the measurement of 
 value are 1 cent, 1 dime, 1 dollar, and 1 eagle. 
 
 'te 
 
COMPOUND QUANTITIES 
 
 173 
 
 The value of a postage-stamp used to mail a letter to any 
 part of the United States or Canada is measured by the 
 number 2 and the unit 1 ct. 
 
 The cost of a quart of berries worth 15 ot. is measured by 
 the number 3 and the unit 1 nickel, or by the number 1 
 and the unit 1 dime plus the number 1 and the unit the 
 nickel. 
 
 It may also be measured by the number 15 and the unit 1 
 ct. It may also be measured by the number 1 and the unit 
 1 quarter-dollar less the number 1 and tlie unit 1 dime. 
 
 The unit for measuring oil is 1 gal. That for buying spice 
 by retail is 1 oz. Frequently a quantity is expressed with 
 reference to two or more units. Thus, the length of a table 
 being 1 yd. 2 ft. 6 in., the units are 1 yd., 1 ft., and 1 in. 
 
 Exercise 116 
 
 1. Name instances in which f 1 is the unit of vaKie ; 1 ct. ; 
 1 nickel; 1 dime. >Vliat unit of value is most commonly 
 used ? 
 
 2. Name instances in which the units of weight used are 1 
 oz. ; 1 lb. ; 1 cwt. ; 1 T. 
 
 3. Name quantities whose weight is measured by these units : 
 1 gr., 1 pwt., 1 oz., and 1 lb. 
 
 4. Name quantities which are measured by the units : 1 gr., 
 13, 13, 13,11b. 
 
 5. What quantities are expressed in terms of these units : 1 
 in. ? 1 ft. ? 1 yd. ? 1 rd. ? 1 mi. ? 
 
 6. Name things whose measurement is given in terms of the 
 units 1 pt., 1 qt., 1 gal. 
 
 J . ij 
 
 1^; 
 
 F-l 
 
 % 
 
 h 
 
174 
 
 ARITHMETIC 
 
 7. Name articles whose measurement is expressed in terms of 
 the unit 1 bu. ; 1 pk. ; 1 qt. 
 
 8. In measuring time give instances in which you use as the 
 unit 1 century; 1 yr. ; 1 mo.; 1 wk. ; 1 da.; 1 hr.; 1 min.; 1 sec. 
 
 9. Name articles whose quantity is expressed in terms of the 
 unit 1 doz. ; 1 gross ; 1 great gross ; 1 score. 
 
 10. In measuring paper the following units are used: 1 quire, 
 1 ream. Give instances when 1 quire is used as the unit, and 
 also when 1 ream is used. 
 
 11. What unit of weight connects Avoirdupois, Troy, and 
 Apothecaries' weight? "What number expresses the measure 
 of 1 lb. of each kind in terms of the common unit? Of 
 1 oz. ? 
 
 12. What units are common to Apothecaries' and Troy weight 
 and of equal value ? 
 
 13. What units of length connect Surveyors' Long Measure 
 with Linear Measure ? 
 
 14. Name live units of area which are derived from corre- 
 sponding units of length. Why is 1 A. chosen as a unit of 
 area? Give instances in which 1 A. is used as the unit of 
 area, and also when 1 sq. mi. is the unit. 
 
 15. Name a unit of volume larger than 1 cu. yd. Why have 
 we no units of volume corresponding to the linear units, 1 rd. 
 and 1 mi. ? 
 
 16. Find the number of cu. in. in 1 qt., dry measure, and 
 find how much greater it is than 1 qt., liquid measure. 
 
 17. State the number of days in the years 1500, 1600, 1700, 
 1800, 1900, 2000. 
 
 18. The circumference of a circle is 1,296,000 in. in length. 
 Find the length of 1°; of 1'; of 1". 
 
 \m 
 
COMPOUND QUANTITIES 
 
 175 
 
 * l!*! 
 
 Reduction 
 
 178. Reduction Descending is the process of reducing a 
 quantity expressed in terms of a unit or units of measure- 
 ment to a quantity expressed in terms of a smaller unit, or of 
 smaller units of measurement. 
 
 Reduce 2 mi. 36 rd. 5 yd. 2 ft. to feet. 
 320 
 
 640 
 J6 
 
 676 rd. 
 
 338 
 3380 
 5 
 
 3723 yd. 
 3 
 
 Note. — In this reduction we are to think 
 of the operation as signifying 2 x 320 rd., or 
 by the law of commutation as 320 x 2 rd., and 
 not as representing 320 x 2 mi. 
 
 11169 
 2 
 
 11171 ft. 
 
 2 mi. = 2 X 320 rd. = 040 rd. 
 2 mi. 30 rd. = 040 rd. + 30 rd. = 670 rd. 
 676 rd. = 676 x 6i yd. = 3718 yd. 
 2 mi. 36 rd. 6 yd. = 3718 yd. + 5 yd. = 3723 yd. 
 .3723 yd. = 3723 x 3 ft. = 11,169 ft. 
 .-. 2 mi. 36 rd. 5 yd. 2 ft. = 11,169 ft. + 2 ft. = 11,171 ft. 
 
 Exercise 117 
 
 1. Reduce 5 gal. 3 qt. 1 pt. 2 gi. to gills. 
 
 2. Reduce 18 bii. G pk. 3 qt. 1 pt. to pints. 
 
 3. Reduce £5 12 s. 9d. to pence. 
 
 4. Reduce 16 lb. 8 oz. 15 pwt. 17 gr. to grains. 
 
 5. Reduce 7 T. 18 cwt. 14 lb. 12 oz. to ounces. 
 
 6. Reduce 2 yr. 15 da. 17 min. to minutes. 
 
 7. Reduce 18 rd. 4 yd. 2 ft. C> in. to inches. 
 
 8. Reduce 12 lb. 5 oz. 5 dr. 2 sc. 16 gr. to grains. 
 
 9. Reduce 2 A. 4 sq. rd. 2 sq. yd. 8 sq. ft. to square feet. 
 
 ir\ 
 
 I 
 
 h 
 
 '■1' 
 
^, 
 
 .<tv^ 
 
 
 IMAGE EVALUATION 
 TEST TARGET (MT-3) 
 
 // 
 
 v. 
 
 // ^ ^^^ 
 
 /K/. 
 
 .^ 
 
 f 
 
 
 1.0 
 
 I.I 
 
 128 125 
 
 
 ■ 2.2 
 
 118 
 
 m 
 ■a 
 
 11-25 III 1.4 
 
 12.0 
 
 .^ 
 
 
 V5 
 
 ^;. 
 
 
 
 '/ 
 
 /A 
 
 Photographic 
 
 Sdences 
 
 Corporation 
 
 33 WEST MAIN STRIET 
 
 WltSTIR,N.Y. UStO 
 
 (716) •72-4503 
 
 '%^^^ 
 > 
 

 ^o 
 
176 
 
 ARITHMETIC 
 
 10. Reduce 16 cu. ft. 1374 cu. in. to cubic inches. 
 
 11. Reduce 1 mi. 18 rd. 2 yd. 2 ft. 6 in. to inches. 
 
 12. State how to reduce a quantity from higher to lower de- 
 nominations. 
 
 13. Find the number of A. in a township. 
 
 14. Find the number of cu. in. in a vessel containing 25 gal. 
 
 15. Reduce 59 lb. 7 oz. 14 pwt. 19 gr. to grains. 
 
 16. Reduce 7 T. 15 cwt. 5G lb. to ounces. 
 
 17. Reduce 17 lb. 2 5 2 3 to grains. 
 
 18. Reduce 17 cu. yd. 1001 cu. in. to cubic inches. 
 
 19. Reduce 760 bu. 3 pk. to quarts. 
 
 20. Reduce 56 reams 19 quires 18 sheets to sheets. 
 
 21. Reduce 36 wk. 5 da. 17 hr. to seconds ; and 1 mo. of 30 da. 
 23 hr. 59 sec. to seconds. 
 
 22. Reduce 35 A. 80 sq. rd. 28 sq. yd. to square yards. 
 
 179. Reduction Ascending is the process of reducing a 
 quantity expressed in terms of a unit, or of units, to a quan- 
 tity expressed in terms of a larger unit, or of larger units. 
 
 (1) Reduce 242,337 in. to higher denominations. 
 
 242337 
 
 12 
 3 
 
 11 
 320 
 
 20194 ft. 9 in. 
 
 6731 yd. 1 ft. 
 2 
 
 13462 half yd. 
 
 1223 rd. 9 half yd., i.e. 4 yd. 1 ft. 6 in. 
 
 3 mi. 263 rd. 
 
 .-. 242,337 in. = 3 mi. 263 rd. 4 yd. 1 ft. in. ) 
 
 1 ft. 6 in. J 
 = 3 mi. 203 rd. 5 yd. ft. 3 in. 
 
COMPOUND QUANTITIES 
 
 177 
 
 242,337 in. = 20,194 ft. 9 in. 
 20,194 ft. 9 in. = 6731 yd. 1 ft. 9 in. 
 6731 yd. 1 ft. 9 in. = 1223 rd. 4J yd. 1 ft. 9 in. 
 1223 rd. ^ yd. 1 ft. 9 in. = 3 mi. 203 id. 4 J yd. 1 ft. 9 in. 
 
 = 3 mi. 203 rd. 5 yd. ft. 3 in. 
 .-. 242,337 in. = 3 mi. 263 rd. 5 yd. 3 in. 
 
 (2) To prove this answer correct, reduce 3 mi. 263 rd. 5 yd. 3 in. to 
 in. , by tlie method of the preceding exercise. 
 
 180. We reduce a quantity expressed in terms of a smaller unit to larger 
 units in order to get a more definite idea of its value. 
 
 Thus we form no definite idea of a distance between two points when we 
 are told that it is 242,337 in. ; but we have a definite conception of the 
 same distance when we are told that it is 3 mi. 263 rd. 6 yd. 3 in. 
 
 Exercise 118 
 
 Reduce to higher denominations, and prove every third answer 
 correct : 
 
 4. 4728 cu. ft. to cd. 
 
 5. 18,425 lb. of wheat to bu. 
 
 6. 21,489 d 
 
 1. 678 pt. 
 
 2. 4622 pt. of dry measure. 
 
 3. 483,197 sec. 
 
 7. 93,742 oz. 
 
 8. State how to reduce a quantity expressed in terms of a 
 lower unit to higher units. 
 
 9. Reduce 5420 gr. to oz., Avoir. 
 
 10. 141,728 gr. 14. 1,364,428 in. 
 
 11. 57,893 cu. in. 15. 273,460 sq. yd. 
 
 12. 56,735 d. 16. 6,188,724 sq. in. 
 
 13. 5,838,297 oz. 17. 429,678 in. 
 
 18. 73,940 n^. 
 
 19. 89,673 gr.. Apothecaries' weight. 
 
 20. 7493 units. 
 
 21. Reduce 37,921 in. to ch., rd., etc. 
 
 N 
 
 u 
 
 
 ( ■ 
 
178 
 
 ARITHMETIC 
 
 22. Reduce 121,838 A. to townships, etc. 
 
 23. The Imperial gal. of Great Britain contains 277.274 cu. in. 
 Find, correct to three decimal places, the number which measures 
 the Imperial gal., the gal., liquid measure, being the unit. 
 
 24. Express in bu. and cu. in. the volume of a bin 8 ft. by 5 ft. 
 by 4 ft. 
 
 25. How many cu. in. are there in 4 qt., dry measure ? Show 
 that this is .163 greater than the gal. measure. 
 
 Compound Addition and Subtraction 
 
 181. Add: 
 
 
 
 
 
 ml. 
 
 rd. 
 
 yd- 
 
 ft. 
 
 in. 
 
 2 
 
 27 
 
 1 
 
 2 
 
 8 
 
 1 
 
 146 
 
 2 
 
 1 
 
 6 
 
 8 
 
 91 
 
 2 
 
 
 
 4 
 
 7 
 
 152 
 
 1 
 
 2 
 
 9 
 
 19 
 
 97 
 
 ^ 
 
 1 
 
 3 
 
 
 
 1 - 
 
 "2 - 
 
 = 1 
 
 6 
 
 19 
 
 97 
 
 9 
 
 The sum of the in. column is 27 in., or 2 ft. 3 in. 
 
 The sum of the ft. column, increased by 2 ft., is 7 ft., or 2 yd. 1 ft. 
 
 The sum of the yd. column, increased by 2 yd., is 8 yd., or 1 rd. 2| yd. 
 
 The sum of the rd. column, increased by 1 rd., is 417 rd., or 1 mi. 97 rd. 
 
 The sum of the mi. column, increased by 1 mi., is 19 mi. 
 
 Changing | yd. to 1 ft. 6 in. and adding, we have the sum = 19 mi. 97 rd. 
 2 yd. 2 ft. 9 in. 
 
 As in the problems in addition in Chapter III., we are required in the 
 preceding question to find the whole quantity measured by the four given 
 parts, of which the first is 2 mi. 27 rd. 1 yd. 2 ft. 8 in. What are the other 
 three measured parts ? 
 
 In this question how would the work of writing and adding be diminished 
 if our units of length were arranged according to the decimal system ? 
 
:J 
 
 COMPOUND QUANTITIES 
 
 179 
 
 182. Subtract 53 lb. 5 oz. 18 pwt. from 72 lb. 4 oz. 7 pwt. 
 
 lb. oz. pwt. 
 
 72 4 7 
 
 53 5 18 
 
 18 
 
 10 
 
 9 
 
 Since we cannot take 18 pwt. from 7 pwt., take 1 oz. or 20 pwt. from 4 oz. 
 and add it to the 7 pwt., making 27 pwt. 18 pwt. from 27 pwt. leaves 9 pwt. 
 Since we cannot take 5 oz., from 3 oz., take 1 lb. or 12 oz. from 72 lb. and 
 add it to the 3 oz., making 15 oz. 6 oz. from 15 oz. leaves 10 oz. 53 lb. 
 from 71 lb. leaves 18 lb. 
 
 Hence the difference = 18 lb. 10 oz. 9 pwt. 
 
 As in the problems in subtraction in Chapter IV., we are given in the 
 preceding question the whole quantity measured by 72 lb. 4 oz. 7 pwt., and 
 one part, viz. 53 lb. 5 oz. 18 pwt., and are required to find the part measured 
 by their difference. 
 
 
 
 
 
 Exercise 119 
 
 
 
 
 
 Add: 
 
 
 
 
 
 
 
 
 
 
 bu. 
 
 pk. 
 
 
 qt. 
 
 pt. 
 
 
 T. 
 
 cwt. 
 
 
 lb. 
 
 1. 3 
 
 5 
 
 
 6 
 
 1 
 
 4. 
 
 16 
 
 17 
 
 
 74 
 
 8 
 
 4 
 
 
 1 
 
 
 
 
 13 
 
 10 
 
 
 20 
 
 7 
 
 3 
 
 
 6 
 
 1 
 
 
 17 
 
 15 
 
 
 19 
 
 9 
 
 4 
 
 
 3 
 
 1 
 
 
 84 
 
 
 
 
 87 
 
 
 
 oz. 
 
 11 
 
 
 pwt. 
 
 16 
 
 5. 
 
 11 
 
 11 
 
 
 36 
 
 lb. 
 2. 18 
 
 5 
 
 22 
 
 3 
 3 
 
 3 
 
 2 
 
 19 
 
 16 
 
 
 9 
 
 
 22 
 
 
 56 
 
 
 
 1 
 
 10 
 
 23 
 
 
 8 
 
 
 6 
 
 
 3 
 
 2 
 
 2 
 
 11 
 
 17 
 
 
 6 
 
 
 13 
 
 
 15 
 
 79 
 
 6 
 4 
 
 1 
 1 
 
 9 
 
 
 
 
 
 
 10 
 
 £ 
 
 
 «. 
 
 
 d. 
 
 
 
 
 
 
 3. 5 
 
 
 17 
 
 
 10 
 
 
 cu. j'd. 
 
 cu. ft. 
 
 
 cu. in 
 
 36 
 
 
 
 
 
 11 
 
 6. 
 
 3 
 
 23 
 
 
 171 
 
 7 
 
 
 3 
 
 
 4 
 
 
 17 
 
 17 
 
 
 31 
 
 73 
 
 
 19 
 
 
 8 
 
 
 28 
 
 26 
 
 
 1000 
 
 30 
 
 
 14 
 
 
 5 
 
 
 34 
 
 23 
 
 
 1101 
 
9IMII 
 
 ^■■■■■■■■■■■■i 
 
 ■!■■ 
 
 180 
 
 ARITHMETIC 
 
 7. State how to add compound quantities. 
 Subtract : 
 
 lb. 
 8. 144 
 
 oz. 
 
 8 
 
 pwt. 
 
 14 
 
 
 10. 
 
 lb 
 144 
 
 5 
 9 
 
 3 
 4 
 
 3 
 
 1 
 
 106 
 
 11 
 
 ft. 
 1 
 
 16 
 
 in, 
 
 5 
 
 
 11. 
 
 129 
 
 
 
 7 
 
 3 
 
 yd. 
 9. 15 
 
 lb. 
 5836 
 
 oz, 
 
 
 
 pwt, 
 
 
 
 
 
 13 
 
 2 
 12. 
 
 7 
 
 cu. yd. 
 
 37 
 35 
 
 cu. ft 
 
 18 
 24 
 
 
 4976 
 
 cu. in. 
 
 857 
 1280 
 
 7 
 
 13 
 
 19 
 
 13. State how to subtract one compound quantity from another. 
 
 14. State in what respects addition and subtraction of com- 
 pound quantities are the same as addition and subtraction of 
 numbers, and state how they differ. 
 
 Compound Multiplication and Division 
 
 183. Multiply 5 wk. 6 da. 18 hr. by 11. 
 
 We are here required to find the whole quantity measured by the unit 
 6 wk. 6 da. 18 hr. and the number 11. 
 
 wk. 
 
 da. 
 
 hr. 
 
 5 
 
 6 
 
 18 
 11 
 
 65 
 
 6 
 
 We multiply 18 hr. by 11 and obtain the product 198 hr. or 8 da. 6 hr. 
 Then we multiply 6 da. Jy 11 and obtain the product 66 da., which, 
 increased by 8 da. is 74 da., or 10 wk. 4 da. Multiplying 5 wk. by 
 11, and adding 10 wk., we have 65 wk. Hence the product is 65 wk. 
 6 da. 18 hr. 
 
 What is the ratio of 65 wk. 4 da. 6 hr. to 6 wk. 6 da. 18 hr. ? 
 
COMPOUND QUANTITIES 
 
 181 
 
 184. (1) Divide 88 rd. 3 yd. 1 ft. by 34. 
 
 rd, yd. 
 
 34) 88 3 
 68 
 
 ft. 
 
 1 
 
 rd. 
 (2 
 
 yd. 
 3 
 
 ft. 
 1 
 
 20 rd. 
 
 
 
 
 
 110 yd. 
 3 
 
 113 yd. 
 
 102 
 
 
 
 
 
 11yd. 
 3 
 
 
 
 
 
 33 ft. 
 1 
 
 34 ft. 
 34 
 
 
 
 
 
 Hence the quotient or unit of measure is 2 rd. 3 yd. 1 ft. 
 
 The remainder on dividing 88 rd. by 34 is 20 rd. 20 rd. 3 yd. = 113 yd. 
 The remainder on dividing 113 yd. by 34 is 11 yd. 11 yd. 1 ft. = 34 ft. 
 
 On dividing 34 ft. by 34, there is no remainder. Hence the quotient is 
 2 rd. 3 yd. 1 ft. What part of the dividend is tlie quotient ? 
 
 (2) Divide 73 gal. 1 pt. by 16 gal. 1 qt. 
 
 We are here required to find the number which is the ratio of the quan- 
 tity 73 gal. 1 pt. to the unit 16 gal. 1 qt. 
 
 73 gal. 1 pt. = 685 pt. ; 16 gal. 1 qt. = 130 pt. 
 
 585 pt. H- 130 pt. =41. 
 
 .'. the quotient or ratio is 4^. 
 
 
 Exercise 120 
 
 1. Multiply 7 gal. 3 qt. 1 pt. by 9. 
 
 2. Multiply 8 da. 12 hr. 25 min. by 28. 
 
 3. Divide £199 6 s. 8 d by 13. 
 
 4. Divide 459 lb. 4 oz. 5 pwt. 22 gr. by 29. 
 
 5. Multiply 86 lb. 7 oz. 16 pwt. 11 gr. by 8. 
 
 6. Multiply 5 wk. 6 da. 18 hr. 14 min. by 11. 
 
 
^m 
 
 182 
 
 ARITHMETIC 
 
 7. Divide 1738 cu. yd. 1236 cu. in. by 798. 
 
 8. Divide 684 da. 8 hr. 9 min. by 47. 
 
 9. Multiply 70 yd. 2 ft. 10 in. by 7. 
 
 10. Divide 1 mi. 54 rd. 1 ft. 2 in. by 29. 
 
 11. Multiply 2 hr. 8 min. 9 sec. by 15. 
 
 12. Divide 13° 26' by 15. 
 
 13. Divide 14 yd. 1 ft. 8 in. by 3 yd. 4 in. 
 
 14. Divide 16 bu. 1 pk. 2 qt. by 10 bu. 3 pk. 4 qt. 
 
 15. (a) State how to multiply a compound quantity by a given 
 number, {b) State how to divide a compound quantity by a given 
 number. 
 
 16. State in what respects multiplication and division of com- 
 pound quantities are like multiplication and division of numbers, 
 and how they differ. 
 
 te 
 
 1{ 
 
 Fractions of Simple and Compound Quantities 
 185. (1) Find the value of | of a mi. 
 
 I mi. 
 
 I of 320 rd. = 266 1 rd. 
 
 f rd. =1 of 51 yd. = 3| yd. 
 f yd. = I of 3 ft. = 2 ft. 
 
 .-. f of a mi. = 266 rd. 3 yd. 2 ft. 
 
 (2) Find the value of | bu. 
 
 f pk. 
 
 f bu. = |of 4pk. =2| pk. 
 2| pk. - I pk. = Iff pk. 
 If pk. = ft of 8 qt. = 4/3 qt. 
 
 .-. f bu. -f pk. = lpk. 4^\qt. 
 
 Exercise 121 
 
 Find the value of : 
 
 1. |of £1; J of £2. 
 
 2. y3^ of a da. ; j\ of a mi. 
 
 3. f of 3 T. ; 7f lb. Avoir. 
 
 t< 
 
IH^Hi 
 
 ■^PBPWffpP' 
 
 COMPOUND QUANTITIES 
 
 183 
 
 II 
 
 4. f bu. - 4 pk. ; I lb. Troy + f lb. Troy - f oz. Troy. 
 
 5. I A. ; I A. — I sq. rd. ; | sq. mi. 
 
 6. f of 5 lb. 8 oz. 6 pwt. 
 
 7. f of 2 mi. 38 rd. 4 yd. 2 ft. 2 in. 
 
 8. State how to express a fraction of a unit of measure in 
 terms of smaller units. 
 
 Express .854 of an A. in lower denominations. 
 
 .854 A. 
 160 
 
 61240 
 854 
 
 136.640 sq. rd. 
 30^ 
 
 16 
 1920 
 
 19.36 sq. yd. 
 9 
 
 3.24 sq. ft. 
 144 
 
 " 96 
 96 
 24 
 
 .-. .864 A. = 136 sq. rd. 19 sq. yd. 3 sq. ft. 34.66 sq. in. 
 Adapt note, § 178, to this problem. 
 
 34.66 sq. in. 
 
 Find the value of : 
 
 1. .84 of a da. 
 
 2. .045 of a mi. 
 
 3. .6 of a lb. Troy. 
 
 Exercise 122 
 
 4. 5.923 mi. - 75.18 rd. 
 
 5. £75.43 -16.76 s. 
 
 6. 4.7 A. - 2.93 sq. rd. 
 
 186. Reduce 213 rd. 5 ft. 6 in. to the fraction of 3 mi. 
 
 213 rd. 5 ft. 6 in. = 42,240 in. 
 
 3 mi. = 3 X 63,360 in. =: 190,080. 
 
 .-. 213 rd. 5 ft. 6 in. = t%%\%, or f of 3 mi. 
 
 The G. C. M. of 42,240 and 190,080 is 21,120, which divides the numera- 
 tor twice and the denominator 9 times. 
 
 ■I?: 
 
 f >i 
 
 Hi 
 
 
184 
 
 ARITHMETIC 
 
 Exercise 123 
 
 1. Reduce £1 7 s. 6d. to the fraction of £ 2. 
 
 2. Reduce 22 rd. 5 yd. 2 ft. 6 in. to the fraction of a mi. 
 
 3. Reduce 8 hr. 3 inin. to the fraction of a da. 
 
 4. Reduce 4 mo. 3 da. to the fraction of a yr. (30 da. to a 
 mo. and 360 da. to a yr.). 
 
 5. Reduce 11 mo. 18 da. to the fraction of a yr. 
 
 6. Reduce 1^ in. to the fraction of 1^ yd. 
 
 7. Reduce f lb. Avoir, to the fraction of 2 lb. Troy. 
 
 8. Express 213 rd. 1 yd. 2 ft. 6 in. as a fraction of 145 rd. 
 2 yd. 1 ft. 6 in. 
 
 187. Express 58 rd. 2 yd. 7.2 in. as a decimal of a mi. 
 
 7.2 in. 
 
 12 
 
 3 
 
 6.5 
 
 320 
 
 .Oft. 
 
 2.2 yd. 
 
 58.4 rd. 
 
 .1825 mi. 
 
 7.2 in. = .6 ft. = .2 yd. 
 2 yd. + .2 yd. = 2.2 yd. = .4 rd. 
 
 58.4 rd. =r .1825 mi. 
 .-. 58 rd. 2 yd. 7.e in. = .1825 mi. 
 Prove this answer correct by reducing .1825 mi. to lower denominations. 
 
 Exercise 124 
 
 1. Reduce 8 oz. 15.2 pwt. to the decimal of a lb. 
 
 2. Reduce 21 hr. 57 min. 36 sec. to the decimal of a da. 
 
 3. Reduce 147 rd. 1 yd. 3.6 in. to the decimal of a mi. 
 
 4. 25 sq. mi. 128 A. is what decimal of a township ? 
 
 5. Reduce 67 sq. rd. 6 sq. yd. 64.8 sq. in. to the decimal of 
 
 an A. 
 
COMPOUND QUANTITIES 
 
 185 
 
 BoAKD Measure 
 
 188. Boards 1 inch or less in thickness are sold by the 
 square foot. 
 
 Thus a board 18 ft. long, 14 in. wide, and 1 in. thick or less contains 
 18 X If, or 21 ft., board measure. 
 
 To find the number of feet, board measure, in lumber more 
 
 than 1 inch thick, we find the number of square feet in the 
 
 surface of the board and multiply this result by the number 
 
 of inches that the lumber is thick. 
 
 Thus a board 15 ft. long, 8 in. wide, and 2J in. thick contains ISx^^x |, 
 or 25 board feet. 
 
 Exercise 125 
 
 How many feet, board measure, in : 
 
 1. A board 20 ft. long, 9 in. wide, and 1 in. thick ? | in. thick ? 
 
 2. A board 18 ft. long, 8 in. wide, and 2^ in. thick ? 
 
 3. A scantling 16 ft. long, 3 in. wide, and 4 in. thick ? 
 
 4. Twenty scantlings, 24 ft. long, 5 in. wide, and 7 in. thick ? 
 
 5. A stick of timber 33 ft. long, and 14 in. square ? 
 
 6. One cu. ft. ? 
 
 7. What is the cost of 25 joists each 6 in. by 4 in. by 15 ft. 
 at $ 22 per M. ? 
 
 8. What is the cost of 24 joists each 5 in. by 7 in. by 10 ft. 
 at $ 21 per M. ? 
 
 9. How much will it cost to enclose a rectangular lot 50 ft. 
 wide and 100 ft. deep with a tight board fence 16 ft. high with 
 boards that cost $ 18 per M. ? 
 
 (it 
 
 i 
 
 m 
 
 Longitude and Time 
 
 189. Turn to your Geography and find several meridian 
 lines. Find the prime meridian which passes through Green- 
 wich, England. 
 
186 
 
 ARITHMETIC 
 
 The imaginary lines drawn on the earth's surface from 
 pole to pole are called meridians. The meridian passing 
 through Greenwich, a town near London, England, hav- 
 ing the royal observatory, is called the prime or standard 
 meridian. 
 
 Places west of the prime meridian are in west longitude, 
 and places east of the prime meridian are in east longitude. 
 Thus, Washington is 77° 7' west longitude, and Paris 2° 20' 
 east longitude. 
 
 190. Find from the maps in your geographies to the 
 nearest degree the longitude of these cities : New York, 
 Pittsburg, Richmond, Atlanta, Chicago, Denver, Salt Lake 
 City, San Francisco. 
 
 Find also the longitude of Rome, Stockholm, Athens, Con- 
 stantinople, St. Petersburg, and Moscow. 
 
 191. The difference in longitude between Philadelphia, 
 which is 75° 9' west longitude, and Portland, which is 70° 15' 
 west longitude, is 4° 54'. 
 
 The difference between the longitude of Philadelphia and 
 that of Paris, which is 2° 20' east longitude, is 77° 29', and is 
 obtained by finding the sum of the longitudes. 
 
 192. Find on the map of the United States and name the 
 meridians that denote Eastern time. Central time. Mountain 
 time, and Pacific time. How many degrees are there between 
 these meridian lines ? What is the difference in time between 
 places situated on these meridians? 
 
 193. As the sun rises in the east, it is sunrise in New York 
 earlier than in Chicago ; consequently at any time during 
 the day the clock time in New York is later than in Chicago, 
 
II' 
 
 COMPOUND QUANTITIES 
 
 187 
 
 Similarly, clock time in San Francisco, which is west of 
 Chicago, is earlier than in the latter city. 
 
 194. Since the sun appears to move in a circle about the 
 earth, i.e, through 300° in 24 hr., we have the following; 
 
 In 24 hr. the sun passes through 360°. 
 
 In 1 hr. the sun passes through 15°. 
 
 In 1 min. the sun passes through ^j^ of 15° = ^° = 15'. 
 
 In 1 sec. the sun passes through ^^ of 15' = ^' = 15". 
 
 Hence^ to reduce longitude expressed in time to longitude 
 expressed in degrees^ we multiply by 15. and to reduce longi- 
 tude expressed in degrees to longitude expressed in time^ we 
 divide by 15> 
 
 195. Make the multiplication table of 15 and memorize it, so as to be 
 able to work questions in longitude and time by short multiplication and 
 division. 
 
 Exercise 126 
 
 1. What is the difference in longitude between two places 
 whose difference in time is 1 hr. ? 2 hr. ? 4 hr. ? 2 min. ? 
 3 min. ? 1 sec. ? 3 sec. ? 
 
 2. What is the difference in longitude between two places 
 whose difference in time is 1 hr. 2 min. ? 1 hr. 3 min. 2 sec. ? 
 
 3. What is the difference in time between two places whose 
 difference in longitude is 30° V 75°? 120'? 90'? 135"? 105"? 
 
 4. What is the difference in time between two places whose 
 difference in longitude is 15° 45' 30" ? 75° 15' 45" ? 
 
 5. Find the difference in longitude between the following 
 places, and illustrate your answers by diagrams: 
 
 Washington 77° west longitude and Helena 112° west longitude ? 
 Washington 77° west longitude and Hamburg 10° east longitude ? 
 Cairo 32° east longitude and Hamburg 10° east longitude ? 
 
 i,:jl 
 
 i tJ 
 
 Ml 
 
 if 
 
 i u 
 
188 
 
 ARITHMETIC 
 
 6. What is the difference in time between two places : 
 
 (1) One 64° west longitude, the other 34° east longitude ? 
 
 (2) One 64° west longitude, the other 26° east longitude ? 
 
 (3) One 64° east longitude, the other 34° east longitude ? 
 
 7. When it is 6 a.m. at San Francisco, what time is it at 
 a place 45° east of San Francisco? 30° east? 15° 45' east? 
 30° 15' 45" east ? 
 
 8. When it is 11 a.m. at Chicago, what time is it at a place 
 60° west of Chicago ? 30° west ? 45' west ? 15° 45' ? 
 
 196. (1) Find the difference in time between St. Louis 90° 19' 
 26" west longitude and Sacramento 121° 25' 41" west longitude. 
 
 121° 25' 41" 
 
 90 19 26 
 
 15 )31° 6' 16 '^ difference in longitude 
 
 2 hr. 4 min 25 sec. difference in time 
 
 A difference in longitude of 31^ gives a difference of 31 -f- 15, or 2 hr. of 
 time, with a remainder of 1°. A difference in longitude of 1° 6', or 66', gives 
 a difference of 66-4-15, or 4 min. of time, witli remainder 6'. A differ- 
 ence of 6' 15", or 375", gives a difference of 375 -r- 15, or 25 sec. of time. 
 
 (2) Berlin is 13° 23' 53" east longitude and Boston is 71° 4' 9" 
 
 west longitude. When it is 1.15 p.m. at Boston, what time is it at 
 
 Berlin ? 
 
 13° 23' 53'' 
 
 71 4 9 
 
 16 )84° 28' 2" difference in longitude 
 
 5 hr. 37 min. 52 ^^^ sec. difference in time 
 
 1 hr. 15 min. 
 
 time in Boston 
 
 6 hr. 52 min. 52 j\ sec. time in Berlin 
 .*. it is 52 min. 52^^ sec. after 6 p.m., or 7 min. 7|| sec. to 7 p.m. 
 
 Ezerciae 127 
 
 Find the difference in time between the following cities : 
 
 1. Brooklyn 73° 58' W. and Omaha 95° 28' W. 
 
 2. St. Paul 93° 3' 45" W. and Cleveland 81° 39' W. 
 
 S 
 
COMPOUND QUANTITIES 
 
 189 
 
 4. 
 5. 
 6. 
 
 7. 
 8. 
 
 Indianapolis 86°6'57" W. and San Francisco 122°26'12" W. 
 Cincinnati 84°28'36" W. and Glasgow 4°17' 6" W. 
 Detroit 83° 5' 7" W. and Vienna 16° 22' 22" E. 
 Pillsbury 79° 55' 43" W. and Amsterdam 4° 52' 13" E. 
 Newark 74° 9' 12" W. and Rome 12° 27' 58'' E. 
 
 When it is 11 a.m. at Cleveland, what o'clock is it at 
 St. Paul ? 
 
 9. What time is it at Indianapolis at the opening of school 
 at 9 A.M. in San Francisco ? 
 
 10. When it is 7 a.m. at Cincinnati, what time is it at Glasgow ? 
 
 11. When it is 8 a.m. at Omaha, what time is it at Brooklyn ? 
 
 12. A man travels until his watch is 1 hr. 5 min. 16 sec. 
 slow. Does he travel east or west, and how many degrees has 
 he gone ? 
 
 13. A vessel sailed from a port directly on a line of latitude 
 a certain distance, and then due north to port, where the captain 
 found that his chronometer was 40 min. slow. In which direction 
 did he sail at first, and how many degrees ? 
 
 14. What is the difference in longitude between two places 
 whose difference in time is : 
 
 {a) 2 hr. 33 min. 18 sec. ? 
 (6) 4 hr. 27 min. 46 sec. ? 
 (c) 6 hr. 12 min. 29 sec. ? 
 
 15. Buffalo is 78°57'48" W. and Constantinople is 28°59'3" E. 
 What time is it in Constantinople when it is 20 min. after 6 a.m., 
 July 6, in Buffalo? 
 
 16. What time is it in Buffalo when it is 20 min. after 6 a.m., 
 July 6, in Constantinople ? 
 
 17. Given the longitude of two places, state how to find the 
 time in the place east at a given time in the place west. 
 
 f.. 
 
 
 ''il 
 
 '•' II 
 
190 
 
 ARITHMETIC 
 
 Ezerciae 128 
 
 1. Find values of the quantity measured by the number 6 
 according as the unit is £ 2 5 s. or 6 oz. 10 pwt. 16 gr. 
 
 2. Find the vahie of a pile of cord wood 13' 4" long by 
 3' 9" high at $4.50 a cd. 
 
 3. How many gr. are there in 9 oz. 17 pwt. 22 gr., and how 
 many A., etc., in 167,412,715 sq. in. ? 
 
 4. In 161,384 in. how many mi. ? 
 
 5. If a sovereign weigh 123.274 gr., how many sovereigns 
 will weigh 21 lb. 4 oz. 16 pwt. 10 gr. ? 
 
 6. The working wheel of a locomotive is 226 in. in circum- 
 ference. It turns 91 times in 1 min. Through how many rd., 
 etc., does it draw the train in 1 min. ? 
 
 7. How much must be paid for 360 ft. of boards at $ 12.00 
 per M., 250 shingles at $ 2.50 per M., and 760 ft. of timber at 
 $ 1.00 per C. ? 
 
 8. How many min. are there in y'j yr. + ^^ wk. + ^ hr. ? 
 
 9. A horse trotted 1 mi. in 2 min. 12 sec. Taking his stride 
 at 16 ft., how many times per sec. did his feet touch the ground ? 
 
 10. A bicyclist rode 39 mi. in 3 hr. 15 min. What was his 
 rate in mi. per hr., in yd. per min., and in ft. per sec. ? 
 
 11. Keduce || of £ 20 to the decimal of £ 100. 
 
 12. How many oz. of gold are worth £ 23 16 s. when .63 oz. 
 is worth 44 s. 2d.? 
 
 13. If wire fencing cost 6^ per yd., find what must be paid 
 for enclosing a field 305 yd. long and 156 yd. wide, there being 
 4 rows of wire. 
 
 14. Convert £296 16 s. sterling into dollars and cents, £1 
 being worth $ 4.8665. 
 
 pr 
 
 act 
 to 
 
COMPOUND QUANTITIES 
 
 191 
 
 
 15. Make out the following account neatly, accurately, and in 
 proper form. All fractions are to be retained. 
 
 John Wilson bought from you to-day : 
 
 7^ lb. cheese @ 12^^ per lb. ; 
 6 J lb. butter @ 23^ per lb. ; 
 2J lb. tea @ 55^ per lb. ; 
 27 lb. sugar @ f 1 per 18 lb. 
 
 16. If a yard measure is \ of an inch too long, what is the 
 actual distance between two points which is found by this measure 
 to be 500 yd. 2 ft. 6 in. ? 
 
 17. Express 1 lb. Troy as the fraction of 1 lb. Avoirdupois ; and, 
 conversely, express 1 lb. Avoirdupois as the fraction of 1 lb. Troy. 
 
 18. Express as the fraction of a ft. the remainder after .012 
 of a yd. has been subtracted as often as it is possible from 
 1.087 yd. 
 
 19. Which is the heavier, a pound of gold or a pound of 
 feathers, and an ounce of gold or an ounce of feathers ? By how 
 much in each case ? 
 
 20. How many silver spoons, each weighing 2 oz. 16 pwt., 
 could be made out of a bar of silver, the weight of which is 50 oz. 
 8 pwt. ? 
 
 21. If the unit of Troy weight, called the pennyweight, 
 contained 14.25 gr. instead of 2i, how many gr. would make 
 1 lb. Troy ? 
 
 22. A man bought a quantity of tea supposed to be done up in 
 packages of 1 lb. each, for which he was to pay $ 64 ; on weigh- 
 ing them, however, it was found that each package was 1 oz. too 
 light. How much should he pay for the tea ? 
 
 23. Required the mean of the following observations of temper- 
 ature: 41° 29'; 41° 271'; 39° 13'; 41° 33'; 37° 47^'; 44° 28'; 40° 13'. 
 
 24. Sold 20,900 ft. of lumber for $ 331.62|, gaining thereby 
 ^ 78.37^. What had it cost per C. ? 
 
 
192 
 
 ARITHMETIC 
 
 25. A lot 150 ft. long and 100 ft. wide is to be surrounded 
 by a close board fence 6 ft. high. What will the boards cost at 
 $ 12.50 per thousand ft. ? 
 
 26. If a room be 12 ft. square, what must its height be in order 
 that the area of the walls may amount to 60 sq. yd. ? 
 
 27. Find the value of a rectangular field 330 yd. by 156 yd. @ 
 $ 36.50 per A. 
 
 28. Find the surface area and the volume of a rectangular 
 block 3' 9" X 2' 4" xl' 3". 
 
 29. Express as a fraction of an A. the sum of the following : 
 i of f of If of 1 A. ; I of if of f f of 100 sq. rd. ; and |f of 2\ 
 times 605 sq. yd. 
 
 30. The Manufacturers and Liberal Arts Building of the 
 Columbian Fair was in the form of a rectangle and covered an 
 area of 30 A. 76 sq. rd. 19 sq. yd. 7 sq. ft. The building was 
 787 ft. wide. How many ft. in length was it ? 
 
 31. A 200-acre farm is sown with grain as follows : Peas, 25 A. 
 126 sq. rd. 10 sq. yd. ; oats, 46 A. 134 sq. rd. 15 sq. yd. ; wheat, 
 75 A. 125 sq. rd. 25 sq. yd. The buildings, garden, and orchard 
 occupy 12 A., and the rest is pasture. How many A. of pasture 
 are there ? 
 
 32. If a road is 4 rd. wide, how many mi. of it will make 
 10 A. ? 
 
 33. A map is drawn to a scale of half an inch to a mile. How 
 many acres are represented by a square inch on the map ? 
 
 34. After drawing off 124 gal. of water from a cistern, ^j of 
 the water still remained. How many gal. did the cistern at 
 first contain ? How many gal. were left in it ? 
 
 35. Some Atlantic liners consume 200 T. of coal per day. 
 They average 8 da. out and 8 back. For fear of accidents they 
 carry a supply for 4 da. extra. How many cu. yd. of the hold 
 of such a steamer will be occupied with coal for her round trip 
 if each ton is 33 cu. f t. ? 
 
'y\- f^ 
 
 COMPOUND QUANTITIES 
 
 193 
 
 Vi 
 
 36. A load of wood 10 ft. long, 3 ft. 8 in. wide, and 3 ft. high 
 was sold for $ 3. 
 
 (a) What was the price per cd. ? 
 
 (6) At $ 4 per cd., what would the load be worth ? 
 
 37. Find the value of a pile of tan bark 180 ft. long, 48 ft. 
 wide, and 16 ft. high at f 2.25 per cd. 
 
 38. Find the amount of the following bill, retaining all 
 
 fractions : 
 
 3f lb. tea @ 80^ ; 
 
 300 lb. sugar @ 4}^ ; 
 
 45 yd. print @ llJjZf ; 
 
 2J gal. syrup @ 65^ ; 
 
 12^ yd. towelling @ 12i^ ; 
 
 f doz. knives and forks @ $ 2.50 ; 
 
 27 lb. cheese @ 15^ ; 
 
 1 lb. 10 oz. lemon peel @ 32 ^ per lb. 
 
 39. A train 80 yd. long crossed a bridge 140 yd. long in 221- 
 sec. Find the average speed of the train while crossing. 
 
 40. Find the length of a bridge which a train 100 yd. long 
 required 1 min. 15 sec. to cross, running at a speed of 15 mi. per 
 hour. 
 
 41. Find the value of 2 J cwt. + 37f lb. + lOf oz. 
 
 42. Find the weight of a bar 3 yd. 1 ft. 9 in. long, of which a 
 yard weighs 15 lb. 
 
 43. Find the cost price of lead per cwt., if the sale of 48 cwt. 
 for f 218.70 gives a profit of ^ of the original price. 
 
 44. Find the expense of fencing a railway (both sides) 73 mi. 
 in length, at the rate of $ 5.50 per rd. 
 
 45. If a wheel makes 260 revolutions in passing over 1 mi. 
 520 yd. 2 ft., what is its circumference ? 
 
 46. A block of stone is 4 ft. long, 2 ft. 6 in. broad, and 1 ft. 
 3 in. thick; it weighs 27 cwt. Find the weight of 50 cu. in. 
 of the stone. 
 
 ■!v 
 
 :i, f, 
 
 ' u\ 
 
 p. -I 
 
 o 
 
 
194 
 
 ARITHMETIC 
 
 47. A rectangular lot 45 ft. front by 99 ft. deep was sold for 
 $ 3150. What was the price per ft. frontage, and what the price 
 per A. at the rate of the selling price of the lot ? 
 
 48. If I buy 147 gal. of molasses at 19^ a gal., and use 33 gal. 
 of it, at how much must I sell the remainder per gal. so as to 
 receive as much as the whole cost ? 
 
 49. When 1 oz. of gold costs $ 19.45, what is the cost of 
 .04 lb. ? 
 
 50. A bushel of wheat weighs 60 lb. and a barrel of flour 
 weighs 196 lb. If 3 lb. of wheat make 2 lb. of flour, how many 
 bbl. of flour can be made from 343 bu. of wheat ? 
 
 51. A grocer receives f 9.60 for a bill of goods weighed on 
 scales that gave only 15\ oz. to the pound. How many cents' 
 worth did he cheat his customer ? 
 
 52. If a cow gives 12 qt. 1 pt. of milk every day and 1 lb. 8 oz. 
 of butter can be made from 25 qt. of milk, how many pounds of 
 butter can be made in one week from the milk of 16 cows ? 
 
 53. A man can run 100 yd. in 10 sec. How many mi. will a 
 steamboat go in 5 2 da. at the same rate ? 
 
 54. How many mi. must be travelled by a team in plough- 
 ing lengthwise a piece of land 60 rd. long and 40 rd. wide, if 
 each furrow is 10 in. wide ? 
 
 55. A farmer exchanges 3J T. of wheat at 642^ per bu. for 
 coal at $ 6.75 per T. How much coal does he get ? 
 
 56. How much wheat is necessary to sow a field containing 4| 
 A., if f oz. is sown on every sq. yd. ? 
 
 57. Two clocks point to 2 at the same instant; one loses 3h 
 sec. and the other gains 4 sec. in 12 hr. When will one be half 
 an hour before the other, and what time will each clock then 
 show ? 
 
 58. A railway company pays $24.75 per A. for a portion of 
 road 100 mi. long and 94 J ft. wide. Find the whole amount 
 paid. 
 
COMPOUND QUANTITIES 
 
 195 
 
 59. A man mowing grass walks at the rate of .35 mi. an 
 hour, and in 70 min. mows a grass plot of 1056 sq. yd. How 
 broad does he mow ? 
 
 60. Find the expense of sodding a plot of ground, which is 40 
 yd. long and 100 ft. wide, with sods each a yd. in length and a 
 ft. in breadth ; the sods when laid costing 75^ per hundred. 
 
 61. A cd. of wood and 100 bu. of grain fill equal spaces. A 
 cubic bin whose edge is 12 ft. contains 45,900 lb. of grain. Find 
 the weight of 1 bu. of this grain. 
 
 62. Make out the following bill neatly and accurately. John 
 Smith, a merchant of Chicago, sold to William Jones, on June 
 15, 1895 : 
 
 5 lb. 8 oz. of butter 
 2 lb. 10 oz. of tea 
 4 doz. lemons 
 8 lb. coffee 
 1 bu. 3 pk. chestnuts 
 11 doz. penholders 
 
 @ 160 per lb. ; 
 
 @ 3P an oz. ; 
 
 @ 4^ for 3 lemons. ; 
 
 @ 37^^ per lb.; 
 
 @ 10^ per qt. ; 
 
 @ 1^^ each. 
 
 63. Find cost of digging a cellar 48 ft. long, 30 ft. wide, and 
 6 ft. deep, at 20^ per cu. yd., and flooring it with Portland cement 
 at 10 ^ per sq. yd. 
 
 64. A piece of land is surrounded by a stone wall 8 ft. high 
 and 2 ft. thick ; the land inside the wall is 100 ft. long and 50 ft. 
 wide. How many cu. ft. of stone does the wall contain ? 
 
 65. How many bushels of potatoes can be sold out of a garden 
 in which there are 160 rows of potatoes, in each row 240 hills, 
 and on an average 10 potatoes in each hill, if 6 potatoes make 
 3 pt. ? 
 
 
 ''ill 
 
 IM 
 
 'ill 
 
 
 I H 
 
196 
 
 ARITHMETIC 
 
 1 1 
 If 
 
 66. Farmer B. sold to a merchant the following articles to 
 apply on an overdue account of $ 54.45 : 
 
 1680 lb. of hay 
 
 3 J cd. of wood 
 
 4 bbl. of apples 
 350 lb. of flour 
 
 30 lb. 10 oz. butter 
 
 @ 1 15 per T. ; 
 @ $ 4.80 per cd. ; 
 @ $ 2.75 per bbl. ; 
 @ 12.50 per cwt.; 
 
 @16ji^per lb. 
 
 Make out the account neatly, showing the balance and to whom 
 due. 
 
CHAPTER XIII 
 
 
 PERCENTAGE 
 
 197. The expressions $ y§^ and $ .05 denote that the quan- 
 tity !jj5 1 is conceived as made up of 100 equal parts or units, 
 and that 5 of these parts or units have been taken to measure 
 the quantity denoted by j^ or .05. 
 
 The phrase per cent means hundredths. Thus the fraction 
 j^Q and the decimal .05 are also written 5 per cent or 5%. 
 Hence j^q, .05, and 5% of any quantity are equal. 
 
 198. (1) Express J as hundredths and also as per cent. 
 
 | = .33ior ^ = 331%. 
 
 (2) A horse dealer who had 600 horses sold 480 of them. 
 What per cent of his horses did he sell ? 
 
 We are here given the measured quantity or 480 horses to compare with 
 the quantity 600 horses, and we are required to find the per cent which is the 
 number. 
 
 The number sold = f f g or f or 80 % of the whole number. 
 
 199. The term per cent is used constantly in busine iS. 
 The merchant gains 20%, meaning that he gains $20 on 
 every $100 he has invested in goods. The insurance com- 
 pany charges 2% for insuring furniture, meaning that 1 2 is 
 charged on every $ 100 worth of furniture insured. A man 
 borrows money at 5%, meaning that he is to pay $5 interest 
 on every f 100 borrowed. The commission merchant charges 
 
 197 
 
 
198 
 
 ARITHMETIC 
 
 2% of the buying or selling price. The broker charges |% for 
 buying stocks, and so on. 
 
 Exercise 129 
 
 Express as hundredths and also as per cent : 
 
 !• h h h h h h h h iV> A> A) TUf ^\» ytt* ih ihf ish ts' 
 
 9 1 2 1 5 1 3 5 7 JL 1 1 15 17 14 41 
 -*• 7> ^> ^J ^» ¥» f' 'Sf t' T2> To' T¥J T^> TSr? ^T» t?' 
 
 3. Out of a class of 25 pupils 5 are absent. What part of the 
 class is absent ? How many hundredths ? What per cent of the 
 class ? 
 
 « 
 
 4. A merchant paid $3 for hats which he sold for f 4. What 
 fraction of the cost price did he gain ? How many hundredths ? 
 What per cent of the cost ? 
 
 5. A person bought a house for f 5000 and afterward sold it 
 for f 4000. The loss was what fraction of the cost ? How many 
 hundredths ? What per cent ? 
 
 6. A fruit dealer bought strawberries for f 1.75 a crate and sold 
 them for $ 2.25 a crate. What per cent did he gain ? 
 
 7. A man bought a horse for $ 234 and afterwards sold it for 
 $ 273. What per cent of the cost did he gain ? 
 
 8. The population of a town of 32,000 inhabitants increases 
 1120 in one year. What is the per cent of increase ? 
 
 9. How do you find the gain per cent when you are given the 
 cost price and the selling price of an article ? 
 
 200. Express 
 terms. 
 
 25% and 37 J % as fractions in their lowest 
 
 25o/„ = ^^5^=i. 
 *'" 100 200 8 
 
PERCENTAGE 
 
 199 
 
 Exercise 130 
 
 What fractions in their lowest terms are equivalent to the 
 following : 
 
 1. 1%,4%,5%,10%,20%,25%,30%,40%,80%,90%,100%? 
 
 2. 60%, 75%, 35%, 24%, 70%, 10%, 85%? 
 
 3. Vl\%, 'm,<^c, 021%, 87i%? 
 
 4. 6i%, 8i%,0S%, 10|%,83i%? 
 
 5. lli%, 14f%,9JT%? 
 
 6. lOf %, 5A%, lf|%? 
 
 7. 100%, 120%, 125%, 175%, 250%, 325%? 
 
 8. A horse which cost $ 120 was sold at a gain of 25%. The 
 gain is equal to what part of the cost ? How much was gained ? 
 What was the selling price ? 
 
 9. Cloth which cost GOj^ per yd. was sold at a loss of 16|%. 
 The loss was what fraction of the cost ? What was the loss on 
 each yd. ? What was the selling price per yd. ? 
 
 10. An article costing % 4.20 was sold at a gain of 8|%. Find 
 the gain. Find the selling price. 
 
 11. Tea is bought for 84^ per lb. and sold at an advance of 
 14^%. What was the selling price of each lb. ? 
 
 12. A drover sold 400 sheep at a gain of 10%. He gained the 
 cost price of how many sheep ? 
 
 13. How do you find the gain on an article when you are given 
 the cost price and the gain per cent ? How do you find the selling 
 price ? 
 
 
 \ I'll 
 % "ur 
 
 
 i;i 
 
 *•=; 
 
 «: 
 
 201. The following results should be memorized so that 
 the fractions or the per cent can be given rapidly in any 
 order : 
 
 ^^'1 
 
200 
 
 ARITHMETIC 
 
 
 20% = i 
 
 33i% = i 
 
 37i% = f 
 
 100% = 1 
 
 40% = J 
 
 66i% = | 
 
 50% = i 
 
 6i% = iV 
 
 60% = f 
 
 100% = 1 
 
 62J%=| 
 
 6i% = iV 
 
 80% = i 
 
 12|%=i 
 
 75% = 1 
 
 8i% = tV 
 
 100% = 1 
 
 25% = i 
 
 87i% = J 
 
 16§% = J 
 
 Note. —The expression 20% = ^ signifies that 20% of a quantity = J of 
 it. Thus, 20 % of 80 A. = ^ of 80 A. 
 
 Exercise 131 
 
 Read the following decimals as per cents : 
 
 1. .25, .16|, .40, .75, .03^, 1.20. 
 
 2. .15, .37^, .45J, .05^, .00^, 2.40. 
 
 202. What is the quantity which is measured by the unit 
 $ 840 and the number 16-|% ? 
 
 The quantity = 16 1% of $840 = ^ of $840 = $140. 
 
 Exercise 132 
 
 1. What is the quantity which is measured by the unit ^720 
 and the number 8J%? 
 
 2. What is the quantity which is measured by the unit $465 
 and the number 20%? 
 
 3. What is the gain that is measured by the cost $893 and 
 the number 6%? 
 
 4. What is the loss that is measured by the cost $1268 and 
 the ratio 32%? 
 
 5. What is the quantity that is measured by the sum of the 
 unit $ 468.25 and 8% of it ? 
 
 6. What is the quantity that is measured by the difference 
 between the unit $ 4397.50 and 6% of it ? 
 
PEKCENTAGE 
 
 201 
 
 7. If the gain on selling is measured by the number 7% and 
 the cost .f»4r)(), find the gain. 
 
 8. Find the selling price of an article which cost $600, the 
 loss on selling being measured by the number 5% and the cost 
 price. 
 
 203. A speculator bought a house for $ 2349 and sold it 
 at a gain of 17%. Find the selling price. 
 
 In this question the selling price is the sum of the cost, which is known, 
 and the gain, which is unknown. The gain is measured by the number 17 % 
 or .17, and the cost price $2349. 
 
 $2349 cost 
 
 ^n number 
 
 16443 
 2349 
 $399.33 gain 
 
 The gain = 17 % of $ 2349 = $ 399.33. 
 .-. the selling price = $2349 + $399.33 = $2748.33. 
 
 Exercise 133 
 
 1. Write as decimals: 17%, 13%, 37%, 23^%, 146%, 346%, 
 6%, 8%, 81%, 11%, 1%, i%. 
 
 2. Find 34% of $893. 
 
 3. Find 27% of 6594 bu. of wheat. 
 
 4. If 39% of a cargo of flour, consisting of 8492 bbl., was 
 damaged, how many bbl. were damaged ? 
 
 5. A farmer who sold his crop of Avheat in 1892 for $ 967.20, 
 received 13% less the next year. How much less did he receive 
 for his crop in 1893 than in 1892 ? 
 
 6. A grain dealer invested $6459 in wheat, and 23% of that 
 amount in oats. How much did he invest in oats ? 
 
 7. What does a bill for $ 1896 become after a reduction of 3%? 
 
 
202 
 
 ARITHMETIC 
 
 8. What is the selling price of an article costing $ 18, and 
 sold at a loss of 9%? 
 
 9. What is the selling price of an article costing ^7, and 
 sold at a gain of 7%? 
 
 204. Express |% as a decimal and also as a common frac- 
 tion in its lowest terms. 
 
 441 
 
 4 0/ — _5_ — /* — * - 
 
 100 600 125 
 
 Exercise 134 
 
 Express as decimals and also as common fractions in their 
 lowest terms: 
 
 2- f %. Wo, i%, i%> i%, i%, i%, t\%- 
 
 3. tV%. j\%, -h% \%% -hlo, \\% A%> 11%. 
 
 4. What part of 1% is i%? f %? |%? yV%'^ 
 
 5. If ^% is charged for sending money from Chicago to New 
 York, what is charged for sending % 1200 ? 
 
 6. If 1% is charged for sending money to St. Louis, find how 
 much is charged when f 1632 is sent. 
 
 7. If 1^% is charged as commission for buying stock, what is 
 the commission on buying % 2400 stock ? 
 
 8. If 1% is charged for selling stock, find the commission 
 charged for selling % 1600 stock. 
 
 9. 6% per annum is what per cent for 1 month ? |% a month 
 is what per cent per annum ? 
 
 10. What is the cost of insuring 550 bbl. of flour, worth 
 ^4 per bbl., the cost of insurance being ^% of the value of the 
 flour? 
 
PERCENTAGE 
 
 203 
 
 205. (1) Express 102| % as a decimal. 
 
 1021% = ^= 1.021 = 1.025. 
 
 (2) I send my agent f 5100, which is 102% of the money 
 which he invested for me in cotton. What does my agent 
 pay for cotton? 
 
 102% or 1.02 of the cost of the cotton = $5100. 
 .-. the cost of the cotton = $5100 -^ 1.02 = $5000. 
 
 Exercise 135 
 
 1. Express as decimals: 103%, 104%, 101%, 102%. 
 
 2. Express as decimals: 97%, 90%, 99%, 98%. 
 
 3. Express as decimals : 1031%, 104}%, 102-J%, I01i%. 
 
 4. Express as decimals: 97^%, 961%, 99^%, 98f%. 
 
 5. A real estate broker in St. Louis received f 5047, which was 
 103% of the sum he was to invest in real estate. What sum did 
 he invest in real estate ? 
 
 6. My agent sent me $3395, which was 97% of the selling 
 price of some railway stock. What did the stock sell for ? 
 
 7. My agent on selling a quantity of wheat sent me 981% of 
 the proceeds. If I received $ 3546, what did the wheat sell for ? 
 
 8. I sent my agent $4545.30, which was 104}% of the sum 
 he was to invest in buying silk. What did he pay for the silk ? 
 
 206. (1) A house is sold for 116,400, and 25% of the 
 purchase money is paid down, the balance to remain on mort- 
 gage. How much remains on mortgage ? 
 
 In this problem we are given the measured whole, i.e. the selling price, 
 and the number or 25 % of it. We are required to find the balance which is 
 the difference between the selling price and the sum paid. 
 
 The sum paid = 25% or ^ of $ 10,400 = $4100. 
 
 .-. the balance = $16,400 - $4100 = $ 12,300. 
 
 I 
 
 .if 
 
204 
 
 ARITHMETIC 
 
 m 
 
 fe 
 
 (2) On Jan. 10 a merchant buys goods invoiced at f 876.40, 
 on the following terms: 4 months, or less 6% if paid within 
 10 days. What sum will pay the debt on Jan. 15? 
 
 Since Jan. 15 is less than 10 days after Jan. 10, the sum due will be 6% 
 less than 1 876.40. 
 
 The discount = 6 % or .06 of $ 876.40 = $ 52. 584. 
 .-. the sum due = ^ 876.40 - $52.58 = $823.82. 
 
 >i> 
 
 I J' 
 
 Exercise 136 
 
 1. A maltster malts 7500 bu. of barley, which in the process 
 increases 12'] %. How many bu. of malt has he ? 
 
 2. Certain books are bought at f 1.75 each. At what must 
 they be sold to gain 12% ? 
 
 3. A merchant asked 30% advance on goods which cost $120, 
 but finally took 25% less than the price asked. What did he sell 
 them for ? 
 
 4. A merchant bought apples at 60^ per bu., and sold them 
 at a gain of 25%. Find the selling price per bu. How many 
 bu. did he sell if he received all together f 37.50 ? 
 
 5. Bought f 64 worth of apples at 80j^ per bu., part of which 
 being damaged and rendered worthless, I sold the remainder 
 at an advance of 50%, receiving $76.80. How many bu. were 
 damaged ? 
 
 6. If 10% of an army of 23,400 men were slain in battle, and 
 5 per cent of the remainder were mortally wounded, find the 
 sum of the killed and mortally wounded. 
 
 7. The population of a town of 64,000 inhabitants increases at 
 the rate of 2^% in each year. Find its population 1, 2, and 
 3 years hence. 
 
 8. The population of a city is a million ; it increases 1^% for 
 3 years successively. Find the population at the end of 3 years. 
 
PERCENTAGE 
 
 205 
 
 9. A lawyer collected $287.50, and charged 5% for his ser- 
 vices. How much did he retain and how much did he pay 
 over? What per cent is the amount paid over of the amount 
 collected ? 
 
 10. The cost price of a book is .^1.60, the expense of sale 5% 
 upon the cost price, and the profit 25% upon the whole outlay. 
 Find the selling price of the book. 
 
 11. The cattle on a certain stock farm increase at the rate of 
 18f % per annum. If there are 4096 cattle in 1894, how many 
 will there be in 1896 ? 
 
 12. A man bought a store and contents for $ 4720 ; he sold the 
 same for 12^% less than he gave, and then lost 15% of the sell- 
 ing price in bad debts. Find his entire loss. 
 
 13. A person having bought goods for f 40 sells half of them 
 at a gain of 5%. For how much must he sell the remainder so 
 as to gain 20% on the whole ? 
 
 14. A grocer mixes two kinds of tea which cost him 38^ and 
 44^ per lb. respectively. What must be the selling price of the 
 mixture in order that he may gain 15 % on his outlay ? 
 
 15. A grain dealer expended $ 2150 in the purchase of wheat, 
 one-half as much again in the purchase of barley, and twice as 
 much in the purchase of corn ; he sold the wheat at a profit of 
 6%, the barley at a loss of 5%, and the corn at a gain of 2%. 
 Find his gain on the whole transaction. 
 
 16. A person gave $ 150 for one horse and $ 225 for another. 
 He sold the first horse at a gain of 20%, and the second at a loss 
 of 20%. Find the selling price of each horse and the gain or loss 
 on the whole transaction. 
 
 17. A sells goods to B which cost him $465, at a gain of 6%, 
 B sells them to C at a loss of 3%, and C sells them to D, gaining 
 10%. What did D give for the goods ? 
 
 m 
 
 m 
 
 if,. 
 
 id 
 
 
W^OM 
 
 206 
 
 AKTTHMETIC 
 
 18. A man having bought a lot of goods for ^ 450 sells J at 
 a loss of 5%, 4 at a gain of 7%, and the remainder at a gain of 
 2%. Find the total gain. 
 
 19. A merchant began business with a capital of $ 30,000. He 
 gained 16|% the first year, which he added to his capital, and 
 12-|^% the second year, which he added to his capital. In the 
 third year he lost 20%. Find his capital at the end of the third 
 year. 
 
 20. Sugar being composed of 49.856% of oxygen, 43.265% of 
 carbon, and the remainder hydrogen, find how many lb. of 
 each of these materials there are in 1 T. of sugar. 
 
 21. A merchant buys a bill of dry goods, April 16, amounting 
 to $6377.84, on the following terms : 4 mo., or less 5% if paid 
 within 30 da. How much would settle the account on May 16 ? 
 The amount paid May 16 is what per cent of the full amount 
 of the bill ? 
 
 22. Water is composed of 88.9% of oxygen and 11.1% of 
 hydrogen. How many lb. are there of each in 1 cu. yd. of w^ater ? 
 (A cu. ft. of water weighs 1000 oz.) 
 
 207. If a gain of $ 24 is measured by a certain number and 
 the cost price, which is $ 64, find the number as a rate per 
 cent. 
 
 The required number x $ 64 = .$ 24. 
 .*. the required number = $fi = 2.| _ 3 _ 37 1%. 
 
 Eziercise 137 
 
 1. If the quantity of land sold, which is 50 A., is measured 
 by a certain number and a unit of 200 A., find the number as a 
 rate per cent. 
 
 2. If a gain of |>45 is measured by a certain number and the 
 cost $ 225, find the number as a rate per cent. 
 
PERCENTAGE 
 
 207 
 
 3. If a loss f 36 is measured by a certain rate per cent and the 
 cost, which is $ 108, find the rate per cent. 
 
 4. If the sum of money paid for fruit, which is $ 6, is measured 
 by a certain number and the unit $ 72, find the number as a rate 
 per cent. 
 
 208. (1) A merchant sold 60 yd. of cloth from a web con- 
 taining 150 yd. What per cent of the web did he sell ? 
 
 We are here given the measured part and the measured whole, and we are 
 required to find the number expressing the ratio of the part to the whole. 
 The quantity sold = j^^^ or f or 40 % of the web. 
 
 (2) An article which cost 13.60 was sold for 14.32. Find 
 the gain per cent. 
 
 The gain = $4.32 - $ 3.60 = 72 cents. 
 .'. the gain = /g% or I or 20 % of the cost. 
 
 Exercise 138 
 
 1. The cost price of an article is $64, and the gain on selling 
 $ 16. Find the gain per cent. 
 
 2. Tea is bought at 84^ per lb., and sold at 98^. Find gain 
 per cent. 
 
 3. Out of 48 eggs, 6 were broken. What per cent of the 
 whole number was broken ? 
 
 4. The cost price of an article was $ 56, and the selling price 
 1 49. Find the loss per cent. 
 
 5. What per cent is 1 in. of 1 ft. ? 1 ft. of 1 yd. ? 1 yd. of 
 1 rd. ? 1 rd. of 1 mi. ? 
 
 6. What per cent is 1 min. of 1 hr. ? 1 hr. of 1 da. ? 1 da. 
 of 1 wk. ? 1 wk. of 1 yr. ? 
 
 7. What per cent is 1 pt. of 1 qt. ? 1 qt. of 1 gal. ? 
 
 8. What per cent of 1 pk. is 1 qt. ? Of 1 bu. is 1 pk. ? 
 
 fisri 
 
208 
 
 ARITHMETIC 
 
 9. A merchant by selling 1 lb. of butter gains the cost price 
 of 1 oz. What is his gain per cent ? 
 
 10. One lb. Troy is what per cent of 1 lb. Avoir. ? One lb. 
 Avoir, is what per cent of 1 lb. Troy ? 
 
 11. The volume of 1 gal. is what per cent of 1 cu. ft.? A 
 cu. ft. is what per cent of 1 gal. ? 
 
 12. An area containing 1 sq. yd. is increased by 4 sq. ft. Find 
 the per cent of increase. 
 
 13. If £ 1 is worth $ 4.866, what per cent of £ 1 is ^1 ? 
 
 Ii 
 
 f 
 
 Exercise 139 
 
 1. A paymaster receives $ 150,000 from the treasury, but 
 fails to account for $ 2250. What is the percentage of loss to the 
 government ? 
 
 2. A city of 17,000 inhabitants increases in a given time to 
 20,000. Find the increase per cent. 
 
 3. $ 640 increased by a certain per cent of itself equals $ 720. 
 Required the rate per cent. 
 
 4. A house worth f 3500 rents for f 400. For what per cent 
 of its value does it rent ? 
 
 • 5. If a tradesman gain $ 1.32 on an article which he sells for 
 $ 5.28, what is his gain per cent ? 
 
 6. An article which cost 84^ is sold for 93^. Find the gain 
 per cent. 
 
 7. A city gained 2467 in population in 5 years. If its popu- 
 lation was 14,802 five years ago, what was the gain per cent ? 
 
 8. In a certain year the number of graduates of a school was 
 70. Ten years later it was 213. Find the rate per cent of 
 increase. 
 
 9. A tea merchant mixes 40 lb. of tea at 45^ per lb. with 
 50 lb. at 27^ per lb., and sells the mixture at 42^ per lb. What 
 per cent profit does he make ? 
 
PERCENTAGE 
 
 209 
 
 10. Paid f 664.25 for transportation on an invoice of goods 
 amounting to $ 8866. What per cent must be added to the invoice 
 price to make a profit of 20% on the full cost ? 
 
 11. A grocer uses for a 1 lb. weight one which only weighs 
 15.75 oz. What does he gain per cent by his dishonesty ? 
 
 12. I bought 500 sheep at $ 6 a head ; their food cost me $ 1.25 
 a head ; I then sold them at $ 10 a head. Find my whole gain, 
 and also my gain per cent. 
 
 13. A man's income is derived from the proceeds of f 4550 at 
 a certain rate per cent ; and $ 5420 at 1 % more than the former 
 rate. His whole income being $ 453, find the rates. 
 
 14. If a commodity be bought for $ 16.42 a cwt. and sold for 
 18^ a lb., find the rate of profit per cent. 
 
 15. The police returns for a certain year give 1350 male 
 offenders, and 1150 female; the next year's returns show a 
 decrease of 6% in the number of male criminals, and an increase 
 of 8% in number of female. Find increase or decrease per cent 
 in whole number of criminals. 
 
 16. The area of North America is 8,000,000 sq. mi., and of the 
 Mississippi and St. Lawrence basins respectively 1,250,000 and 
 350,000 sq. mi. Find what per cent of the area of the continent 
 is drained by each of these rivers. 
 
 17. Turn to your geography and find the area of the basins of 
 the principal rivers, and then find what per cent of the area of 
 North America is drained by each river and by all of them 
 together. 
 
 18. The length of the Missouri-Mississippi River is 4200 mi., 
 and of the St. Lawrence 2000 mi. The length of the St. Law- 
 rence is what per cent of that of the Missouri-Mississippi ? The 
 area of the St. Lawrence basin is what per cent of the area of the 
 Mississippi basin ? 
 
 ■i 
 
 1 
 
 ..!lfj 
 
210 
 
 ARITHMETIC 
 
 209. If a gain of $36 is measured by the number 22|% and 
 the cost, find the cost. 
 
 The gain = 22 1 % or fy of the cost. 
 I of the cost = $36. 
 I of the cost = $ 18. 
 f of the cost = $ 162. 
 .-. the cost = $162. 
 
 Exercise 140 
 
 1. If the quantity $18 is measured by the number 6% and 
 a certain unit, find the unit. 
 
 2. If the gain $32 is measured by the number 8% and the 
 cost, find the cost. 
 
 3. If the loss $60 is measured by the ratio 12% and the cost, 
 find the cost. 
 
 4. If the selling price $3810 is measured by the number 
 127% and the cost price, find the cost price. 
 
 5. If the selling price $5640 is measured by the number 94% 
 and the cost, find the cost.. 
 
 210. A trader sold a horse at an advance of 12%, gaining 
 $18. Find the cost price of the horse. 
 
 12 % or 3j\ of the cost = $ 18. 
 ^^5 of the cost = $ 6. 
 If of the cost = $ 150. 
 .-. the cost = $150. 
 
 Exercise 141 
 
 1. A quantity of sugar was sold at an advance of 12J%. If 
 the gain was $ 17, what was the cost ? 
 
 2. A yd. of cloth was sold at a loss of 37|^%. If the loss was 
 36 ^ a yd., what was the cost ? 
 
PERCENTAGE 
 
 211 
 
 3. Forty-five per cent of a piece of cloth was sold. If 135 
 yd. were sold, how many yd. were in the piece at first ? How 
 many remain unsold? What per cent remains unsold ? 
 
 4. If 3% more be gained by selling a horse for $333 than by 
 selling him for $ 324, find his original price. 
 
 5. A man bought a horse which he sold at a loss of 8%. 
 If he had received $24 more, he would have gained 7%. What 
 did the horse cost him ? 
 
 6. A clerk pays 16% of his salary each year for board. If his 
 board costs him f 208 a year, what is his salary ? 
 
 7. A man sold a field consisting of 15 A., which was 6^% of 
 his farm. How many A. were in the farm at first ? 
 
 8. If 25% of my money is invested in bank stock and the 
 remainder in business, what per cent of my money is invested 
 in business ? My bank stock is worth what part of my business 
 capital ? If I have f 4800 invested in business, what is the value 
 of my bank stock ? 
 
 9. Twenty per cent of my money is invested in business, and 
 the remainder, which is $ 12,800, in real estate. How much have 
 I invested in business ? 
 
 10. I invested 25% of my money in business, and put 6|% of 
 the remainder in the bank. If I put $ 600 in the bank, how 
 much money did I have at fiist? 
 
 11. Twenty-eight per cent of a sum of money was invested in 
 business, and 12|^% of the remainder in real estate. If the sum 
 invested in business exceeds that invested in real estate by $ 1900, 
 find the amount of money I had at first. 
 
 H 
 
 il ;* 
 
 211. (1) A man invests 77 J % of his capital in bank stock, 
 and has $ 29,367 left. What is his capital ? 
 
 The amount left = 100% - 77 J % or 22 1% of his capital. 
 22|% or ^% of his capital = $29,367. 
 ^V of his capital = $ 8263. 
 |§ of his capital = $ 130,520. 
 .'. his capital = $ 130,520. 
 
"■' ?■ 
 
 212 
 
 ARITHMETIC 
 
 li 
 
 II 
 
 (2) If a profit of 17% is made by selling an article at an 
 advance of $ 24.50, what would have been the selling price if 
 the loss had been 8% ? 
 
 17% of the cost - 
 
 1 % of the cost = 
 
 100% of the cost = 
 
 The loss = 8% of the cost = 
 
 .-. the second selling price = 
 
 24.50. 
 
 1.4412. 
 
 144.12. 
 
 11.53. 
 
 144.12 - 1 11. Sr. = 11132.59. 
 
 Exercise 142 
 
 1. A man invests 42% of his capital in real estate, and has 
 $ 53,070 left. What is his capital ? 
 
 2. A bankrupt's assets are f 23,625, and he pays 40% of his 
 liabilities. What are his liabilities ? 
 
 3. A merchant loses G|% of the cost price by selling an 
 article at a loss of $ 27.30. Find the cost price, and also at what 
 he must sell it to gain 7.}%. 
 
 4. By selling a house at a loss of $ 150, a real estate dealer 
 loses 6f % of the cost. Find the cost and also the gain per cent 
 if it had been sold for f 2625. 
 
 5. I sold a lot at a gain of 8J%, thereby gaining $113. 
 What should I have sold it for to gain 9%? 
 
 6. Coals are 20% cheaper this year than last. If the price 
 were to rise f 1 per T., they would still be 50^ per T. cheaper 
 than last year. Find the price last year. 
 
 7. A person asked for a lot of land 40% more than it cost him, 
 but finally reduced his price 15%, gaining on the whole f 1000. 
 For how much did he sell the land ? 
 
 8. A merchant sold | of a quantity of cloth at a gain of 20%, 
 and the remainder at cost. His gain was what per cent of the 
 cost ? If he gained f 7.29, what was the cost of the goods ? 
 
 9. A merchant sold | of a quantity of tea at a gain of 12%, 
 and the remainder at a gain of 9%, gaining all together $2.75. 
 Find the cost of the tea. 
 
PERCENTAGE 
 
 213 
 
 10. A speculator gained 20% on J of his investment, and lost 
 24% on the remainder. All together he made $270. Find the 
 amount of his investment. 
 
 11. A business firm's resources consist of notes, merchandise, 
 personal accounts, etc., to the amount of $ 9117.61, and a balance, 
 which is 44% of their entire capital on deposit in bank. How 
 much is on deposit ? 
 
 12. Ten per cent of an army were slain on the field of battle, 
 and 5 per cent of the remainder were mortally wounded. The 
 difference between the killed and mortally wounded was 1100. 
 How many men went into battle? 
 
 212. (1) A horse was sold for 1117, which was 8^% more 
 than it cost. Find the cost price. 
 
 The gain = Sl% or y'j of the cost. 
 
 The selling price = f| of the cost. 
 
 II of the cost = i 117. 
 1 
 
 T2 
 
 J of the cost = $ 9. 
 \l of the cost = ^ 108, 
 .-. the cost = $ 108. 
 
 (2) A horse was sold for 1154, which was 8J% less than 
 it cost. Find the cost price. 
 
 The loss = 81%, or j\ of the cost. 
 The selling price = || of the cost, 
 fi- of the cost = $ 154. 
 
 1 of the cost = .f 14. 
 
 T2 
 
 f I of the cost = $ 168. 
 the cost - $ 168. 
 
 III! 
 
 Exercise 143 
 
 1. A house and lot were sold for f 3600, which was 20% more 
 than they cost. Find the cost price. 
 
 2. A house and lot were sold for $4200, which was 25% less 
 than they cost. Find the cost price. 
 
214 
 
 AIlITIIMETrC 
 
 1, 
 
 3. A speculator gained 7% by selling wheat for $2140. Find 
 the cost price. 
 
 4. Eggs are sold at the rate of 15^ per doz., a profit of 25% 
 being made. What is the cost price per doz. ? 
 
 5. I sold a book for 42^, gaining 10if%. Find the cost price. 
 How much would be gained by selling at a gain of 25%? What 
 would then be the selling price ? 
 
 6. By selling hats at 00^ each, a merchant gains 3.'ijr%. 
 Find the cost price. What would have been the actual loss, and 
 what the loss per cent, if they had been sold at 3G ^ each ? 
 
 7. 1 sold a lot of land for $ GOO, thereby gaining 20%. Find 
 the cost price. 
 
 8. I sold a lot of land for $G00, thereby losing 20%. Find 
 the cost price. 
 
 9. What is the cost of both lots in questions 7 and 8 ? What 
 is their selling price ? How nuich is the cost of both greater than 
 their selling price ? 
 
 10. A dealer sold two bicycles for $ 45 each, losing 25% on one 
 and gaining 25% on the other. How much did he lose on the 
 whole transaction ? 
 
 213. (1) If a debt, after a reduction of 3%, becomes 
 $1008.80, what would it become after a reduction of 4% ? 
 
 After a reduction of 3 %, the amount owed = 07 % of the original debt, and 
 after a reduction of 4 % it becomes 90 % of the original debt. 
 
 97 % of the debt = $1008.80. 
 
 1 % of the debt = $ 10.40. 
 90 % of the debt = !$ 998.40. 
 .'. after a reduction of 4 % the debt = $998.40. 
 
 (2) The population of a city increases 2% yearly. It now 
 has 132,651 inhabitants. How many had it 1, 2, and 3 years 
 ago? 
 
PERCENTAGE 
 
 215 
 
 The population now = 102% or 1.02 of that 1 year ago. 
 
 1.02 of the popuhition I year ago = 132, (5ol. 
 
 .'. the population 1 year ago = 132,051 -r 1.02 
 .'. the population 2 years ago = 130,050 ■- 1.02 
 . •. the population 3 years ago = 127,500 -i- 1.02 
 
 Prove these answers correct. 
 
 130,050. 
 127,500. 
 125,000. 
 
 Exercise 144 
 
 1. A horse was sold for $058, which was 1C)|% more than its 
 cost. How much did it cost ? 
 
 2. A speculator gained 3% by celling wheat for $6437.50. 
 Find the cost price. 
 
 3. A merchant, after a business of 5 years, found his capital 
 increased to $28,000, showing a gain of 60% on his original capi- 
 tal. Find that capital. 
 
 4. Eggs are sold at the rate of 5 for 6^, a profit of 20% 
 being made. Find the price at which they are bought. 
 
 5. In 1896 a city has a population of 28,000 inhabitants. If 
 its population increased 17}|% in the two years previous, what 
 was it in 1894 ? If its population decreased 17i|% in the two 
 years previous, what was it in 1894 ? 
 
 6. By selling an article for f 2.64 a merchant loses 12%. 
 What was the cost, and for what must he sell it to gain 16|%? 
 
 7. A merchant sells tea at 75^ per lb., thereby losing 5%. 
 What was the cost, and at what price per lb. must it be sold to 
 gain 41 %? 
 
 8. Flour is sold for $6 per bbl., at a loss of 17%. Find 
 what selling price would give 16%. 
 
 9. If, by selling an article for f 25.30, 8% be lost, what per 
 cent is gained or lost if it be sold for $ 38 ? 
 
 10. I sold goods at 1 21.60 per cwt., thereby gaining 14f%. 
 Find the cost per lb. 
 
 I I 
 ; I 
 
 0M 
 W 
 
 
 ti '!« 
 
216 
 
 ARITHMETIC 
 
 11. A farmer sold his crop of wheat in 1871 for 8% more than 
 he obtained for his crop of the preceding year; he received for 
 both crops $2(500; how much did he get for his crop of 1870? 
 
 12. I sold two houses, receiving $ 2400 for each. On the first 
 I gained 25%, and on the second lost 25%. Find the actual loss 
 and also the loss per cent. 
 
 13. By selling a lot of land for $600, thereby gaining 20% ; a 
 second for $ 600, losing 20% ; and a third at an advance of 20% 
 on cost ; I find I have made $ 75 on the whole transaction. Find 
 the cost of each lot. 
 
 14. A ship depreciates in value each year at the rate of 10% 
 of its value at the beginning of the year, and its value at the end 
 of three years is $ 14,580. What was its original value ? 
 
 15. The cattle on a stock farm increase at the rate of 18|% 
 per annum. In 1889 there were 6859 head of cattle on the farm ; 
 how many were there in 1888 ? In 1887 ? In 1886 ? 
 
 16. An importer pays for freight and duty 10% on cost price, 
 and sells to the retailer at a profit of 20% ; the retailer sells to 
 the consumer at a profit of 25%. Find the amount paid by the 
 consumer for goods which cost the importer $ 7500. 
 
 I 
 
 I 
 
 Profit and Loss 
 
 214. The Profit is the amount by which the selling price 
 exceeds the buying price. 
 
 The rate of profit is usually expressed as a certain per cent 
 of the cost price. 
 
 The Loss is the amount by which the selling price falls 
 short of the cost price. 
 
 The rate of loss is usually expressed as a certain per cent 
 of the cost price. 
 
 «ii 
 
1 
 
 PERCENTAGE 
 
 217 
 
 215. (1) At a forced sale a bankrupt's house was sold for 
 f 8000, which was 20% less than its real value. If the house 
 had been sold for ^12,000, what per cent of its real value 
 would it have brought ? 
 
 80 % of the value of the house = $ 8000. 
 1 % of the vakie of the house = 100. 
 100 % of the vahie of the house = 10,000. 
 .'. the second selling price = ] o^^^ of the value 
 
 = 120 % of tlie value. 
 
 (2) The manufacturer of an article makes a profit of 25%, 
 the wholesale dealer makes a profit of 20%, and the retail 
 dealer makes a profit of 30%. What is the cost to the manu- 
 facturer of an article that retails at $15.60? 
 
 Let the cost to the manufacturer 
 
 The selling price of the manufacturer 
 
 The gain of the wholesale dealer 
 
 The selling price of the wholesale dealer 
 
 The gain of the retail dealer 
 
 The selling price of the retail dealer 
 
 195 units of money = .* 
 1 unit of money = 
 
 100 units of money = 
 .•. the prime cost = 
 
 = 100 units of money. 
 = 125 units of money. 
 = 25 units of money. 
 = 150 units of money. 
 = 45 units of money. 
 = 195 units of money. 
 
 115.00. 
 
 .08. 
 
 8.00. 
 
 8.00. 
 
 General Statement of Solution 
 
 (3) Represent the cost to the manufacturer by 100 units 
 of money, and then find the number of units representing 
 respectively the selling prices of the manufacturer, the whole- 
 sale dealer, and the retail dealer. Put the number of units 
 of money which represent the retail price equal to $15.60 
 and find the value of 100 units of money, which is the cost 
 of manufacturing. 
 
 tif 
 
 '! -.' I 
 
 '! ,; 
 .it 
 
 ,'< i' 
 
 •llM. 
 
 
 
 n 
 
218 
 
 ARITHMETIC 
 
 Question in Proof 
 
 (4) The manufacturer of an article makes a profit of 25%, 
 the wholesale dealer a profit of 20%, and the retail dealer a 
 profit of 30%. What is the retail price of an article which 
 cost the manufacturer $S? 
 
 Proof 
 
 The manufacturer's gain = 25 % of $ 8 = $ 2. 
 
 The manufacturer's selling price = $ 10. 
 
 The wholesale dealer's gain = 20 % of $ 10 = $ 2. 
 
 The wholesale dealer's selling price = $ 12. 
 
 The retail dealer's gain = 30 % of $ 12 = $ 3.60. 
 
 The retail dealer's selling price = ^ 15.60. 
 
 .-. $ 8 is the correct answer to the previous question. 
 
 Making Questions 
 
 (6) Make a question in which you are given the selling 
 price and the gain per cent, and are required to fiind the cost 
 price. 
 
 Making 
 
 Let the cost of a house = $ 6250. 
 
 Let the gain per cent on selling = 37^ %. 
 
 The gain = 37 1 % of $ 6250 = $ 2343.75. 
 
 The selling price = $6250 + $2343.75 = $8593.75. 
 
 Problem 
 
 A house was sold at 37|% above cost. If the selling pnce 
 was $8593.75, find the cost price. 
 
 Other questions may also be written down from the same 
 making, thus : 
 
 A house which cost $6250 was sold at a gain of 37J%. 
 Find the selling price. 
 
 A house which cost $6250 was sold for $8593.75. Find 
 the gain per cent. 
 
PERCENTAGE 
 
 219 
 
 A house which cost $6250 was sold at a gain of 82343.75. 
 Find the gain per cent. 
 
 In the following exercise state in general terms how to 
 solve each question. Prove some of your answers correct, 
 framing the question in proof. Make questions similar to 
 problems in the exercise. 
 
 E:sercise 145 
 
 1. A lot of dry goods was sold at an advance of 18%. If the 
 gain was f 436.50, what was the cost ? 
 
 2. I made a mixture of wine consisting of 1 gal. at 50^, 3 at 
 90^, 4 at $ 1.20, and 12 at 40^. I sell the mixture at $ 1.60 a gal. 
 Find my gain per cent. 
 
 3. A merchant's price is 25% above cost. If he allow a customer 
 a discount of 12% on his bill, what per cent x^rolit does he make? 
 
 4. If cloth when sold at a loss of 25% brings ^5 a yd., what 
 would be the gain or loss per cent if sold at $ 6.40 a yd. ? 
 
 5. Eggs are bought at 27^ a doz., and sold at the rate of 8 for 
 25^. Find the rate of profit. 
 
 6. A merchant sells goods to a customer at a profit of 60%, 
 but the buyer becomes bankrupt and pays only 70 cents on the 
 dollar ; what per cent does the merchant gain or lose on the sale ? 
 
 7. A man bought a horse which he sold again at a loss of 
 10%. If he had received $ 45 more for him he would have gained 
 12 J%. Find the cost of the horse. 
 
 8. A bookseller sold a book at 17% below cost, but had he 
 charged 50 cents more for it, he would have gained 7%. Find 
 the cost of the book to the bookseller, and the price at which he 
 sold it. 
 
 9. A tradesman bought goods foi- $ 1200 and sold i of them at 
 a loss of 10%. For how much must he sell the remainder to gain 
 20% on the whole? 
 
220 
 
 ARITHMETIC 
 
 
 10. A man bought a house and lot for $ 4750. After spending 
 $ 1143 on repairs and improvements, and paying f 128 for taxes 
 and other expenses, he sold tha property for $ 6800. What rate 
 per cent of profit did his investment yield him ? 
 
 11. By selling cloth at f 1.20 per yd., a tradesman lost 6^% 
 on his outlay. At what price must he sell it to gain 12^%? 
 
 12. If a manufacturer sells an article of which the first cost is 
 f 400, to a wholesale dealer at 10% profit, the wholesale dealer 
 to the retailer at 15% profit, and the retailer to the consumer at 
 30% profit, what sum is paid by the consumer as profits in addi- 
 tion to the first cost of the article ? 
 
 13. A grocer sold, at 51^ per lb., a portion of a stock of tea, 
 incurring a loss of 15% and a total loss of $18 on the quantity 
 sold. How many lb. did he sell ? 
 
 14. A merchant marks his goods so that he may allow a dis- 
 count of 5%, and still make a profit of 15%. Find the marked 
 price of broadcloth that cost him $ 3.80 a yd. 
 
 15. A drover bought 400 sheep at a certain price per head. 
 He sold I of them at a gain of 20%, y^^ of them at a gain of 15%, 
 
 •and the remainder at a loss of 10%, gaining on the whole $217. 
 How much did he pay for the 400 sheep ? 
 
 16. A grain dealer expended a certain sum of money in the 
 purchase of wheat, half as much again in the purchase of barley, 
 and twice as much in the purchase of oats ; he sold the wheat at 
 a profit of 5%, the barley at a profit of 8%, and the oats at a 
 profit of 10%, receiving all together $9740. Find the sum laid 
 out in each grain. 
 
 17. A person sold two horses at $ 160 each, losing 20% on one 
 and gaining 20% on the other. Did he gain or lose on the whole 
 transaction, and how much ? 
 
 18. A man bought 360 bu. of wheat at a certain price per 
 bu. and sold J of it at a gain of 10%, J at a loss of 25%, and 
 
 ■.-, I ii'tr ii 
 
 ■I 
 
1 
 
 PERCENTAGE 
 
 221 
 
 the remainder at a gain of 45%, and by so doing realized $594 
 for the whole lot. What was the cost price per bii. ? 
 
 19. A speculator paid $ 1400 for two lots, the price of one of 
 them being 40% that of the other. He sold the cheaper lot at a 
 gain of 50%, and the dearer one at a loss of 30%. Find his gain 
 or loss per cent on the whole transaction. 
 
 20. A merchant buys 3150 yd. of cloth. He sells | of it at a 
 gain of 6%, | at a gain of 8%, | at a gain of 12%, and the 
 remainder at a loss of 3%. Had he sold the whole at a gain of 
 5% he would have received $28.98 more than he did. Find the 
 prime cost of 1 yd. 
 
 21. The manufacturer of an article charged 20% profit, the 
 wholesale dealer charged 25% of an advance on the manufactu- 
 rer's price, and the retail dealer charged 30% of an advance on 
 the wholesale price. Find the cost to the manufacturer of an 
 article for which the retail dealer charged $ 23.40. 
 
 22. I buy two cows for $55-^ if I sell the first at a loss of 5%, 
 and the second at a gain of 5%, I should gain yY%. What was 
 the price of each cow ? 
 
 
 Commercial or Trade Discount 
 
 216. Commercial discount is an allowance made by mer- 
 chants upon their catalogue prices. 
 
 The commercial discount is reckoned at a certain rate per 
 cent. 
 
 Sometimes several discounts are allowed to a purchaser. 
 
 In such a case, the first discount is to be deducted, and 
 then the second discount is to be reckoned upon the re- 
 mainder and then deducted, and so on for each successive 
 discount. 
 
 .1. . 
 
 -,' I, 
 
 
222 
 
 ARITHMETIC 
 
 217. What is the net amount of a bill for $ 720 subject to 
 discounts of 20% and 6%? Find a single discount equiva- 
 lent to these successive discounts. 
 
 The first discount = 20 % of $ 720 = $ 144. 
 The first remainder = $ 720 - $ 144 = § 576. 
 The second discount = 6 % of !;? 576 = $ 34.56. 
 .-. the net amount = $ 576 - f 34.56 = $ 541.44. 
 Again, the single equivalent discount = ^ 720 - $ 541.44 = $ 178.56. 
 .-. the rate of a single discount = $ 178.56 -^ $ 720 = .248 = 24.8 %. 
 
 k' 
 
 
 Exercise 146 
 
 1. An invoice was $650, trade discounts 20% and 8% off. 
 Find the cost of the goods. 
 
 2. What is the net amount of a bill of goods, the list price of 
 which is $ 245, trade discounts 18% and 5% off for cash ? 
 
 3. What is the difference on an invoice of $ 540, between 
 40% direct discount, and discounts of 25% and 15%? 
 
 4. A dealer buys a book, list price 80j^, at a discount of 25% ; 
 he sells the book for 80^. What per cent is the profit ? 
 
 5. What is the net amount of a bill of $ 480, discounts being 
 12J% and 8%? Find a single discount equivalent to these suc- 
 cessive discounts. 
 
 6. A man paid $380 for goods, at discounts of 20% and 5%. 
 Find the list price of the goods. 
 
 7. A dealer paid .1^299.20 for goods at 15% and 12% off. 
 Find the list price of the goods. 
 
 8. Find the net cash amount of a bill for $ 1266, subject to 
 discounts of 33^%, 10%, and 5%, for cash. 
 
 9. Find the difference between a single discount of 40%, and 
 successive discounts of 25% and 15%. 
 
 10. Find the net amount of a bill of f 340, discounts being 30, 
 15, and 6. Find a single discount equivalent to these three dis- 
 counts. 
 
PERCENTAGE 
 
 223 
 
 11. Find the net cash amount of a bill of f 254, discounts being 
 25%, 12.1%, 5%, j^ind a single discount equivalent to these 
 three successive discounts. 
 
 12. A merchant who receives successive discounts of 20%, 15%, 
 and 10%, on a bill of $ 750, sells at an advance of 33i %. What 
 does he sell his goods for ? His selling price is what per cent 
 less than the list price ? 
 
 13. What is the difference between discounting a bill of $ 3000 
 at 40%, and then taking a discount oif the remainder of 5% for 
 cash, and discounting the whole at 45%? 
 
 14. A merchant buys goods at 40 and 20 off the list price and 
 sells them at 30 and 10 off the list price. What is his gain per 
 cent ? 
 
 15. An invoice of crockery, amounting to f 1473.20, was sold 
 Jan. 3, at 90 days, subject to 40% and 10% discount, with an 
 additional discount of 6% if paid within 20 days. How much 
 will be required to pay the bill on Jan. 21 ? 
 
 Commission and Bkokeiiage 
 
 218. A Commission Merchant is one who buys or sells goods 
 for other persons by their authority. Commission merchants 
 are usually placed in possession of the goods bought. 
 
 A Broker is a person who, in the name of his principal, 
 effects contracts to buy or sell. 
 
 The broker is not in general placed in possession of the 
 goods bought or sold. 
 
 The title Broker is also applied to persons who deal in 
 stocks, bonds, bills of exchange, promissory notes, etc., and 
 to mercantile agents, who transact the business for a ship in 
 port. 
 
 Commission is the charge made by an agent for transacting 
 business. 
 
 II 
 

 224 
 
 ARITHMETIC 
 
 In buying, the commission is reckoned on the cost price ; 
 in selling, the commission is reckoned on the selling price. 
 
 219. (1) A commission merchant sold 270 bbl. of flour at 
 $6 a bbl., and received 5% commission. What was his com- 
 mission? How much did he remit to his employer? 
 
 The selling price = 270 x $0 = $ 1620. 
 .-. the commission = 5 % of $ 1620 = ^ 81. 
 .-. the amount remitted = ••$ 1620 - $ 81 = $ 1539. 
 
 (2) A commission of i 242.58 was charged for selling 
 f 3772 worth of goods. What was the rate of commission ? 
 
 242.58 
 
 The commission = 
 
 of the selling price 
 
 3772 
 
 = .0643 of the selling price. 
 .•. the rate of commission = 6.43 %. 
 
 (3) A grain dealer charged 3J% for selling a quantity of 
 
 wheat, and received for his commission t| 218.40. For how 
 
 much did he sell the wheat ? 
 
 The commission = 3i % or .035 of the selling price. 
 
 .035 of the selling price = $218.40. 
 
 .-. the selling price = 218.40 -=- .035 = $6240. 
 
 (4) If 1612.50 include the price paid for certain goods, 
 
 and 21% commission to the agent, how much money does 
 
 the agent expend in purchasing the goods ? 
 
 Let the cost price of the goods = 100 units of money. 
 
 Then the commission = 2.| units of money. 
 
 The amount sent to the commission merchant = 102 ,• units of money. 
 
 102 1 units = $512.50. 
 
 1 unit = 512.50 -=- 102.5 = $5. 
 100 units = $ 500. 
 .-. the cost of the goods = $500. 
 
 As in Exercise 145, give the general statements of solu- 
 tions, prove answers, and make questions. Do this also in 
 each of the following exercises. 
 

 PERCENTAGE 
 
 225 
 
 Exercise 147 
 
 1. A commission merchant sold 480 bbl. of flour at $3.50 a 
 bbl. on a commission of 2%. What was his commission ? How 
 much did he remit to his employer ? The amount remitted was 
 what per cent of the selling price ? 
 
 2. My agent sold coffee to the amount of f 850 on a commis- 
 sion of 3%. Find his commission and also the amount remitted 
 to his employer. The amount remitted is what per cent of the 
 selling price ? 
 
 3. An agent sold 210 bu. of oats at 60^ a bu., and charged 
 f 3.78 for doing so. Find his rate of commission. 
 
 4. On a debt of $1725 a creditor receives a dividend of 60%, 
 on which he allows his attorney 5%. He receives a further 
 dividend of 25%, on which he allows his attorney 6%. What is 
 the net amount that he receives ? 
 
 5. If a commission of $212.94 is paid for buying $6552 
 worth of goods, find the rate per cent of commission. 
 
 6. An agent received $40.62.^ for selling a house worth 
 $ 1625. Find his rate per cent of commission. 
 
 7. An agent, who is paid a commission on what he invests, 
 received $4896, a,nd invests $4800. Find his rate per cent of 
 commission. 
 
 8. An agent received $ 56 for selling grain on a commission 
 of 4%. Find value of grain sold. 
 
 9. A commission merchant charged 2J-% for selling a quantity 
 of pork, and received for his commission $64.82. Find the 
 selling price of the pork. 
 
 10. The owner of a house offered an agent $ 500 commission, 
 if the agent could sell the house for $10,500. What rate per 
 cent commission was the owner offering ? Had the owner offered 
 5% commission, what would have been the commission on 
 $ 10,500 ? 
 
 Ill 
 
 'i 
 
 H m 
 
 m 
 
r 
 
 226 
 
 ARITHMETIC 
 
 11. I bought a bicycle for $ 70, which was J- of my commission 
 at 3J-% for selling a quantity of land. For how much was the 
 land sold? 
 
 12. A real estate dealer sold land for 100 units of money, on 
 a commission of 4%. How many units of money did he keep 
 for his commission ? 11 jw many units of money did he send his 
 employer ? If his employer received f 2tS80, what did the land 
 sell for ? What was the agent's commission ? 
 
 13. An agent remits f 4850 to his employer after taking out 
 his commission of 3%. Find the selling price. 
 
 14. My agent sent me as my share of the selling price of flour 
 $ 2038.40. If the flour sold for 1 3.25 a bbl., and the agent's 
 commission was 2%, how numy bbl. did he sell ? 
 
 15. My agent bought a quantity of goods for me on a com- 
 mission of 2 % . If the cost of the goods was 100 units of money, 
 how many units of money did his commission equal? How many 
 units did I have to send him to cover the cost of the goods and 
 his commission ? 
 
 16. A merchant in Buffalo sends a commission merchant in 
 New York $ 3120, instructing him to purchase goods, reserving 
 his commission at 4%. Find his commission. 
 
 17. A merchant sent f 3238.30 to New Orleans to be expended 
 in cotton. The broker in New Orleans charged 6% commission. 
 What sum was paid for the cotton ? 
 
 18. Sent to a commission merchant in Chicago $2080.80 to 
 invest in flour, his commission being 2% on the amount expended. 
 How many bbl. of flour could be purchased at $ 4.25 a bbl. ? 
 
 19. A real estate broker sold a house on 3J% commission, and 
 sent to the owner $ 6176. What was the broker's commission, 
 and what sum did he receive for the house ? 
 
 20. I send $ 5250 to a commission merchant in St. Louis, who 
 charges 5% for investing, with instructions to purchase certain 
 
PERCENTAGE 
 
 51011 
 
 out 
 
 227 
 
 <^oo(Is, deducting his commission from tlie amount of money sent 
 him. Find his commission. 
 
 21. (^<) Received .|I41{)() from my jigent, who had deducted his 
 commission at 5% as proceeds of sale of goods. What were the 
 goods sohl at? (b) Kemitted $4100, inchuling commission, to 
 my agent to invest for me on commission of 5%. What was his 
 commission ? 
 
 22. An agent sokl a quantity of flour for f 5100 on commission 
 of 3%, and invested the remainder in tea, first taking out his 
 commission of L}% on tlie price paid for the tea. Find the amount 
 paid for the tea, and also the total commission. 
 
 Insurance 
 
 220. Insurance is a contract by which a person whose 
 property is insured receives security against loss by fire or 
 accident in consideration of a sum of money paid to the 
 insurance company. 
 
 The Premium is the sum paid for insurance. It is always 
 a certain per cent of tlie stem insured. 
 
 The Policy is the written contract of insurance. 
 
 221. (1) A factory valued at 835,000 was insured for | of 
 its value, the rate of insurance being |% for one year. What 
 was the amount of the premium ? 
 
 The premium = |% of | of $35,000. 
 
 = Ax§x^^^^ = $131.25. 
 800 6 1 
 
 (2) The sum of $ 285 was paid for the insurance at | of 
 its value of a ship worth 1 50,000. What was the rate per 
 cent of premium if $ 3.75 was charged for the policy and the 
 preliminary survey ? 
 
k'l 
 
 228 aritiimi:tic 
 
 The promium --= 828;') - 83.75 = 8281.25. 
 •The ainoiiMt insured =: •' of 8r)0,0«)() = iJiloTjSOO. 
 The rate of nisiiraiice = 8281.25 -=- 837,500 = .0075. 
 .*. the rate per cent = YJ'^^ % or :| %. 
 
 (3) For what sum must a cargo worth $ 33,950 be insured 
 
 at 3% so tliat in case of k)ss the owner may recover both the 
 
 value of the cargo and the premium ? 
 
 Let the amount of insurance = 100 units of money. 
 Then the premium = ^ units of money. 
 The vahie of the goods = 07 units of money. 
 07 units of money = 8 •33,050. 
 
 1 unit of money = 8 350. 
 100 units of money = 8^5,000. 
 .'. the cargo must be insured for 8 35,000. 
 
 Exercise 148. 
 
 1. A warehouse vahied at $ G2,o()0 was insured for f of its 
 vahie. The rate of insurance was 1\% for three years, and the 
 cost of the policy and the agent's expenses wer $ 2.50. What 
 was the amount paid for the insurance ? 
 
 2. An insurance company took a risk at 2J%, and reinsured f 
 of the risk at 2%, The premium received exceeded the premium 
 paid by $ 42. Find the amount of the risk. 
 
 3. What will be the cost of insuring a cargo of 24,000 bu. of 
 wheat valued at $ 1.05 per bu., the insurance covering 4 of the 
 value of the cargo, the premium rate being 1|%, and the other 
 expenses of the insurance being 2^% of the premium? 
 
 4. A merchant's stock was insured for $ 42,000, ^ of this 
 amount being at |%, | of the remainder at |%, and the re- 
 mainder at 1%. Find the total amount of premium paid. 
 
 5. A merchant insured his stock for $ 33,000 for one year at 
 ^fo- Six months thereafter the policy was cancelled at the re- 
 quest of the insured. Find the amount of premium returned, the 
 short rate for six months being |%. 
 
PERCENTAGE 
 
 229 
 
 6. A factory and the machinery therein is insured for 
 f r)r),000; f of this sum is at i|% premium, ;ind the reuuiiu(h^r is 
 at 5%. What is the average rate per cent of premium paid on 
 the whole ? 
 
 7. A tire insurance company charged $ 190.88 for insuring a 
 house for f 17,500. What was the rate per cent of insurance ? 
 
 8. A merchant's stock was worth f 120,000. He insured it at 
 I its value, paying ^ 700 premium. What was the rate per cent 
 of insurance ? What was the rate in cents per f 100 ? 
 
 9. A shipment of goods is insured for $7500, and f 18.75 is 
 paid as premium. At that rate, what would be the amount of the 
 premium on $ 18,750 ? 
 
 10. For what sum was a house insured if the premium paid 
 was f 17.50 and the rate of insurance ^%? 
 
 11. For what sum was a shop insured if the rate of insurance 
 was 65^ per $ 100 and the premium paid was $ 81.25 ? 
 
 12. A fire insui lUce company received $ 350 for insuring a 
 factory at 1^% premium, and charged ^% for insuring a less haz- 
 ardous property of the same valuation as the factory. What was 
 the amount of the premium paid on the second property ? 
 
 13. A building and contents are insured as follows: $12,000 
 in the first. $ 8000 in the second, and $ 5000 in the third insur- 
 ance company. Were a loss to the extent of $ 3500 to occur 
 through fire, what portion of the loss should each company bear ? 
 
 14. Merchandise valued at $ 63,000 was insured in the first 
 insurance company for f 15,000, in the second for f 12,000, and 
 in the third for f 8000. If the merchandise is damaged by fire to 
 the extent of $ 10,500, how much of the damage should each 
 company pay ? 
 
 15. A fire insurance company insured a building for $ 60,000 
 at B% premium, and reinsured i of the risk in another company 
 at 1%, and ^ of the risk in a third company at f%. What 
 
 ii III 
 
1 
 
 230 
 
 ARITHMETIC 
 
 n"t 
 
 amount and what rate of premium did the company net on the 
 remainder of their risk ? 
 
 16. A steamboat worth $60,000 is insured in three companies; 
 in two to the amount of $ 15,000 each, and in the third to the 
 amount of $20,000. For what sum wouhl each company be 
 liable if the vessel were to sustain damage to the extent of 
 $ 6600 ? 
 
 17. For what amount must property worth f 7600 be insured, 
 at 5%, so that in case of loss both the premium and the value of 
 the goods may be recovered ? 
 
 i I 
 
 Taxes and Duties 
 
 222. A Tax is a sum of money assessed on persons or prop- 
 erty for public purposes. 
 
 The tax on property is reckoned at a certain rate per cent 
 of the assessed value of the property. 
 
 Direct taxes are levied by the state, county, township, city, 
 or the school district. 
 
 Some states levy a tax upon each voter, independent of 
 the property he owns. Such a tax is called a Poll-tax, and 
 as a rule does not exceed '12 a year. 
 
 Indirect taxes, called Duties, are levied by the general gov- 
 ernment on imported goods. 
 
 An Ad Valorem Duty is reckoned at a certain rate per cent 
 of the cost of the goods in the country from which they have 
 been imported. 
 
 A Specific Duty is a fixed charge on the quantity of goods 
 without reference to their cost, as a specific tax of one cent 
 a pound. 
 
 223. The people of a school section wish to build a new 
 school-house, which will cost $2850. The taxable property 
 
PERCENTAGE 
 
 231 
 
 of the section is valued at 8190,000; what will be the rate 
 of taxation, and what will be a man's tax whose property is 
 valued at $7500? 
 
 The tax on $ 190,000 = $2850. 
 .'. the rate of taxation = $2850 -^ $ 190,000 = .015 or 1J%. 
 . •. the tax on $ 7500 = 1| % of § 7500 = $ 112.50. 
 
 
 Exercise 149 
 
 1. State expenses which the government meets by taxation ; 
 the county ; the township ; the village or city ; the school dif;- 
 trict. 
 
 2. What is the tax on property assessed at $ G400, the rate of 
 taxation being 1^%? 
 
 3. In a school section a tax of f 4000 is to be raised. If the 
 assessed valuation of the property is $ 250,000, what will be the 
 tax on the dollar, and what is A's tax, whose property is valued 
 at $ 1800 ? 
 
 4. What is the assessed value of property taxed $37.50, at 
 the rate of 15 mills on the dollar ? 
 
 5. What is the assessed value of property taxed $37.80, at 
 the rate of 41 mills on the dollar ? 
 
 6. A person, after paying an income tax of 16 mills on the 
 dollar, has $ 8265.60 left. What is his income ? 
 
 7. What amount must a town be taxed so that after allowing 
 the collector 3% the net amount realized may be $ 24,250 ? 
 
 8. What sum must be assessed to raise $ 12,250, the collec- 
 tor's commission being 2%? 
 
 9. If it costs 2% to collect, and 5% of the tax assessed is 
 non-collectible, what amount must be levied in order to raise 
 $ 27,930 ? 
 
 10. The municipal rates being reduced from 192 mills to 171 
 mills on the dollar, my taxes are lowered by $4.05. For how 
 much am I assessed ? 
 
 M 
 
 i>m 
 
 
232 
 
 ARITHMETIC 
 
 11. In a certain section a school-house is to be built at an ex- 
 pense of $ 8400, to be defrayed by a tax upon property valued 
 at $ 700,000. What is the rate of taxation to cover both the cost 
 of the school-house and the collector's commission at 4%? 
 
 12. The assessed valuation of a town is $972,250, and the 
 town has 320 polls paying $ 1.50 each; what is the rate of taxa- 
 tion when the tax levy is f 19,925? What tax must a person 
 pay whose property is assessed for $ 7500, and who pays for one 
 poll ? 
 
 13. If the assessed value of a town is $1,260,000, and the 
 town has 420 polls paying f 1.25 each, what is the rate of taxa- 
 tion on property when the tax levy is $ 29,925 ? What does A 
 pay, whose property is assessed at $ 8500 and who pays one poll ? 
 
 14. A house assessed at $ 2200 was rented for f 23 a month, 
 the tenant to pay taxes and water-rates. The taxes were 17| 
 mills on the dollar, and the water-rates were $ 5 per quarter year. 
 How much all together did the tenant pay per year for the house ? 
 If the property had cost the landlord $ 2500, what rate per cent 
 per year was he receiving on his investment ? 
 
 15. Paid 30% duty on a watch, and sold it at a loss of 5% ; 
 but had it been sold for $ 21.06 more, there would have been a 
 gain of 8|%. Find the cost price. 
 
 16. Distinguish specific and ad valorem duties. A quantity of 
 raisins invoiced at $ 877 cost $ 990.25 in store, after paying duty 
 and $ 16.12 for freight. Find rate of duty. 
 
 17. An importer purchased goods, paying freight 10% and 
 duty 20% on the original outlay; he was obliged to sell the 
 goods at a loss of 20%, but had he received f 585 more than he 
 actually sold them for, he would have made a profit of 4%. Find 
 the original cost of the goods. 
 
 18. What is the duty on 600 yd. of cloth invoiced at 6 francs 
 per yd., the duty being 30%? (1 franc = 19.3^.) 
 
PERCENTAGE 
 
 233 
 
 Miscellaneous Exercise 150 
 
 1. If I buy an article for f 3.60 and sell it for 1 4.20, what is 
 my gain per cent ? 
 
 2. If I sell goods for ^3360 and gain 12%, what was the cost 
 price ? 
 
 3. If 425 yd. of silk be sold for 1 1657.50, and 20% profit be 
 made, what did it cost per yd. ? 
 
 4. If, by selling goods for $1088, I lose 16%, how much 
 per cent should I have lost or gained if I had sold them for 
 f 1344 ? 
 
 5. A tradesman's prices are 25% above cost price. If he 
 allow a customer 8% on his bill, what profit does he make ? 
 
 6. If 8% be gained by selling a piece of ground for $8251.20, 
 what would be gained per cent by selling it for $ 8404? 
 
 7. Find the brokerage on $ 1324 at J%. 
 
 8. Find the brokerage on $375 at 5%. 
 
 9. What amount of money was invested, when the broker's 
 charges at 1|% amounted to $150? 
 
 10. My agent has purchased real estate, on my account, to the 
 amount of $ 19,384. What is his commission at 1^%? 
 
 11. I send my agent $3654, with instructions to deduct his 
 commission at 1^%, and invest the balance in tea. How much 
 did he invest ? 
 
 12. Gave $ 15,037.50 to a broker to invest, with instructions, 
 after deducting brokerage at |%, to invest the balance in govern- 
 ment bonds. What will be the sum invested, and how much will 
 be the brokerage ? 
 
 13. What will be the premium of insurance on the furniture 
 of a house valued at $ 1500, at 2i%? 
 
 14. What is the premium for insuring a cargo, valued at 
 $16,450, at 3^%? 
 
 
 mi 
 
 
 iiJ' 
 
 ii 
 
 I; if 
 
 I. 
 
 I 
 
234 
 
 ARITHMETIC 
 
 15. A person at the age of 40 insures his life in each of two 
 offices for $4500, the premiums being at the rate of 3^ and 3J% 
 respectively. Find his annual payment. 
 
 16. A trader gets 600 bbl. of flour insured for 80% of its cost, 
 at 2J%, paying $37.80 premium. At what price per bbl. did he 
 purchase the flour ? 
 
 17. A shipment of dry goods was insured at 1|% to cover J of 
 its value. The premium was f 28. What were the goods worth? 
 
 18. A man who owns f 12,750 worth of property pays a tax of 
 f 216.75. Find the rate on the dollar. 
 
 19. A certain town has property assessed at $520,000, and 
 levies a tax of $7800. What should B pay, whose property is 
 assessed at $2500? 
 
 20. A town has levied a tax of $ 7690, which sum includes the 
 amount voted for building a town hall and the collector's fees, at 
 3%. What was expended on the town hall ? 
 
 21. What is the rate per cent of commission when I receive $ 5 
 for selling goods to the value of $ 180 ? 
 
 22. I sold a quantity of goods for $ 273.68, on a commission of 
 2|%. Find my commission. 
 
 23. A and B insure their houses against fire, and A has to pay 
 $ 7.50 more than B, who pays $ 28.75. Find the value of their 
 houses, the rate of insurance being |%. 
 
 24. A. B. bought goods amounting to $ 7460 subject to 25 and 5 
 off, $ 3730 subject to 30 off, and $ 1492 subject to 20 and 10 off. 
 Find the net cost of the goods. Were the invoice-clerk to bill A. B. 
 with goods amounting to $ 12,682 subject to 30 off, what would 
 be the amount of the error in the net cost of the goods ? 
 
 25. A mixture of coffee and chiccory in the proportion of 8 
 parts of coffee to 1 part of chiccory is sold at 35^ per lb., being an 
 advance of 40% on the cost. The chiccory cost 9^ per lb. Find the 
 cost of the coffee per lb. 
 
PERCENTAGE 
 
 235 
 
 26. A man bought a house and lot for $ 4750. After spending 
 $ 1143 on repairs and improvements, and paying f 128 for taxes 
 and other expenses, he sold the property for $ 6800. What rate 
 per cent of profit did his investment yield him ? 
 
 27. In an examination A obtained 78% of the full number of 
 marks, beating B by 16% of the full number. If A received 975 
 marks, how many did B receive ? What percentage of B's num- 
 ber was A's ? 
 
 28. By selling a certain book for $3.96, I would lose 12% of 
 the cost. What advance on this proposed selling price would 
 give a profit of 12% of the cost? What rate per cent on the 
 proposed selling price would this advance be ? 
 
 29. Goods are sold at a loss of 15% on the cost. By what 
 percentage of itself should the selling price be advanced to yield 
 a profit of 15% on the cost ? 
 
 30. A man having bought a certain quantity of goods for $ 150, 
 sells J of them at a loss of 4%. By what increase per cent must 
 he raise that selling price that by selling the whole at that in- 
 creased rate he may gain 4% on his entire outlay? 
 
 31. The cost of freight and insurance on a certain quantity of 
 goods was 15%, and that of duty 10% on the original outlay. 
 The goods were sold at a loss of 5 % , but had they brought f 3 
 more there would have been a gain of 1 % . How much did they 
 cost? 
 
 32. A man began business with a certain capital ; he gained 
 20% the first year, which he added to his capital, and 37-J% the 
 second year, which he added to his capital ; in the third year he 
 lost 40% ; had he received f 600 more for the goods sold the last 
 year, he would have cleared in the three years 2 per cent of his 
 original capital. Find the capital with which he commenced 
 business. 
 
 33. A merchant bought 400 lb. of tea and 1600 lb. of coffee, 
 the cost of the latter per lb. being 16|% that of the former; 
 
 a 
 
 I?; 
 
 m 
 
 lit 
 
 ii 
 
 ■i 
 
236 
 
 ARITHMETIC 
 
 I 
 
 he sold the tea at a profit of 33|%, and the coffee at a loss of 
 20%, gaining, however, on the whole f 60. Find his buying 
 prices and his selling prices. 
 
 34. A sells a quantity of wheat at $ 1 per bu. and gains 20%. 
 Afterwards he sold some of the same wheat to the amount of 
 ^37.50 and gained 50%. How many bu. were there in the last 
 lot and at what rate per bu. did he sell it ? 
 
 35. A person marks his goods so that he may allow a discount 
 of 4%, and still make a profit of 15%. What must be the marked 
 price of an article that cost him $ 4.80 ? 
 
 A manufacturer who employed men at f 1.60 a day found 
 
 36. 
 
 What wages were 
 
 that he could save 15% by employing women. 
 
 paid the latter, supposing a man could do ^ more than a woman 
 
 in the same time ? 
 
 37. A merchant buys goods; the cost of freight is 8%, and 
 that of insurance 12% on the original outlay ; he is obliged to sell 
 them at a loss of 7% ; but if he had received f 5.10 more for them 
 he would have gained 1 J%. Find the original outlay. 
 
 38. A merchant sells 50 yd. of broadcloth at a gain of 15%, 
 and 75 yd., which cost the same per yd., at a gain of 10%, and 
 finds that if he had sold the whole at a uniform gain of 12i%, he 
 would have received $ 2.25 more than he actually did receive. 
 W^hat was the cost price per yd. ? 
 
 and marks J 
 
 of 
 
 39. A man buys goods for a certain sum 
 them at a profit of 24%, and | of them at a profit of .>uyc ; 
 but had he marked | of them at 24% gain, and ^ at 36% gain, 
 he would have realized $ 240 less than before. Find the cost of 
 the goods. 
 
 40. A wheat buyer sold \ of his wheat at a certain gain per 
 cent, J of it at a gain of twice the former rate per cent, and the 
 remainder at a gain per cent of 3 times the first gain. If the 
 gain on the entire stock was 26%, what did he gain on each part ? 
 
PERCENTAGE 
 
 237 
 
 5 . 
 
 of 
 ng 
 
 If he gained 5% on the first part, what was the entire gain per 
 cent ? 
 
 41. A merchant wishes to mark some goods which cost $ 1.20 
 per yd., so that after making a reduction of 20% off the marked 
 prices, he may yet gain 10%. At what price per yd. must he 
 mark the goods ? 
 
 42. Sold goods to a certain amount on a commission of 5%, and 
 having remitted the net proceeds to the owner, received for prompt 
 payment i%, which amounted to f 24.22|. What was the selling 
 price of the goods ? 
 
 43. My agent sold a quantity of flour for $ 2550 on a commis- 
 sion of 3%, and invested the proceeds (after taking out his com- 
 mission) in tea on a commission of 2% on the price paid for the 
 tea. Find how much he paid for the tea and also his total 
 commission. 
 
 44. At 2i%, for what must property worth $ 3600 be insured, 
 so that in the event of loss the worth of the goods and the premium 
 of insurance may be recovered ? 
 
 45. What sum must be insured on a house worth $ 665, so that 
 in case of loss the owner may receive ^ of this sum, and also | of 
 the premium, which was at 6%? 
 
 46. What single discount is equivalent to successive discounts 
 of 20% and 10%? 
 
 47. Show that successive discounts of specified rates may be 
 taken off a list price in any order without affecting the net price. 
 Thus 20 and 10 off is equivalent to 10 and 20 off, so also 30 and 
 10 and 5 off, 10 and 30 and 5 off, and 5 and 30 and 10 off are all 
 equivalent. 
 
 48. An agent sold 6 mowing-machines at f 120 each, and 12 at 
 f 140 each. He paid for transportation $ 72, and, after deducting 
 his commission, remitted $ 2208 to his employer. What was the 
 rate of commission ? 
 
 k 
 
i 
 
 238 
 
 ARITHMETIC 
 
 49. A man allows his agent 5% of his gross rentals, and receives 
 a net rental of $ 3488.40. If the gross rental is 6% of the value 
 of the property, what is the value of the property ? 
 
 50. A shipment of goods is insured for $ 6000, which sum 
 covers the value of the goods, the premium at 1|% and $2.50 for 
 expenses. What was the value of the goods ? 
 
 [mil 
 i.'f 
 
 i m 
 
 

 ires 
 lue 
 
 If 
 
 im 
 for 
 
 CHAPTER XIV 
 
 DfTEEEST 
 
 224. Interest is money paid for the use of money. 
 The Principal is the sum loaned. 
 
 The Amount is the sum of the principal and interest. 
 The Rate of Interest is always expressed as a rate per cent 
 of the principal. 
 
 The unit of time is 1 yr. 
 
 225. (1) What is the interest on $ 638 for 1 yr. at 6% ? 
 
 $ 638 principal 
 .06 rate per unit 
 $ 38.28 interest for 1 yr. 
 
 .-. the interest on $638 for 1 yr. at 6% = 6% of $638 = $38.28. 
 
 (2) Find the interest on and the amount of $ 473.28 for 
 81 da. at 7%. 
 
 $473.28 principal 
 
 .07 rate 
 
 $33.1296 interest for 1 yr. 
 
 $33.13 interest for 1 yr. 
 9 
 
 40 ) $298. 17 
 
 7.45 interest for 81 da. 
 473.28 
 $480.73 amount 
 
 The interest for 1 yr. = 7 % of $473.28 = $33.13. 
 
 The interest for 81 da. = gVu or tu o^ $33.13 = $7.45. 
 
 The amount = $ 473.28 + 7.45 = $ 480.73. 
 
 239 
 
240 
 
 ARITHMETIC 
 
 :ii 
 
 !i 
 
 (3) Find the amount of * 385.35, from July 7, 1895, to 
 Oct. 13, 1895, at 7i%. 
 
 mo. da. 
 Oct. 13 = 10 13 
 
 July 7=7 7 
 
 The time =3 6 = 3^ mo. = ^^ yr. 
 
 The rate = /5X V% = 2%. 
 
 The interest = 2 % of $ 385.35 = $ 6.71. 
 
 The amount = $385.35 + 16.71 = $392.06. 
 
 226. Six Per Gent Method. 
 
 The interest at 6% for 1 yr. = .06 of the principal. 
 
 The interest at 6% for 1 mo. = ^.^ of .06 or .005 of the principal. 
 
 The interest at 6% for 1 da. = g^^ of .005 or .000^ of the principal. 
 
 Find the interest on $435 for 9 mo. 24 da. at 6%. 
 The interest for mo. = 9 x .005 = .045 
 
 The interest for 24 da. 
 
 = 24 X .000 » = .004 
 
 The interest for 9 mo. 24 da. = .049 
 
 .-. the interest = .049 x $435 = $21,315. 
 
 To find the interest at any other rate per cent, divide the 
 interest at 6% by 6 and multiply by the given rate per cent. 
 
 The interest at 7i % may be found by increasing the interest at 6 % by 
 |i 1 ' 11 
 
 -2 or - ; that at 5i % by diminishing the interest at 6 % by - or — 
 6 4 2 /o J o ''^ 6 12 
 
 By what fraction must the interest at 6 % be increased in order to give 
 the interest at each of the following rates : 7 %, 8 %, 9 %, 6.^ %, 6.2 % ? 
 
 By what fraction must the interest at 6 % be diminished in order to give 
 the interest at each of the following rates : 5 % 4 %, 3 %, 4^ %, 4^ %, 5f % ? 
 
 Exercise 151 
 
 Find the interest on : 
 
 1. ^449 fori yr. at 5%. 
 
 2. $757 fori yr. at 4%. 
 
 3. $643.17 foil yr. at 7%. 
 
 4. $725 for 4 mo. at 8%. 
 
 5. $587.50 for 5 mo. at uy^. 
 
 6. $ 628.90 for 9 mo. at 4i-%. 
 
 7. $323.75 for 60 da. at 8%. 
 
 8. $ 958.50 for 90 da. at M %. 
 
INTEREST 
 
 241 
 
 9. $2805 for 33 da. at 0%. 12. f 225.90 for 03 da. at 7%. 
 
 10. $312.80 for 93 da. at 0%. 13. f 390.50 for 93 da. at 0%. 
 
 11. $ 012.94 for 33 da. at 7 i>%. 14. $8390.40 for 123 da. at 8%. 
 
 15. $ 4087.50 for 1 mo. 3 da. at 9%. 
 
 16. $ 1405.53 for 3 mo. 3 da. at 5%. 
 
 17. $ 1350 for 3 mo. 21 da. at 7%. 
 
 18. $295.30 for 57 da. at 0.2%. 
 
 19. $ 1200 from May 7 to June at 7%. 
 
 20. $ 975.05 from Sept. 10 to Dec. 8 at 01%. 
 
 21. $450 from Sept. 4 to Oct. 27 at 7%. 
 
 22. $ 79.50 from Dec. 23 to Feb. 20 at 7^ %. 
 
 23. $580.07 from Jan. 15, 1892, to May 1, 1892, at 8%. 
 
 24. State how to find the simple interest when the principal, 
 rate per cent, and time are given. 
 
 25. Name the terms in problems in Profit and Loss, Commis- 
 sion and Insurance, which correspond to principal and rate per 
 cent of interest. 
 
 26. Find the relation between the interest, principal, and 
 amount, when the time is 3 mo., and rate 8% ; time 120 da., 
 rate 9%. 
 
 27. Find the amount of $473.28 for 3 mo. at ^% per month. 
 
 28. Find the amount of $ 885.85 for 1 mo. 15 da. at 5%. 
 
 29. Find the amount of $ 028.25 for 185 da. at 4] %. 
 
 30. Find the amount of $ 935.08 for 00 da. at 0] %. 
 
 31. Find the amount of $ 147.50 for 93 da. at 7%. 
 
 32. Find the amount of $ 250 from July 9 to Aug. 18 at 8%. 
 
 33. Find the amount of $2394 from May 8 to Sept. 21 at 4%. 
 
 34. Find the amount of $ 5240 from March 1 to Aug. 3 at 5%. 
 
 35. Find the amount of $ 230.80 from Jan. 4, 1890, to June 23, 
 1890, at 0%. 
 
 36. Find the amount of $ 057.00 from Aug. 9 to Dec. 5 at 8%. 
 
 . I' 
 
242 
 
 ARITHMETIC 
 
 37. A person loaned $ 480 for 2 mo. and 13 da. at 9%. What 
 interest did he receive ? 
 
 38. On March 20, a merchant sold goods to the value of $ 1168, 
 and received a note, due June 8, next, for that sum with interest 
 at 7% per annum. For what amount was the note drawn? 
 
 39. A debt of $ 175 became due on June 13, after which date 
 interest was charged at the rate of 8%. What must be paid to 
 settle the debt Sept. 14 ? 
 
 40. A owes J$ 15,000 bearing interest at 5% per annum; he 
 pays at the end of each year for interest and part payment of 
 principal $ 2500. Find the amount of his debt at the end of the 
 third year. 
 
 41. A man engaged in business with a capital of $10,920 is 
 making 12J% per annum on his capital, but on account of ill 
 health he quits the business and loans his money at 7|%. How 
 much does he lose by the change in 2 yr. 5J mo. ? 
 
 42. f420. Chicago, June 4, 1895. 
 
 Sixty days from date I promise to pay Samuel Jones, or order, 
 four hundred and twenty dollars, with interest at six per cent, 
 value received. Richard Walsh. 
 
 What is the amount of this note on maturity, 63 da. after 
 June 4? 
 
 Exact Interest 
 
 227. In order to find the exact interest we must reckon 
 365 da. to a year. Exact interest is used by the United 
 States Government and sometimes in business transactions. 
 
 228. The exact interest at 5 % for 1 da. is ^f j, or ^^j of the principal. 
 The common interest is ^1^ or ^\ of the principal. Therefore the exact 
 
 interest is ^^j -r- ^^ or ^| of the common interest. Hence the exact interest is 
 equal to the common interest diminished by ^^ of itself. 
 
 229. Find the exact interest on 1^4250 from May 12 to 
 Oct. 3 at 1%. 
 
INTEREST 
 
 243 
 
 The number of days froiii May 12 to Oct. 3 = 19 + 30 + 31 -|- 31 + 30 + 
 3 = 144. 
 
 The interest on .$4250 at 7% for 1 yr. = $297.50. 
 
 The interest on $ 4250 at 7 % for 144 da. = J|| of $ 297.60 = $ 117.37. 
 
 : I! 
 
 li 
 
 Exercise 152 
 
 Find the exact interest on : 
 
 1. $ 2450 for 146 da. at G%. 
 
 2. $3475 for 292 da. at 7%. 
 
 3. $ 1560 for 60 da. at 5%. 
 
 4. 1^629 for 113 da. at 6%. 
 
 5. $ 1400 from July 6 to Dec. 4 at 5^%. 
 
 6. $ 1850 from March 1 to Aug. 6 at 6^%. 
 
 230. In Exercise 151 we were given the principal, rate 
 per cent, and time, and were required to find the interest or 
 the amount. In the following exercise we shall have given 
 the principal, interest or amount, and the time, and will be 
 required to find the rate per cent. 
 
 231. (1) At what rate per cent will $480 yield 118.20 
 interest in 7 mo. ? 
 
 The interest on $480 for 7 mo. = $18.20. 
 
 The interest on $ 480 for 1 mo. = $2.60. 
 
 The interest on $480 for 1 yr. = $31.20. 
 
 .-. the rate per cent = $31.20 -4- $480 = .06^ or 6J%. 
 
 (2) At what rate per cent must I loan $ 2840 for 145 da. 
 to amount to $ 2922.86 ? 
 
 The interest = $2922.36 - $2840 = $82.36. 
 The interest for 145 da. or ^ yr. = $ 82.36. 
 The interest for 1 yr. = $82.36 x ^ = $204.48. 
 .-. the rate per cent = $ 204.48 ^ $2840 = .072 or 7^ %. 
 
 ^1 
 
 
 I 
 
244 
 
 ARITHMETIC 
 
 ■ 11 
 
 Exercise 153 
 
 1. A gentleman invests $2500 for his son in order to give 
 him a yearly income of $ 175. Find the rate per cent. 
 
 Find the rate per cent : 
 
 2. When the interest on $ 1200 for 8 mo. is |48. 
 
 3. When the interest on $ 640 for 9 mo. is f 35. 
 
 4. When the interest on $ 585 for 1 mo. 18 da. is $4.94. 
 
 5. When a loan of $600 amounts to $626.75 in 7 mo. 4 da., 
 what rate per cent is charged ? 
 
 6. A man pays $14.91 for the use of $568 for 4 mo. 15 da. 
 Find the rate per cent. 
 
 7. A man lent $4800 for 6 mo. 6 da., and at the expiration 
 of the time received in payment of interest and principal $4955. 
 Find the rate per cent. 
 
 8. At what rate will $438 borrowed on April 17 amount at 
 simple interest to $ 445.519 on July 29 next following, if exact 
 interest is reckoned ? 
 
 232. To find the time when we are given the principal, 
 interest, or amount, and rate per cent. 
 
 In what time will the interest on $845 be $32.95| at 61% ? 
 
 The interest on $845 at 6i% for 1 yr. =$54.92^. 
 Therefore the fraction of a year = $32,955 -f- .^54.925 = f. 
 .-. the time = 2 yr. or 7 mo. 6 da. 
 
 Exercise 154 
 
 1. In what time will the interest on $ 750 at 7% equal $ 26.25 ? 
 
 2. In what time will $400 amount to $415 at 5% per annum ? 
 
 3. A man invests $4760 at 8%, in order that his son may 
 have an income of $ 95.20 at certain equal intervals of time. How 
 often is the interest payable a year ? 
 
INTEREST 
 
 245 
 
 4. A gentleman gives his note for $ 350, together with interest 
 at ^%. If he pays f 362.(30 to settle the note, when is it paid ? 
 
 5. A person leaves nnpaid a sum of money, on which he pays 
 7|% interest, until the interest equals -^^ of the principal. Find 
 the time. 
 
 6. A principal of $1200 was loaned May 12, 1892, at S%. 
 At what date did it amount to $ 1216.80 ? 
 
 7. In what time will $273.85 yield $8.86 simple interest at 
 
 8. In how many da. will $ 733.65 amount to $ 743.70 at 5% 
 simple interest ? 
 
 9. A debt of $ 175 became due on June 13, after which date 
 interest was charged at 7% per annum ; when the debt was paid 
 the interest accrued on it was $4.10. When was the debt paid? 
 
 10. In what time will $ 143 amount to $ 150 at 7% interest ? 
 
 233. To find the principal when the interest, time, and 
 rate are given. 
 
 Find the principal that will produce 140.77 in 9 mo. at 
 
 The interest for mo. or ;] yr. = $40.77. 
 The interest for \ yr. = ^ 13.50. 
 
 The interest for 1 yr. = $54.36. 
 
 8 % or .08 of the principal = $ 54.36. 
 .-. the principal = $54.30 ^ .08 = $079.50. 
 
 Exercise 155 
 
 1. What principal will produce $ 60 in 216 da., at 7^% per 
 annum ? 
 
 2. A man borrowed money at 6%, and paid $323.70 interest 
 a yr. Find what sum he borrowed. 
 
246 
 
 ARITHMETIC 
 
 
 i!^l 
 
 3. A man loans money at 8% per annum, interest payable 
 semiannually. If his semiannual interest is $38.40, find the 
 sum loaned. 
 
 4. A man left to his wife a yearly income of $ 1750, to the 
 oldest son $ 1540 yearly, and to the youngest $ 1260 yearly. Find 
 what sum must be invested at 6^% to produce these amounts. 
 
 5. What principal will yield $ 43.25 interest in \ yr. at 5^%? 
 
 234. To find the principal when the amount, time, and 
 rate per cent are given. 
 
 Find the principal that will amount to $ 1312.50 in 8 mo. 
 
 at 7J%. 
 
 The interest on $ 1 for 8 mo. at 7 J % = $ .05. 
 The amount of $ 1 for 8 mo. at 7^ % = $ 1.05 = 1.05 of $ 1. 
 Hence $ 1312.50 is the amount of $ 1312.50 -=- 1.05 = $ 1250. 
 .'. the principal = $ 1250. 
 
 Exercise 156 
 
 1. Find what sum would pay now a debt of $450 due in 
 6 mo. at 6% per annum. 
 
 2. What must be paid now to cancel a debt of $ 1368.25 9 mo. 
 before it is due, money being worth 7%? 
 
 3. Which is cheaper, lumber bought at $ 35 a thousand on 
 9 mo. credit, or at $ 34.30 on 6 mo. credit, money being worth 
 
 4. I bought a lot, paying f 400 cash, and the balance $ 800 
 in 9 mo. What is the cash value of the lot, money bringing 6%? 
 
 5. What principal will amount to f 1000 in 4 mo. at 4 J ? 
 
 6. What principal will amount to $ 73.56 in 63 da. at 8%? 
 
 7. A debt due on March 3 was not paid, and interest at 6J% 
 was charged on it from that date. On June 9 following, the debt 
 amounted to f 100. What was the sum due on March 3 ? 
 
fl 
 
 INTEREST 
 
 247 
 
 8. A merchant bought 500 bbl. of flour at $ 6.25 a bbl. on a 
 credit of 8 mo. He sold it at f 6.50 a bbl. on a credit of 4 mo. 
 What was his net cash gain, money being worth 6%? 
 
 9. A merchant borrows $ 1600 for 1 yr. at 7%. Find what 
 he owes at the end of the yr. In case he pays only $ 12 inter- 
 est, how much will he owe at the beginning of the next yr. ? 
 What will he owe at the end of the yr. ? 
 
 10. If I borrowed f 1200 Jan. 1, 1894, at 6%, what would 
 I owe Jan. 1, 1895 ? If I kept the money until Jan. 1, 1896, 
 what would I then owe ? 
 
 Bank Discount 
 
 235. A mer3hant, who desires to obtain a loan of $800 
 for 90 da., makes a note and takes it to the bank, which 
 deducts the interest on $ 800 for 93 da. at a certain rate per 
 cent, which varies from time to time. This bank gives him 
 the proceeds, and collects the $800 at the end of 93 da. 
 
 The 3 da. added to the specified time are called days of 
 grace, which must elapse before payment is due. 
 
 California, Idaho, New Jersey, New York, Oregon, Utah, 
 Vermont, and Wisconsin have abolished dai/s of grace. 
 
 236. Bank Discount is, therefore, simple interest collected 
 m advance upon the sum due on a note at its maturity. 
 
 Nearly all notes specify the place of payment. In case the 
 place of payment is not specified in the note, it is to be paid 
 at the business office of the maker of the note. 
 
 237. $450.75. Chicago, July 3, 1896. 
 
 Sixty days after date I promise to pay to the order of James 
 
 Smith, four hundred fifty and ^^^ dollars at the First National 
 
 Bank. Value received. 
 
 Horace Ward. 
 
 Discounted July 3, at G%. Find proceeds. 
 
 I 
 
 I * i 
 
 ' 
 
248 
 
 ARITHMETIC 
 
 The day of maturity = 63 da. after July 3 = Sept. 4. 
 
 The discount = the interest on $ 450.75 at G % for 63 da. = $ 4.73. 
 
 The proceeds = $ 450. 75 - -fi 4. 73 = $ 446.02. 
 
 238. The Day of Maturity is the day on which the note 
 becomes legally due. 
 
 The Proceeds of a Note is the sum of money received for it 
 when discounted. 
 
 It is found by subtracting the discount from the value of 
 the note at maturity. 
 
 The Time to run is the time between the day on which the 
 note is discounted and the day of maturity. 
 
 Zixercise 157 
 
 1. $600. Chicago, July 6, 1896. 
 Thirty days after date I promise to pay to George Boies, or 
 
 order, six hundred dollars, value received. 
 
 EoBERT Brown. 
 Discounted at 7 %, July 6, 1896. Find proceeds. 
 
 Face of Note Date of Note Time Rate of Discount 
 
 2. 1312.80 ; May 13, 1895 ; 90 da. ; 6%. Find proceeds. 
 
 3. $ 225.90 ; June 14, 1896 ; 2 mo. ; 7%. Find proceeds. 
 
 4. 1100.00; Feb. 12, 1896 ; 30 da. ; 5%. Find proceeds. 
 
 5. State how to find the proceeds of any note discounted at 
 once. 
 
 Face of Note Date of Note Time Rate of Discount 
 
 6. $ 1.00 ; Jan. 7, 1894 ; 57 da. ; 6%. Find proceeds. 
 
 7. In question 6 what face value would give f 99 proceeds ? 
 $ 198 ? f 495 ? Prove your face value correct by discounting. 
 
 8. Write the notes corresponding to questions 2 and 3. 
 
 9. $390j%V Si-RiNOFiELD, III., May 1, 1890. 
 Three months after date I promise to pay to the order of 
 
 Thomas A. Stuart, three hundred ninety and y% dollars. Value 
 received. 
 
 James Hknderson. 
 
 Discounted May 1, 1890, at 6%. Find proceeds. 
 
INTEREST 
 
 249 
 
 239. (1) fl 712.65. Chicago, July 6, 1895. 
 
 Sixty days from date I promise to pay George Wilson, 
 
 or order, seven hundred twelve and -^^^ dollars, for value 
 
 received 
 
 Samuel Jones. 
 
 Discounted at 7%, Aug. 6, 1895. 
 
 In the above note, find the dai/ of maturity^ the time to run, 
 the di8cou7it, and the proceeds. 
 
 The day of maturity = 63 da. after July C = Sei)t. 7, 1895. 
 
 The time to run = the number of days between Aug. 6 and Sept. 7. 
 
 = 32 da. = ^^, yr. 
 The discount = the interest on $ 712.65 for 32 da. at 7 % = $ 4.43. 
 
 The proceeds = $712.65 - $4.43 = $708.22. 
 
 (2) $450.76. St. Louis, May 5, 1895. 
 
 Three months after date, for value received, I promise to 
 pay Thomas King, or order, four hundred fifty and ^^^ dollars, 
 at the First National Bank, with interest at 6%. 
 
 AiiTHUR Hill. 
 
 Discounted July 1, 1895, at 8%. 
 
 The day of maturity = 3 mo. 3 da. after May 5 = Aug. 8, 1895. 
 
 The amount of the note, Aug. 8, 1895 = the amount of §450.76 for 3 mo. 
 
 3 da. at 6% = $457.75. 
 The time to run = the number of days between July 1 and Aug. 8 = 38 da. 
 The discount = the interest on '$ 457.75 for .^8 da. at S%=:^ 3.87. 
 The proceeds = ^ 457.75 - $ 3.87 = $ 453.88. 
 
 In the following exercise find the day of maturity, the time 
 to run, the discount, and the proceeds. 
 
 If 
 
 Exercise 158 
 
 1. f 2400. Clkvkland, 0., March 3, 1894. 
 
 Three months after date I promise to pay Ralph Barker, or 
 order, twenty-four hundred dollars, value received. 
 
 ROBKKT PeTEKSON. 
 
 Discounted at 1%^ May 7. 
 
250 
 
 ARITHMETIC 
 
 2. A note for $ 572.80 drawn on June 13, and payable 4 mo. 
 after date, was discounted at 7% on June 27. Find the pro- 
 ceeds. 
 
 3. $2400. Cleveland, O., March 3, 1894. 
 Three months after date I promise to pay Kalph Barker, or 
 
 order, twenty-four hundred dollars, for value received, with inter- 
 est at 6%. T5 -r> 
 
 ' Robert Peterson. 
 
 Discounted at 7%, May 7. 
 
 4. State business transactions which may have preceded the 
 giving of the notes in questions 1, 2, and 3. 
 
 5. On July 7, James Monroe bought a farm from John 
 Harris, paying f 2000 cash, and giving his note, without interest, 
 for $ 1200, payable in 60 da. Write the note. 
 
 Face of Note Date of Note 
 
 6. 3 312.80; May 13, 1890 
 
 7. 3 975.65 ; Sept. 5, 1892 
 
 8. f 450.00; Aug. 28,1891 
 
 9. $ 79.50 ; Pec. 17, 1889 
 
 10. $ 586.67 ; Dec. 28, 1891 
 
 11. f2480. 
 
 Time Date of Disc. 
 
 90 da. ; May 13 ; 
 
 3 mo. ; Sept. 16 ; 
 
 60 da. ; Sept. 4 ; 
 
 2 mo. ; Dec. 23 ; 
 
 Rate of Disc. 
 
 7%. 
 7%. 
 
 4 mo. ; Jan. 15, 1892 ; 
 
 Buffalo, N. Y., Nov. 19, 1892. 
 
 Six months after date I promise to pay Alfred Jameson, or 
 order, two thousand four hundred and eighty dollars, value 
 received, with interest at 5%. 
 
 William O'Connor. 
 Discounted at 6%, Jan. 4, 1893. 
 
 12. $ 2065.76. New Orleans, June 4, 1895. 
 
 Ninety days after date I promise to pay to the order of Edgar 
 Johnston, two thousand sixty-five and ^^^ dollars, for value 
 received, with interest at 6%. 
 
 Alexander Grant. 
 
 Discounted at 8%, July 4, 1896. 
 
INTEREST 
 
 251 
 
 13. State how to find the proceeds of a note, not bearing inter- 
 est, when discounted. VVliat change is to be made in the solution 
 when the note bears interest ? 
 
 14. Find the proceeds of a note payable in 87 da., whose face 
 value is $ 1, discounted immediately at 8%. 
 
 15. In question 14, what would have been the face value of 
 the note if the proceeds had been ^dS? $ 980 ? $ 196 ? $ 392 ? 
 
 240. (1) Find the face of a note payable in 60 da., that 
 will realize $840 when discounted at 6J%. 
 Let us consider a similar note whose face is $1. 
 The discount on $ 1 for 63 da. at 6J % = $ .011375. 
 The proceeds of a note whose face is $ 1 = $ 1 - .011375 = $ .988625. 
 Hence .988625 of the face = $ 840. 
 
 .-. the face = $840 -- .088625 = $849.66. 
 
 Puoop 
 The discount on $849.66 for 63 da. at 6.i % = $9.66. 
 The proceeds = $ 840.66 - $ 9.66 = $ 840. 
 .-. $849.66 Is the correct answer. 
 
 (2) In solving the preceding question it is unnecessary to cany out the 
 discount on $ 1 beyond the fourth figure in the decimal. 
 Thus the discount on $ 1 = $ .0114. 
 The proceeds of $ 1 = $ .9886. 
 
 The face of the note = $ 840 -^ ,9886 = $ 849.68. 
 
 This is correct within 2 f. 
 
 'A 
 
 P 
 
 H 
 
 Exercise 159 
 
 1. Find the face value of a note for 27 da. that will realize 
 $ 1990 when discounted at 0%. 
 
 2. Write the note corresponding to question 1, and prove that 
 the face as written in the note is correct. 
 
 3. For how much must a 2 months' note be drawn so that 
 when discounted at 7% it may yield $ 500 ? Prove your answer 
 correct and write the note. 
 
 4i^ 
 
 J 
 
252 
 
 ARITHMETIC 
 
 I' 
 
 4. A gentleman wishes to borrow $ 800 at a bank. For what 
 sum must he give a 60 days' note which is discounted at 7^% ? 
 
 5. What must be the face of a 4 months' note discounted at 
 8%, to realize $89.50? 
 
 6. For what sum must a note be drawn in order that if dis- 
 counted 89 da. before maturity, the proceeds may be $ 425 ; the 
 rate of discount being 7%? 
 
 7. What must be the face of a note so that when discounted 
 at a bank for 4 mo. and 9 da. at 9%, it will give f 240 ? 
 
 8. State how to find the face value of a note, when the term 
 of discount, the rate per cent, and the proceeds are given. 
 
 9. Make a question in which it is required to find the face of 
 a note — given the proceeds, rate of discount, and term of dis- 
 count. 
 
 10. I owe a man $ 575, and gave him a note at 60 da. What 
 must be the face of the note to pay him the exact debt, when 
 discounted at (bank discount) 1J% a month? 
 
 11. A sold B a bill of goods amounting to $ 7600, but B, not 
 having the money, gave A a note for 3 mo., which, when discounted 
 at the bank at 8%, paid the debt. Required the face value of 
 the note. 
 
 Miscellaneous Exercise 160 
 
 1. Find the interest on f 794.35 for 188 da. at 5%. 
 
 2. To what sum would $87.68 amount in 97 da. at 6^% 
 interest ? 
 
 3. A certain sum amounts to $ 1488 in 8 mo., and $ 1530 in 
 15 mo., simple interest. What is the rate per cent ? 
 
 4. A merchant borrowed $ 1680 June 16, and $ 1728 Sept. 28 ; 
 the merchant repays the whole sum, with interest, Jan. 2 nex.t. 
 Find the amount repaid, interest being 7^% per annum. 
 
IISJEREST 
 
 253 
 
 1 
 
 5. Bought 9000 bii. of wheat at 75^ a bu., payable in G mo. ; 
 I sold it immediately for 72^ a bu., cash, and put the money at 
 interest at 6%. At the end of the G mo. I paid for the wheat. 
 Did I gain or lose by the transaction, and how much ? 
 
 6. What is the bank discount and proceeds of a note of $ 11G8, 
 drawn Jan. 18 at 11 mo., discounted at the bank May 20 at 6%? 
 
 7. I owe a bill amounting to $ 219.75, and I give my note for 
 60 da. How must I draw it to cover the discount at G^%? 
 
 8. A. B. has a note of $ 800 to pay at the Merchants' National 
 Bank. At the time of its maturity he pays ^ 200, and gives a 
 note for 3 mo., days of grace included, for the balance. The rate 
 of discount being 8 % per annum, what was the face of the note ? 
 
 9. A note for f 1750 was drawn July 10 for 4 mo. It was 
 discounted Sept. 2 at the Granite Bank at 7^% per annum. What 
 sum was received for it ? 
 
 10. A note is drawn for 3 mo., days of grace included, and 
 when discounted at 7% per annum at the Farmers' ]>ank it 
 realizes $ 4.55 less than its face value. What is the face value 
 of the note? 
 
 11. Find the proceeds of the following joint note discounted 
 in New York Dec. 18, 1896, at 7^%. 
 
 ^ 3473%V New Yokk, Dec. 18, 1896. 
 
 Ninety days after date we jointly and severally promise to pay 
 to the order of Jno. Locke & Co., three hundred and forty-seven 
 ■f^jf dollars, at the Standard Bank. Value received. 
 
 Isaac Harper. 
 
 A. C. Early. 
 
 12. For how much must a 90-day note be drawn to realize 
 $ 190 when discounted at 6%? 
 
 13. Find the proceeds of a note whose face value is $950, 
 payable in 3 mo. from Feb. 1, 1896, and discounted Feb. 6, 1896, 
 at 7%. 
 
 m 
 
 nr 
 
254 
 
 ARITHMETIC 
 
 14. A note for $3G0 was discounted 40 da. before maturity 
 and the proceeds were f 350.80. What was the rate of discount, 
 there being no exchange ? 
 
 15. The proceeds of a note for f 137.50, discounted 40 da. 
 before maturity, were $136.40. What was the rate of discount 
 charged on the face of the note ? 
 
 16. June 18, 1895, a merchant purchased goods amounting per 
 catalogue prices to |> 647.80, subject to 25 and 5 off. He was 
 allowed 3 months' credit, after which he was charged interest 
 at 8%. Find the amount of the account Feb. 21, 1896. 
 
 * Partial Payments 
 
 241. A Partial Payment is a payment of only a part of a 
 debt. 
 
 An Indorsement is an acknowledgment of the receipt of a 
 partial payment, written on the back of a note, stating the 
 amount and the date of payment. 
 
 242. The following is the method of solving questions in 
 partial payments when the note is paid in full in a year 
 or less: 
 
 What amount is due Dec. 20, 1895, on a note for $1600, 
 dated Jan. 14, 1895, with interest at 6%, on which the 
 following payments are indorsed : 
 
 Feb. 20, 1895, $400; May 5, 1895, $200; Aug. 2, 1895, 
 $600. 
 
 The time between Jan. 14 and Dec. 20 = 11 mo. da. ($1600). 
 The time between Feb. 20 and Dec. 20 = 10 mo. ($400). 
 Tlie time between May 5 and Dec. 20 = 7 mo. 15 da. ($200). 
 The time between Aug. 2 and Dec. 20 == 4 mo. 18 da. ($000). 
 The amount of $ 1000 for 11 mo. da. @ % = $ 1089.00. 
 The amount of $400 for 10 mo. @ 0% = $420. 
 The amount of $200 for 7 mo. 16 da. @ 6 % = $207.60. 
 
INTEREST 
 
 255 
 
 The amount of .$000 for 4 mo. 18 da. @ 0% = $013.80. 
 The total amount of the payments Dec. 20 = $420 + $207.60 + $613.80 
 = $1241.30. 
 
 .-. the amount due Dec. 20 = $1089.00 - $1241.30 = $448.30. 
 
 243. This solution is in accordance with the Merchants' 
 Rule, wliieli is as follows : When the 7iote is paid in full in a 
 year or less^ find the amoutit of the note and of each payment 
 at the date of settlement. From the amount of the note subtract 
 the sum of the amounts of the payments. The difference thus 
 found is the sum due on the date of settlement. 
 
 Exercise 161 
 
 1. What is due Nov. 1, 189o, on a note for f 2000, dated 
 Feb. 1, 1895, with interest at 0%, on which the following pay- 
 ments are indorsed: Jnly 1, 1895, f 600; Sept. 1, 1895, $800? 
 
 2. On a note for $ 2400, dated Sept. 8, 1892, and drawing? ()<^, 
 the following payments were indorsed: Oct. 8, 1892, $(500; 
 Jan. 8, 1893, $ 250 ; July 15, 1893, $ 850. What was due on 
 the note Sept. 8, 1893 ? 
 
 3. A note for f 1500, dated April 14, 1894, with interest at 
 5%, bears the following indorsements: June 8, 1894, $450; 
 Oct. 29, 1894, $ 750. Wliat is due April 14, 1895 ? 
 
 4. On a note for f 1050, dated Oct. 0, 1893, and bearing interest 
 at 5J%, the following payments were made: Dec. 5, 1893, $240; 
 Feb." 25, 1894, $320; June 30, 1894, $300. AVhat is due Aug. 
 15, 1894? 
 
 5. How much was due on the following note, on Dec. 31, 
 1889 ? 
 
 I 950. Nkw Yohk, Jan. 2, 1889. 
 
 For value received, I promise to i)ay James IJrown or order, on 
 
 demand, nine hundred and fifty dollars, with interest from date 
 
 at 6% per annum. 
 
 George Thompson. 
 
 I 
 
 <i,ii 
 
 n 
 
256 
 
 ARITHMETIC 
 
 On this note the foUowing payments were indorsed : Feb. 22, 
 1889, 1225; May 22, 1880, 185; July 21, 188t), ii;i25; Sept. 21, 
 1889, $ 325. 
 
 244. The following is the method of solving questions in 
 partial payments when the note runs longer than a year: 
 
 A note for .ii<2400, dated March 15, 1893, and drawing 
 interest at 6%, had the following payments indorsed upon it: 
 June 30, 1893, $250; Sept. 12, 1893, |25; April 9, 1894, 
 1450; Sept. 14, 1894, !i 84.50. How much was due on the 
 note March 4, 1895? 
 
 yr. 
 1893 
 
 1893 
 
 mo. 
 6 
 3 
 
 da. 
 30 
 15 
 
 (June 30) 
 (March 15) 
 
 
 yr. mo. 
 1894 4 
 1893 9 
 
 da. 
 
 9 
 
 12 
 
 (April 9) 
 (Sept. 12) 
 
 
 3 
 
 15 
 
 ($250) 
 
 
 
 
 27 
 
 (§450) 
 
 1893 
 1893 
 
 9 
 6 
 
 2 
 
 12 
 30 
 
 12 
 
 (Sept. 12) 
 (June 30) 
 
 (§25) 
 
 
 1894 9 
 1894 4 
 
 14 
 9 
 
 (Sept. 14) 
 (April 9) 
 
 
 6 
 
 5 
 
 (§84.50) 
 
 
 
 
 yr. mo. 
 1895 3 
 1894 9 
 
 da. 
 29 
 14 
 
 (March 29) 
 (Sept. 14) 
 
 
 
 
 
 
 6 
 
 15 
 
 
 
 
 The interest on $2400 from March 15, 1893, to June 30, 1893 = $42. 
 
 The amount due June 30, 1893 = 8 2400 + $42 -$250 = .§2102. 
 
 The interest on $2192 from June 30, 1893, to Sept. 12, 1893 = $26.30. 
 
 The interest on §2192 from Sept. 12, 1893, to April 9, 1894 = $75.G2. 
 
 The amount due April 9, 1894 = §2192 + §20.30 + $75.02 -§25 -§450= 
 1818.92. 
 
 The interest on § 1818.92 from April 9, 1894, to Sept. 14, 1894 = $ 40.99. 
 
 The amount due Sept. 14, 1894 = $1818.92+ §40.99 - $84.50 = $1781.41. 
 
 The hiterest on § 1781.41 from Sept. 14, 1894, to March 29, 1896 = $57.90. 
 
 .-. the amount due March 3, 1895 = § 1781.41 + $ 57.90 = § 1839.31. 
 
INTKIIKST 
 
 257 
 
 Note. — In the solution of the above example, the interest on ^2102 from 
 June 30, 1803, to Sept. 12, 1803, ia found to be !$ 20.30, wiiiiMi is greater than 
 the payment uuule on Sept. 12 ; conseiiuently, the interest is again eomputed 
 on ."$2102 from Sept. 12, 1803, to April 0, 1804, and the interest is found to be 
 $75.02. The sum of the two payments ,$2r) and $450 is now greater than 
 the sum of the two interests .$20.30 and ;$ 75.02, and tlie solution proceeds as 
 given above. 
 
 245. The following is the rule for finding the amount due 
 on a note at a given date, when the sum is not paid within a 
 year. It is known as the United States Rule because it is 
 adopted by the Supreme Court of the United States. F'utd the 
 amount of the principal until the time of a pat/ment. Subtract 
 the payment made from this amount^ and use the remainder for 
 a new principal. Continue this process until the time of settle- 
 ment^ ivheyi the last amount is the sum due. If however^ any 
 payment is less than the accrued interest^ compute the interest 
 for the next period on the same principal as before^ and do this 
 until the sum of the payments equals or is yreater than the sum 
 of the interests. 
 
 Exercise 162 
 
 1. On a note for $ 2000, dated Jan. 24, 1890, and drawing G% 
 interest, are indorsed the following payments: May 24, 1890, 
 f 440; Aug. 24, 1890, $324; Feb. 24, 1891, $139. How much 
 is due July 24, 1891 ? 
 
 2. A note for f 1200, dated Jan. 18, 1889, and drawing interest 
 at 5%, had payments indorsed upon it as follows: March 18, 
 1889, $ 210 ; Sept. 18, 1889, $ 15 ; Feb. 12, 1890, $ 2G0. Find the 
 balance due May 9, 1890. 
 
 3. On a mortgage for f 3750, dated May 10, 1887, and bearing 
 interest at 0%, there were paid May 16, 1888, $350; Sept. 18, 
 1888, $280; Jan. 22, 1889, $750; May 10, 1889, $925. What 
 sum was due on the mortgage Oct. 31, 1889 ? 
 
 \ 
 
 iit 
 
258 
 
 ARITHMETIC 
 
 4. How much was due on the following note, Oct. 30, 1889 ? 
 
 f 850. New York, Oct. 30, 1887. 
 
 For value received, I promise to pay Alex, ''.''hompson or order, 
 on demand, eight hundred and fifty dollars, with interest from 
 
 date at 6%. 
 
 John Stuart. 
 
 On this note the following payments were indorsed : April 20, 
 
 1888, 1125; Nov. 20, 1888, 1125; Jan. 20, 1889, $75; July 20, 
 
 1889, 1 425. 
 
 Compound Interest 
 
 246. Compound Interest is interest which is found for stated 
 periods and added at the end of each period to the principal, 
 the sum of the principal and interest becoming the new prin- 
 cipal. 
 
 The unit of time is 1 yr., although the interest may be 
 compounded annually, semiannually, quarterly, and so on. 
 
 Thus 6% compounded semiannually means that each new 
 principal is increased each 6 mo. by 3% of itself. 
 
 247. If 15000 deposited at a savings bank draws interest 
 at 4%, semiannually, the interest due at the end of the first 
 half-year will be 2% of $5000 or !|100. 
 
 If this $ 100 is not drawn, it is placed to the credit of the 
 depositor, who has now f 5100 on deposit. 
 
 The interest for the second half-year is 2% of 15100 or 
 1102. 
 
 If this is not drawn, it is placed to the credit of the depos- 
 itor, making his deposit 15202. 
 
 The interest for the third half-year is 2% of $5202 or 
 $ 104.04. 
 
 If this is not drawn, it is placed to the credit of the 
 
 d 
 
 el 
 
 w 
 
 m 
 
INTEREST 
 
 259 
 
 depositor, making his deposit ^1 5306.04 at the end of 1 yr. 
 6 mo. 
 
 Tlius i>5000 at 4% interest, compounded sc^miannually, 
 will in 1 yr. 6 mo. amount to «t5806.04; and the compound 
 interest for that time will be *5300.04 - -f 5000 = *306.04. 
 
 
 § 5000 original principal 
 
 
 1.02 
 
 
 10000 
 
 
 50000 
 
 
 $5100.00 amount at the end of the first 
 
 period 
 
 1.02 
 
 
 10200 
 
 
 51000 
 
 
 $5202.00 amount at the end of the second 
 
 period 
 
 i.02 
 
 
 10404 
 
 
 52020 
 
 
 $5300.04 amount at the end of the third period 
 
 248. Find the compound interest on '1}«5000 for 1 yr. 
 10 mo. 15 da. at 4%, payable semiannually. 
 
 As in the last paragraph, find the amount of 15000 for 
 1 yr. 6 mo., and then complete tlie work thus : 
 
 The rate per cent for 4 mo. 15 da. or I yr. = 1x4 % = 1 J %. 
 
 $ 5300.04 amount at t';ie end of the third period 
 l.OU 
 
 2C6302 
 530604 
 6306040 
 
 $5385.0306 amount at the end of the fourth period 
 
 /. the compound interest = $5385.63 - $5000 = $385.63. 
 
 Instead of finding the rate for 4 mo. 15 da., wc might have found the 
 interest on $5306.04 for 1 yr. at 4%, and then have taken I of it to find 
 the interest for the last period, since 4 mo. 15 da. is | of 1 yr. 
 
 n 
 
260 
 
 ARITHMETIC 
 
 i I' 
 
 249. Find what principal will in 1 yr. 10 mo. and 15 da. 
 amount to -15385.63. 
 
 From a study of the preceding paragraph it will appear 
 tliat to find the principal at the beginning of the fourth 
 period, or the amount at the end of the third period, we 
 must divide $5385.63 by LOIJ or 1.015, since the fourth 
 amount is found by multiplying the fourth principal by 
 1.011. 
 
 The principal at tlie beginning of tlie fourth period = $6385.63 ~- 1.016 
 
 = -15300.04. 
 The principal at the beginning of tlie third period = $ 5306.04 ~- 1.02 
 
 = $ 5202. 
 The principal at the beginning of the second period = $5202 ~ 1.02 
 
 = $ 5100. 
 The principal at the beginning of the first period = $ 5100 -^ 1.02 
 
 = .$ 5000. 
 .-. the principal = $5000. 
 
 In practice it is often more convenient to divide by the 1.02 three times 
 in succession and then by 1.015 for the last division. 
 
 Exercise 163 
 
 Find tlie amount and the; compound interest of: 
 
 1. f 800 for ;^ yr. at 5%, compounded annually. 
 
 2. .'i)4jjr) for 4 yr. at 4%, compounded annually. 
 
 3. $250 for 2 yr. at 6%, compounded semiannually. 
 
 4. Find t\w. amount and also the compound interest on 1 1000 
 for ?y yr. at 5*/^;. 
 
 5. In question 4 what Avonld the amount have been at simple 
 interest? How much has co be paid as interest on interest? 
 What fraction is it of the i'l-st year's interest ? AVhat per cent? 
 
 6. Find the amount of f ,300 for 2 yr. at G%, interest payable 
 semiannually. 
 
INTEREST 
 
 261 
 
 7. Eind the amount of f>650 for 1 yr. 3 mo., interest payable 
 quarterly at 4% per annum. 
 
 8. Find the compound interest on $8240 for 2 yr. at 5%, 
 payable semiannually. 
 
 9. State how to find the amount of a sum of money at com- 
 pound interest, for a given time and rate. 
 
 10. Find the amount and also the compound interest on 1^2500 
 for 1 yr. 10 mo. and 15 da. at 6%, payable semiannually. 
 
 11. What principal will amount to $ 2247.20 in 2 yr. at 6%? 
 
 12. What sum of money put out at compound interest for 2 yr. 
 at 7% Avill amount to $100? 
 
 13. What sum of money put out for 2 yr. at 5%, payable half- 
 yearly, will amount to $ 600 ? 
 
 14. State how to find the principal that amounts to a given 
 sum of money at compound interest for a given time and rate. 
 
 15. Find the difference between the simple and compound 
 interest of $ 1050 for 3 yr. at 4%. 
 
 16. A sum of money put out at simple interest for 2 yr. at 0% 
 amoimted to $ 896. To what sum would it have amounted had it 
 been lent at compound interest ? 
 
 17. The simple interest on a sum of money for 3 yr. at 7% is 
 f 420. What is the compound interest of the same sum for the 
 same time ? 
 
 18. A man deposits in the savings bank $1500, on which the 
 interest at 3% per annum is to be added to the principal every 
 6 mo. How much money has the man in the bank at the end 
 of 2 yr. ? 
 
 19. What will be the amount, compound interest, of |2400 for 
 1^ yr. at 10% per annum, paid half-yearly, and at wliat rate, simple 
 interest, will it amount to the same sum in the same time ? 
 
 '1 
 
 '4 
 
 y 
 1 
 
262 
 
 ARITIBIETIC 
 
 * Annual Intkrest 
 
 250. Annual Interest is the sum of the simple interest on 
 the principal and on each year's interest, if unpaid, from the 
 time it is due until the date of settlement. 
 
 Annual Interest is charged when the words "interest pay- 
 able annually " are in the note. 
 
 251. Find the amount duo Oct. 9, 1895, on a note for 
 11200, dated July 5, 1891, with interest payable annually at 
 6%. 
 
 (1) 
 
 }>•• 
 
 IIIO. 
 
 (la. 
 
 1895 
 
 10 
 
 9 
 
 1801 
 
 7 
 
 5 
 
 4 3 4 
 
 The interest on § 1200 f(«i- 1 yr. at 0% = $ 72. 
 The interest on $ 1200 for 4 yr. a nio. 4 (hi. at 0% = .<i;;J06.80. 
 
 (2) The interest clue July 5, 1892, bears interest for l] yr. .'3 nio. 4 da. 
 The interest due July o, 189.">, bears interest for 2 yr. il nio. 4 da. 
 The interest due July 5, 1894, bears interest for 1 yr. 3 mo. 4 da. 
 The interest due July 5, 1895, bears interest for 3 nio. 4 da. 
 
 •. the interest on $ 1200 for 1 yr., i.e. $ 72, bears interest for 7 yr. 10 da. 
 The interest on $ 72 for 7 yr. 10 da. (a) 0% = .<^ 30.43. 
 .-. the amount due = $ 1200 + ^ 300.80 + $ 30.43 = $ 1537.23. 
 
 Exercise 164 
 
 1. Find the amount due July 10, 1S9G, on a note for f 900, 
 dated July IG, 1892, with interest payable annually at 0%. 
 
 2. Find the amount due March 9, 1890, on a note dated Jan. 3, 
 1892, for 11500 at 5%, interest payable annually. 
 
 3. Find the amount due Sept. 26, 1890, on a note for f 280, 
 dated June 5, 1893, witli interest at 4] %, payable annually. 
 
 4. Find the amount due June 8, 1890, on a note dated Aug. 12, 
 1892, for $ 712.50, with interest at i>%, payable annually. 
 
INTEREST 
 
 263 
 
 5. Find the amount due Nov. 4, 1896, on a note dated Dec. 24, 
 1892, for $842, with interest at 4^%, payable annually. 
 
 6. Find the simple, annual, and compound interest on a note 
 for $ 1000, dated Aug. 12, 1892, and due Aug. 12, 1896, interest 
 at 6%. 
 
 7. Compare the methods of finding the simple, annual, and 
 compound interest on a sum of money, for a given time, at a 
 given rate per cent. 
 
 
 * Stocks and Bonds 
 
 252. The capital of a bank or other public company is 
 called Stock. 
 
 It is usually divided into a definite number of eqaal parts 
 
 or Shares. 
 
 The original value of a share, generally |100, $50, or $25, 
 
 is called its Par Value. 
 
 253. The Market Value of a share is the sum for which it 
 can be sold. 
 
 Stock is said to be above par, or at a premium, when the 
 market value is greater than its par value; it is said to be 
 below par, or at a discount, when the market value of the 
 share is less than its par value. 
 
 Thus if $100 stock sells for $112 money, the stock is at 
 12% premium, and it is said to sell at 112. 
 
 If $100 stock sells for $96 money, the stock is at 4% dis- 
 count, and is quoted at 96. 
 
 254. A Stock Broker is a person who buys or sells stocks, 
 bonds, or similar securities. His commission, called Brokerage, 
 is reckoned at a certain rate per cent, which varies, the most 
 common rate being J of 1% or |%. 
 
 I ; 
 
264 
 
 ARITHMETIC 
 
 255. A Bond is a note bearing interest issued by a govern- 
 ment or corporation. There are two kinds of bonds, 
 
 registered and coupon bonds. 
 
 A Registered Stock or Bond is one which is registered on 
 the books of tlie company or government issuing it, and 
 which cannot be sold or transferred except in writing at the 
 office of the treasurer. 
 
 An Interest Coupon is an interest certificate payable to the 
 bearer, which is attached to the bond, and which is detached 
 when the interest becomes due. 
 
 One coupon is attached to the bond for each instalment 
 of interest to be paid on it. 
 
 256. The following is the quotation of U. S. bonds in the 
 market of July 8, 1896: 
 
 
 
 Bill 
 
 A sked 
 
 Registered 
 
 2's 
 
 95 
 
 • • • 
 
 Kegistered 
 
 4's 
 
 108 
 
 108 Ji^ 
 
 Coupon 
 
 4's 
 
 100 
 
 109| 
 
 New Coupon 
 
 4's 
 
 iiov 
 
 116f 
 
 Registered 
 
 5's 
 
 112| 
 
 113 
 
 257. The following is the quotation of stock in the market 
 of July 8, 1896 : 
 
 Stocks 
 Am. Sugar 
 Am. Sugar pfd. 
 C. B. & Q. 
 C. R. I. & P. 
 Michigan Central 
 Manhattan 
 Del. & Hud. 
 
 OjK'iiinff 
 
 lOlf 
 
 63 
 90 
 
 97.1- 
 124^ 
 
 Iliffliest 
 lllJ- 
 101} 
 
 ni 
 
 63| 
 96 
 97^ 
 124| 
 
 Lowest 
 
 109| 
 
 101| 
 
 71| 
 
 62| 
 
 96 
 
 96 .V 
 
 124^ 
 
 Julys 
 
 110 
 lOlf 
 
 72J 
 62 f 
 96 
 96 1 
 1241 
 
 Closings 
 
 July 7 
 110] 
 
 lou 
 
 721 
 63} 
 
 97 
 
 124^ 
 
 Ezercisd 165 
 
 1. At what different prices is Am. Sugar stock quoted at 
 July 8, 1896 ? 
 
INTEREST 
 
 265 
 
 2. What will a seller receive from his broker for 1 share of 
 C. B. & Q. stock, July 8, 1896, at each of the quoted prices, 
 brokerage being J%? What from 1 share of Am. Sugar pfd. ? 
 
 3. What will a buyer have to pay for 1 share of Manhattan 
 stock at each (quotation, July 8, 18*.)6, brokerage l^? What 
 for C. li. I. & P. ? 
 
 4. At what per cent premium are the different quotations for 
 Am. Sugar, Am. Sugar pfd., and Del. & Hud. stock, July 8, 1896 ? 
 
 5. At what per cent discount are the different quotations for 
 C. B. & Q., C. R. I. & P., Michigan Central, and Manhattan stock, 
 July 8, 1896 ? 
 
 6. What would I receive for 1 share of ])el. & Hud., July 
 8, 1896, sold at the highest price, brokerage J-%? What for 10 
 shares ? What for 100 shares ? 
 
 7. What would I have to pay for 1 share of C. B. & Q. 
 stock, July 8, 1896, bought at the opening price, brokerage ^%? 
 What for 10 shares ? What for 100 shares ? 
 
 8. What would I have to pay for 1 share of Am. Sugar 
 stock, July 8, 1896, at the lowest (j noted pri(;e, brokerage ^%? 
 What for io shares ? What for 100 shares ? 
 
 9. What would I receive for 1 share of C. B. & Q. R. R. 
 stock, July 8, 1896, sold at the lowest quotation, brokerage J-%? 
 What for 10 shares ? Wliat for 100 shares ? 
 
 10. What will 1 share of C. B. & Q. stock cost, July 8, 1896, 
 at the opening price, brokerage j^%? JEow many shares can I 
 buy for .| 144 ? For f 216 ? For .fi 360 ? 
 
 11. What will 1 share of C. R. T. & P. stock cost, July 8, 
 1896, at the lowest (piotation, brokerage |%? How many shares 
 can be bought for $ 125 ? For f 625 ? 
 
 12. What is the difference between the highest and lowest 
 quotations of Manhattan stock, July 8, 1896 ? 
 
 li 
 
266 
 
 ARITHMETIC 
 
 ;> 
 
 13. What is the difference between the closing prices of Del. & 
 Hud. stock, July 7 and July 8, 1896 ? 
 
 14. What is the difference between the opening and closing 
 prices of C. K & Q. stock, July 8, 1896 ? 
 
 258. (1) How much will be realized by selling out 66 
 shares of N. Y. Central H. H. stock at 95|, brokerage |%? 
 
 1 share of stock sells for $95' - $ ^ or $OC)'i money. 
 .-. 66 shares of stock sell for 60 x ii!()r>3 or .$0294.75 money. 
 Note. — The brokerage = 66 x ^ | = $8.26. 
 
 (2) How many shares of Del. & Hud. stock at 124^, 
 brokerage |%, can I buy Tor 15591.25? 
 
 1 share costs $ 124i + $ J or $ 124.25. 
 
 .-. the number of shares = $ 5591.21 ~ $ 124.25 = 45. 
 
 Note. —The brokerage = 45 x $ | = $ 5. 62 J. 
 
 (3) A broker realizes $7.25 from a sale of stock, brokerage 
 J%. What was the par value of the stock sold? 
 
 I % of the par value = .$ 7.25. 
 1 % of the par value = $ 58. 
 100 % of the par value = $ 5800. 
 .•. the par value = $ 5800. 
 
 (4) I sold through my broker 95 shares of Chicago and 
 Northwestern R. R. stock, receiving for it $ 9476.25, brokerage 
 1%. Find at what price the stock was quoted. 
 
 05 shares sell for 39476.25. 
 1 share sells for .«> 9476.25 ^ 05 = $ 09.75 ; 
 i.e. excluding brokerage, the selling price = 99|. 
 .-. stock is quoted at 99| + i or 99|. 
 
 (5) What annual income will be realized from $3828.12 J, 
 invested in the U. S. 4's at 109|, brokerage |%? 
 
 1 share costs $ 109| + $ i = ^ 109| = $ 109.375. 
 
 The number of shares = $3828.125 -f- $109,375 = 35. 
 .-. the income = 35 x $ 4 = $ 140. 
 
tl 
 
 INTEREST 
 
 267 
 
 (6) What amount of money must be invested in 6% stock, 
 at 119|, brokerage J%, to realize an income of $978? 
 
 1 share yields an income of ^0. 
 The number of shares = ^ 1)78 -=- )$ = 103. 
 1 share costs $ 11»:| + )$ ^ = $ 119 J. 
 1G3 shares cost 103 x !$ IIU^ = .«$ 19,639.02 J. 
 .-. $19,539.02^ must be invested. 
 
 (7) If 6% stock is bought at 109|, what per cent does it 
 pay on the investment, brokerage l%? 
 
 1 share costs $ 109 J + $ J = $ 1 10. 
 
 $ 110 yields an income of $6. 
 
 .•. the rate per cent = Yis or 5^% of the investment. 
 
 (8) What must I pay for 8% stock to realize an income of 
 6% on the investment, brokerage |%? 
 
 8 % of the cost of 1 share = .I! 0. 
 1 % of the cost of 1 share = !$ |. 
 100 % of the cost of 1 share = $75 ; 
 i.e. excluding brokerage, the cost price is $75. 
 .'. stock is quoted at 74J. 
 
 i 
 
 Exercise 166 
 
 1. What will 25 shares of Adams Express stock cost at 148, 
 brokerage J%? 
 
 2. What is realized from the sale of 208 shares of C. B. & Q. 
 K R. stock at 71^, brokerage |%? 
 
 3. What did I pay for 39 shares Chicago and Northwestern, 
 July 8, 1896, stock selling at 99^ and brokerage being 1%? 
 
 4. Find what I received from the sale of 84 shares of Western 
 Union stock at 82|, brokerage *-%. 
 
 5. What is the cost of 1 20,000 U. S. 4's at 112f, broker- 
 age i%? 
 
 6. Find the cost of f 24,000 U. S. 4's at 116f, brokerage 
 
268 
 
 ARITHMETIC 
 
 If 
 
 7. July 8, 189C, 40 shares of Chicago City Railway, reg. at 
 220, were sold, brokerage |%. Find what was received by the 
 owner of the stock. 
 
 8. How many shares of Manhattan R. R. stock at \)7\ can I 
 buy for $3505.50, brokerage i%? 
 
 9. July 8, 1896, Am. Sugar pfd. stock was quoted at 101^. 
 How many shares were bought for f 4257.75, brokerage ^%? 
 
 10. A stockholder sold 1). L. and W. R. R. stock at 157J, receiv- 
 ing all togetljer $ 3771. How many shares did he sell, brokerage 
 being ^%? 
 
 11. How many shares of Wells Fargo Express stock must I 
 sell at 95, brokerage ^%, to receive f 9677.25? 
 
 12. If from my sales of New York Central R. R. stock at 95J, I 
 receive $6278.25, how much stock did I sell, brokerage being ^%? 
 
 13. How many shares of D. L. and W. R. R. stock at 158^ can 
 be bought for $2855.25, brokerage ^%? 
 
 14. A broker sells 24 shares of stock on a commission of J|,%. 
 How much does he realize ? 
 
 15. How many shares of stock does a broker sell to realize a 
 commission of $16.25, brokerage i%? 
 
 16. A broker realizes $ 12.50 from the sale of stock, brokerage 
 i%- ^Vhat was the par value of the stock sold and what did it 
 sell for at 70f ? 
 
 17. A broker received $46.50 for buying stock on a commis- 
 sion of f %. How much stock did he buy ? 
 
 18. I sold through my broker 40 shares of stock, receiving for 
 it $4860, brokerage J%. At what price was the stock quoted? 
 
 19. A person received $6053.12^ for $6500 stock after paying 
 his broker i%. Find at what per cent discount the stock was sold. 
 
 20. July 7, 1896, $ 1654.25 was paid for 26 shares of Rock 
 Island R. R. stock, brokerage ^%. At what was Rock Island 
 stock quoted, July 7 ? 
 
1 
 
 INTKHEST 
 
 2G9 
 
 21. What annual income will be. ()l)tainod from $ ()()71, invested 
 in U. S. 4's coup, of i\)'2r> at 11()|, l)r()kcra<;e J%? 
 
 22. A person paid .f SaTS.no for U. S. 4's at 112}, brokerage 
 i<,. TA'^hat was his income from the bonds? 
 
 23. If 1 invest fSoiS.*).!;") in stock at Oo^, paying 5% dividend, 
 what will be my income, brokerage i%? 
 
 24. What income will be realized fron\ .1$ 9229.50 invested in 
 stock at 109 J, brokerage J%, paying a dividend of 5.^%? 
 
 25. What amount of money must be invested in S% stock at 
 158^, brokerage J%, to realize an income of $ 109() ? 
 
 26. What sum must T invest in 4h% stock at 99|, to produce 
 an annual income of $ 1038, brokerage ]i %? 
 
 27. How much must T invest in U. S. 5's at 112J to realize 
 an annual income of Ji;4r)0, brokerage ^%? 
 
 28. If street railway stock bought at 232 yields a half-yearly 
 dividend of 0^%, how much must I invest to obtain a semiannual 
 income of $325, brokerage ' %? 
 
 29. If I buy stock through a broker who charges J,%, how 
 much must I invest in stock at 153, paying 9% dividends, to 
 secure an income of f 1350 ? 
 
 30. If 4|% stock is bought at 74|, brokerage i%, what per 
 cent does it pay on the investment ? 
 
 31. If 8% stock is bought at 159|, what per cent does it pay 
 on the investment, brokerage J%? 
 
 32. What must I pay for 4% stock to pay 5% on the invest- 
 ment, brokerage i%? 
 
 33. What rate: of interest do I realize on an investment in 0% 
 stock at 107^, brokerage i%'.' 
 
 34. What must I pay for 5% stock to yield an income of G% 
 on my investment ? 
 
 35. A person receives $000 from an 8% bank dividend. How 
 much stock does he own ? 
 
 I i 
 

 N^ 
 
 IMAGE EVALUATION 
 TEST TARGET (MT-3) 
 
 1.0 
 
 I.I 
 
 11.25 
 
 L£|2j8 |25 
 
 ■^ I2i2 122 
 
 :^ u£ 12.0 
 
 ■luu 
 
 Hiotographic 
 
 Sciences 
 
 Corporation 
 
 23 WIST MAIN STRUT 
 
 WnSTIR.N.Y. 14510 
 
 (716)a72-4S03 
 

 Sf. 
 
 ^ 
 
270 
 
 ARITHMETIC 
 
 36. A person having $5000 bank stock sells out when it is at 
 40% premium. What amount of money does he receive, brokerage 
 being i%? 
 
 37. Bought through a broker 1600 shares (f 100) R. R. stock at 
 69^, brokerage i%. What was the gross cost of the stock ? 
 
 38. A speculator bought 36,500 shares (f 100) R. R. stock at 
 39 1, and sold them at 40§. What was his gain, brokerage i% on 
 both transactions ? 
 
 39. A bank declared a dividend of S^%. How much should a 
 stockholder owning 120 shares (f 50) receive ? 
 
 40. One company guarantees to pay 6% on shares of f 100 
 each; another guarantees at the rate of 5|% on shares of f 30 
 each ; the price of the former is $ 124.50, and of the latter f 34. 
 Find the rates of interest which they return to the purchaser. 
 
 41. A broker receives f 42,100 to invest in U.S. 5-20 bonds, 
 after reserving ^% on the par value of the amount purchased. 
 What was his commission, the bonds being at a premium of 5%? 
 
 42. A man bought through a broker 1900 shares (f 100) R. R. 
 stock at 54} and sold them at 55|. What was his net profit on 
 the transaction, brokerage each way |%? 
 
 43. An insurance company declared a dividend of 9%. What 
 rate is that on the market value of the shares which are at 185 ? 
 
 44. Compare the rates on the cash values of 6% on stock at 
 216 and 3^% on stock at 125. 
 
 45. Sold 37 shares (f 25) B. and L. Association stock, receiving 
 therefor f 1019.81. At what rate was the stock sold ? 
 
 46. Bought through a broker 750 shares (f 50) in the Farmers' 
 Loan and Savings Society', paying therefor $ 43,968.75. At what 
 quotation were they bought, brokerage |^%? 
 
 47. Bought stock at 197f and sold it at 194f, having mean- 
 while received a dividend of 6% on it. My net gain on the 
 transaction after paying |% brokerage each way is $336. How 
 many shares ($ 40) did I buy ? 
 
INTEREST 
 
 271 
 
 48. How many railway shares ($100) at 40% discount must 
 be sold in order that the proceeds invested in bank stock, which 
 is 4% below par, and pays a dividend of 7%, may yield an income 
 of f 1680, brokerage included? 
 
 49. Explain the terms: Stocks, Shares, Dividends. When is 
 stock at par ? At a premium ? At a discount ? 
 
 50. When the 3^ per cents are at 98, what must be the price 
 of another stock yielding 4|^%, so that the latter may be as profit-, 
 able as the former, brokerage included? 
 
 Exchange 
 
 259. If A of Chicago owes B of St. Paul a sum of money, 
 he can discharge the debt in any one of several ways. He 
 can buy a post-office order at the Chicago post-office payable 
 to B at the post-office at St. Paul; he can buy an express 
 order at the office of an express company, payable to B at 
 any office of the same company ; or he can buy a draft at a 
 bank payable to B at a bank in St. Paul. 
 
 Give some reasons why it is better to discharge a debt by 
 means of a post-office order, express order, or draft than by 
 sending the money in a registered letter or by express or 
 check. 
 
 260. Tlie following are the rates charged for express orders 
 to any part of the United States or Canada : 
 
 Rates for orders not over 
 
 $5.00, 5^. 
 
 140.00, 
 
 n^. 
 
 1 10.00, 8^. 
 
 60.00, 
 
 20^. 
 
 20.00, 10 ^. 
 
 75.00, 
 
 25^. 
 
 30.00, 15)^. 
 
 100.00, 
 
 30 <2^. 
 
 Over 1 100 at above rate. 
 
 
 
272 
 
 ARITHMETIC 
 
 261. The rates charged for post-office orders to any part of 
 the United States are the same as for express orders up to 
 $ 100, but orders not exceeding $ 2.50 are sold at a charge of 
 3^. Single post-office orders are not issued for more than 
 $ 100, and for larger amounts additional orders are issued. 
 
 Exercise 167 
 
 . 1. What is the cost of an express order for $25? $44? 
 f73? $78? 
 
 2. What is the cost of a post-office order for $80? $32? 
 $95? $1.50? 
 
 3. What is the cost of an express order for $ 75 ? $ 100 ? 
 
 4. What is the cost of a post-office order for $ 75 ? $ 100 ? 
 
 5. What is the cost of a draft for $75? $100? $150? 
 $240? $325? $180? The charge in each case is \% and 
 the least charge 25 ^. 
 
 6. By which of the three methods given in questions 3, 4, and 5 
 is it cheaper to send money in sums greater than $ 75 ? In sums 
 less than $ 75, if 25 ^ is the smallest charge for a draft ? 
 
 262. Exchange is generally conducted through bankers, 
 who issue drafts directing a second bank to pay a specified 
 sum of money to the order of the person named in the draft. 
 
 A Time Draft is one payable at a specified time after sight 
 or date. 
 
 If A in Chicago owe B in St. Paul a sum of money, B may send a draft to 
 A for the amount. If A accepts the draft, he writes tlie word "accepted" 
 with the date across the face and signs his name. 
 
 Exchange is the system of paying debts to persons in dis- 
 tant places without actually sending the money, by means 
 of money orders and drafts. 
 
4 
 
 fM 
 
 INTEREST 
 
 273 
 
 )f 
 
 10 
 
 f 
 
 263. (1) Find the cost of a draft on New York for 1 600, 
 when exchange is ^% premium. 
 
 The premium = \ % of $000 = $ 1.50. 
 .-. the cost = $000 + $ 1.50 = $601.50. 
 
 (2) Find the cost of a draft on New Orleans for 11200, 
 payable 60 da. after date, exchange being \% discount, and 
 interest 6%. 
 
 The discount = \ % of $ 1200 = $ 3.00. 
 The discount for 03 da. = % of $ 1200 for 03 da. = $ 12.60. 
 .-. the cost = $ 1200 - % 3.00 - % 12.60 = $ 1184.40. 
 Show that if exchange had been \ % premium, the cost would have been 
 $ 1190.40. 
 
 Exercise 168 
 
 1. Find the cost of a draft for % 900 at i% premium. 
 
 2. Find the cost of a draft for % 1600 at ^% discount. 
 
 3. Find the cost of a draft for | 4500 at f % discount. 
 
 4. Find the cost of a draft for $ 2800 at f of 1% premium. 
 
 5. Find the cost of a draft for % 1000, payable in 60 da., 
 exchange being \fJo premium, and interest 6%. 
 
 6. Find the cost of a draft for f 360, payable in 30 da., 
 exchange being i% discount, and interest 5%. 
 
 7. Find the cost of a draft for % 1250, payable in 60 da., 
 exchange being }% premium, and interest 4^%. 
 
 8. Find the cost of a draft for f 1800, payable in 30 da., when 
 exchange is at par, and interest 4%. 
 
 9. Find the cost of a bill of exchange on London for £ 600, 
 when exchange is quoted at % 4.88. 
 
 10. Find the cost of a 60-da. draft on Liverpool for £ 750, 
 exchange at 60 da. being $ 4.86. 
 
 11. What will be the cost of a bill of exchange in Paris for 
 2400 francs at 5.16^ francs per $ 1 ? 
 
 12. What will be the cost of a bill of exchange on Berlin, for 
 2400 marks, the rate of exchange being 95| ct. for 4 marks ? 
 
1:': 
 
 CHAPTER XV 
 
 EATIO AND PROPORTION 
 
 264. If two quantities be expressed in terms of the same 
 unit, their Batio is the quotient obtained by dividing the num- 
 ber measuring the first quantity by the number measuring 
 the second quantity. 
 
 Thus the ratio of $ 3 to f 5 = |, or, as it is frequently 
 written, 3:5. 
 
 The first term of a ratio is called the Antecedent, and the 
 second the Consequent. 
 
 Since a ratio may be expressed as a fraction, both terms 
 of a ratio may be multiplied or divided by the same number 
 without changing its value. 
 
 lifCf 
 
 265. (1) If 15 bbl. of flour cost $ 111, what will 35 bbl. cost ? 
 Multiply $ 111 by f f . •.• 36 bbl. will cost f | of what 16 bbl. cost. 
 
 X = 
 
 — «5 111 
 
 i X 
 
 35 _ 
 T5 — 
 
 $269. 
 
 Or the question may be solved thus : 
 
 15 bbl. cost $111. 
 1 bbl. costs! VV- 
 
 .-. 36 bbl. cost ^^^ ^^^^ = $ 269. 
 
 16 
 
 (2) If 56 men do a piece of work in 21 da., how long will 
 24 men require to do it ? 
 
 Multiply 21 da. by |f. •.• 24 men will take || as long as 56 men. 
 
 da. 
 
 X = V X II = 49 da. 
 274 
 
RATIO AND PROPORTION 
 
 275 
 
 !i 
 
 I 
 
 Or thus : 
 
 56 men do the work in 21 da. 
 1 man can do the work in 56 x 21 da. 
 
 .•.24 men can do the work in ^i^Lil-fii or 49 da. 
 
 24 
 
 ( f 
 
 
 Exercise 169 
 
 1. If 6 articles cost $ 14.30, how much will 13 cost at the 
 same rate ? 
 
 2. If 25 lb. of tea cost f 16, how many lb. can be bought for 
 $56? 
 
 3. If the 4-lb. loaf costs 11^ when flour is $6 a bbl., find 
 its cost when flour is f 7|^ a bbl. 
 
 4. A bankrupt owes $ 3000 ; his assets are f 1740. What 
 sum will a creditor receive whose claim is $ 350 ? 
 
 5. The expense of carpeting a room was $ 100 ; if the breadth 
 of the room had been 4 ft. greater, the expense would have been 
 $ 120. Find the breadth. 
 
 6. If a man working 9-J hr. per da. finishes a piece of work in 
 6 da., in what time would he have finished it if he had worked 
 8 J hr. per da. ? 
 
 7. If a garrison of 1500 men have provisions for 13 mo., how 
 long will their provisions last if it be increased to 2200 ? 
 
 8. If 4 men or 6 women can do a piece of v;ork in 20 da., how 
 long will it take 3 men and 15 women to do the same work ? 
 
 9. A creditor receives f 1.50 for every f 4 of what was due 
 to him, and thereby loses f 301.05. What was the sum due ? 
 
 10. In a certain business one partner, whose share is f\ of the 
 whole, receives from it a profit of $ 859.20. What share is owned 
 by another, whose profit is $ 1969 ? 
 
 11. A person contracts to do a piece of work in 30 da., and 
 employs 15 men upon it ; the work is half finished in 24 da. How 
 
276 
 
 ARITHMETIC 
 
 many additional workmen must be then introduced in order to 
 perform the contract ? 
 
 12. The profits of a garden for 2 yr, were $ 1456 ; the profits 
 of the second yr. being |J of those of the first. Find the profits 
 of each yr. 
 
 13. If 10 men can do a piece of work in 12 da., how soon after 
 beginning must they be joined by 3 more so as to finish the work 
 in 10 da. ? 
 
 14. If $ 120 gain $ 5.81 in 126 da., find the gain in 360 da. 
 
 15. A bankrupt who is paying 37^^ on the dollar divides 
 among his creditors $ 6300. What do his debts amount to ? 
 
 16. If 3 men or 5 boys can do a piece of work in 17 da., in 
 how many da. will 5 men and 3 boys do a piece of work 3 times 
 as great ? 
 
 17. If 3 men can do as much work in a da. as 4 boys, how long 
 will it take 64 boys to finish a piece of work of which 12 men 
 have done J in 16 da. ? 
 
 18. If a debt after a deduction of 3% becomes $ 1008.80, what 
 would it have become after a deduction of 4 % had been made ? 
 
 19. Six sheets of paper measuring 8 in. by 10 in. weigh an 
 ounce. Find the weight of 120 sheets of the same kind of paper, 
 each sheet measuring 11 in. by 17 in. 
 
 20. A person walks from his house to his office at the rate of 
 4 mi. per hr. ; but finding he has forgotten something, returns at 
 the rate of 5 mi. per hr. Compare the time spent in going with 
 that spent in returning. 
 
 21. One train travels 8 J mi. in 20 min., and a second train 9 mi. 
 in 15 min. Compare their rates per hr. 
 
 22. A man can row 6 mi. an hr. in still water. Compare his 
 rate of rowing down a stream which flows at the rate of 2^ mi. an 
 hr. with his rate of rowing up. 
 
 
RATIO AND PROPORTION 
 
 277 
 
 P 
 
 23. One water pipe discharges 141 gal. per hr., another dis- 
 charges 235 gal. per hr. Compare their rates of discharge (a) per 
 hr. ; (b) per min. ; (c) per sec. ; (d) per da. ; (e) per seventh of 
 a da. Also compare the times in which the pipes would each 
 discharge (a) 705 gal. ; (b) 705 qt. ; (c) 705 pt. 
 
 24. Two taps when both open discharge water at the rate of 
 481 gal. per hr. ; the discharge of the smaller of the two being 
 at the rate of 148 gal. per hr. Compare the volume discharged 
 by the larger tap in any given time with the volume discharged 
 by the smaller tap in the same time. Compare also the time in 
 which the larger tap will discharge a given number of gal. with 
 the time required by the smaller to discharge the same number 
 of gal. 
 
 25. A greyhound pursuing a hare takes 3 leaps to every 4 the 
 hare takes; but 2 leaps of the hound are equal in length to 3 
 leaps of the hare. Compare the speed of the hound with that of 
 the hare. 
 
 26. Milk is worth 20^ a gal., but by watering it the value is 
 reduced to 15^ a gal. Find the proportion of water to milk in 
 the mixture. 
 
 27. Two men receive ^15 for doing a certain piece of work. 
 Now one man had worked but 3 da. while the other had worked 5 
 da. on the job. If the money is to be divided in proportion to the 
 lengths of time the men worked, how much should each receive ? 
 
 * CoMrouND Proportion 
 
 266. (1) If 20 men can dig 60 yd. of earth in 4 da., how 
 many yards can 30 men dig in 9 da. ? 
 
 Men 
 
 Yd. 
 
 Da 
 
 20 
 
 60 
 
 4 
 
 30 
 
 X 
 
 9 
 
 Multiply 60 yd. by |§. 
 Multiply the result by f . 
 
 30 men can dig f ^ as many yd. as 20 men. 
 •.• in 9 da. 30 men can dig | as much as in 4 da. 
 
 yd 
 
 II 
 
 .-. X = -"yO- X ig X I = 202^ yd. 
 
278 
 
 ARITHMETIC 
 
 (2) If 120 bu. of oats last 14 horses 56 da., in how many 
 da. will 6 horses consume 90 bu. ? 
 
 Bu. 
 
 Horses 
 
 Da. 
 
 120 
 
 14 
 
 56 
 
 90 
 
 6 
 
 X 
 
 Multiply 66 da. by j^.£^. •.■ 90 bu. will last ^:^(f as many da. as 120 bu. 
 Multiply the result by J/. •.• 90 bu. will last 6 horses ^ as long as 14 
 horses. 
 
 da. 
 
 .-. X = Y X ^2% X V- = 98 da. 
 
 267. In each of the above two solutions, in order that the 
 ratio may easily be seen, the items in the question have been 
 written in horizontal lines. In number (1) we are required 
 to find the number of yards, and the problem is to determine 
 the ratio resulting from each comparison, and how it affects 
 the number of yards. 
 
 In number (2) we'are required to find the number of days, 
 and the problem is to determine the ratios, and how they 
 affect the number of dai/s. 
 
 \'*< 
 
 268. Both problems may, if preferred, be solved by the 
 unitary method, thus : 
 
 20 men in 4 da. dig 60 yd. 
 1 man in 4 da. digs |§ yd. 
 
 0) 
 
 (2) 
 
 1 man in 1 da. digs 
 
 60 
 
 20 X 4 
 
 yd. 
 
 30 men in 1 da. dig ^^_><_§2 yd. 
 
 20 X 4 
 
 OA • n A J. 60 X 30 X 9 
 30 men m 9 da. dig — ^^^— 
 
 ° 20 X 4 
 14 horses eat 120 bu. in 56 da. 
 
 56 X 14 
 
 = 202^ yd. 
 
 1 horse eats 1 bu. in 
 6 horses eat 90 bu. in 
 
 da. 
 120 
 
 56 X 14 X 90 
 
 120x6 
 
 = 98 da. 
 
RATIO AND PllOrORTrON 
 
 279 
 
 14 
 
 To prove the answer correct, substitute the answer in place of x in the 
 horizontal li:^*^ and omit one of the quantities, frame the question and then 
 solve. 
 
 Bu. Horses Da. 
 
 120 14 60 
 
 90 a; 98 
 
 If 120 bu. of oats last 14 horses for 56 da., how many 
 horses will 90 bu. last 98 da. ? 
 
 On solving, % will be found equal to 6, which proves the former solution 
 correct. How many questions can be made from the numbers in the two 
 lines, including the original one ? 
 
 Solve the following questions. State one or more questions 
 in proof for each problem, and prove your answers correct. 
 
 Exercise 170 
 
 1. If 7 horses are kept 20 da. for $ 14, how many will be kept 
 
 7 da. for f 28 ? 
 
 2. If 3 men e?rn $75 in 20 da., how many men will earn 
 $ 78.75 in 9 da. at the same rate ? 
 
 3. If 16 horses eat 96 bu. of corn in 42 da., in how many days 
 will 7 horses eat 66 bu. ? 
 
 4. If 16 horses can plough 1280 A. in 8 da., how many A. will 
 12 horses plough in 5 da. ? 
 
 5. If 20 men can perform a piece of work in 12 da., find the 
 number of men who could perform another piece of work 3 times 
 as great in \ of the time. 
 
 6. If 252 men can dig a trench 210 yd. long, 3 wide, and 2 
 deep, in 5 da. of 11 hr. each, in how many days of 9 hr. each 
 will 22 men dig a trench of 420 yd. long, 5 wide, and 3 deep ? 
 
 7. If 10 men can reap a field of 1\ A. in 3 da. of 12 hr. each, 
 how long will it take 8 men to reap 9 A., working 16 hr. a day ? 
 
 I 
 
280 
 
 ARTTHMETTC 
 
 1. 1 
 
 8. If 25 men can do a piece of work in 24 da., working 8 hr. 
 a day, how many hours a day woiihl ,*^0 men have to work in order 
 to do the same piece of work in 16 da. ? 
 
 9. A town which is defended by 1200 men, with provisions 
 enoM'^h to sustain them 42 da., supposing each man to receive 
 18 oz. a day, obtains an increase of 200 men to its garrison. 
 What must now be the allowance to each man, in order that the 
 provisions may serve the whole garrison for 54 da. ? 
 
 10. If 560 flagstones, each IJ ft. square, will pave a court- 
 yard, how many will be required for a yard twice the size, each 
 flagstone being 14 in. by 9 in. ? 
 
 11. If 20 men in 3 wk. earn f 900, in what time will 12 men 
 earn f 1500 ? 
 
 12. If y\ of a meadow be mown by 12 men in 6 da., find in 
 what time the remainder could be mown by 10 men. 
 
 13. If 36 men, working 16 da., can dig a trench 72 yd. long, 
 18 yd. wide, and 12 yd. deep, how many men can dig a trench 
 64 yd. long, 27 yd. wide, and 18 yd. deep in 24 da. ? 
 
 14. If 25 men build a wall 15 ft. high, 2 ft. thick, and 50 ft. 
 long, in 12 da. of 9 hr. each, how many hours per day must 40 
 men work to build a wall 60 ft. long, 3 ft. thick, and 20 ft. high 
 in 25 da. ? 
 
 15. Twenty men can do a piece of work in 12 da. Find how 
 many men will do half as much again in one-fifth part of the 
 time, supposing that they work the same number of hours in the 
 day, and that 2 of the second set can do as much work in an 
 hour as 3 of the first set. 
 
 16. If 12 men do a piece of work in 21 da., in what time will 
 10 men do a piece of work 1 j as great, if 3 of the first set do as 
 much in an hour as 4 of the second set ? 
 
 17. A miller has a bin 8 ft. long, 4 J ft. wide, and 2 J ft. deep, 
 holding 75 bu. How deep must he make another bin which is to 
 be 18 ft. long and 3f ft. wide, so that its capacity may be 450 bu. ? 
 
 tLi/V.iil->,'i; 
 
 lirtaSi.f 
 
RATIO AND PROPORTION 
 
 281 
 
 hr. 
 [•der 
 
 Ions 
 
 }ive 
 
 Bon. 
 
 the 
 
 in 
 
 
 18. What is the weight of a block of stone 12 ft. 6 in. long, 
 6 ft. 6 in. broad, and 8 ft. 3 in. deep, when a block of the same 
 stone 5 ft. long, 3 ft. 9 in. broad, and 2 ft. 6 in. deep, weighs 
 7500 lb. ? 
 
 Propoktional Parts 
 
 269. (1) Divide 1720 into parts proportional to 4, 5, and 6. 
 
 The total number of parts = 4 + 5 + = 16. 
 
 .-. the first part = *^ of $ 720 = $ 192, 
 the second part = f\ of $ 720 = $ 240, 
 the third part = {^ of $ 720 = # 288. 
 
 (2) Divide 316 lb. into parts proportional to |^, ^, and \. 
 
 Multiplying i, |, and | by their L. C. M. 120, we have the parts propor- 
 tional to 40, 24, and 16. 
 
 The total number of parts = 40 + 24 + 15 = 79. 
 
 .'. the parts are respectively ^%, |a, and || of 316 lb. = IGO, 96, and 60 lb. 
 
 Proof. — Dividing 160, 96, and 60 by 480, the denominator which reduces 
 160 to ^, we have J, \, J, which proves the results found to be correct. 
 
 Exercise 171 
 
 1. Divide 1331 into parts proportional to 2, 4, 5. 
 
 2. Divide $73.50 into parts proportional to \, J, J. 
 
 3. Divide 19 T. 1104 lb. into parts proportional to \, \, \. 
 
 4. Divide $ 1064 into parts proportional to 2, 2\, 2|. 
 
 5. Divide 180 lb. into parts proportional to 3.3, .7, .5. 
 
 6. Divide f 4500 between two persons in proportion to their 
 ages, which are 21 and 24 yr. 
 
 7. Two men receive $15 for doing a certain piece of work. 
 Now one man had worked only 3 da., while the other had worked 
 5 da. on the job. If the money is to be divided in proportion 
 to the lengths of time the men worked, how much should each 
 receive ? 
 
282 
 
 ARITHMETIC 
 
 
 -! ' 
 
 8. Divide 4472 into parts which shall be to each other in the 
 ratio of 3, 5, 7, 11. 
 
 9. Divide $84.42 into two parts which shall be to each other 
 as 5 : 16. 
 
 10. A company of militia consisting of 72 men is to be raised 
 from 3 towns which contain respectively 1500, 7000, and 9500 
 men. How many must each town provide ? 
 
 11. Sugar is composed of 49.856 parts oxygen, 43.265 carbon, 
 and 6.879 hydrogen. How many lb. of each are there in 1300 lb. 
 of sugar ? 
 
 12. Gunpowder is composed of nitre, charcoal, and sulphur in 
 the proportion of 33, 7, and 5. 
 
 (1) How many lb. of sulphur are there in 180 lb. of powder ? 
 
 (2) How many lb. of powder can be made with 30 lb. of 
 sulphur ? 
 
 (3) How much nitre and sulphur must be mixed with 112 lb. 
 of charcoal to form gunpowder ? 
 
 13. A man divides $3300 amongst his three sons, whose ages 
 are 16, 19, and 25 yr., in sums proportional to their ages; 2 yr. 
 afterwards he similarly divides an equal sum, and again after 3 yr. 
 more. How much does each receive in all ? 
 
 14. Two persons travelling together agree to pay expenses 
 in the ratio of 7 to 5. The first (who contributes the greater 
 sum) pays on the whole 1 103.40, the second $63.40. What 
 must one pay the other to settle their expenses according to 
 agreement ? 
 
 15. Divide $480 among A, B, C, and D, so that B may receive 
 as much as A : C as much as A and B together ; and D as much 
 as A, B, and C together. 
 
lb. 
 
 RATIO AND PROPORTION 
 
 Partnership 
 
 283 
 
 270. In Simple Partnership the capital of each partner is 
 supposed to be invested for the same time. 
 
 In Compound Partnership the time is taken into account as 
 well as the capital in determining the gain or loss of each 
 partner. 
 
 271. A, B, and C engage in business. A furnishes 87500, 
 B 15000, and C 14500. If they gain 12380, what is each 
 one's share ? 
 
 Dividing their capitals by $ 500, we have their capitals, and therefore their 
 gains proportional to 15, 10, and 9. 
 
 The total number of parts = 15 + 10 + 9 = 34. 
 
 .-. their respective gains are if, fj, and 3^ of |2380 = $ 1050, f 700, and 
 {^630. 
 
 Exercise 172 
 
 1. Two merchants, A and B, form a joint capital. A puts in 
 $ 1200 and B $ 1800. They gain |400. How ought the gain to 
 be divided between them ? 
 
 2. A bankrupt owes three creditors, A, B, and C, $175, $210, 
 and $265 respectively; his property is worth $422.50. What 
 ought each to receive ? 
 
 3. A, B, and C entered into partnership. A puts in $ 6000, 
 B $4000, and C $2000. They gained $2250. What is each 
 one's share of the gain ? 
 
 4. Two men purchase a house for $3600, the first contributing 
 $1600 and the second $2000. If it rents so as to pay 12% on 
 its value, what share of the rent should each receive ? 
 
 5. Two persons have gained in trade $ 3456 ; the one put in 
 $ 10,560 and the other $ 8640. What is each person's share of 
 the profits ? 
 
 r, 
 
\\ 
 
 284 
 
 ARITHMETIC 
 
 I 
 
 ;i 1^ 
 
 6. R. Stuart and G. Armstrong enter into partnership. Stuart 
 contributes $ 4500 to the partnership and Armstrong contributes 
 $ 7500. Their net gain at the end of the year is $ 1750. How 
 much of the sum should each partner receive ? 
 
 7. Three partners invest respectively f 7800, $5750, and 
 $ 9450 in business. At the end of the first year they find their 
 net gain to be $3156. What is the amount of each partner's 
 share of this gain ? 
 
 8. A, B, and C form a partnership with a capital of f 20,000. 
 A contributes f 5000, B $ 7000, and C the remainder. They gain 
 20% of the total capital. Find each man's share of the profits. 
 
 9. T. Allan and E. Jamieson engage in business with a joint 
 capital of $ 19,200, and agree to share gains and losses in propor- 
 tion to their investments. At the end of a year Allan receives a 
 dividend of $1100, and Jamieson a dividend of $1300. What 
 was the amount of the investment of each ? 
 
 10. D. Rowan, F. Galbraith, and J. Munro enter into partner- 
 ship. They gain $ 7500, of which Rowan receives $ 2100, Gal- 
 braith $3100, and Munro the balance. How much did Rowan 
 and Galbraith respectively invest if the amount of Munro's 
 investment was $18,000? 
 
 11. A, B, and C pay $37.80 as rent for a pasture. A puts in 
 5 horses, B 12 cows, and C 60 sheep. If 1 horse eats as much as 
 2 cows, and 1 cov,' as much as 3 sheep, what rent should each 
 pay? 
 
 272. (1) A, B, and C enter into partnership. A puts in 
 
 1700 for 12 mo., B $500 for 9 mo., and C $600 for 8 mo. 
 
 Divide a profit of $2066 equitably among them. 
 
 The gain on 1 700 for 12 mo. = the gain on $ 8400 for 1 mo. 
 The gain on $ 500 for 9 mo. = the gain on $ 4500 for 1 mo. 
 The gain on 1 600 for 8 mo. = the gain on $ 4800 for 1 mo. 
 .*. the proportional parts representing the gains are 84, 45, and 48, or 28, 
 15, and 16. 28 + 15 + 16 = 59. 
 
 .'.the respective gains are f f , ^|, and ^| of $ 2065 = $980, $ 625, and $ 560. 
 
 '...-t^U. 
 
 J:*»ij 
 
 
RATIO AND PROPORTION 
 
 285 
 
 I 
 
 .{., 
 
 (2) A commenced business with 14000 stock ; 3 mo. after, 
 he took in B with a capital of $2000; and 4 mo. after B 
 became a partner, he took in C with a capital of $600; at 
 the end of the year the firm had gained 13450. Find the 
 share of each. 
 
 A's capital = $4000 for 12 mo. = $48,000 for 1 mo. 
 B's capital = $2000 for 9 mo. = $ 18,000 for 1 mo. 
 C's capital = $ 600 for 5 mo. = $ 3000 for 1 mo. 
 .-. the respective gains are proportional to 48, 18, and S ; i.e. 16, 6, and 1. 
 
 16 + 6 + 1 = 23. 
 
 .-. the respective shares are if, /g, and j^j of $3450 = $2400, $900, and 
 $150. 
 
 , 
 
 E:sercise 173 
 
 1. D and E enter into partnership ; D puts in $480 for 3 mo., 
 and E $ 900 for 4 mo. They gain $ 840. What is each man's 
 share in the gain ? 
 
 2. A, B, and C entered into partnership ; A put in $ 1200 for 
 
 8 mo., B $800 for 10 mo., and C $400 for 12 mo. They gained 
 $ 3920. What was each man's share of the gain ? 
 
 3. A, B, and C are partners ; A puts in $ 5000 for 6 mo., 
 B $6000 for 8 mo., and C $900 for 11 mo. The profit is 
 $ 5575.50. What is the share of each ? 
 
 4. Three graziers hire a pasture for their common use, for 
 which they pay $ 318. One puts in 20 oxen for 6 mo., another 
 24 oxen for 8 mo., and the third 28 oxen for 4 mo. How much 
 of the rent should each pay ? 
 
 5. A and B enter into partnership ; A contributes $ 15,000 for 
 
 9 mo., and B $ 12,000 for 6 mo. They gain $ 5750. Find each 
 man's share of the gain. 
 
 6. A, B, and C rent a field for $ 56.50 ; A puts in 70 cattle 
 for 6 mo., B 40 for 9 mo., and C 50 for 7 mo. What ought each 
 to pay ? 
 
 1=' 
 
 I'l 
 
286 
 
 ARITHMETIC 
 
 IT 
 li 
 
 
 7. Three merchants enter into partnership; the first invests 
 $ 1855 for 7 mo., the second invests $ 887.50 for 10 mo., and the 
 third invests $ 770 for 11 mo., and they gain f 434. What should 
 be each partner's share of the gain ? 
 
 8. L, M, and N entered into partnership and invested respec- 
 tively $ 19,200, $ 22,500, and $ 28,300. At the end of 5 mo. L 
 invested f 3800 additional, M 1 2500, and N 1 3700. At the end 
 of a year the net gain of the firm was found to be $ 7850. What 
 was each partner's share of this ? 
 
 9. A and B enter into partnership ; A puts in $ 400 at first, 
 and $ 500 at the end of 2 mo. ; B puts in $ 300 at first, and 
 $ 600 at the end of 3 mo. The profit at the end of the year 
 is $ 470. How should this be divided ? 
 
 10. A and B engage in trade ; A invests $ 6000, and at the 
 end of 5 mo. withdraws f 2000 ; B puts into the business $ 4000, 
 and at the end of 7 mo. $ 6000 more. Divide a gain of $ 6800 at 
 the end of the year. 
 
 11. A, B, and C form a partnership with a joint stock of 
 $15,600; A's stock continues in trade 6 mo., B's 8 mo., and 
 C's 12 mo. A's gain is $ 1200, B's $ 2400, and C's f 1800. What 
 stock did each put in ? 
 
 12. Two men complete in a fortnight a piece of work for which 
 they are paid f 46.75 ; one of them works alternately 9 hr. and 
 8 hr. a day, the other works 8J hr. for 5 da. in the week, and 
 does nothing on the remaining day. What part of the sum 
 should each receive ? 
 
 13. A and B are partners ; A's capital is to B's as 4 to 9. At 
 the end of 4 mo. A withdraws i of his capital, and B | of his. 
 At the end of the year their whole gain is $ 4600. How much 
 belongs to each ? 
 
 14. A, B, and C rent a pasture for $ 92 ; A puts in 6 horses for 
 8 wk., B 12 oxen for 10 wk., C 50 cows for 12 wk. If 5 cows are 
 
 rl 
 
 SI 
 
 HSmwa 
 
RATIO AND PROPORTION 
 
 287 
 
 reckoned equivalent to 3 oxen, and 4 oxen to 3 horses, what 
 should each pay? 
 
 15. Three men, working respectively 8, 9, and 10 hr. a day, 
 receive the same daily wages. After working thus for 3 da., each 
 works 1 hr. a day longer, and the work is finished in 3 da. more. 
 If $ 114 is paid for the work, how much should each man receive ? 
 
 16. Three workmen, A, B, and C, did a certain piece of work 
 and were paid daily wages according to their several degrees of 
 skill. A's efficiency was to B's as 4 to 3, and C's to B's as 5 to 6 ; 
 A worked 5 da., B 6 da., and C 8 da. The whole amount paid 
 for the work was $36.25. Find each man's daily wages. 
 
 h 
 
CHAPTER XVI 
 
 't ;: 
 
 1 
 
 POWERS AND EOOTS 
 Square Root* 
 273. (1) Find the square root of 17.3056. 
 
 81 
 826 
 
 17 .30'66 [4.16 
 16 
 
 180 
 81 
 
 4956 
 4956 
 
 to be 
 
 bellZT' "" """ ""'"''• '■'""•'^ *•'" ''"' ""^ ^^™" ""' •^e found 
 
 (2) Extract the square root of 35 to three decimal places. 
 
 35 I 5.916 
 25 
 
 109 
 
 1181 
 
 11826 
 
 1000 
 981 
 
 1900 
 1181 
 
 71900 
 70956 
 
 (3) Extract the square roots of H |A s 
 
 ^49 V49~7* 
 
 * See Chapter VI II. 
 
 288 
 
POWERS AND ROOTS 
 
 289 
 
 /35 _ \/35 ^ 
 
 = .845. 
 
 \/35_ 5.916 
 
 V49 
 
 Vf = V1J25 = .7905. 
 
 Which denominator is not a perfect square ? Why reduce f to a decimal 
 before extracting the square root ? 
 
 Exercise 174 
 
 Find the square root of : 
 
 1. 40.96; 65.61; 2.1025. 
 
 2. 167.9616; 28.8369; 57648.01. 
 
 3. .042849; .00139876; .00203401. 
 
 4. 5774409; 5.774409. 
 
 5. 10.3041; 2321.3124; .0050367409. 
 
 6. V2; V20; VI; VlOOO to four decimal places. 
 
 7 144. 3 24. fil 
 
 ft 20 1 • 1 5 6 .1.2209 
 
 q 3.5. 7_ 
 
 * Cube Root 
 
 274. The product of 3 x 3 x 3 is 27 ; of 5 x 5 x 5 is 125. 
 The cubes whose sides measure 3 and 5 units of length con- 
 tain 27 and 125 units of volume. We say that 27 is the 
 cube of 3 and that 125 is the cube of 5 ; that 3 is the cube 
 root of 27, and that 5 is the cube root of 125. The cube of 
 5 is written 5^, and the cube root of 5 is indicated thus : V5. 
 
 5^ is also called the third power of 5. 
 
 275. The cubes of 
 
 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 
 are 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000. 
 
 r' 
 
290 
 
 ARITHMETIC 
 
 1 ;;iii 
 I 
 
 276. The cube roots of 
 
 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 
 are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. 
 
 277. These two paragraphs should be mastered by the 
 pupils as the corresponding paragraphs in square root. 
 
 278. The product 4 x 4 x 4 is written 4^. 
 
 The product 4.6 x 4.6 x 4.6 is written (4.6)8. 
 The product | x | x | is written (|)8. 
 The cube root of 4 is written V4. 
 The cube root of .4 is written V3. 
 The cube root of # is written Vl. 
 
 1 
 
 Exercise 175 
 
 Write the following products as powers : 
 
 1. 2x2x2. 
 
 2. 3x3. 
 
 3. 6x5x5. 
 
 4. 5x5x5x5. 
 
 5. 5x5x5x5x5. 
 
 6. 5x5x5x5x5x5. 
 
 7. ixfxf 
 
 8. Jxjxf. 
 
 9. 2.3 X 2.3 X 2.3. 
 
 10. 3.12 X 3.12 X 3.12. 
 
 11. .12 X .12 X .12. 
 
 12. .1 X .1 X .1. 
 
 13. .02 X .02 X .02. 
 
 Write the following powers as products and find their values : 
 14. 43. 15. 123. ^Q 2.53. 17. (J)3. 18. {ly. 
 
 19. .021 20. .1*. 
 
POWERS AND ROOTS 
 
 291 
 
 Prove the following statements : 
 21. v/2l6 = 6. 
 
 22. V15625 = 25. 
 
 23. ^/15S25 = 2.5. 
 
 24. V1.728 = 1.2. 
 
 25. S/:008 = .2. 
 
 26. V.064 = .4. 
 
 27. </^ = l 
 
 3 
 
 30. SyTV^ = t. 
 
 Ezercise 176 
 
 1. Find the ler'^th of one edge of a cube containing 612 cu. 
 in. Find the length of all its edges. Find the area of one of its 
 faces. Of all its faces. 
 
 2. Find the area of one face of a cube containing 729 cu. in. 
 
 3. Find the number of units of length in a cube containing 
 343 units of volume. Find the number of units of area in one 
 face. 
 
 4. Find the edge of a cube one of whose faces contains 144 
 sq. in. Find its volume. 
 
 5. Find the volume of a cube one of whose faces contains 
 225 sq. in. 
 
 6. What is the edge of a cube whose volume is 8 units of vol- 
 ume ? 27 units of volume ? 
 
 7. The ratio of the volumes of two cubes is ^j. What is the 
 ratio of their edges ? 
 
 8. The ratio of the volumes of two cubes is 64 : 125. What is 
 the ratio of their edges ? 
 
 9. The ratio of the edges of two cubes is f. What is the 
 ratio of their volumes ? 
 
 10. The edges of two cubes are as 7 : 9. What is the ratio of 
 their volumes ? 
 
292 
 
 ARITHMETIC 
 
 ii 
 
 ! 1; 
 
 Exercise 177 
 
 1. Find the cubes of 14, 25, 36, 54, 75, and 99. 
 
 2. From the results in § 275, state how many digits there are 
 in the cube of a number of 1 digit. 
 
 3. From the results in question 1, state how many digits 
 there are in the cube of a number of 2 digits. 
 
 4. In long division how many figures form a group ? In 
 square root ? In cube root ? 
 
 5. How many digits are there in the cube root of 512? 64? 
 8 ? What are they ? 
 
 6. Divide 389,017 into groups of figures ; 29,791 ; 3375. How 
 many figures are there in the cube root of each of these numbers ? 
 
 7. Cube 73, 31, and 15. 
 
 8. Judging from the results given in § 276, state the number 
 of digits in the cube root of a number containing 1, .2, or 3 digits. 
 
 9. Write the cubes of 10, 20, 30, 40, 50, 60, 70, 80, and 90. 
 
 10. State how many digits there are in the cube root of a num- 
 ber containing 4, 5, or 6 digits. 
 
 11. What is the first digit in the cube root of 2744 ? 39,304? 
 
 279. To find the cube root of a number, we shall first see 
 
 how the cube of a number is found. 
 
 Since 64 = 50 + 4, we can cube 54 thus : 
 
 50 + 4 
 
 50 + 4 
 
 (4 X 50) + 42 
 502 4- (4 X 50) 
 502 + 2(4 X 50) + 42 
 
 50 + 4 
 
 (4 X 502)+ 2(42 X 50) + 48 
 503 + 2(4x502)+ (42x50) 
 503 + 3(4 X 502) + 3(42 ^ 60) + 4^ 
 
 Since 4 divides each of the last three terms, we can put this result 
 = 503 + 4{3 X 502 + 3 X 4 X 50 + 42}. 
 
POWERS AND ROOTS 
 
 293 
 
 We now wish to recover from such a number as 157,404 its cube root. 
 Plainly the tens' digit of the root is 5, i.e. the first part of the root is 50. 
 
 50 
 
 157 '464 50 + 4 
 
 
 125000 
 
 3 X 502 = 7500 
 
 32464 
 
 3 X 4 X 50 = GOO 
 
 
 42= 16 
 
 
 8110 
 
 32464 
 
 To find the second term, note that in the expression 
 
 4 ^3 X 50-2 + 3 X 4 X 50 + 42} 
 the number 4 is the second digit in the number 54 that was cubed ; hence in 
 the work of taking the cube root, the other factor {3 x 502+ 3 x 4 x 50 4- 42 ^ 
 will be the real divisor and 3 x 502 ^^e trial divisor. 
 
 Therefore, squaring 50 and multiplying by 3, we have 7500. Dividing 7500 
 into 32,464, we find the quotient to be 4 ; completing the divisor by adding 
 3 X 4 X 50 or 600, and 42 or 16, we find the divisor to be 7500 + 600 + 16 
 or 8116. Multiplying this by 4, we have 32,464. Hence we conclude that 
 the cube root of 157,464 is 54. To prove the result correct, cube 64. 
 
 280. The work of extracting the cube root may be shortened 
 
 thus: 
 
 157 '464 [54 
 
 300 X 52 = 7500 
 
 30 X 5 X 4 = 600 
 
 42 = 16 
 
 8116 
 
 125 
 
 32464 
 
 32464 
 
 Find the cube root of 926,859,375. 
 
 926'859'375|975 
 
 300 X 92 = 24300 
 30 X 9 X 7 = 1890 
 
 72= 49 
 
 26239 
 
 300 X 972 = 2822700 
 
 30 X 97 X 5 = 14550 
 
 62 = 25 
 
 2837276 
 
 729 
 
 197859 
 
 183673 
 
 14186375 
 
 14186375 
 
294 
 
 ARITHMETIC 
 
 J-,.;- 
 
 k 
 
 281. The cube of 3.19 is equal to 32.461759. From this 
 it is evident that corresponding to the two figures in the 
 decimal part of the number, viz. 19, we have two groups of 
 three figures, viz. 461 and 759, in the decimal part of the 
 cube. Hence in pointing off, begin at the decimal point and 
 mark the number off into periods of three figures each to the 
 right of the decimal and then again to the left. 
 
 (1) Find the cube root of 95.448993. 
 
 95.443'993 | 4.67 
 
 300 X 42 = 4800 
 
 30 X 4 X 6 = 000 
 
 62 = _25 
 
 5425 
 
 300 X 452 = 607500 
 
 30 X 45 X 7 = 9450 
 
 72 = 49 
 
 610999 
 
 64 
 
 31443 
 
 27126 
 
 4318993 
 
 4318993 
 
 (2) Extract the cube root of 16. 
 
 300 X 22 = 1200 
 
 30 X 2 X 5 = 300 
 
 52= _25 
 
 1526 
 
 300 X 262 _ 187500 
 
 30 X 25 X 1 = 750 
 
 12= 1 
 
 
 
 
 188251 
 
 
 300 X 
 
 2512 = 
 
 18900300 
 
 30 
 
 X261 
 
 x9 = 
 
 667700 
 
 
 
 92 = 
 
 81 
 
 19568081 
 
 16 1 2.619 
 
 8 
 
 8000 
 
 7626 
 
 375000 
 
 188261 
 
 186749000 
 
 176112729 
 
 (3) To extract the cube root of such a number as 843.7295, 
 add ciphers thus, 843. 729'500, and extract the cube root. 
 
POWERS AND ROOTS 
 
 295 
 
 (4) Extract the cube root of 3Y3. 
 
 ■IfK = _^ - i 
 
 ^343 ^343 7* 
 
 (5) Extract the cube root of ^^^. 
 
 To" v^ 2.619 
 
 '343 »/343 7 
 
 (6) Extract the cube root of J|. 
 
 = .369. 
 
 i^ = M or .640. 
 
 V.640 = .861. 
 
 .-. v^f« = .861. 
 
 Which of the three denominators is not a perfect cube ? When should 
 a fraction be reduced to a decimal before extracting its cube root ? Why ? 
 
 Exercise 178 
 
 Find the cube root of : 
 
 1. 29,791. 2. 64,872. 3. 110,592. 
 
 4. 804,357. 5. 941,192. 
 
 6. Compare the processes of long division, of extracting the 
 square root, and the cube root of a number. Note in what 
 respect the}^ are similar and in what respect they are different. 
 
 7. 2,406,104. 13. .001906624. 
 
 8. 69,426,531. 14. 3, .3, .03, .003, .0003. 
 
 9. 8,365,427. 
 
 10. 389.017. 
 
 11. 32.461759. x,. 5, ^y^, j. 
 
 12. .000912673. 18. 3f, 405^2^^^ ^f 
 
 19. A cubical block of stone contains 50,653 cu. ft. What 
 is the area of its side ? 
 
 20. A cube contains 56 cu. ft. 568 cu. in. Find its edge. 
 
 IK 8 125 5 8 32 
 IR 250 135 
 17. 2 _4_9_ 5 
 
 "I 
 
296 
 
 ARITHMETIC 
 
 21. One gal. contains 231 cu. in. Find the edge of a cube 
 equal to it. 
 
 22. Find the length of the inside edge of a cubical vessel which 
 will just hold 10 gal. 
 
 23. Three cubes of lead, measuring respectively ^, |, and f of 
 an in. on the edge, were melted together and cast into a single 
 cube. Find the length of the edge of the cube thus formed, 
 neglecting loss of lead in melting and casting. 
 
 24. Four cubes of lead, measuring respectively 6, 7, 8, and 9 in. 
 on the edge, were melted together and cast into a single cube. 
 Find the length of the edge of the cube thus formed, if 4% of 
 the lead was lost in melting and casting. 
 
 25. Find the volume of a cube, the area of whose surface is 
 100.86 sq. in. 
 
 26. A cube measures 5 in. on the edge. A second cube is 
 three times the volume of the first. By how much does the 
 length of an edge of the second cube exceed that of an edge of 
 the first cube ? 
 
 27. By raising the temperature of a cube of iron, the length 
 of each of its edges was increased by 5%. Find correct to four 
 decimals the ratio of increase in the volume of the cube. 
 
 28. Each edge of a cube is diminished by ^^ of its length. 
 By what fraction of itself is the volume diminished ? By what 
 traction of itself is the area of the surface diminished ? 
 
CHAPTER XVII 
 
 MENSURATION 
 
 282. The rectangle has been treated of in preceding para- 
 graphs. 
 
 Exercise 179 
 
 1. The units of lengths being 1 in., 2 in., 3 in., 4 in., 5 in., 6 in., 
 what are the units of area ? Make mental pictures of the units 
 of area. 
 
 2. If the length and breadth of a rectangle are respectively 
 7 and 5 times the unit of length, how many times the unit of area 
 is the area of the rectangle ? If the unit of length is 6 in., what 
 is the area of the rectangle ? 
 
 3. If the ratios of the length and breadth of a rectangle to the 
 unit of length are respectively 5 and 3, what is the ratio of the 
 area of the rectangle to the unit of area ? 
 
 4. How do you find the number of units of area in a rectangle ? 
 
 5. The measure of the area of a rectangle is 48, the area being 
 432 sq. in: What is the unit of area ? Of length ? 
 
 6. Draw a rectangle 2 in. by 3 in. in one corner of a rectangle 
 6 in. by 9 in. Are the figures similar in shape ? Find the area 
 of each rectangle. What is the ratio of the two breadths ? Of 
 the two lengths ? Of the two areas ? What is the relation 
 between the ratio of the areas and the ratio of the lengths ? 
 
 7. Draw a rectangle 3 in. by 4 in. in one corner of a rectangle 
 6 in. by 8 in. Are the figures similar ? What is the ratio of the 
 breadths? Of the lengths? What, then, is the ratio of the 
 areas? Find the area of each rectangle and prove your last 
 result correct. 
 
 297 
 
 jl 
 
 k 
 
m 
 
 298 
 
 ARITHMETIC 
 
 •* :. 
 
 8. The sides of a rectangle are 4 in. and 6 in. respectively. 
 How often will it measure another rectangle whose sides are 
 twice as long ? 3 times as long ? 4 times ? 5 times ? 
 
 9. The unit of area is a rectangle 3 in. by 5 in. How often 
 will it measure a rectangle each of whose sides is 6 times as long ? 
 7 times ? 8 times ? 9 times ? 
 
 10. Two rectangles are similar in shape, and the second is 4 
 times as large as the first. Compare the lengths of their sides. 
 Illustrate by a drawing and make a mental picture. 
 
 11. The areas of two similar rectangles are as 9 : 1. What is the 
 ratio of their sides ? 
 
 12. The ratio of the areas of two similar rectangles is 9:4. 
 What is the ratio of their sides ? 
 
 13. Make a rule showing how to find the ratio of the sides of 
 two similar rectangles when you know the ratio of their areas. 
 
 14. What are the ratios of the areas of the following pairs of 
 similar rectangles, the ratios of the sides being 4 to 1 ; 5 to 1 ; 
 3 to 4 ; 7 to 2 ? If the area of the smaller rectangle is 16 sq. ft., 
 what is that of the larger in each case ? 
 
 15. What are the ratios of the sides of the following pairs of 
 similar rectangles, the ratios of the areas being 64 to 1 ; 121 to 1 ; 
 49 to 25 ; 144 to 100 ? If the sides of the smaller rectangle are 
 20 and 30 in., what are the sides of the larger rectangles ? 
 
 283. (1) Find to the nearest inch the side of a square 
 whose area is 1 A. 
 
 1 A. = 6,272,640 sq. in. ^ 
 The measure of the area = 6,272,640. 
 The measure of the side = 2504.6+ . 
 .*. the side of the square, to the nearest in. = 2606 in. = 69 yd. 1 ft. 9 in. 
 
 (2) The lengths of the sides of a rectangular piece of land 
 are as 3 to 8 and its area is 60 A. Find the length of its sides. 
 
MENSURATION 
 
 299 
 
 Imagine the length of the field to be divided into 8 equal parts and the 
 breadth into 3, and the field be divided into 24 equal squares by lines drawn 
 through these points of division parallel to the sides of the field. Consider 
 one of these squares the unit of area and one of its sides the unit of length. 
 Then 24 x the unit of area = 60 A. or 600 sq. ch. 
 The unit of area = 25 sq. ch. 
 The unit of length = 5 ch. 
 .'. the lengths of the sides = 15 ch. and 40 ch. 
 
 Exercise 180 
 
 1. A square field contains exactly 8 A. Determine the length 
 of a side of the field, correct to the nearest link. 
 
 2. The area of a chess-board marked in 8 rows of 8 squares 
 each is 100 sq. in. Find the length of a side of a square. 
 
 3. On a certain map it is found that an area of 16,000 A. is 
 represented by an area of 6.25 sq. in. Give the scale of the map 
 in miles to the inch. 
 
 4. A rectangle measures 18' by 30'. Find the difference be- 
 tween its area and that of a square of equal perimeter. 
 
 5 . Two rectangular fields are of equal area. On© field measures 
 15 ch. by 20 ch. ; the other is square. Find the length of a side 
 of the latter field, correct to the nearest link. 
 
 6. How many stalks of wheat could grow on 1 A. of ground, 
 allowing each stalk a rectangular space of 2" by 3" ? 
 
 7. How many pieces of turf 3' 6" by 1' 3" will be required to 
 sod a rectangular lawn 28' by 60'? 
 
 8. Sidewalks 12 ft. wide are laid on both sides of a street 440 
 yd. long. Find the cost of the sidewalks at $ 1.35 per sq. yd. for 
 the pavement and 75^ per lineal yd. for curbing; deducting 
 three crossings of 54 ft. each on both sides of the street. 
 
 9. The area of a rectangular field is 15 A. ; the length of the 
 field is double the width. Find the length of the field in chains. 
 
 10. What length must be cut off a board, which is 7^ in. broad, 
 so that the area may contain 3 sq. ft. ? 
 
. liiiWfT'f ^'?fS!»J#^;«w?. .mw 
 
 ^'^Ift'pfpw 
 
 300 
 
 ARITHMETIC 
 
 f'l' '■ 
 
 I .ill 
 
 11. Find the area of each of the following rectangles, whose 
 dimensions are : 1. jL = 36 ft., ii = 13 ft. 2. Z = 20 ft. 3 in., 
 ^ = 20 in. 3. L = S ft. 9 in., ^ = 3 ft. 8 in. 
 
 12. Find the solid contents of the following : 1. Z = 13 ft. 4 in., 
 B = 7 ft. 6 in., ^= 3 ft. 10 in. 2. L = 20 ft., B=l ft. 6 in., 
 /f=lft. 2in. 
 
 13. Find the length of the following rectangles: 1. Area = 
 40 sq. yd., 5 = 20 ft. 2. Area = 6 sq. ft., ^ = 9 in. 
 
 14. Find the aiea of the four walls of the following rooms : 
 1. i = 32 ft., B=1S ft., H=ll ft. 2. L = 29 ft., B = 23^ ft., 
 
 15. Find the cost of painting a surface : 1. 19 ft. 6 in. by 83 ft. 
 4 in., at 40^ a sq. ft. 2. 25 ft. 8 in. long, and 10 ft. 9 in. wide, at 
 65^ a sq. ft. 
 
 16. A gentleman wishes to set out a rectangular orchard of 
 1260 trees, so placed that the number of rows shall be to the 
 number of trees in a row as 5 to 7. Find the number of rows 
 and also the number of trees in a row. 
 
 17. The sides of a rectangular field containing 27 A. 48 sq. rd. 
 are as 21 to 13. Find the perimeter of the field 
 
 284. A Quadrilateral is a plane figure having four sides. 
 
 A Parallelogram is a quadrilateral whose opposite sides are 
 parallel. 
 
MENSURATION 
 
 301 
 
 m., 
 
 285. To find the area of a parallelogram : 
 
 Let perpendiculars be drawn from C and D perpendicular to AB. Then 
 it is evident that the triangles marked a are equal. Adding to each the 
 quadrilateral marked c, it is evident that the parallelogram ABGD is equal 
 to the rectangle upon the base CD. 
 
 Hence^ to find the measure of the area of a parallelogram^ 
 
 multiply the measure of its base by the measure of its altitude. 
 
 Draw any parallelogram and draw the perpendiculars as in the figure. 
 Cut out the triangles and place them on each other, showing that they are 
 equal. 
 
 286. A Trapezoid is a quadrilateral two of whose sides are 
 parallel. 
 
 The parallel sides are called Bases and the perpendicular 
 distance between the two bases is called the Altitude. 
 
 D E C 
 
 Thus, in the trapezoid, AB and CD are the bases, and AU 
 the altitude. 
 
 287. To find the area of a trapezoid : 
 
 A B M 
 
 i 
 
 3t 
 
 D O 
 
 N C 
 
' I'm 
 
 302 
 
 ARITHMETIC 
 
 I, r 1 '3 
 
 lh'i::i 
 
 m 
 
 Let ABCD be a trapezoid, and let perpendiculars be drawn through E 
 and F, the middle points of AD and BC, to AB and CZ>. Then it is evident 
 that the triangles a and a' are equal and also c and c'. 
 
 To a' and c' and also to a and c add the figure ABFNOE^ and we have 
 
 the trapezoid equal to the rectangle LMNO. 
 
 Again, 
 
 EF = AB -\- AL -^ BM, 
 
 EF=CD-DO-NC. 
 
 Adding, 2 EF = AB -{■ CD, since 
 
 AL = DO and BM= NO, 
 
 i.e. EF=\{AB^ CD). 
 
 .'. ON=l(AB-\-CD). 
 
 Therefore, to measure the area of the trapezoid, we multiply the measure 
 of ON, i.e. of ^(AB + CD), by that of the altitude. 
 
 Hence the area of a trapezoid is found hy multiplying the 
 measure of one-half the sum of its parallel sides by the measure 
 of its altitude, 
 
 288. Find the area of a trapezoid whose parallel sides 
 are 12' 7" and 19' 3" respectively, the perpendicular dis- 
 tance between them being 8' 5". 
 
 The sum of the bases = 12' 7" + 19' 3" = 31' 10". 
 \ the sum of the bases = 15' 11" = 191". 
 The altitude = 8' 6" = 101". 
 .*. the area = 101 x 191 sq. in. = 19,291 sq. in. = 133 sq. ft. 139 sq. in. 
 
 Exercise 181 
 
 1. Fold a sheet of paper in the form of a trapezoid. Draw 
 the lines as given in the figure in § 287, cut out the triangles 
 a and c, and, by placing them on a' and c' respectively, show 
 that a = a' and c = c'. 
 
 2. In the figure in question 1, produce DG to G, making 
 CG = AB. Measure along DG twice with a line equal to EF, 
 thus showing that twice EF = CD -\- AB. 
 
MENSURATION 
 
 303 
 
 3. The length of the base of a parallelogram is 45 ft. ; the 
 length of the perpendicular on the base from the opposite side 
 is 28 ft. Find the area. 
 
 4. The adjacent sides of a parallelogram measure 132 ft. and 
 84 ft. respectively, and the area of the parallelogram is f of that 
 of a square of equal perimeter. Find the perpendicular distance 
 between each pair of parallel sides. 
 
 5. The lengths of the parallel sides of a trapezoid are 12 ft. 
 and 17 ft., and the perpendicular distance between these sides is 
 8 ft. Find its area. 
 
 6. If the parallel sides of a garden are 84 and 92 ft. respec- 
 tively, and their perpendicular distance 82J- ft., what did it cost 
 at $ 1200 an A. ? 
 
 7. The area of a trapezoidal field is 3 J A., and the sum of 
 the lengths of the parallel sides is 440 yd. Find the perpen- 
 dicular distance between these sides. The lengths of the parallel 
 sides being in the ratio of 5 to 6, find these lengths. 
 
 8. The area of the trapezoid is 9750 sq. yd., and the perpen- 
 dicular distance between the parallel sides is 234 ft. If the 
 length of one of the parallel sides be 410 ft., what will be the 
 length of the other parallel side ? 
 
 n 
 
304 
 
 ARITHMETIC 
 
 289. Let ABC be a triangle, and let the rectangle BCDE be drawn. 
 Then it is evident that the triangles a and a' and c and c' are equal. Hence 
 the triangle ABC is one-half of the rectangle BCDE. Hence^ to find the 
 area of a triangle^ multiply one-half the measure of the base by that of the 
 altitude. 
 
 290. To find the area of a triangle when the lengths of the 
 sides are given : 
 
 Find one-half of the sum of the measures of the sides; subtract 
 from this the measure of each side separately. The square 
 root of the product of these four results will give the measure of 
 the area of the triangle. 
 
 Exercise 182 
 
 1. Find the area of a triangle whose base is 45 ft., and alti- 
 tude 17 ft. 
 
 2. A triangular piece of ground containing 4^ A. has a base 
 of 135 yd. Find its altitude. 
 
 Find the areas of the triangles the lengths of whose sides are 
 respectively : 
 
 3. 13 yd., 10 yd., and 13 yd. 
 
 4. 13 yd., 24 yd., and 13 yd. 
 
 5. 13 ft., 4 ft., and 15 ft. 
 
 6. 13 ft., 14 ft., and 15 ft. 
 
 7. 13 in., 11 in., and 20 in. 
 
 8. 13 in., 21 in., and 20 in. 
 
 9. 1.23 ch., 5.95 ch., and 6.76 ch. 
 
 10. 73.2 ch., 45.5 ch., and 87.6 ch. 
 
 11. Find the number of A., etc., that there are in a triangu- 
 lar field of which the sides are 7 ch. 60 1., 9 ch. 50 1., and 5 ch. 
 70 1. 
 
MENSURATION 
 
 305 
 
 12. The three sides of a triangle are 33 ft., 56 ft., and 65 ft. 
 Find the measure of the triangle cut off by joining the points of 
 bisections of the two greater sides. 
 
 13. Multiply the length of each side of the triangle in question 
 3 by 3. Eind the area of the resulting triangle. What is the 
 ratio of its area to that of the A in question 3? 
 
 14. Similarly, multiply each side of the A in question 4 by 5, 
 and find its area. Find the area of the resulting triangle. What 
 is the ratio of the areas ? 
 
 15. If the sides of the triangle in question 5 be multiplied by 
 the number 6, what will be the ratio of its area to that of the tri- 
 angle in question 5 ? Hence find its area. 
 
 16. If each side of the triangle in question 7 be made 10 times 
 as long, how much greater will be the area of the triangle ? 
 
 17. State in what ratio the area of a triangle is increased by 
 increasing the length of each side. 
 
 291. To find the area of a cii'cle : 
 
w 
 
 ,J»UmrVWl«»l"«|"»iif|/Jli!HIJ« t,«J|<^ 
 
 306 
 
 ARITHMETIC 
 
 Draw a circle on cardboard and cut it out — the larger the better. Divide 
 each half of the circle as the semicircle in the figure is divided, the arcs 
 Ay By C, D, etc., being as nearly equal as possible. Cut the circle into two 
 equal parts along the line AOM. 
 
 Cut along OB, OC, etc., cutting nearly to the points B, C, D, but not 
 separating the parts entirely at these points. Spread the resulting figure out 
 as in the darker part of the figure below. 
 
 
 
 A B -C D E 
 
 Then cut up the other semicircle in the same way ; spread open the parts 
 and fit the two semicircles together as in the figure. The resulting figures will 
 be nearly a rectangle. The smaller the arcs AB, BC, etc., the more nearly 
 the area will be to a rectangle whose base is equal to one-half the circumfer- 
 ence and whose altitude is equal to the radius of the circle. 
 
 Hence the measure of the area of a circle is one-half the 
 product of the measures of the circumference and the radius. It 
 may also he expressed thus : 
 
 The measure of the area = \ cr. 
 Again, since c = 3.1416 x 2 r, 
 the measure of the area = 3.1416 r^. 
 
 Both formulas are useful. 
 
 The last rule may be read: The measure of the area of 
 a circle is found by multiplying the square of the measure 
 of the radius hy 3.1416. 
 
MENSURATION 
 
 307 
 
 292. (1) Find the area of a circle whose diameter is 7| in. 
 
 The measure of the area = 3.1410 r^. 
 The radius = 7^ in. -f- 2 = 3.76 in. 
 The measure of the area = 3.1416 x 3.762 = 44.18. 
 .-. the area = 44.18 sq. in. 
 
 (2) If the arc of a circle is 2 ft. and the radius 6 ft., 
 find how many degrees there are in the arc. 
 
 The length of the circumference = 3.1410 x 12 ft. = 37.6992. 
 37.6992 ft. of circumference contain 360^. 
 
 360° 
 
 1 ft. of circumference contains 
 
 2 ft. of circumference contain 
 
 37.6992 
 720° 
 
 or 19^ 6' 64". 
 
 37.6992 
 .-. the arc contains 19° 5' 64". 
 
 (3) Find the length of a radius of a circle whose area 
 is 4 A. 
 
 4 A. = 19,360 sq. yd. 
 The measure of the area = 3.1410 r". 
 .-. r2 = 19,300 - 3.1416 = 6162.46. 
 
 .-. r = V6162.46 = 78.6 yd. 
 
 
 Exercise 183 
 
 1. Find the area of a circle 7 ft. in diameter. 
 
 2. Find the area of a quadrant whose radius is 4 rd. 
 
 3. Find the length of the radius of a circle whose area is 
 1 A. 
 
 4. Find the length of the diameter of a circle whose area 
 is 1 sq. mi. 
 
 5. Find the length of the radius of a circle whose area is 
 equal to the sum of the areas of four circles of 10 in., 15 in., 18 in., 
 and 24 in. radius respectively. 
 
 6. Find the total pressure on a plate 25 in. in diameter, the 
 pressure per sq. in. being 65 lb. 
 
 'V 
 
308 
 
 ARITHMETIC 
 
 7. A circular hole is cut in a circular metal plate of 7 in. 
 radius, so that the weight of the plate is reduced by 40%. Find 
 the length of the radius of the hole. 
 
 8. The area of a semicircle is 13.1 sq. in. Find the length 
 of its perimeter. 
 
 9. The lengths of the sides of a triangle are 13 ft, 14 ft., and 
 15 ft. respectively. Find the difference between the area of the 
 triangle and that of a circle of equal perimeter. 
 
 10. The perimeters of a circle, a square, and an equilateral tri- 
 angle are each 17 ft. in length. Find by how much the area of 
 the circle exceeds the area of the other figures. 
 
 11. Find the length of the diameter of a circle whose area 
 is equal to that of a square whose sides are each 12 ft. long. 
 
 12. Out of a circle of radius 3 ft. is taken a circle of radius 
 2 ft. Find the area of the remainder. 
 
 13. Find the length of the arc which subtends an angle of 60° 
 at the centre of a circle of 10 in. radius. 
 
 14. Find the length of the arc which subtends an angle of 36° 
 at the centre of a circle of 25 in. radius. 
 
 15. How many degrees are there in the angle which an arc 
 whose length is 1 ft. subtends at the centre of a circle of 2 ft. 
 radius ? 
 
 16. The length of the radius of a circle is 8 in. Find the 
 length of the arc of which the angle is (a) 90°, (b) 270°. 
 
 17. There is a circular fish-pond of 90 ft. radius, surrounded 
 by a walk 25 ft. in breadth. Find the area of the walk. 
 
 18. Within a circular garden 70 ch. in circumference is a 
 circular pond 70 rd. in circumference. Find the width of the 
 ring of land which surrounds the pond. 
 
 19. The radii of four circles are respectively 2 ft., 3 ft., 4 ft., 
 5 ft. Show that their areas are as the numbers 4, 9, 16, and 25. * 
 
 Ml 
 
MENSURATION 
 
 809 
 
 293. If we wrap a rectangular sheet of paper about a 
 cylinder, we find that the area of the curved surface of the 
 cylinder is a rectangle whose base is the circumference of 
 the cylinder and altitude the height of the cylinder. Hence 
 we can find its area. 
 
 294. To find the measure of the volume of a cylinder, take 
 the product of the measures of the area of the base and the 
 altitude. 
 
 295. The curved surface of a cone can be unwrapped into 
 a portion of a circle. 
 
 Hence the measure of the curved surface is one-half the 
 product of the measure of its base by its slayit height. 
 
 296. Make a cylinder out of paper and also a right circular 
 cone having the same altitude and base. Fill the cone with 
 some dry material and empty it into the cylinder. Do this 
 three times and the cylinder will be just filled. Hence the 
 volume of a right circular cone is one-third that of a cylinder 
 of equal base and altitude. Hence, to find the volume of a 
 right circular cone, multiply one-third the measure of the area 
 of the base by the measure of the altitude. 
 
 \ 
 
 \ 
 
 i 
 
310 
 
 ARITHMETIC 
 
 297. (1) The length of the radius of a right circular 
 cylinder is 5 in. and its altitude is 8 in. Find its volume 
 and the area of its curved surface. 
 
 The measure of the area of the base = 3.1416 x 26 = 78.64. 
 The measure of the volume of the cylinder = 8 x 78.64 = 628.32. 
 
 .-. the volume = 628.32 cu. in. 
 
 Again, 
 
 The measure of the circumference of the base = 3.1416 x 10 = 31.416. 
 The measure of the area of the curved surface = 8 x 31.416 = 251.328. 
 
 /. the area = 261.328 sq. in. 
 
 (2) Find the area of the curved surface and also the 
 
 volume of a cone whose altitude is 8 in. and whose base is 
 
 12 in. in diameter. 
 
 A 
 
 Since the altitude ^O is perpendicular to the diameter BCD^ the triangle 
 ACB is a right triangle. 
 Hence AB is 10 in. 
 
 Also the circumference of the base = 3.1416 x 12 in. = 37.6992 in. 
 The measure of the area = i x ^ x 37.6992 = 188.496. 
 
 .*. the area = 188.496 sq. in. 
 
 Again, 
 
 The altitude of the cone = 8 in. 
 The area of the base = 3.1416 x 62 or 113.0976 sq. in. 
 The measure of the volume = i x 8 x 113.0976 = 301.6936. 
 .•. the volume = 301.69 cu. in. 
 
MENSURATION 
 
 311 
 
 Exercise 184 
 
 1. Make a cone and cylinder each of whose bases is 9 in. in 
 circumference and altitudes 6 in. 
 
 Fill the cone and empty it three times, in succession, into the 
 cylinder. What is the result ? 
 
 2. Find the volume of a cylinder the radius of whose base is 
 10 in., the altitude being 18 in. 
 
 3. Find the volume of a cone the radius of whose base is 10 
 in., the altitude being 18 in. 
 
 4. How often can the cone in question 3 be filled and emptied 
 into the cylinder in question 2 ? 
 
 5. The length of the radius of the base of a right circular 
 cylinder is 9 in. and its altitude is 16 in. Find the volume. 
 
 6. Find the area of the curved surface of the cylinder in 
 question 5. Find the area of its entire surface. 
 
 7. Find the volume of a cone whose altitude is 15 in., and 
 whose base is a circle 10 in. in diameter. 
 
 8. Find the volume of a cone whose altitude is 12 in., and the 
 diameter of whose base is 5 in. 
 
 9. Find the area of the curved surface of a cone whose alti- 
 tude is 20 in., and the radius of whose base is 15 in. Find also 
 its total area. 
 
 10. What must be the height of a cylindrical column of 
 marble, the radius of whose base is 9 in., in order that it may 
 contain 5| cu. ft. ? 
 
 11. If the diameter of a cylindrical well be 5 ft., and its depth 
 27 ft., how many cu. yd. of earth were removed in order to 
 form it ? 
 
 i ; 
 
 298. It can be shown that the area of the curved surface of 
 a hemisphere is equal to twice the area of its flat surface ; 
 
312 
 
 ARITHMETIC 
 
 hence the area of the surface of the sphere is equal to 4 
 
 times the area of this flat surface. 
 
 Thus the measure of the area of the surface of a sphere is 4 times the 
 product of 3.1416 and the square of the measure of the length of the radius. 
 
 ^ = 4 x3.1416r2. 
 
 299. If we imagine a sphere to be divided into a large 
 number of small cones, as in § 291 we divided the circle 
 into triangles, the centre of the sphere being the vertex of 
 each cone, and a small portion of the circumference being 
 its base, we can think of the volume of the sphere as being 
 equal to the sum of the volumes of the cones. The altitude 
 of each cone is equal to the radius of the sphere, and the 
 total area of their bases is equal to the area of its surface. 
 Hence the volume of the sphere is given by the formula : 
 
 r=Jr(4x 3.1416 xr2) 
 = |x 3.1416 r3. 
 
 Hence the measure of the volume of the sphere is ^ o/ 3.1416 
 times the cube of the measure of the radius. 
 
 300. (1) Find the surface of a sphere whose radius is 6 in. 
 
 The measure of the area = 4 x 3.1416 x 62 = 452.3904. 
 .-. the area = 452.39 sq. in. 
 
 (2) Find the volume of a sphere whose diameter is 8 in. 
 
 The measure of the volume = | x 3.1416 x 48 = 268.0832. 
 .*. the volume = 268.08 cu. in. 
 
 Exercise 185 
 
 1. Find the surface of a sphere whose radius is 3 in. 
 
 2. Find the surface of a sphere 12 in. in diameter. 
 
 3. Find the volumes of the spheres given in questions 1 
 and 2. 
 
w 
 
 MENSURATION 
 
 313 
 
 4. Find the surface of a sphere 5 ft. in diameter. 
 
 5. Find the volume of a sphere whose diameter is 16 ft. 
 
 6. Place a croquet or base ball between two chalk boxes. 
 Place a foot measure in line with one edge of each box. What is 
 the diameter of the ball? What is the area of its surface? 
 What is its volume ? 
 
 7. With a pair of compasses draw a circle with the diameter 
 found in question 6. Cut out this circle and pass the ball through 
 the hole. 
 
 8. If the pressure of the air is equal to 15 lb. a sq. in., what 
 is the pressure on the surface of a sphere 6 in. in diameter ? 
 
CHAPTER XVIII 
 
 THE METRIC SYSTEM OF WEIGHTS AND MEASURES 
 
 mi 
 
 301. The French or Metric System of Weights and Meas- 
 ures is based upon the decimal system. It is used in scien- 
 tific treatises and has been adopted by most of the nations of 
 Europe and South America. It is also in partial use in the 
 United States and Canada. 
 
 302. The fundamental unit of the metric system is the 
 Meter, which is 39.37 in. long. The original standard meter 
 is a platinum rod, called the French Standard Meter, which 
 is deposited in the Archives at Paris. 
 
 FOUR INCHES IN SIXTEENTHS OF AN INCH 
 
 lll|lll|lllll|l|lilllll|lll|lll|ltl|iM|MI|l|l|lll|lli|llllJII 
 
 m 
 
 m 
 
 ii'iiiiHU'ii'imiiiiiiii'iii'iiiiiiitT 
 
 8 
 
 DUQn 
 
 10 
 
 □I 
 
 ONE DECIMETER IN MILLIMETERS 
 
 The length of the measure in the diagram is -^ of a 
 meter, and is called a decimeter. It is divided into 10 equal 
 parts, each of which is called a centimeter. Each of these is 
 divided into 10 equal parts, each of which is called a milli- 
 meter. 
 
 303. In order that pupils may study this system of weights and meas- 
 ures to the best advantage, the school should be provided with a system of 
 
 314 
 
w 
 
 1 
 
 METRIC SYSTEM 
 
 315 
 
 Meas- 
 scien- 
 ms of 
 n the 
 
 s the 
 meter 
 ;vhich 
 
 nrpr 
 
 10 
 
 of a 
 
 equal 
 
 lese is 
 
 milli- 
 
 l meas- 
 stem of 
 
 metric weights and measures, and each pupil with a foot rule on which the 
 decimeter, centimeter, and millimeter are marked. 
 
 An intelligent study of the system can, however, be made if the teacher 
 has a metric stick and a liter for reference. 
 
 304. The names of the higher or lower units in the metric 
 system are formed by attaching certain prefixes to the names 
 of the standard units, thus : 
 
 Deca signifies 10 times the unit. 
 Hecto signifies 100 times the unit. 
 Kilo signifies 1000 times the unit. 
 
 Deci signifies the 10th part or .1 of the unit. 
 Oenti signifies the 100th part or .01 of the unit. 
 Milli signifies the 1000th part or .001 of the unit. 
 
 Units of Length 
 
 1 millimeter (mm.) = .001 meter 
 1 centimeter (cm.) = .01 meter 
 1 decimeter (dm.) = ,1 meter 
 1 meter (m.) = Standard unit 
 
 1 decameter (Dm.) = 10 meters 
 1 hectometer (Hm.) = 100 meters 
 1 kilometer (Km.) = 1000 meters 
 1 myriameter (Mm.) = 10,000 meters 
 
 The units in common use are the millimeter, centimeter, meter, and kilo- 
 meter. 
 
 305. Write out the table of the units of length, thus : 
 
 (a) 10 millimeters = 1 centimeter, etc. 
 (h) 1 millimeter = j^q centimeter, etc. 
 
 Exercise 186 
 
 1. Measure with a yardstick in the school-yard a distance 
 of 11 yd. Measure along the same distance 10 times with the 
 meter. What is the difference in inches? 
 

 316 
 
 ARITHMETIC 
 
 
 :1 " 
 
 2. Reduce 11 yd. and also 10 m. to in., and verify the result 
 obtained in the first question. 
 
 3. Measure the length of the schoolroom in m. and decimals 
 of a m. 
 
 4. Find the hypotenuse of a right triangle whose sides are 
 (1) 6 m. and 8 m., (2) 24 cm. and 45 cm. 
 
 5. Librarians frequently use the centimeter as the unit in 
 registering the heights of books. Express in cm. the height of 
 your (a) Arithmetic, (b) History, (c) Geography, (d) and also of 
 several other books. 
 
 6. Measure and express in terms of the cm. the distances 
 between the lines on ruled paper. 
 
 7. Measure and express in terms of the cm. the height of the 
 schoolroom thermometer. 
 
 8. How many cm. are there in a full line of this book ? 
 
 9. Press tightly together the leaves of this book. Make a 
 layer just 1 cm. thick. Count the leaves and find the thickness 
 of 1 leaf as a decimal of a mm. 
 
 10. Cut a slit 2 mm. wide, by 3 cm. long, in a sheet of paper. 
 
 11. How many mm. are there in the width of your pencil ? 
 
 12. Find the number of in. in 1 Km., reduce the result to a 
 decimal of a mi., and show that 1 Km. is nearly equal to J 
 of a mi. 
 
 13. If a train travels at the rate of 20 m. a sec, what is the 
 rate in Km. per hr. ? 
 
 14. Show that 5 in. are very nearly equal to 127 mm. 
 
 1ft. For what do we use the in., the yd., and the mi. ? For what 
 are the mm., the cm., the m., and the Km. respectively used ? 
 
METRIC SYSTEM 
 
 317 
 
 Units op Area 
 
 306. The principal units for measuring land are the square 
 meter, called the Centare (ca.), the square decameter called the 
 Are (a.), and the square hektometer called the Hectare (ha.). 
 
 1 sq. 
 
 
 cm. 
 
 
 1 sq. dm. 
 
 Each side of this square measures 
 
 1 dm., or 
 
 3j-f in., very nearly. 
 
 A liter is a cube each face of which has the dimensions of this 
 
 square. 
 
 A gram is the weight of a cubic centimeter (see small square 
 
 above) of distilled water, weighed in vacuo at temperature of 
 
 maximum density, 39.1 F. A liter or cubic decimeter of such 
 
 water weighs 1 kg. or 2^ lb. nearly. 
 
 307. How many units of length are there in the side of 
 a square meter, the decimeter being the unit ? 
 
 How many units of area are there in a square meter, the 
 square decimeter being the unit ? 
 
 i 
 i 
 
 14 . 
 
m 
 
 ,11 
 
 ,f|i 
 
 318 
 
 ARITHMETIC 
 
 How many units of area are there in a square decimeter, 
 the square centimeter being the unit of area ? 
 
 What part of a square meter is a square decimeter? 
 
 Write a square decimeter as a decimal of a square meter. 
 
 Write a square centimeter as a decimal of a square deci- 
 meter. As a decimal of a square meter. What is the ratio 
 of each unit of area in the metric system to the next smaller, 
 and also to the next higher ? 
 
 Units of Area 
 
 1 square millimeter (q. mm.)= .000001 of a square meter 
 1 square centimeter (q. cm.) = .0001 of a square meter 
 1 square decimeter = .01 of a square meter 
 
 1 square meter (q. m.) = standard unit 
 
 1 square decameter 
 
 = 100 square meters 
 
 meters 
 
 1 square hektometer = 10,000 square meters 
 
 1 square kilometer (q. Km.) = 1,000,000 squarfe meti 
 1 centare (ca.) = .01 are (a.) 
 
 1 hectare (ha.) = 100 ares 
 
 Note that the square meter is 1| times as large as the square yard. 
 
 E:sercise 187 
 
 1. Cut out of paper a q. dm. and also a q. cm. What is the 
 ratio of the two areas ? 
 
 2. Draw a sq. ft. and measure it with the q. dm. of paper as 
 the unit of area. 
 
 3. A q. dm. contains .10764 sq. ft. Find the number of 
 q. dm. contained in a sq. ft. correct to two decimal places, and 
 compare the result with that obtained in question 2. 
 
 4. Make a drawing of a q. m., and draw a sq. yd. within it. 
 
 5. Mark out an are on the school-ground. 
 
 6. State the table of area, expressing each unit of area as 
 equal to 100 times the next lower. 
 
 7. Find the area of a page of this book in q. cm. 
 
 8. Find the area of the room in ca. 
 
ter, 
 
 jr. 
 
 leci- 
 fatio 
 Her, 
 
 METRIC SYSTEM 
 
 Units op Volume 
 
 319 
 
 308. The principal units of volume are the cubic meter^ 
 also called the Stere, and the cubic decimeter, called the Liter. 
 
 309. How many units of length are there in a side of 
 a meter, the decimeter being the unit ? 
 
 How many units of volume are there in the cubic meter 
 or stere, the cubic decimeter or liter being the unit ? 
 
 What is the ratio of each unit of volume in the metric 
 system to the next smaller unit? To the next larger? 
 
 Units of Volume 
 
 1 cubic millimeter (c. mm.)= .000000001 of a cubic meter 
 1 cubic centimeter (c. cm.) = .000001 of a cubic meter 
 1 cubic decimeter = .001 of a cubic meter 
 
 1 cubic meter (c. m.) = standard unit 
 
 Exercise 188 
 
 1. Make a liter out of paper. 
 
 2. Fill a qt., liquid measure, with sand and empty it into 
 your liter. Which of the two measures is larger ? 
 
 3. rill a liter with sand and empty it into a qt., dry meas- 
 ure. Which of the two is larger ? 
 
 4. Fill a gal. measure, using the liter as a dipper, and note 
 how many liters are equivalent to the gal. 
 
 5. If 1 liter is equal to .264 gal., find correct to two decimal 
 places the number of liters in a gallon. Compare this result 
 with that obtained in question 4. 
 
 6. State some purposes for which the liter is used. 
 
 7. Make a c. cm. out of paper. 
 
 8. Make the necessary measurements and compute the volume 
 of the room in c. m. 
 
 '. -ii 
 
 « ; 
 
^. «jr"i^T.--l'Ji,»'«^ 'jr- 
 
 320 
 
 ARITHMETIC 
 
 9. Make the necessary measurements and compute the volume 
 of a box in liters. 
 
 10. Express as a decimal part of a c. m. the volume of a beam 
 3 m. long, 10 cm. wide, and 5 cm. thick. 
 
 11. A cylindrical vessel having a base of a q. m. is filled with 
 water to the depth of 2 m. How many liters of water does 
 it contain? 
 
 12. How many liters of water may be held by a vessel meas- 
 uring 25 X 35 X 75 cm. ? 
 
 13. What will it cost to build a wall 1 Hm. long, ^ dm. thick, 
 and 1 m. high, at f 5 a c. m. ? 
 
 Units of Weight 
 
 310. The principal units of weight are the Oram and the 
 Kilogram. 
 
 The Oram is the weight of a cubic centimeter of distilled 
 water at 40°, at which temperature water is at its maximum 
 density. 
 
 A nickel weighs 5 g. 
 
 A Hter of distilled water at 40° weighs 1 kg. The kilogram is nearly 
 equal to 2^ lb. Avoir. 
 
 A cubic meter of water at 40° weighs a metric ton (1000 kg.). 
 
 W( 
 
 Exercise 189 
 
 1. One gram is equal to 15.432 gr. Show that 1 kg. is 
 approximately equal to 2^ lb. Avoir. 
 
 2. A cubic meter of distilled water at 40° weighs how many 
 kg. ? If 1 kg. is equal to 2^ lb. Avoir., how many pounds does 
 1 c. m. of water weigh ? 
 
 3. Show that a metric ton weighs about 10% more than our 
 short ton. 
 
-wr 
 
 METRIC SYSTEM 
 
 821 
 
 ne 
 
 m 
 
 4. If sulphiu'ic acid is 1.8 times as heavy as water, what 
 weight of the acid will a two-liter bottle contain ? 
 
 5. What part of a liter is 750 g. of water ? 
 
 6 What is the weight of 1 deciliter of water ? 
 
 7. If alcohol is 80% as heavy as water, what will 375 c. cm. 
 of alcohol weigh ? 
 
 8. If 20 c. cm. of lead weighs 227 g., what is the ratio of the 
 weight of lead to that of an equal volume of water ? 
 
 9. If a quantity of iron weighs 7.8 times as much as an equal 
 quantity of water, what is the weight of an iron bar 75 x 4 x 3 cm. ? 
 
 10. A body weighing 512 g. in air weighs 428 g. in water. 
 What per cent of its weight is lost ? 
 
 11. A liter flask was two-fifths filled with water; the remain- 
 ing space being filled with sand, the weight was found to be 
 2050 g. Kequired the weight of a liter of sand. 
 
 12. If the pressure of the air on the surface of a body is 1 kg. 
 to the q. cm., what is the pressure of the air on the surface of a 
 sphere whoso radius is 10 cm. ? 
 
 13. A cubical block of ice measures 3 dm. along its edge. 
 What will be its weight if ice weighs 94% as much as an equal 
 volume of water ? 
 
 14. What is the weight of air in a room 5 m. long, 3 m. wide, 
 4 m. high, if 1 c. dm. of air weighs .0018 kg. ? 
 
 '! 
 
CHAPTER XIX 
 
 Miscellaneous Exercise 190 
 
 1. Write the following in figures : 
 
 (a) Fifty thousand nine hundred and nine. 
 
 (b) Nine hundred thousand and ninety. 
 
 (c) Six hundred and fifty thousand seven hundred. 
 
 (d) Eight hundred and seven thousand and eight. 
 
 (e) Seven hundred and seventy thousand and sixty-seven. 
 (/) Nine million ninety thousand and ninety-nine. 
 
 (g) Eighty million nine hundred thousand and thirty. 
 
 (h) Nine hundred and seventy million eight hundred and 
 eighty-seven thousand. 
 
 (i) Six hundred and seventeen million and n^ 'y-three. 
 
 (j) Nine hundred and nineteen thousand f^ hundred and 
 
 eleven. 
 
 2. Write in figures : 
 
 Twenty-five thousand four hundred and ninety ; ninety-nine 
 thousand nine hundred and seventeen ; nine hundred and seven 
 thousand six hundred and six ; one million ; MDCCCXCV. And 
 in words : 9009 ; 16,060 ; 7018 ; 207,509 ; 75,115. 
 
 3. (a) Define and give examples of quantity, unit, and number. 
 (b) Explain the basis of our system of numeration. 
 
 4. Write in figures (placed for addition): Nine hundred and 
 nineteen ; three hundred and eleven ; seven hundred and seventy ; 
 eight hundred and ninety -seven ; six hundred and eight ; XCVII. j 
 LXVII. J CXIX. ; CDL. j and DCXL. 
 
 322 
 
MISCELLANEOUS EXERCISE 
 
 323 
 
 5. Add: 4/)()789()12.3 
 
 ()78<)()1L>;U5 
 
 8y()i2.'Mr)()7 
 oi)i2;Mr)(;78 
 
 6598()9r>;^26 
 8a9087(>549 
 7788995566 
 3453453456 
 
 6. Write down neatly the following statement of six weeks' 
 cash receipts ; add the amounts vertically and horizontally, and 
 prove the correctness of the work by adding your results : 
 
 
 MON. 
 
 TUE8. 
 
 Wed. 
 
 TlIUR. 
 
 Fri. 
 
 Sat. 
 
 Total 
 
 1st 
 
 $28.79 
 
 $34.71 
 
 $35.33 
 
 $30.10 
 
 $27.97 
 
 $47.81 
 
 
 2d 
 
 23.87 
 
 30.03 
 
 29.38 
 
 33.84 
 
 26.77 
 
 48.77 
 
 
 8d 
 
 16.99 
 
 27.09 
 
 28.77 
 
 30.10 
 
 24.95 
 
 43.07 
 
 
 4th 
 
 29.13 
 
 33.72 
 
 30.81 
 
 39.17 
 
 28.47 
 
 50.05 
 
 
 6th 
 
 18.47 
 
 :J2.29 
 
 26.73 
 
 34.45 
 
 28.88 
 
 54.39 
 
 
 6th 
 
 19.02 
 
 27.06 
 
 29.04 
 
 29.89 
 
 29.61 
 
 61.93 
 
 
 Total 
 
 
 
 
 
 
 
 
 7. Solve, as in question 6 : 
 
 
 MON. 
 
 Tubs. 
 
 Wed. 
 
 TlIUR. 
 
 Fri. 
 
 Sat. 
 
 Total 
 
 1st 
 
 2d 
 
 3d 
 
 4th 
 
 5th 
 
 6th 
 
 $65.96 
 58.71 
 47.68 
 29.69 
 81.45 
 42.63 
 
 $24.89 
 41.65 
 99.57 
 70.80 
 66.93 
 68.77 
 
 $ 79.79 
 24.67 
 50.60 
 87.91 
 64.82 
 81.79 
 
 $40.78 
 94.20 
 80.71 
 74.93 
 96.57 
 60.86 
 
 $37.50 
 70.26 
 91.82 
 36.63 
 12.72 
 31.87 
 
 $89.61 
 42.51 
 89.76 
 21.90 
 96.67 
 75.82 
 
 
 Total 
 
 
 
 
 
 
 
 
 I 
 
.^'^"' P^T" )°'''^T^ -'!- ■^V'^V!X:;,Tr\ 
 
 ■r^Tz^ I 
 
 324 
 
 ARITHMETIC 
 
 8. Mr. Jones bought one house for $865 and another for 
 $ 984, and sold them both for f 1900. How much did he gain ? 
 
 9. John has $149, Will has $87 more than John, and Sam 
 has $ 115 more than both. How many dollars has Sam ? 
 
 10. Ned sold two bags of potatoes. With the money he got 
 for them he bought a knife for 25^, a saw for 65^j a hatchet for 
 75^, and had 45^ left. How much did he get for his potatoes ? 
 
 11. Two men together receive $ 97.75, but one receives $ 18.25 
 more than the other. How much does each receive ? 
 
 12. A and B start together and walk in the same direction, A 
 at the rate of 4 mi. an hr., and B at the rate of 3 mi. an hr. At 
 the end of 7 hr. A turns and goes back. How many mi. will 
 B have gone when he meets A ? 
 
 13. In a factory, 12 men, 16 women, and 30 boys are employed. 
 At the end of a week they receive $330. A man is paid as 
 much as 2 women, and a woman as much as 3 boys. What is the 
 share of each ? 
 
 14. A man bought a number of cows for $1080; he sold half 
 of them for $ 810, thereby gaining $ 15 on each one sold. What 
 did each cow cost ? 
 
 15. A clerk received a salary of $ 650 a yr. He spent 50^ 
 a da. the first yr., $ 4 a wk. the second yr., and $ 22 a mo. the 
 third yr. How much did he save in three years ? 
 
 16. The subtrahend is 9564, the remainder is 1965. What is 
 the minuend? The multiplier is 96 and the product is 82,848. 
 What is the multiplicand ? 
 
 17. The dividend is 1800, the quotient is 17, and the remainder 
 66. What is the divisor ? 
 
 18. How many times can 506 be subtracted from the product 
 of 6072 and 13,986 ? 
 
 19. The quotient of a division is 834. What quotient would 
 have been obtained if both dividend and divisor had been first 
 multiplied by 13 ? Why ? 
 
MISCELLANEOUS EXERCISE 
 
 325 
 
 for 
 ISam 
 
 got 
 
 for 
 9 
 
 20. Subtract 847.1 J from 1003/^, explaining fully each step. ^ 
 
 21. Simplify J - | of | + y^-, and find how many times the 
 result is contained in f h- ( J of y\ — i). 
 
 22. Divide the sum of | of 8| and 2f of 5| by the difference i 
 between f of 3i and J of i of 2§. 
 
 23. Prove: (1) 
 
 24. Simplify 3L+p^ 
 
 2 of 2 _ 4 . /9\ 
 
 10 
 
 of 
 
 2 of 2 _ 2 of 2 
 
 K 
 
 25. A boy's age now is \ of his father's. In 6 yr. it will be /- 
 \ his father's present age. How old is he ? 
 
 26. A house and lot are together worth $2100; \ of the value y 
 of the house is equal to \ of the value of the lot. Find the 
 value of each. 
 
 27. The circumference of a wheel is '^-^- of its diameter. Find 
 the diameter of a wagon wheel which makes 360 revolutions in 
 going a mile. 
 
 28. A man owned a |-interest in a mill, and sold ^ of his 
 interest to one man, and \ of his interest to another. What part 
 of the mill did each of the three men then own ? 
 
 29. If to a certain number its |, ^, and \ be added, the sura 
 will be 122 ; required the number. 
 
 30. Find the number which is 207 more than the sum of \ and 
 I of itself. 
 
 31. A man spent -^ of his money for a house, \ of the remain- 
 der for cattle, and the rest for a farm. If the farm cost him 
 $357 less than the house and cattle together, what did he pay 
 for all? 
 
 32. A legacy of f 9500 is to be divided among A, B, and C, 
 so that A will get j% of the whole, and B will get J as much as 
 C. Find the shares of each. 
 
 33. A man spent f of his; money for provisions, f of the/^ 
 remainder for clothing, ^^ of the remainder for charity, and had 
 
 % 9.10 left. How much did he have at first ? 
 
 ^ 
 
 ^ 
 
 ■ ■■ 
 
 if 
 
•■W''W'''!^}WWVi' 
 
 326 
 
 ARITHMETIC 
 
 34. Nathan Curd sells a merchant 752 lb. of cheese at 11|^ 
 per lb., and receives the following goods in exchange : 
 
 11 yd. silk @ $ 2.25 ; 96 lb. nails @ 3f )2^ ; 
 
 400 lb. sugar @ 4 J^ ; 56 yd. gray cotton @ 9|^ ; 
 
 12 lb. raisins @ 11^ ; H yd. white cotton @ 10(i^ ; 
 
 3 pr. gloves @ 75^. 
 
 i Find the balance due Nathan Curd. 
 
 35. A man owns a horse and saddle; \ of the value of the 
 horse is equal to 4 times the value of the saddle ; the horse and 
 saddle together are worth ^ 170. Find the value of each. 
 
 36. A man bought a horse and carriage for $ 280, and | of 
 the cost of the carriage was equal to | of the cost of the horse. 
 What was the cost of each ? 
 
 37. Divide the product of .037 and .0025 by the sum of .9, .02, 
 and .005. 
 
 38. Divide 6 by .000725, correct to four decimal places. 
 
 39. Add together 1.302, 3.2589, and 40.93. Multiply the sum 
 by .00297 and divide the product by 90.09. 
 
 ^ 40. Multiply 350.4 by .0105 and divide the product by .0000219. 
 
 41. What decimal must be taken from the sum of 69 J, 8.2, 
 5.445, .065, and 20y*^, so that it will contain 6.05 an exact number 
 of times ? 
 
 ^ 42. A drover lost .065 of his flock by wolves, .105 by disease, 
 and .27 by theft. He then sold .75 of what remained, and has 
 280 sheep left. Find the number in his original flock. 
 
 43. Find the amount of the following bill : 
 1328 ft. siding, at $ 1.62^ per C. ; 
 48,480 cu. ft. timber, at $ 59.37^ per M. ; 
 7400 fence rails, at ^ 7.75 per C. ; 
 8400 fence pickets, at $ 15.00 per M. ; 
 6680 lb. hay, at f 12.50 per T. 
 
MISCELLANEOUS EXERCISE 
 
 327 
 
 44. A cooper paid $78.32 for 16,488 bbl. staves. Eequired 
 the price per M. 
 
 45. A rectangular field is 7 ch. 75 1. long and 4 ch. 871 l. wide. -J^ 
 How many rd. of fencing are required to enclose it ? ' 
 
 46. How many mi. of road, 3 rd. wide will contain 8 A. of ^^ 
 land ? 
 
 47. Make a drawing that will show the number of sq. yd. 
 in a sq. rd. 
 
 48. Find the value of a piece of land 20 ft. x 40 rd., at $ 1000 
 per A. 
 
 49. A certain map is drawn on a scale of 8 mi. to an inch. On 
 this map the township of Scott measures 1/^ in. in length and 
 
 11 in. in width. How many A. does it contain? 
 
 50. Find the expense of sodding a plot of gronnd which is 
 40 yd. long and 100 ft. wide, with sods each 1 yd. in length and 
 1 ft. in breadth, the sods, when laid, costing 75^ per hmidred. 
 
 51. A floor 16 ft. 8 in. by 14 ft. 2 in. is to be laid with square 
 tiles. Find the dimensions of the largest tiles that can be used 
 without cutting or fitting. „^,i>^"^ 
 
 52. Find the cost of papering a room 24 ft. long, 21 ft. wide, f^ 
 
 12 ft. high, at 25^ a roll, 12 yd. long and 21 in. wide. 
 
 53. How much will it cost to plaster the walls and ceiling 
 of a room 15 ft. long, 12 ft. wide, and 11 ft. high, at 32^^ per *" 
 sq. yd. ? , 
 
 54. A room 18 ft. by 16 ft. is carpeted with carpet f yd. wide, 
 and the smallest possible number of yd. of the carpet is used. 
 Find (a) the number of breadths, (h) the number of yd. 
 
 55. How many thousand shingles, 18 in. long and 4 in. wide, 
 lying J to the weather, are required to shingle the roof of a build- 
 
 f 
 
 I- 
 
■(r.-"-y'> •''^'■, pyT- 
 
 328 
 
 ARITHMETIC 
 
 ing 54 ft. long, with rafters 22 ft. long, the first row of shingles 
 being double ? 
 
 56. A schoolroom is 30 ft. long, 24 ft. wide, and 10 ft. high 
 above the wainscoting. The trustees pay $ 20 per M. for a new 
 floor, f 15 per M. for a new board ceiling, 10^ per sq. yd. for 
 painting the ceiling, 4^ per sq. yd. for tinting the walls, and 
 $ 2 per da. for 6 da. labor. Find the total cost. 
 
 57. A cubical cistern is 5 ft. deep. How many gal. of water 
 will it hold if 231 cu. in. make a gal. ? 
 
 58. How many cubical blocks, each edge of which is J ft., are 
 equivalent to a block of wood 8 ft. long, 4 ft. wide, and 2 ft. 
 thick ? 
 
 59. If the ceiling of a square room is 15 ft. high, how many 
 sq. ft. of floor must it have in order that 50 pupils and the 
 teacher may each have 300 cu. ft. of air? 
 
 60. Four-foot wood piled 5^ ft. high requires how many ft. in 
 length of the pile for 2 J cd. ? 
 
 61. What is the value of a pile of wood 360 ft. long, 12 ft. 
 wide, 6 ft. high, at $ 3.20 per cd. ? 
 
 62. A square plot of ground that contains ^^^ A. is covered 
 with cordwood (4 ft. long) to an average height of 12 ft. What 
 is tlie wood worth at $ 4.12 per cd. ? 
 
 63. Required the cost of 35 pieces of scantling, 18 ft. long, 
 4 in. wide, and 2 in. thick, at $ 14 per M., board measure. 
 
 64. How many board feet are there in 12 scantlings 16 ft. by 
 4 in. by 2 in. ? 
 
 65. It is required to build a sidewalk \ mi. in length, 8 ft. 
 wide, and 2 in. thick, supported by three continuous lines of 
 scantling 4 in. square. What will the lumber cost at $ 17 per M. ? 
 
 \m 
 
WSl^'^P"- ■ A'*'-™': ^J"--*" v.< ■-;.>T HFT--' 
 
 MISCELLANEOUS EXERCISE 309 
 
 66. Fiud the v<aliie of the fonowing hiinber at $ 15 per M. : 
 
 20 pieces 2 x 4, 18 ft. long; 
 20 pieces 4x4, 12 ft. long ; 
 20 pieces 3 x 10, 10 ft. long. 
 
 67. A farmer sold a lot of barley, weighing 2712 lb., when 
 barley was 40<^ per bu. In weighing the grain, the dealer made 
 a mistake and took it as rye, and paid for it at 40)^ per bu. How 
 much did the farmer gain or lose by the mistake ? 
 
 68. The weight of a cu. ft. of water is G2\ lb., and 1 gal. 
 contains 231 cu. in. Find the weight in oz. of 1 pt. of water. 
 
 69. A lake whose area is 45 A. is covered with ice 3 in. 
 thick. Find the weight of the ice in T., if 1 cu. ft. weighs 
 920 oz. Avoir. 
 
 70. In what time would a field, 80 by CO rd., pay for under- 
 draining lengthwise, at 2^ per ft., if the field yield 2 bu., at 60^, 
 per A. more than before draining? The drains are 4 rd. apart, 
 and the first drain runs down the centre of the field. 
 
 71. Find the amount of the following bill : 
 
 June 1, 1890, G. Murray & Co. sold to John Scott, 4880 bu. 
 36 lb. wheat @ 58^ per bii., 4532 lb. peas @ 52^ per bu., 38 bu. 3 pk. 
 barley fa). 50 per bu., 465 lb. flour (a) 3 1.50 per cwt., 4685 lb. bran 
 @ $ 15 per T. Write out a receipt in full for payment of account, 
 June 26. 
 
 72. Find the length of the shortest line that can be exactly 
 measured by a yard measure, a ten-foot pole, or a two-rod chain. 
 
 73. Required the cost of 1 doz. silver spoons, each weighing 
 18 pwt. 18 gr., at $ 1.15 per oz. 
 
 74. Reduce 7 gal. 3 qt. 1 pt. to the fraction of a bbl. 
 
 75. I sow 11 bu. 2 pk. 4 qt. of wheat, and raise therefrom 
 215 bu. 2 qt. How much is the average yield per bu. of seed ? 
 
 t 
 
 I , 
 
330 
 
 ARITHMETIC 
 
 76. The running time of the Empire State Express from New- 
 York to Buffalo is 8 hr. 30 min., and the distance is 440 mi. If 
 stops of 5 min. each are made at Albany, Utica, Syracuse, and 
 Rochester, what is its average speed per hr. ? 
 
 77. Find (a) the exact number of da. from Jan. 17, 1896, to 
 April 5, 1896; (b) the difference in time by subtraction of dates. 
 
 78. A railroad train moves 1 mi. in (j5 sec. What is its speed 
 per hr. ? 
 
 79. A note given Aug. 15, 1896, for 90 da., will mature 
 when ? 
 
 80. A can walk 3^ mi. in 50 min., and I> can walk 2J mi. in 
 36 min. How many yd. will A be ahead of B when A has gone 
 6 mi., if they start together ? 
 
 81. A farmer delivered at a warehouse four loads of wheat 
 weighing respectively 2113 lb., 2310 lb., 2270 lb., and 2091 lb. 
 How much should he have received at 72^ per bu. ? 
 
 82. The difference in longitude between two places being 
 9° 34' 25", what is the difference in time ? 
 
 83. A man has a salary of $400 a yr., and has $500 in the 
 bank. If he spends $ 500 a yr., in what time will his money be 
 all gone ? 
 
 84. What is the shortest stick that can be cut into pieces, 
 9 in., 12 in., or 15 in. in length, with nothing remaining ? 
 
 85. (a) What is meant by a Common Multiple of two or more 
 fractions ? 
 
 (b) Find the L. C. M. of 2}, 3|, 3-^%, Uj%. 
 
 86. (a) What is meant by the prime factors of a number ? 
 
 (b) Find the prime factors of 13,230, 22,050, and 23,625 ; and 
 
 (c) By means of the prime factors find their G. C. M. and 
 L. C. M. 
 
MISCELLANEOUS EXERCISE 
 
 331 
 
 87. Resolve 16,335 and 18,018 into their prime factors, and 
 from inspection of these write the prime factors of their (a) 
 L.C.M. and (b) G.C.M. 
 
 88. A farmer bought a number of horses and cows for $ 2000. 
 There were 3 times as many cows as horses, and a horse costs 
 twice as much as a cow. If each horse cost f 80, how many- 
 cows did he buy ? 
 
 89. The difference in weight of two chests of tea is 25 lb. ; the 
 value of both at ()5^ per lb. is $ 113.75. How many lb. of tea are 
 in each chest? 
 
 90. What is the smallest sum of money with which you can 
 buy chickens at 25^, or geese at 50^, or turkeys at 75^, or lambs 
 at $ 3, or sheep at $ 5, or pigs at $ 7, or cows at $ 35, or horses 
 at $ 140, and have exactly f 15 left for expenses ? 
 
 91. Ten cents will buy 3 oranges, 4 lemons, or 5 apples. How 
 many apples are worth as much as 5 doz. oranges and 7 doz. 
 lemons ? 
 
 92. One workman charges $3 for a day's work of 8 hr., and 
 another $3.50 for a day's work of 9 hr. Which had I better 
 employ, and how much shall I have to pay him for work that he 
 can do in a fortnight, working 6 hr. a day ? 
 
 93. A can do a piece of work in | of a da., and B in i of a 
 da. In what time can both together do it ? If f 1.40 be paid 
 for the work, how much should A receive ? 
 
 94. How many oranges must a boy buy and sell to make a 
 profit of f 9.30, if he buys at the rate of 5 for 3^, and sells at the 
 rate of 4 for 3^ ? 
 
 95. A and B dig a ditch in 50 hr. With C's help they could 
 
 have done it in 18J hr. 
 alone ? 
 
 In what time could C do | of the work 
 
 96. Three men can dig a certain drain in 8 da. They work 
 at it for 5 da., when one of them falls ill, and the other two finish 
 
 ■I 
 
 
 
 U 
 
332 
 
 ARITHMETIC 
 
 the work in 5 da. more. How iiiucli of the work did the first 
 man do before he fell ill ? 
 
 97. A boy can run 6 times around a circular plot of ground 
 in 52 sec. ; another boy can run 9 times around the same plot in 
 80 sec. If they start from the same place at the same time, and 
 run in the same direction, how many rounds will each make 
 before the faster boy overtakes the slower ? 
 
 98. Express in the form of a vulgar fraction the average of 
 
 3 A 
 
 1, .4|, and 
 
 .486J. 
 
 99. In a granary there are 4 bins, each 10 ft. long and 5 ft. 
 wide. How high must they be boarded in front to be capable of 
 holding 860 bu. ? 
 
 100. The outfit of a livery stable is worth | .SOOO ; \ the value 
 of the horses is equal to \ the value of vehicles, harness, etc. 
 Find the value of the horses. 
 
 101. A farmer agreed to pay his hired man 10 sheep and f 160 
 for 1 yr. labor. The man quit work at the end of 7 mo., receiving 
 the sheep and f 60 as a fair settlement. Find the value of each 
 sheep. 
 
 3 02. Divide $1200 among A, B, and C, so that A may have 
 $ 70 more than B, and twice as much as C. 
 
 103. A train going 25 mi. an hr. starts at 1 o'clock p.m. on 
 a trip of 280 mi. ; another going 37 mi. an hr. starts for the same 
 place at 12 min. past 4 o'clock p.m. When and where will the 
 former be overtaken? 
 
 104. In the number, 28,672, the value expressed by the first 
 two digits from the left is how many times the value expressed 
 by the fourth digit from the left ? 
 
 y 105. A town whose population was 10,000 increased 10% 
 every year for 3 yr. What was its population at the end of 
 that period? 
 
MISCELLANEOUS EXERCISE 
 
 333 
 
 106. A house and lot was sold for f 7030, at a loss of 16|% 
 of its cost. Find the cost. 
 
 107. Five men in a factory accomplish as much as 8 boys. 
 What per cent of a man's work does a boy do ? What per cent 
 of a boy's work does a man do ? 
 
 108. Forty-five per cent of a carload of melons were sold to 
 one dealer, and 33 J % of those left to another. How many were 
 there in the car before any were sold, if after the second sale 
 there remained 110 ? 
 
 109. In a certain school 48% of the pupils are boys, and there ^ 
 are 39 girls. Find the number of boys. 
 
 110. How many lb. of flour will be required to make 1000 lb. 
 of bread, if the bread weigh 30% more than the flour used? 
 
 111. 93 lb. 6 oz. is what per cent of 43 lb. 12 oz. ? 
 
 112. Water, in freezing, expands 10%. If 1 cu. ft. of water i^ 
 weighs 1000 oz., find the weight of 1 cu. ft. of ice. 
 
 113. Give answers to the following: 
 
 (a) 15|% of 660 = (d) .2% of 40 = 
 
 (b) 660 is 15}% of what number ? (e) 40 is .2% of what number ? 
 
 (c) f is what per cent of | ? 
 
 (/) What per cent of itself must be added to a number so that 
 the sum diminished by 10% of itself may be 17% more than the 
 original number ? 
 
 114. Brooms are bought wholesale at $20 a gross. What per 
 cent profit will be made by selling them at 20^ each ? 
 
 115. A merchant purchases sugar at $4.50 per cwt. At what 
 price per lb. must he sell it in order to gain 5|%? 
 
 116. I bought a house for $4000 and spent 40% of the cost 
 in repairs. What must I rent it for a month in order to make 
 a clear gain of 5% of the total cost, taxes and repairs amounting 
 to $ 72 yearly ? 
 
 ii> 
 
 >ii ! 
 
 1^ 
 
 
334 
 
 ARITHMETIC 
 
 117. By selling a piano for $260, a dealer loses 20%. How 
 much should he have sold it for to gain 5%? 
 
 118. A man having lost 20% of his capital is worth exactly 
 as much as another who has just gained 15% on his capital. 
 The second man's capital was originally $ 9000. What was the 
 first man's capital ? 
 
 119. A dealer sold an article for $8.10 and lost 10%. At 
 what selling price would he have gained 10%? 
 
 120. A bookseller deducts 10% from the market price of his 
 books, and after this has a gain of 25%. He sells a book for 
 $7.20. Find the cost price of the book, and what per cent the 
 marked price is in advance of the cost price. 
 
 121. A merchant bought 1000 yd. of carpet at GO^ a yard, and 
 sold f of it at a profit of 30%, J at a profit of 20%, and the rest 
 at a loss of 20%. How much did he receive for the carpet ? 
 
 122. A sells goods to B at a gain of 12%, and B sells the same 
 goods to C at a gain of 7|-%. C paid $3762.50 for the goods. 
 How much did A pay for them ? 
 
 123. A machinist sold two seed-drills for equal sums of money. 
 He gained 25% on the one and lost 25% on the other. His total 
 loss was $ 9.60. Find the cost of each drill. 
 
 124. R purchased a house and lot for $3300, paid $975 for 
 repairs, and now rents the premises for $30 a month. If he 
 expends annually for taxes $48.70, and for incidental repairs 
 $ 35, what is his per cent of annual income on his investment ? 
 
 125. A merchant closed out a stock of cloaks for $311.04, 
 at a loss of 28%. Required the loss by the transaction. 
 
 126. By selling my cloth at $ 1.26 per yd. I gain 11^ more than 
 I lose by selling it at $ 1 .05 per yd. What would I gain by sell- 
 ing 800 yd. at $ 1.40 per yd ? 
 
 127. A merchant marks his goods at 40% in advance of cost, 
 and in selling uses a lb. weight i oz. too light. If he throws off 
 10% of his marked price, find his gain per cent. 
 
MISCELLANEOUS EXERCLSE 
 
 335 
 
 128. State the relation between 1 lb. Troy and 1 lb. Avoir. 
 What is the gain per cent when the selling price per oz. Avoir, 
 is the same as the cost per oz. Troy ? 
 
 129. A man bought a bankrupt stock at 60^ on the dollar of 
 the invoice price, which was $4840. He sold half of it at 10% 
 advance on invoice price, half the remainder at 20% below invoice 
 price, and the balance at 50% of invoice price. His expenses 
 were 10% of his investment. Find his loss or gain (a) in money, 
 and {b) in rate per cent. 
 
 130. The list price of an article is f 150. If trade discounts 
 of 25% and 16|% are allowed, what is the net price ? 
 
 131. If a dealer buys stoves at a discount of 22% from list 
 price, and sells them at list price, what is his per cent of gross 
 profit on the investment ? 
 
 132. Required the net price of an article listed at $400, 30%, 
 10%, and 5% off. 
 
 133. From the list price of a line of goods a purchaser is 
 allowed a trade discount of 20% ; a further discount of 10% 
 off the trade price for taking a quantity, and a still further dis- 
 count of 5% off his bill for cash. Find his gain per cent by sell- 
 ing at 10% less than the list price. 
 
 134. The net price of a reaper is $ 158.40, and the trade dis- 
 counts allowed are 20% and 10%. Find the list price. 
 
 135. A commission merchant sold coffee for me and remitted 
 $1960, after deducting his commission of 2%. What is the 
 value of the coffee ? 
 
 136. If an agent receives $1092 to buy pork, how many lb., 
 at 6J^ per lb., can he buy and retain his commission of 5% for 
 buying ? 
 
 137. A commission merchant sold 1014 bu. of oats, at il^ 
 per bu., paid $33.74 freight charges, and retained 3J% commis- 
 sion. How much should he remit to the consignor ? 
 
386 
 
 AKITIIMETIC 
 
 138. A lad earned f 21.1(5 eolleeting accounts for a physician. 
 He was allowed 5J%. What amount did he collect ? 
 
 139. Find the premium paid to insure a house worth .^7500 
 for I of its value for 3 yr., the rate bciag f % of the policy for 
 each year. 
 
 140. What premium must be paid to insure a cargo of 4880 bu. 
 of wheat, valued at 78^ per bu., at 1|%, the policy being for only 
 J of its value ? 
 
 141. A building is insured for .*i>400 more than f of its cost at 
 4%. If destroyed, the loss will be $216. Find the cost of the 
 building. 
 
 142. A dealer shipped 200 bbl. of apples to Liverpool; the 
 average cost of the apples was $ 3.75 per bbl. For what sum must 
 he have the apples insured at J% premium to guard against all 
 loss, in case of shipwreck, his other expenses being $ 75 ? 
 
 143. If in a certain town $3093.75 was raised from a j% tax, 
 what was the assessed valuation of the property in the town ? 
 
 144. A tax of $24,750 is levied on a town, the assessed valua- 
 tion being 1.5 mills on a dollar. What tax does a man pay on an 
 income of $ 1100, of which $ 400 is exempted ? 
 
 145. A farmer whose property is assessed at $ 9600 pays on 
 the dollar, IJ mills for township rates, 1^ for county rates, IJ 
 for railway bonus, and 2^ for school rate. How much does he 
 pay in all ? 
 
 146. IVs tax was $ 86.2755 when the rate was 7.635 mills on a 
 dollar. What was the assessed valuation of his property ? 
 
 147. A certain school section is assessed for $150,000. The 
 trustees have built a schoolhouse costing $ 1800. 
 
 (a) What will the schoolhouse cost a ratepayer whose property 
 is assessed for $ 4500 ? 
 
 (b) What would be the rate of taxation per annum on the 
 whole section if the house were paid for in six equal annual pay- 
 ments, without interest ? 
 
MISCELLANEOUS EXERCISE 
 
 837 
 
 148. A clerk pays f 7.50 taxes on liis salary. What is his 
 total salary if $400 of it is exempt from taxation and a 2.V% rate 
 is levied on the remainder ? 
 
 149. What per cent mnst be assessed on $ 1,500,000 to produce 
 129,400 after paying 2% for (collecting? 
 
 150. An importer receives an invoice of kid gloves billed at 
 $680, pays a duty of 50% ad valorem, and sells them at an 
 advance of 33^% on their gross cost to him. How does the price 
 paid by the purchaser compare with the exporter's price ? 
 
 151. A merchant imports 75 cases of indigo, gross weight 196 
 lb. each, allowing 15% for tare. What was the duty at 5^ per lb. ? 
 
 152. What will $1 amount to in 3 yr. 216 da., at 7|% per 
 annum, simple interest? 
 
 153. Find the simple interest on f 597.50 for 2 yr. 5 mo. 
 12 da., at 8% per annum. 
 
 154. How long will it take $450, at 8%, to yield $21.30 
 interest ? 
 
 155. What amount will be due July 1, 1896, on a note of $80, 
 drawn Feb. 6, 1896, and bearing interest at 5|% per annum, 
 exact interest ? 
 
 156. Find the sum due Sept. 2, 1893, on a note for $ 147.33, 
 given Jan. 13, 1893, and beaming interest at 4% per annum. 
 
 157. Find the exact interest on $225 from July 13, 1893, to 
 Sept. 3, 1893, at 6%. 
 
 158. Find the interest on $ 1, at 71% per annum, from Jan. 1, 
 1895, to June 3, 1895. (Complete answer required.) 
 
 159. What sum will amount to $354.09 in 7 mo., at 3% perj^ 
 annum ? 
 
 160. In what time will $ 1350 earn $ 31.88 at 5% per annum ? 
 
 161. Find the face of a draft that cost $434.70, at |% 
 premium. 
 
 :i.r 
 
338 
 
 ARITHMETIC 
 
 162. If the interest is f 12.57, the time 8 mo. 2 da., and the 
 rate per annum 5^%, what is the principal? 
 
 163. Find the exact interest on $ 150 from July 16 to Dec. 9, 
 at 5% per annum. 
 
 164. A person borrows money for 6 yr. at 3^%, simple interest, 
 and repays at the end of the time, as principal and interest, $847. 
 How much did he borrow ? 
 
 165. Find the simple interest on f 912.50, at 8%, from Feb. 
 13, 1895, to Dec. 19, 1896. 
 
 166. A note of f 360, drawn April 20, 1895, is paid July 2, 
 1896, with interest at 7|% per annam. Find the amount paid, 
 simple interest. 
 
 167. Oct. 15, 1895, a young man deposited in the savings 
 bank the sum of 1 860.75. May 20, 1896, he withdrew the prin- 
 cipal and simple interest at 4% per annum. What amount did 
 he withdraw? 
 
 168. Bought a horse for $ 160, and gave in payment my note 
 dated Aug. 15, 1896, with interest at 7^% per annum until paid. 
 Jan. 9, 1897, I sold the horse for $ 200 cash, and paid my note. 
 What was my net gain ? 
 
 169. If for $ 7 I can have the use of $ 35 for 3 yr. 4 mo., how 
 much a month shall I have to pay for the use of $8750? 
 
 170. Jan. 1, 1894, a person borrowed $2417.50 at 6|%, 
 simple interest, promising to return it as soon as it amounted 
 to $ 2582.50. On what day did the loan expire ? (365 da. = 
 
 1 yi'.) 
 
 171. March 1, 1896, a storekeeper bought goods amounting, 
 at catalogue prices, to $840, on which he was allowed succes- 
 sive discounts of 33^% and 5%. The account is payable in 
 60 da., after which time interest is to be charged at 7% i)er 
 annum. June 1, 1896, he paid $ 100. How much is due July 1, 
 1896 ? 
 
mrm' 
 
 MISCELLANEOUS EXERCISE 
 
 339 
 
 172. Find the proceeds of a note for |>200 given at Albany, 
 N.Y., for 3 mo., and discounted at bank the day it was made at 
 6%. 
 
 173. Find the proceeds of a note for .Ij; 1G8 due Oct. 20, 180(), 
 and discounted Sept. 25, 1890, at a Brooklyn, N.Y., bank, at 0% 
 per annum. 
 
 174. f 12;Uj%. St. Louis, Jan. 15, 1894. 
 Ninety days after date, I promise to pay A. Dee, or order, the 
 
 sum of one thousand two hundred and thirty-four jW- dollars, at 
 
 the Bank of C'ommerce here. Vahie received. 
 
 C. Dke. 
 
 This note was discounted Feb. 10, 1894, at G% per annum. 
 Find the proceeds. 
 
 175. A note for $230, drawn Jan. 2, 189(5, at ,') mo., and bear- 
 ing interest at 8% per annum, is discounted Feb. 1 at 7%. Find 
 the proceeds. 
 
 176. Find the proceeds of the following note : 
 
 $2400. Hamilton, Ohio, Feb. 8, 1800. 
 
 Five months after date, value received, I promise to pay 
 Thomas Cowan, or oi'der, the sum of two tliousand four hundred 
 dollars, at the Bank of Hamilton, with interest at 0% per annum. 
 
 Van(je Allen. 
 
 Discounted May 22. 1890, at 7%. 
 
 177. The discount on a note for $3000, wliich matured A])ril 
 21, 1890, and was discounted Feb. 24, 1890, was $45.00. Find 
 the rate of discount. 
 
 178. A buys 000 yd. of silk at 95^^ per yd., and sells it at once, 
 receiving in payment a 90-day note for $700, whicli he at once 
 discounts at a bank at (>% per annum. Find l\\e gain. 
 
 179. For what sum must a note be drawn June 1, 1890, pay- 
 able in 90 da., so that when discounted June 14, at 8%, the i^ro- 
 ceeds will be $ 717.20 ? 
 
340 
 
 ARITHMETIC 
 
 180. What rate of interest is made by a bank which discounts 
 a 90-day note at 6% per annum ? 
 
 181. Jan. 1, A owes a bank $15,000. He offers for discount 
 certain notes: $2500 due Feb. 15, 13700 due March 13, and 
 $7500 due April 1. If these are discounted at 8% per annum, 
 how much cash must he pay ? 
 
 182. What must be the face of a note so that when discounted 
 at a bank for 90 da. at 0%, the proceeds will be $ 1969? 
 
 183. What is the present worth of a note for $ 540, due in 90 
 da., drawing interest at 0%, discounted at 8%, true discount? 
 
 184. Find the proceeds of a note for $ 292.73, discounted at 
 bank, for 35 da., at G% per annum, exact interest method. 
 
 185. Find the value of (1.03)^ 
 
 186. A man has the choice of loaning his money at 7|%, com- 
 pound interest, or at 8%, simple interest, money and interest to 
 be paid at end of 3 yr. Show which is the better investment. 
 
 . 187. An annual deposit of $ 250 is made with a loan company 
 which pays 4% per annum on deposits, compounded half-yearly. 
 Find the amount of all these deposits when the fourth has been 
 made. 
 
 188. June 30, 1890, I borrow $16.50, to be returned April 
 30, 1892. With compound interest at 6^%, what amount must 
 I then pay ? 
 
 189. A man puts $350 in a savings bank each year, making 
 his first deposit Dec. 31, 1893. How much will there be to his 
 credit Jan. 1, 1897, the bank adding 4% per annum ? 
 
 190. A owes B $400 due in 1 yr., $300 due in 2 yr., $200 
 due in 3 yr. What sum paid now would cancel the debt, money 
 being worth 5% per annum, compound interest? 
 
MISCELLANEOUS EXERCISE 
 
 341 
 
 191. A lent a sum of money for 2 yr., at 10% per annum, 
 interest compounded yearly. B lent an equal sum for the same 
 time at 10% per annum, interest compounded half-yearly. B 
 gained $ 220.25 more than A. Find the sum each lent. 
 
 192. A man rents a farm for two years at $ 441 per annum, the 
 rent for any year being supposed to be paid at end of that year. 
 Money being worth 5% per annum, compound interest, find what 
 sum would, in advance, pay the two years' rent. 
 
 193. I invest f 39,900 in per cents at 95. What is my 
 income ? 
 
 (a) What sum invested in 8% bonds at 33^% premium will 
 yield an income of $ 1200 ? 
 
 (Jj) What if the bonds were at 33 1% discount ? 
 
 194. Find the cost of 04 shares of railroad stock at 107 f, 
 brokerage i%. 
 
 195. Required the gain on 28 shares of stock bought at 97 .V 
 and sold at 103 J. 
 
 196. April 4, 1896, 20 shares of Chicago City Railway stock, 
 quoted at 210, were sold, brokerage i%. Find what was received 
 by the owner of the stock. 
 
 197. A railroad company declared a dividend of 1J% for the 
 quarter ending Sept. 30, 1894. If the stock was quoted at 105, 
 what was the rate of income per annum on an investment in the 
 stock of that company ? 
 
 198. How much must be invested in U. S. 5's, at 113 J, to secure 
 an annual income of f 175 ? 
 
 199. A man invests ^ 12,000 in 3% stock at 75. He sells out 
 at 80 and invests ^ of the . proceeds in Sh % stock at 96, and the 
 remainder at 5% par. Find the change in his income. 
 
 200. A man owned $8940 bank stock, which paid a yearly 
 dividend of ^%. He sold out 102 1, and invested the proceeds 
 
■.,r-.».Tcr'.''|-,^.y;,^'^T;'f 
 
 342 
 
 ARITHMETIC 
 
 in Michigan Central stock at 74J, paying a yearly dividend of 
 3%. By how nuicli was his yearly income changed by the 
 transfer ? 
 
 201. What must be the market value of 6% stock, so that 
 after paying an income tax of 16 mills on the dollar, it may 
 yield 5% on the investment ? 
 
 202. A person bought stock at 95^, and after receiving a half- 
 yearly dividend of 7% per annum, sold out at 92 J, brokerage 
 each way being ^%. If his net gain was $25, how much stock 
 did he buy ? 
 
 203. If a 5% stock sells at 105, how much must be invested 
 in it to yield a yearly income of $ 794, after paying an income 
 tax of 15 mills on the dollar, $400 of income being exempted 
 from taxation ? 
 
 204. A man invests f 0000 in 5% stock at 120. At the end of 
 1 yr., having just received the yearly dividend, he sells at 121|^. 
 How much better oif is he than if he had loaned his money at 
 5% per annum? 
 
 205. I own $0000 of bank stock, paying an annual dividend 
 of 5%. How much will my annual revenue from the bank stock 
 be reduced by selling enough of it at 72 to pay a note of $3735 
 9 mo. before it is due, reckoning true discount at 5% per annum ? 
 
 206. A man received $ 495 as dividend on his bank stock. 
 He sold 40 shares (f 100) at 143J^, and the remainder at 1441, 
 paying ^% brokerage on each transaction. What were the net 
 proceeds of the sales ? 
 
 207. $ 1200 is to be divided between two persons, A and B, so 
 that A's share is to B's share as 2 to 7. 
 
 208. What is the ratio of 3J to | ? Answer in per cent. 
 
 209. Divide 102(5 into four parts that shall be in the ratio 
 of 3, 11, 17, and 23. 
 
MISCELLANEOUS EXERCISE 
 
 343 
 
 210. All upright pole 16 ft. long casts a shadow 5 ft. 4 in. 
 long, and at the same hour the shadow of a tree is found to 
 be 2(j ft. 9 in. Required the height of the tree. 
 
 211. The sum of three numbers is 940. The first iiumber 
 equals | of the second, and the second equals j\ of the third. 
 Find the numbers. 
 
 212. If 18 men do | of a piece of work in 30 da. of 10 hr., in 
 what time should 15 men do the whole work, working 9 hr. a da. ? 
 
 213. If 10 yd. of muslin, IJ yd. wide, cost $ 1.30, what is the 
 cost of 12 yd., Vg yd. wide ? 
 
 214. One-sixth of the square of a certain number is 384. Find 
 the number. 
 
 215. Find the square root of .0 correct to three decimal places. 
 
 1 
 
 '2' 
 
 iet 
 
 50 
 
 O 
 
 216. F^ind, within one inch, the side of a square whose area is 
 5 A. 
 
 217. A rectangular field whose length is J of its width contains 
 
 2 A. 112 s(p rd. Find the length of a diagonal. 
 
 218. lle(iuired the base of a right-angled triangle whose liypote- 
 niise is 1()J ft., and perpendicular 9| ft. 
 
 219. A ladder 78 ft. long stands perpendicularly against a 
 building. How far must it be pulled out at the foot that the top 
 may be lowered ft. ? 
 
 220. A I'oad runs round a circular pond; the outer circum- 
 ference is 440 yd., and the width of the road is 20 yd. Find the 
 area of the pond. 
 
 221. In order to drain a swamp a ditch was dug 1 mi. long, 
 
 3 ft. dev\), (> ft. wide, at the surface, and 4 ft. wide at the bottom. 
 Find the total cost at 9^ per cu. yd. 
 
'-frw'T?:TTr^r9!llf^ffnrrr^ 
 
 344 
 
 ARITHMETIC 
 
 222. How many gal. in a circular cistern G ft. in diameter and 
 7 ft. deep ? 
 
 223. The surface of a cube is 432 sq. ft. What is its volume ? 
 
 224. (a) A circular cistern, 8 ft. in diameter and 9 ft. in depth, 
 is tilled with water to the height of ft. How many gal. of 
 water in the cistern ? (1 cu. ft. = 7.48 gal.) 
 
 (b) If a sphere whose diameter is 4 ft. is submerged in the 
 water in the cistern, how high will it cause the water to rise ? 
 
 225. How many cd. are there in a cylindrical log 20 ft. long 
 and 3 ft. G in. in diameter? 
 
 226. Find the diameter of a circle whose area is equal to the 
 sum of the areas of two circles whose diameters are 12 in. and 
 IG in. respectively. 
 
 227. Find the area of the curved surface of a right circular 
 cone the radius of whose base is 3.5 in., and whose altitude is 
 7 in. 
 
 228. A chord of a circle, whose radius is 12 in., subtends a 
 right angle at the centre of the circle. Find the area of the 
 smaller segment cut oft" by this chord. 
 
 229. A spherical shell, internal diameter 14 in., is filled with 
 water. Its contents are poured into a cylindrical vessel whose 
 internal radius is 14 in. Find the depth of the water in the 
 cylinder. 
 
 230. The sides of a triangle are 40, 45, and 50 ft. respectively. 
 Find the length of the perpendicular from the vertex to the side 
 45 ft. 
 
 231. The diameter of a circular plate of lead is 13 in. From 
 this is cut out a circular plate of radius G in., and the remainder 
 of the lead is moulded into the form of a circular plate, with J of 
 the former thickness. . Find the diameter of this plate. 
 
 232. The sides of a triangle are 13, 14, and 15 ft. Find its 
 area and the length of the three perpendiculars from the angles 
 on the opposite sides. 
 
MISCKLLANEOUS EXERCISE 
 
 345 
 
 233. The external dimensions of a rectangular covered box, 
 made of inch stuff, are 7, 8, and 9 ft. Find the capacity of the 
 box and the quantity of lumber in it. 
 
 234. A ball of yarn 3 in. in diameter makes one mitten. How 
 many similar mittens will a ball 6 in. in diameter make ? 
 
 235. A farmer employs a number of men and 8 boys; he pays 
 the boys f .65 and the men f 1.10 per day. The amount that he 
 paid to all was as much as if each had received $ .92 per day. 
 How many men were employed ? 
 
 236. Two men start from the same point at the same time to 
 walk in the same direction around a block of land 1\ mi. on each 
 
 side. A goes at the rate of 4 mi. and B 3 mi. an hr. 
 will A walk before he overtakes B ? 
 
 How far 
 
 237. A commission merchant sells a consignment of wheat for 
 $27,500, on a commission of 2^%. He pays $250 for freight 
 and storage, and with the net proceeds buys pork at $ 6.25 per 
 cwt., charging 21% for buying. How many cwt. of pork does he 
 buy, and what is the amount of his two commissions ? 
 
 238. Find the cost of the material required to fence 2 J mi. of 
 railway (both sides), posts placed 8 ft. apart, an 8-in. base 1 in. 
 thick, a 2 X 4 in. rail at top, and G strands of wire. The posts 
 cost 12^^ each, the lumber f 14 per M., and the wire 4^ per lb. 
 (A lb. of wire stretches 1 rd.) 
 
 239. A number of two digits is multiplied by 3, and the product 
 placed to the left of the original number. Show that the number 
 so formed is always exactly divisible by 7. 
 
 240. A merchant reduces the marked price of an article by 
 a certain per cent. He gives the same per cent off this reduced 
 price for cash. The cash price is now J| of the original marked 
 price. Find the rate per cent. 
 
340 
 
 ARITHMETIC 
 
 241. Divide ,|l 910 a,„o„,, A ,> ,„ 
 
 242. A starts to walk f,- / ""'' ""'"' ^''^^ "^ C'«- 
 -a 1 I,,.. afte..an:t .t; ^I V! T ■"'' "^ ^ -'• » ^ 
 VValk,„s on, B arrives at Q o ,?' l ''"'l overtakes A i„ 4 hr 
 f.om P to Q. ' « - '"• 'before A. Fina the distance 
 
 .livi:f.,eVT'^;;f '^'""^ 'y « i^ "- s™ of its digits is 
 - -- e;Sar r:r J- -?- - - ^anda 
 
A's shcare 
 ' of C's. 
 
 »i- an hr., 
 in 4 hr. 
 distance 
 
 digits is 
 e hands