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Tous les autres exemplaires originaux sont fiimis en commenpant par la premlAre page qui comporte une empreinte d'impression ou d'illustration et en terminant par la dernidre page qui comporte une telle empreinte. Un des symboles suivants apparaTtra sur la derniire image de cheque microfiche, selon le cas: le symbols — ► signifie "A SUIVRE ", le symbols V signifie "FIN". Les cartas, planches, tableaux, etc., peuvent dtre film^s d des taux de reduction diffdrents. Lcrsque le document est trop grand pour dtre reproduit en un seul clich6, il est film6 d partir de I'argie sup^riour gauche, de gauche d droite. et de haut en bas, en prenant le nombre d'images n6cessaire. Les diagrammes suivants illustrent la mdthode. 1 2 3 1 2 3 4 5 6 / i THE PUBLIC SCHOOL MENTAL ARITHMETIC ■0 S f •The m THE PUBLIC SCHOOL MENTAL ARITHMETIC BASED ON McLELLAN AND DEWEY'S " PSYCHOLOGY OF NUMBER »» BT J. A. McLELLAN, A.:\L, LL.D. PUESIDEST ONTAUIO NOIIMAI, (< •I.I.KC K AUTHOR (with DR. DEWEV) OV "THE I'SYCIK.Um; V OK SI MltEIl," "APPLIED PsvcnoLOUY," "klements of ai.<;i:i:ka," etc, AND A. F. AMES, A.B. HONOR graduate IX MATHEMATICS; FOKMEUI.Y MATHEMATICAL MASTEU, ST. THOMAS COI,I,E<; I ATE INSTITUTE SUPERINTENDENT OF SCHOOLS, RIVERSIDE, ILL. TOPvONTO THE COPP CLAUIv COM PAN V, Limited COPTRIGHT, 1899, By the macmillan company. Set up and electrotyped January, 1899. Reprinted August, 1899 ; September, 1901. NorS3oxilPrf8i ress J. 8. Gushing & Co. — BcrwK V k Smith Norwood Mass. U.S.A. ! PREFACE •| i| It has recently been stated by a well-known col- lege professor that *'boys enter college or training school at eigliteen, after having spent fiom one sixtli to one fourth of their entire school life in siudying mathematics. Yet they know very little matlie- matics. In their examination the asking of even three questions shows that they haven't the dimmest idea of what it is all about." This statement, if true, does not prove — as the professor seems to think it does — the justice of the Ilamiltonian on- slaught on mathematical study. It simply proves that the prevailing methods of teaching arithmetic are radically wrong. The serious defects in existing methods are mainly due to the fact that they take no account of the real nature of number, and of how the child's mind works in grasping tiie concepts of number and numerical relations. In other words, arithmetic has never been "psychologized." If there is a science of education and rational methods founded upon it, there must be a psychology of arithmetic, a psychology of language, etc. The one- sided theory that education is concerned only with fitting the child for existing civilization has made VI PR K FACE tlie so-cjiIKmI '^ pnicliciir' Jiiins juid metliods domi- iijiut in scliuol work. These metliods are, and ever inust be, essentially defeetive, inasmneh as they are founded on a half truth ; they take no account of the powers and capacities of the individual who is to be made an effective instrument in maintaining and perfecting tins civilization into which he is born. The individual, indeed, lives and moves and has his being in society, and therefore there is a social side to education. Sociology must have some- thing to say on the i)roblem of education. But, on the other hand, society lives and moves and has its being only through the development of the individ- ual ; that is to say, education has its ])sycliological side. While the social side, the idea of fitting the individual to play his part in existing civilization, may give a standard for the development of the powers and capacities of the individual, and may point out the subject-matter to be used in this devel- opment, it does not show liow this development of powers, this adaptation, is to be secured. In other words, sociology indicates ivliat is to be done with the individual, but fails to show liow it is to be done. This is a question of psychology. The meaning of this is that we must have both a psy- chology of the individual mind and a psychology of the subjects through which it is to be developed. On this principle the '' Public School Arithmetic," the "Primary Public School Arithmetic," and the I I r UK FACE VI I ^ < 't "Public School Moiital Aritlmiotic " liave been pre- ])iii'e(l. It is believed tliat, by direct tejiclung and }iel[)rMl siif'-i^estion, these books will, in some de^ij^i'ee, c()ntril)ute to the ^tj^rowth of a rational, and therefore economical, method of teachini^. At all events, the series differs from all other text-books in beiiiir based on the Psycholoin^y of Number. This '^ iMental Arithmetic" completes the scries and completes the method. In the methods of the schools — speakiuG^ generally — the ignoring of mental arith- metic or the teaching of it in a haphazard, and there- fore ineffective, way contributes very largely to the present unsatisfactory results, viz. '4ittle mathemat- ics and less training of logical faculty." IMental arithmetic, systematically taught from a I'ationally prepared text-book, is the life and soul of rational method. Tliere is constant adaptation to the nor- mal mental action of the child. During the les- son the teacher is in vital touch with the child's mind ; sees the child's personal self-activity in the making of images and in controlling their move- ments. There is hence the least possible waste for both teacher and pupil. The teacher takes care of the image, and then the concept takes care of itself. From long and varied experience, both in teaching the subject and inspecting the teaching of others, it is firmly held that, compared with " written " arith- metic alone, mental arithmetic, if systematically taught, Avill produce at least twice the knowledge Vlll ITiKFACK jiiid twice tlie rowKii in a ;•<€ QUANTITY — UNIT — NUMBER Lesson 1 1. Make a mental picture of a line 12 in. long and also of a unit of measure 3 in. long. How many- times will the 3-in. unit measure tlie quantity/ 12 in.? 2. The number 4 obtained in example 1 is 4 what? 4 units of 3-in. in what ? 3. This number 4 is often called the ratio of 12 in. to 3 in. What is the ratio of 12 in. to 4 in. ? 4. What part of 12 in. is equal to 3 in. ? ^ is often called the 7*atio of 3 in. to 12 in. What is the ratio of 4 in. to 12 in. ? 5. Find the number of 3-in. units of length in each of these quantities : 15 in. ; 2 f t. ; 30 in. ; 1 yd. ; 1 ft. 6 in. ; 2 ft. 3 in. What is the ratio of each of these quantities to the unit ? Of the unit to each quantity ? 6. In question 5 how did you find the number of 3-in. units in each quantity ? B 1 i MENTAL ARITHMETIC 11 • 7. How would you find the iminber of 2-iii. units in a given quantity? Of 3-in. units? Of 4-in. units ? Of units of any length ? 8. Find the number of 2-in. units in each of tliese quantities: 14 in., 1 ft. 8 in., 24 in.; 2 ft in. ; 1 yd. 4 in. ; 1 yd. 1 ft. 9. In questions 10-15 name the quaiitlti/ and the mnt m each case. How will you find the nuni- ber of units in each quantity ? 10. Into how many hair ribbons, each 9 in. loner can you divide a piece 36 in. ? "" 11. A can contains 12 qt. of milk. How many pitchers, each containing 3 pt., can be filled with its contents ? 12. A furnace burns on the average 5 T. of coal a month. In how many months will it burn 40 T. ? 13. How many pounds of sug-.r at 6^ a lb. will cost 48^ ? How many pounds of sugar at 5^ a lb. wdl cost $ 2 ? 14. How many barrels of flour at $5 a bbl. will cost $ 45 ? 15. How many pounds of raisins at 8 ^ a lb. will cost 50 p ? 16. In questions 10-15 the number tliat you found m eacli case was the nun.her of wliat ? Tliis number IS otton called the ratio of the quantity to the unit. « LESSON 1 8 17. How many widtlis of carpet 3 ft. wide are needed to carpet a room 6 yd. wide? In terms of wliat unit must you express both the quantity 6 yd. and the unit 3 ft. before dividing ? o 18. How many strips of paper 2 ft. wide are needed to cover the walls of a room whose perimeter is 20 yd. ? 19. How many quarts of milk at 6j^ a qt. will cost .1^1. 80? 20. In the following examples what numbers express the measurements ? Quantity 20 da. Jhr. ^4 2bu. Unit 15 5 da. 6 min. 2 dimes 8 qt. Quantity 21b. 10 qt. 1 pt. 4 yd. 2 ft. 3 gal. 3 qt. 16 sq. in. Unit 4 oz. 3pt. 2 ft. 5 qt. 2-in. square 21. What is the ratio of each quantity in exam- ple 20 to its own unit ? What is the ratio of the unit to the quantity ? 22. A man earns |15 a week and spends 111. In liow many weeks will he save ^ 13G ? What is the unit here ? 23. Two boats travel up a river at the rate of 12 and 9 mi. per hour, respectively. If they start together, when will the faster be 36 mi. ahead ? 4 MENTAL AIHTIIMI:TIC 24. A farmer's wife sold 10 lb. of butter at 2^^ a lb. and with the proceeds bought sugar at 5^ a lb. How many pounds did she buy ? What is the quantity here? ^ 25. A man travels 12 hr. at the rate of 4 mi. an hr. and returns to the starting-point at the rate of 6 mi. an hr. How long does it take him to return ? Lesson 2 1. i\Iake a mental picture of a line 18 in. lono* divided into 3 equal parts. JIow long is the unit of measure that divides it into 3 parts ? 2. What unit would divide it into 6 parts? 2 parts? 9 parts? 3. Find the units that divide each of these quantities into 4 parts: 20 in.; 1 ft. 4 in.; |24; 1 bu.; 1 bu. 8 qt.; 2 doz. oranges. 4. In question 3 how did you find the value of the unit in each case? 5. How would you find the size of the unit divid- ing a quantity into 4 parts ? 5 parts ? 6 parts ? Into any number of parts ? 6. In questions 7-13 name the quantity and the number in each case, llow will you find the value ot each unit ? 7. If 3 lb. of coffee cost 96^, what is the price of 1 lb. ? ^ i«•' IUi-anck for examplo 7 in: Tlie Sinn siuMit - .') ^ + 5> ^ (;ivo tlu; statement of Ixdanre. for encli of Ukj following examples of tiiis losson us yon solve tlioni. 7. Miriam paid T)^ for street ear fare and spent 9^ for a yard of ribbon. What did she spend altogether ? 8. A man spent >i!<8 for a coat and ^\ for a vest. What did both eost iiim ? 9. l':thel spent 0^ for apples and 9^ for oranges. How mueh did she spend in all? 10. A hidy paid 87 for a lounge and *5 for a chair. What did 1/oth cost? 11. A grocer buys })ineapples at 9^ apiece and sells them at a gain of 4^ each Find the selling price. 12. Add -4 to (>, 9, 3, 7, 8, 5, G, 4, 7, 6, 9. 13. Add 7 to 8, G, 3, 5, 4, 8, 9, 7, 2, 5, 9. 14. Add 5 to 2, 8, 4, 9, 8, 7, 5, 9, 6, 8, 7. 15. Add 9 to 7, 4, 9, G, 3, 9, 8, 5, 2, 4, 1. 16. Add 3 to 5, 7, 9, G, 4, 5, 7, 3, G, 9, 8. 17. Add 8 to 5, 9, 7, 5, I), 2, 8, G, 9, 7, 4. 18. Add G to 9, 5, 8, 7, 4, 8, 5, 7, 4, 9, 6. Statkment of Balance The weight = 1 times 2 lb. + 3 lb. 19. What is the weight of 4 2-lb. cans of corn and 3 lb. of cheese ? LESSON 4 \) 20. Whiit is tli(3 cost of 2 s[)0()ls of thread lit M apiccu and a paper of needles at r)f^? 21. What lunst I pay for a paper of i)ins at T^ind 2 hiuiehes of tape at 4^ apiece ? 22. Add 5 to each of these numbers. (a) G, 2(>, {\i\ 73, 80, 96, 87, 10, 4<.), r>0. (/>) 3, 82, 18, 68, 48, 88, 58, 24, 78, 98. 0) 8, 08, 40, 28, 58, 38, 77, 18, 80, 98. ((0 4, 54, 14, 44, 09, 84, 94, 78, 29, 84. ie) 7, 17, 88, 57, 24, 87, 05, 97, 40, 77. (f) 5, 25, 84, 05, 75, 90, 15, 45, 57, 89. 0/) 9, 49, 19, 28, 84, 59, 99, 79, 88, 09. (/t) If to any number endin^^ in 9 you add 5, in what iigurc does the sum end ? 23. A(hl in turn each of the foUowing numbers to the numbers given in (question 22 : 2, 0, 8, 7, 1, 9, 4, 10, 8. 24. Saturday Helen had 58^ in the Penny Savings and Monday she put in 7^ more. How much had she then ? 25. I paid *80 for a gohl watch, ^d for a chain, and -ii^ 5 for a gold ring. Find the total cost. Lesson 4 Give the statement of balance in each example. 1. I paid 98^ for a Dewey sailor suit and 8^ for a flair. Find the cost of both. 10 MKNTAL AKITIIMICTIC I. il 11 ( f ii ii 2. Find the cost of 2 yd. of rlhhoii at 12^ a yd. and 1 yd. of cambric at t>^. 3. A man earns 't 1;") a week, tlie older of his two sons ^1, and the yonnj,^^' ^{j. Find their total weekly earnings. 4. Add the followincr : 5. 8 4 G 9 7 8 5 9 8 4 8 9 47 53 75 87 76 94 8 9 4 6 5 2 7 7 8 9 72 37 58 88 43 94 6. Find the cost of 8 bunches of lirecrackers at 4 P apiece and 5 cents' worth of punk. 7. I walked 3 mi. to the railway station, trav- elled 58 mi. by train, and rode G mi. from the station to my destination. Find the total distance. 8. What is the sum of 9 and G? 19 and 6? 6 and 29 ? 16 and 9 ? 9 and 38 ? 6 and 69 ? 86 and 9? 9 and 56 ? 9. What is the sum of 49, 4, 8 ? 5, 62, 4 ? 6, 48,7? 3,58,9? 7,25,6? 6,78,8? 9,78,6? 10. Peoria is 55 mi. north of Springfield. How far apart are two places, one 6 mi. north of Peoria and the other 8 mi. south of Springfield ? 11. I paid 5^ for a basket, 59^ for a tub, and 10^ for a tin cup. Find the total cost. l.KSSON 4 11 12. A milknijui (U-Uvurs milk to five fainilics that take 3 qt. a day, and to four I'aiiiilies takinjjf 2 qt. a day. How much milk do these nine families buy eaeh day ? 13. A boy ridinj^ his bicycle at the rate of H mi. an hour goes from A to B in 2 hr., from H to (' in 3 hr., and from C to D in 2 hr. How far is it from A to 1) ? 14. I paid 5 ten-dollar bills and 1 five-dollar bill for a rug. What did it cost ? 15. I bought an organ, giving for it 6 ten-dollar bills, 3 one-dollar bills, and half a dollar. What did it cost ? 16. A lady gave for a piano 4 fifty-dollar bills, 4 ten-dollar bills, and a five-dollar bill. What did it cost her ? 17. For my hat I paid a two-dollar bill, 3 dimes, and a nickel. What did the hat cost ? 18. A lady preserved 4 2-qt. jars of cherries, 5 2-qt. jars of pears, and 3 2-qt. jars of peaches. How many quarts of fruit did she preserve? 19. What quantity is represented by the sum of 3, 2, 4, the unit of measure being |1 ? ^ 10 ? 'i^ 100 ? 2 mi.? 2 A.? 20. What distance is represented by 4 units of 100 mi. each, 3 units of 10 mi., and 2 units of 1 mi. ? 21. James gave 6 dimes and a nickel for a history and 8^ for a block of paper. Find what all cost. i-v 1 .1 ::i '11 ? 1 12 MENTAL ARITHMETIC \ 22. Richard had 9 marbles, Harry 48, and Thomas 7. How many had all three ? 23. A farmer sold 2 horses for 1 100 each, a cow for $20, and a sheep for $5. What did he get for all? 24. A thermometer registers 8° below zero. This is how many degrees below the freezing point ? 25. I paid 59^ for a walking stick, the price of which had been reduced 25^. Find the original price. Lesson 5 1. Find the sum of the following: 32 and 20, 64 and 80, 86 and 30, 75 and GO, 97 and 50, 58 and 80. 2. Find the sum of 58 and 34. Note. 58 and 30 = 88 ; 88 and 4 = 92. As in example 2, 3. 43 and 25, 57 and 32, 37 and Go^ 84 and 49, 69 and 74, 77 and 26, 55 and 36, 85 and 69, A man paid find the sum of the following : 26 and 73, 41 and 16, 4. 5. 6. 68 and 32, 68 and 29, 94 and 48, 85 and 58, 28 and 93, 78 and 57, 43 and 99, 38 and 56. 43 and 39, 53 and 77. 39 and 96, 42 and 89. 96 and 91, 88 and 33. 7. $ 25 for one for his 167 for a bicycle for himself and son. Find the cost of both. m I LESSON 6 13 8. A farmer who owned 76 A. of land bonglit an adjoining piece containing 24 A. Find the size of his farm after the purchase. 9. A field is 66 rd. long and 48 rd. wide. How far does a person go in walking from one corner of the field to the opposite corner if he walks along the edge of the field ? 10. If there are 39 pupils in the first grade and 44 in the second, how many pupils are there in ihe two grades ? 11. A man lost 1 23 by selling his horse for 198. Find what the horse cost him. 12. A cow is tied to a stake with a rope 44 ft. long. A second cow is tied with a rope 7 yd. longer. Find the length of the second rope. 13. A merchant bought cloth at 98^ a yd. and sold it at a gain of 25 ^ a yd. Find the selling price. 14. A milkman had two cans, one holding 8 gal. and the other 26 qt. Find the total capacity of both cans. 15. Winifred gave ^.96 for a geography and | as much for a reader. What did both cost ? 16. After taking 2 doz. lemons out of a box there remained 19 in the box. How many were in the box at first ? 17. A farmer sold a cow fo]" -1' 35 and 7 lambs at 14 each. What was the total selling price ? I'M,* '. "'ti ■■ Ik , lit ii \ I- ,1 ■3i 14 MENTAL ARITHMETIC 18. One winter morning the thermometer in the sclioolroom registered 40°. In an hour the tempera- ture increased 19°. Wliat did the thermometer then register ? 19. Wliat is tlie distance from Chicago to Mil- wavdvee, if, after travelling 57 mi. from Chicago, I still have 23 mi. further to go to reach Milwaukee ? 20. A rope was cut into two pieces. If the shorter piece was 5 yd. 1 ft. and the longer 45 ft., how many feet long was the rope ? 21. How many quarts of oats are there in a bin containing 2 bu. 28 qt.? How many pounds ? 22. A field is 42 rd. long and 28 rd. wide. How many rods of fencing are required to enclose it ? 23. A bicycle dealer who had 88 bicycles in stock received a shipment of 4 doz. more. How many did he then have in stock ? 24. Find the number of hours in 2 da. 17 hr. Find the number of sheets of paper in 2 quires 12 sheets. 25. I paid 25 units of *100 each for a lot and built a house on it at a cost of 36 units of $100 each. Find the total cost. Lesson 6 1. Find the sum of 28 and 34 ; of 62 and 45. Find the sum of 48 and 27 ; of 75 and 89. LESSON 6 15 2. Find the sum of 5G and 17 ; 73 and 68 ; 50, 17, 68. How do you find the sum of three numbers like 56, 17, 68 ? Find the sum of 45, 24, 53. 3. Find the sum of the following numbers : 120, 32 ; 140, 53 ; 134, 65 ; 128, 76 ; 184, 78 ; 256, 89. 4. Find the sum of : 25, 32, 69 ; 48, 36, 63 ; 77, 56, 99 ; 91, 84, 32 ; 67, 97, 83 ; 54. 69, 27. 5. A lady pays $11 for a bedstead, 861 for a dresser, and $ 21 for a washstand. Find the entire cost. 6. A dealer bought three carloads of coal. The first weighed 24 T., the second 22 T., and the third 19 T. Find the total amount. 7. The sides of a triangular field are respectively 36 rd., 45 rd., and 53 rd. Find its perimeter. 8. Monday a grocer sold goods to the value of 1 56, Tuesday to the value of $ 62, and Wednesday to the value of 147. Find his sales for the first half of the week. 9. A horse cost $ 125 and the harness $ 16. What was the cost of both ? 10. An arithmetic cost 35^, a history 27^ more. Find the cost of the history. Of both. 11. A grocer paid 1 47 for sugar and $ 28 more than that amount for flour. What did both cost him ? 12. A man began business with 38 thousand dol- lars and in ten years cleared 58 thousand dollars. How much was he then worth? (■■'(; :i> ,,'s i < .» . >„ 16 MENTAL ARITHMETIC 13. llow iiiany days are there in tlie tliree spring months ? In tlie tlu-ee summer months ? In the three winter montlis ? 14. A fruit dealer sohl G oranges for 15^, otliers for 24 ^, and a dozen of another kind for 45^. IIow many oranges did he sell and what did he get for them ? 15. State how you solved examples 5-14. State how you will solve the remaining examples in this lesson. 16. A young man earns $1 a week, his older hrother -11^4 more, and his fatlier $G more than tlie older brother. What do all three earn a week ? 17. A man paid ^ 52 for a buggy and i- as much more for a horse. Find the cost of botli. '^ 18. If I paid 25 hundi-ed dollars for one lot, 18 hundred dollars for another, and 22 hundred dollars for a third lot, what was the entire cost ? 19. IIow many nickels are tliere in three piles of which the first contains 28, the second 16, and the third 11. AVhat is their value in dollars and cents ? 20. If it was 15 min. after 9 o'clock 17 min. ago, what time will it be in "^4 min.? 21. A young man entered the navy 16 yr. ago at tlie age of 25. How old will he be 12 yr. from now ? 22. A farmer divided his farm among his three sons, giving to the eldest 85 A., to the second son 76 A., and to the youngest 68 A. What was the size of his farm ? LESSON 6 17 23. A merchant bought ck)thes at 1 15 a suit and overcoats at |12 apiece, and sokl them at an advance of lii^5 eacli. What must a customer pay for a suit of clothes and an overcoat ? 24. A dealer bought lemons at 25)^ a dozen, and oranges at 85 ^ a dozen. If he sold them so as to gain 10^ a dozen, what did he receive for a dozen of each ? 25. Three children buy a hammock. The first contributes 75^, tlie second G9^, and the third -41^. What did the hammock cost ? ,:4 I i 'tf '1 lii !! SUBTEACTION f i, if' I Lesson 7 Statement of Balance Ex. 1. The other part = 9 in. - o in. Ex. 2. The quantity left = 3 gal - 3 qt. Give the statement of balance for each example in this lesson as you work it. 1. A string 9 in. long is cut into two parts one of whicli is 5 in. long. How long is tlie other part ? 2. From a can containing 3 gal. of milk 3 qt. are poured out. How many quarts are left in the can ? 3. If 2 gal. of milk are poured into an empty can that will hold 15 qt., how many quarts are needed to fill it ? 4. In examples 2 and 3 the quantities to be sub- tracted are not expressed in terms of the same unit. Explain this statement. What must be done with (iuantities expressed in terms of different units be- fore you can find their difference ? 5. A boy has 8 nickels and buys 6 oranges at ii nickel each. Find how many cents he has left. 6. Two boys were playing a game of marbles. It one of them got 7 out of 15, how many did the otiier get ? 18 t] I T LESSON 7 19 7. A piece 2 ft. 3 in. long is cut from a rope 6 ft. 11 in. long. Find the length of the other piece. 8. If in paying for a pair of shoes that cost $Q I gave the clerk 2 five-dollar bills, what change should I receive back ? 9. A lady has 7 2-qt. jars of fruit. If her family eats 1 qt. a da. for G da., how many jars of fruit will slie liave at the end of that time ? 10. If the perimeter of an oblong is 18 in. and the length 5 in., what is the width? 11. Two vessels going in opposite directions are 17 mi. apart one hour after they pass each other. If one is going 8 mi, an hour, what is the rate of the other ? 12. A post 12 ft. long is 4 ft. below ground. How long is the part above ground ? 13. From a piece of cloth 5 yd. long a piece 6 ft. long is cut. How many feet of cloth are left ? 14. I paid 2 dimes for a collar and got back a nickel in change. What did the collar cost me ? 15. I paid Ui^ for a pound of butter, giving three different coins in payment. What were they ? 16. A merchant buys berries for 5^ a box and sells them at the rate of 2 for 15^. Find his gain on two boxes. On a crate of 16 boxes. 17. A boy picked 8 qt, of berries in two days. If he picked 3 qt. 1 pt. the first day, how much did he pick the second day ? ; i 20 MKiN'lAL AUITIIMKTIC ^1 tfi 18. From a line that is times the measuring unit I cut off a part that is 4 times as k^ng as the measur- ing unit. How many inches long is the lemainder, the unit heing 3 in.? 19. A liorse wliich travels at the rate of 8 mi. an hour goes from A to B in 3 lir., and from B to C in 5 hr. How much farther is it from li to C than from A to B ? 20. AVMiat quantity is the difference between tl and 7, the unit of measure being 2 in.? $2? 3 ft.? 4qt.? no? 21. Ifow much greater are 2 units of $5 than 3 units of $2. What is the ratio of this difference to ^ 2 ? Of 82 to this difference ? 22. A piece of cloth 3 ft. long and 2 ft. wide is cut from a square yard of cloth. How many square feet in the remainder ? 23. I ordered half a pound of cheese and received 10 oz. This was how many ounces more than were ordered ? 24. If a man sleeps 7 hr. a day, he sleeps how much less than half the time ? i 1 i i Lesson 8 1. Subtract 4 from eacli of the following : (a) 6, 16, 26, 38, 46, 56, 69, 76, 81, 06. (h) 13, 33, 12, 63, 43, 88, 53, 24, 73, 93. (c) 8, 68, 46, 28, 58, 38, 77, 18, 86, 98. LESSON 8 21 Liriiig unit [e ineasur- enuiiiuler, f 8 mi. an r> to C in to C than 3tween tl 2 ? 3 ft.? 15 than 3 difference :t. wide is my square 1 received than were leeps how ^g i I ((f) 4, 54, 14, 44, 63, 84, 94, 72, 24, 34. (e) 7, 17, 38, 57, 23, 87, 61, 97, 46, 77. (/) 10, 20, 33, 60, 74, 90, 41, 80, 72, 50. (g) 5, 25, 34, 65, 75, 96, 15, 45, 57, 89. (^) 11, 51, 63, 21, 31, 41, 93, 82, 61, 71. (0 9, 49, 19, 28, 34, 59, 99, 79, 88, 69. (m) 12, 32, 64, 92, 82, 41, 49, 52, 72, 22. (w) If from any number ending in 2 you subtract 4, in what figure does the difference end ? Apply a similar question to each part of example 1. 2. Subtract in turn, where possible, each of the following numbers from those given in example 1 : 2, 6, 3, 7, 1, 9, 5, 10, 8. 3. Harry takes a step 34 in. long and John 5 in. shorter. How far does John step ? 4. Robert can jump 6 ft. high and James 7 in. less. How high can James jump ? 5. I took 21 min. to walk to the depot and then found that I was 5 min. late for the train. How Ions: before train time did I leave home ? 6. Two boys run a race of 100 yd. When the first completes the course the second is 7 yd. behind. How far did he run ? 7. U it is 8 min. after 9 o'clock, how long before it will be 10 o'clock. 8. A man paid $25 for an overcoat and a pair of shoes. If the shoes cost $(j, what did the over- coat cost ? '.Ill \ :\ ii ) ""^ ' 1 A f , ';i 09 M KNTAL AKirilMF/nC 9 A lady buys a (luire of paper and then writes 7 !(. iters, using a sheet a letter. How niueh paper has she left? 10. Out of a chiss of 43 pui)ils 9 are absent. How iiiaiiv are present? 11. A thermometer that registered 72° fell 5°. What did it then register? 12 A groeej- sold a peek of beans, taking them out of a busliel basket. \i the basket was full of beans, how many quarts were left in it ? 13. Subtract 2 from each of these numbers : (,0 8, 13, 7, 18, 10, 0, 23, U, 5, 14, 18. (/>) 27, 40, 10, 38, 51, 84, 17, 20, 07, 42, 59. (c) 10, 37, 18, 03, 24, 12, 20, 15, 74, 18, BO. (cZ) 15, 18, 34, 21, 45, 82, 83, 30, 29, 13, 36. 00 121, 132, 105, 143, 137, 183, 149, 158. 14. Subtract in turn, where possible, each of the following numbers from those given in example 13 : 3, 0, 1, 7, 4, 9, 8, 10, 5. 15. A man earns 154 a month, and saves *9 a month. How much does he spend? 16. From 12 tens subtract 8 tens. From 20 hun- di'cul subtract 9 imndred. 17. Out of 4 doz. eggs half a dozen were found to be broken. How many eggs were whole ? 18. Subtract 30 from each of these numbers : 45, 07, 81, 93, 76, 124, 159, 110, 142, 138. LESSON 9 a 2a 19. Sul)tract ill turn, wlioro possible, each of tlio following numbers from tliose given in example 18 : 20, GO, 40, 70, 10, DO, 80, 50, 100. 20. Subtract 28 from 7G. Note. 7G - 2U = 50 ; 5(3 - 8 = 18. 21. Subtract 34 from 50 ; 23 from 49 ; 42 from 73 ; 15 from 08 ; 52 from 95 ; G3 from 99. 22. There are 45 pupils in a class of whom 21 are boys. How many girls are there ? 23. If the freezing-point is 32°, how mucli above the freezing-point is the temperature when the ther- mometer registers 67° ? 24.- A man is 59 yr. of age and his son 23 yr. How much older is the father than the son ? 25. In a race between two bicyclists the faster has gone 59 mi. and the slower 47 mi. How far is he behind ? Lesson 9 1. Subtract 48 from 92 ; 68 from 84 ; 37 from 78 ; 58 from 96 ; 32 from 61 ; 54 from 82 ; 33 from 71 ; 46 from QB. 2. Find the difference between 45 and 80 ; 67 and 91 ; 23 and 56 ; 76 and 95 ; 16 and 82 ; 43 and 88 ; 54 and 29 ; 70 and 22. 3. Subtract each of the following numbers from 100: 4 jni, j "" ■ 'I •I ) .jilt' t i' 'I ■s' !r 62, 37, 24, 81, 98, 56, 73, 45, 19. 24 MKNTAL A KITH MIOTIC ! tl n 4. Siibti'iicl ciicli of tluisi; (lUJiiititit's from '^1.20: H7/, llilf^, l:{/, •jr)f', T-Jf^ T)!)/, 1>2/, IS^. 5. I-'iinl tilt; (liffiM't'iico in (H'lits (Hi subinicting 4Sf^ fioiii .sl.i>r,; Tdf' from .^i^l.r)!); 8!!^ from ^ii^ 1.07 ; Jif^l/Jf) from *l.Tt ; '^o.4S from 'ii'"). 6. 'I'Ik; followiiiL,^ hju'gjiius wore iidvortisLMl in ;i ('lii(;jit(o i)a[)(jr. Find tli(j rt'diictioii in price in t'iicli case. 1. IMllow covers 12f, rod need from 40^. 2. Pillows 2:)f^, rednced from 7;")^. ::5. Silk SUA reduced from *l.oO. 4. Silk erepons i'l.oO, reduced from >i<2.25. f). Wool checks -tl.SH, reduced fi'om '^3.50. (). Silk waists -i^l.DH, reduced fi'om '*it'4. 7. l'Md)i'ellas ol*^, reduced from '*ii<1.2r). 8. I'arasols Sll/, reduced from >'2.r)0. [). Hicycle skirts, '^2.7"), reduced from ^5. 10. IJicycle suits, ''iJ'2.4S, reduced from %o. 11. Fancy ribhon 4(}^, reduced from ^1. 12. IJlack silks ()2/, reduced from .^1.25. IJJ. Silkaline 18^, reduced from oO^. 14. l*ortieres '^^2.2'), reduced from -^0. 15. J^ace curtains 80^, reduced from '^52. 7. A drover bought (50 sheep from three farmers. If he bou^"ht 18 from the lirst and ll) from the sec- ond, how many did he buy from the third? 8. A man who has >i^li) in the bank deposits 836 more, and shortly after withdraws -848. How much still remains in the baidv ? i LESSON 9 25 s. c- 9. A juid 15 start from two i)laoes 00 mi. apart and travul toward each otlior. Tho first day A ^^ot'S JiS mi. and 15 25 mi. How far are they tlien apart? How far is A from Wa starting-placu ? How far is H from A's starting-place ? 10. A and 15 start from two places 85 mi. apart and travel in the same direction. Tlie first day A travels 48 mi. and 15 27 mi. How far are they then a[»art ? How far is A from I5's starting-place? How far is 15 from A's starting-i)lace ? Show that there are two solutions to this example. 11. Mr. Stevens paid *()8 for one horse and ^1^) for another, and sold them both for -if IGO. Kind his gam. 12. A farmer sold two bags of apples for -f 1.50. With this money he bought a hoe for 82^, a rake for 45^, and a spade with the remainder. What did the spade cost? 13. A merchant sells cloth at *1.25. He lowers the price 80^, thus losing 12^ a yd. Find what the cloth cost him a yard. 14. A boy shot an arrow 34 yd. up the road and another down the road 45 yd. His brother brought them to him. How far did he have to walk ? 15. A merchant sells silk at -11^2.50 a yd. He lowers the price 75^ and then sells at 18^ above cost. Fhid what the silk cost him a yard. I "' '. '% 'I ■ 'I . Ik ■•|M I ,1 :i { I! it I :!: M III m n i V )i r MULTIPLICATION Lesson 10 1. Review the multiplication table, page 134. 2. What unit of area is used to measure the area of an oblong 9 in. by G in. ? Of a table 3 ft. by 2 ft. ? Of the floor of a room G yd. by 5 yd. ? Of a farm? Of a section of land G mi. long and 4 mi. wide ? 3. Find the area of each obh^ng given in example 2. 4. Find the area of the floor of each of the rooms whose dimensions are, respectively: 5 yd., 4 yd. ; 8 yd., 7 yd. ; 11 yd., 9 yd. ; 12 yd., 10 yd. 5. How do you find the number of units of area in an oblong ? 6. A garden bed G ft. long and 4 ft. wide is sur- rounded by a Avalk 1 ft. wide. What is the entire area of the bed and walk ? 7. A flower garden is surrounded by a walk 2 ft. wide. TJie length of tlie walk and garden is 12 ft. and their width 10 ft. Find the area of the garden. 8. Find the number of square inches in each of the stamped linen squares whose sides are, respectively, 5, 7, 9, and 12 in. 1 ii h ! LESSON 10 27 -J 9. A rug occupies the centre of a square room and its sides are everjAvliere 2 ft. from the wall. If the length of the rug is 8 ft., what is the area of the room ? 10. IIow long is a township ? llow wide ? What is its area ? 11. What is the difference in area between an 8-in. square and 8 sq. in. ? l^etween a square whose side contains 6 units of length and an oblong containing G units of area ? 12. A township contains a forest 3 mi. long and 2 mi. Avide, the rest being divided into farms. How many scpare miles of farm land are there in the township ? 13. A grass plot 10 ft. square has in the centre a square flower bed wliose side is 4 ft. Find hoAv many square feet of giass are in the plot. 14. In a room 14 ft. long and 12 ft. wide is a rug wliose sides are everywhere 2 ft. from the wall. Find its area. Reduce to lower denominations : 15. 8 yd. 2 ft.; 4 yd. 1 ft.; 12 yd. 2 ft.; 9 yd. 2 ft. ; 13 yd. 1 ft. ; 20 yd. 2 ft. How do you redlice yards and feet to feet ? ^ 16. G ft. 3 in. ; 10 ft. 8 in. ; 4 ft. G in. ; 9 ft. 2 in. ; « ft. 11 in.; 12 ft. G in. How do you reduce feet and inches to inches ? i'lr ; ' 'I 1:;*. f. ' 1H»' 1 4 I ! m •u' 1 '' i^! If' 1. 28 MENTAL ARITHMETIC 17. 4 sq. yd. 6 sq. ft. ; 6 sq. yd. 7 sq. ft. ; 12 sq. yd. 4 sq. ft. ; 9 sq. yd. 5 sq. ft. ; 7 sq. yd. 8 sq. ft. ; 12 sq. yd. 3 sq. ft. How do you reduce square yards and square feet to square feet ? 18. 8 qt. 1 pt.; 12 qt. 1 pt.; 7 gal. 3 qt.; 9 gal. 2 qt. ; 4 gal. 6 pt. ; 12 gal. 4 pt. 19. 6 bu. 3 pk. ; 10 bu. 2 pk. ; 9 pk. 1 qt. ; 11 pk. 2 qt.; 13 bu. 1 pk.; 8 pk. 7 qt. 20. 4 wk. 3 da. ; 8 wk. 5 da. ; 12 wk. 4 da. ; 5 hr. 40 niin.; 4 lir. 18 min. ; 7 min. 35 sec. 21. 5 dimes ()^; 9 dimes 3^; 12 dimes 5 j^; 6 m. 4 dm. ; 8 m. 7 dm. ; 11 m. 8 dm. 22. Find the perimeter of the rooms whose dimen- sions are respectively: 5 yd., 4 yd.; 8 yd., 6 yd.; 12 yd., 8 yd.; 22 ft., 18 ft. 23. How do you find the perimeter of a room? Find the perimeter of a room 8 yd. long and 7 yd. wide. If this room is 3 yd. high, find the area of its four walls. 24. Find the area of the four walls of a room 18 ft. long, 12 ft. wide, and 9 ft. high. How do you lii d the area of the four walls of a room ? 25. How many square feet of wall will a roll of wall paper 8 yd. long and 18 in. wide cover? 26. If the width of a farm is measured by 4 units and its length by 6 units, how many units in its perimeter ? LESSON 11 Lesson 11 29 In examples 1 and 2 tlie sUitements of balance are: The weekly profit = G x |!15. The quantity of provisions = the rations of 6 men for 12 da. As you solve the examples of this lesson, give the statements of balance. 1. A merchant overages a profit of ^15 a day. How much is that a week ? 2. A quantity of provisions will last 6 men 12 da. How long will it last 1 man ? 3. If 8 men can do a piece of work in 9 da., how long will it take 2 men to do it ? 4. If 7 men can do a piece of work in 4 da., hew many men can do it in 2 da. ? 5. If two men start from the same place and travel in opposite dir.^ctions, one at the rate of 4 mi. an hr. and the other at the rate of 5 mi. an hr., how far apart will they be in 3 hr.? 6. How far apart will the men in example 5 be in 8 hr. if they travel in the same direction ? 7. Two men start 60 mi. apart and travel toward each other at the rate of 3 mi. an hr., the other at tlie rate of 4 mi. an hr. How far apart will they be in 6 hr.? 8. How far apart will the men in example 7 be in 6 hr. if they travel in the same direction? Show that there are two solutions. ■ "ii! A. ■ 'I ■i •> l!„ 'l\'} r«', i |lt( ^ wm S- fe li ;,' II ' i ;1i u ■ &■ MENTAL AHITIIMETIC 9. I bouglit 7 T. of cojil xt Jjf G a ton, and handed in iiaynient 5 ten-dollar bills. Row mucli change should 1 receive ? 10. For hoAV mucli must I sell 6 bicycles that cost 5ji'28 each to gain $45? Statement of balance: the selling i^rice = the cost + the gain. 11. A farmer sold 8 sheep at f 6 each and 5 lambs at $ 3 each. How much did he get for them all ? 12. A tailor has a piece of cloth containing 25 yd. If he should use 13 yd. and sell the remainder at 1 5 a yd., what would he receive for it ? 13. A and B, travelling toward each other, met in 12 hr. How far apart were their starting-points if A travelled 9 mi. an hr. and B 8 ? 14. I bought 38 cows for $35 each, and sold them for $ 43 each. Find my profit. 15. A man bought a horse for 8 90, paid 1 3 a week for his expenses, and received $5 a week for his Avork. At the end of 8 wk. he sold him for $80. Did he gain or lose by the whole transaction, and how much ? 16. If a man travels from his home 10 hr. at 6 mi. an hr., and then returns at the rate of 7 mi. an hr. for 5 hr., how far is he then from home ? 17. What is the cost of a 5-lb. pail of butter at 19 ^ a lb., and 2 lb. of coffee at 35 ^ a lb.? I LESSON 12 31 18. A man drives 4 mi. to tlie station, rides on the train for 3 lir. at 33 mi. an lir., and then drives from the depot into the country a distance of 8 mi. What is the entire distance ? 19. A crate of raspherries liolds 24 pt. boxes. How many quarts is this and what is a crate Avorth at G ^ a pint box ? 20. Wliat is the cost of 2 crates of currants at 90^ a crate, 2 crates of Wack raspberries at 75 ^ a crate, and a crate of red raspberries at a 1 1.05 ? 21. How long will it take a lady to look over 2 crates of black raspberries containing IG qt. boxes each, if she can do 2 boxes in 9 min.? 22. A grocer buys soap at f 2.15 per box of 60 bars and sells it at 5^ a bar. Find his gain on a dozen boxes. 23. What will 4 rows of wire fencing cost for a lot 100 ft. long and 50 ft. wide at G^ a yard ? 24. A certain map is drawn so that 4 mi. is rep- resented by 1 in. On this map a certain district is a square whose side is 2 in. What is its area? Lesson 12 1. In terms of what unit is the volume of a prism 4 in. long, 3 in. wide, and 2 in. thick expressed? What is its volume ? i. .•i if-X : i' 'I 5.Y J( :i.i 32 MENTiVL AUITIIMKTIC I i 4 V • ri t w 2. Find the volumes of ^^risms of the following dimensions : Length AVidth Thickxess (J in. 5 in. 4 in. 8 in. 6 in. 4 in. 7 in. 5 in. S in. Length Width Thickness 8 ft. 4 ft. 4 ft. 10 ft. 8 ft. 7 ft. 4 yd. 3 yd. 2 yd. How many cubic inches in a brick 8 in. by 4 in. by 2 in.? 3. A gallon of water will just fill a tin box 11 in. long, 7 in. wide, and 3 in. deep. How many cubic inelies in a gallon ? 4. A cubic yard of earth is a wagon load. How many loads of earth will it take to make an excava- tion 18 ft. long, 12 ft. wide, and 6 ft. deep? 5. How many cul)ic inches in a 2-in. cube ? 3-in.? 4-in.? 5-in.? 6-in.? 7-in.? 8-in.? 9-in.? 10-in.? 6. A cp^nc foot of water weighs 1000 oz. A tank () ft. long, 2 ft. wide, and 2 ft. deep contains how many ounces? 7. The distance between two points, measured with a yardstick 1 in. too long, is found to be 6 yd. What is the actual distance? 8. The length of a room, measured with a yard- stick 1 in. too short, is found to be 8 yd. What is the actual leno-th ? 9. Name two factors of 15, 18, 24, 28. Multiply 3(J by 15, using the two factors of 15 as multipliers. Multiply similarly 3G by 18, 24, and 28 respectively. i; I LESSON 13 33 10. Why is it correct in order to multiply a num- ber by 5 to annex zero and divide by 2 ? J>y tliis method multiply the following numbers by 5 : 36, 45, 78, 9G, 124, 2oG. 11. Find the cost of cementing tlie floor of a cellar 21 ft. long and 18 ft. wide at 15^ a sq. yd. 12. Find the cost of dign^imr a cellar 5 yd. loner, 4 yd. wide, and 2 yd. deep a*^^ 24 P n cu. yd. 13. If 3 ft. is the unit of length, what is tlie unit of area? If the measuring unit is a line 3 in. long, find the area of an oblong wliose dimensions are 6 and 4 times the measuring unit. 14. What is the cost of 2 lb. 8 oz. of butter at IG ^ alb.? 15. An electric car makes 15 trips a day, carry- ing on the average 20 passengers a trip at 5^ a fare. Mow much did it earn in 1 mo. of 30 da.? 16. Why is it correct, in order to multiply a num- ber by 25, to annex two zeros and divide by 4 ? By this method multiply the following numbers by 25: 24, 18, 35, 27, 25, 19. 17. I bought 19 cliickens at 25^ apiece. What change should I get out of a five-dollar bill? 18. A merchant bought 10 bu. of peaches at $1.75 a bu., and 1 bu. being spoiled, he sold the remainder at $2 a bu. Find his gain or loss. D "111 Ml. 4 ....ti. m it f.. \ i tl f:i il \ ; « 34 MENTAL ARITHMETIC 19. A liidy bought a remnant of silkaline contain- ing :5] yd., and a remnant of ribbon containing 5% yd. of ribbon. The price of each was 12^ a yd., find the cost. 20. From a piece of ck)th containing 32 yd. there were sold at different times 11 yd., 8 yd., and 9 yd. The remainder was jnit on the remnant counter to sell at 2;)^ il yd. Find the selling jDrice. 21. A lady gave her little girl a dollar bill with which to buy a 2-lb. roll of butter at 22^ a lb., and 1 lb. of coffee at 85^ a lb. What change did she bring home to her mother if she was allowed to buy 5 ^ worth of candy ? 22. A city store advertises 7/ muslin at 3^ a yd. and 25 ^ lace curtains at 10 p a yd. A lady buys 4 yd. of the latter and 6 yd. of the former. How much less does she pay than if she had bought them at the regular price ? 23. If G bbl. of flour cost $83, what will 24 bbl. cost at the same rate ? 24. If 1 pt. of maple syrup costs 12^, what will 1 gal. cost at the same rate? 25. How many feet in 1 rod ? 2 rods ? 20 rods ? Wire fencing is sold at ^4.90 per bale of 20 rd., or at 2^ per ft. How much is saved by buying 20 rd. by the bale instead of by the foot ? f LESSON 13 Lesson 13 35 ^ F c 1. The triangle a is what part of the oblong AFBD? The triangle h is what part of the oblong AFCE ^ The triangle ABO (i.e. a + h') is what part of tlie oblong BBOE> 2. To find the area of a triatir/Ie multiply one-half the measure of its base by that of its altitude, Menio- rize this rule. 3. If ABO represents a triangle whose base BO is 8 in. and altitude AF is 4 in., what is the area of the triangle represented? Draw the triangle. 4. Find the areas of the following triano-les : Base 10 in. IG in. 18 ft. 24 ft. 36 yd. 80 yd. Altitude 6 in. 12 in. 10 ft. 22 ft. 20 yd. 60 yd. Base 2 ft. 2 ft. 4 in. 3 yd. 1 ft. 4 yd. 2 ft. 3 yd. 4 rd. Altitude 1 ft. 8 in. 1 ft. 10 in. 4 yd. 5 yd. 1 ft, 10 in. 8 yd. 5. What is the area of a triangular garden bed whose base is 2 yd. and altitude 1 yd. 2 ft.? '•'I Mill ! il i!iS ;i If f If ft i: I m i1 I'A 3G MENTAL ARITHMETIC 6. Tiie g'dhh and of a barn measuros 20 ft. and the perpeiidicular distjinoe from tlio I'idoo of the barn to tlie base is 11 ft. Ilow many square feet of lum- ber does it contain ? 7. Whicli is greater tlie ti-ian,ir]o a or a^ ? Which is .q-reater the parallelogram ABCD or the rectangle 8. Ilow do you find tlje area of a rectangle such as UFOD ? Of a parallelogram such as ABCD ? 9. To find the area of a parallelogram multiply the measure of the hase (CD) In, the measure of the alti- tude (CV). .^lemorize this rule. 10. Find tlie areas of tlie following parallelograms: Base Altitude Base Altitude » ±t. 6 ft. 16 I'd. 12 rd. i2yd. 8 yd. 24 mi. 20 mi. IJase = 9 units of length. Altitude = 4 units of length. 11. Name three pairs of factors of 20. Give the base and altitude of three parallelograms whose area ni each case is 20 sq. in. 1 ti f ' LIvSSON 13 a7 ; 12.^ Name the two parallel sides of the traperjold ABOD, Which is larger, the triangle a or a' ? h or h' ? 13. If you add the triangles a' and h' to the large central figure c, you get what figure? If you add the triangles a and h to the central figure c, what fig- ure do you get? Which is larger the trapezokl ABOD or tlie rectangle UFGrff? 14. Show by actually measuring that Gil is one- half the sum of AB and OB. AB and OB are called the bases of the trapezoid ABOB, 15. To find the area of a trapezoid multiply the measure of one-half the siwi of its bases or parallel sides hy the measure of its altitude. Memorize this rule. 16. The lengths of the parallel sides of a trapezoid are 12 ft. and 8 ft. and its altitude is 5 ft. Find its area. '*M. 111!' t«tll I'll hi < I *u. .« fc* 38 mi-:ntal AiiiriiMicTic iti I t i'i t \l 17. Find i]\v areas of tliuso trapezoids; iJASKS ALTITlinK UasKS AltITUDE Hill., Din. Till. 1 ft. 7 in., 1 ft. 8 in. 10 in. 14 ft., 8 ft. G ft. 5 yd. 2 ft., l> yd. 1 ft. 3 yd. 12 ft., 9 ft. 4 ft. 2 yd. 1 ft., 1 yd. 2 ft. 2 ft. 13 ft., 10 ft. 8 ft. 4 yd. 2 ft., 3 yCi, 1 ft. 5 ft. 18. Tlio border of a ru^^ consists of 4 ocpial trape- zoids wliose bases are 10 ft. and ft. If tbe widtli of the border is 2 ft., find its area. 19. \\\ each corner of the ceiitrei)iece of a rug is a trapezoid wliose parallel sides are 13 in. and 1) in. and whose altitude is 14 in. Find their area. 20. In the centre of a rug is a hexagon which can ])e divided into two equal trapezoids w' ^e bases are 2 ft. and 1 ft., and altitude 1 ft. 4 m. Find its area. 21. How do you find the area of a rectangle? Triangle ? Parallelogram ? Trapezoid ? I 1 -'t ■ i; • r ! t DIVISION « 'I '•'m ' !.•• Lesson 14 In the following examples give the statements of balance : X. Wliat must 8 mi. ),e multiplied by to give I; '!;; 8 ' ,°^ f'"'' ^' ""■ "" ^ l^'''^'^'^ "t the Lite Of 8 mi. an hr., how many hours is he travellinc. y 2. C is one factor of 42, what is the other ? Jf''G lb. of dates cost 42^, what is the price per Ib.v buy f^'m ? ''■ '""' """'' '"""'^ ''^ "'«"• '^'^» y- 4. A horse trotted 48 mi. in 6 hr. How many nules an hour did he average ? s. In the preceding four questions name the given quantity the measuring unit, and the number of units in the quantity. 6. Reduce to quarts: 18 pt.; 25 pt.; 36 pt.; 43 pt. ; 57 pt. ; 128 pt. ; 141 pt. ^ 7. Find the number of strips of carpet each 2 ft wide required to carpet a room 16 ft. wide; 20 ft' wide; 26 ft. wide; 17 ft. wide. uf/i^f .*" ^''"^''- ^2 **•' 9*5 ft; 105 ft.; 141 ft.; 186 tt.; 204 ft.; 167 ft.; 343 ft.; 196 ft 39 III ■■I , '1 ■ f'.,i r ' i I : i; 40 MENTAL AK ITllMETIC n m tl ill Mi t 9. How many strips of carpet eacli 3 ft. wide wdll be required to carpet a room 24 ft. wide ? 33 ft. Avide? 20 ft. wide? 10. Reduce to gallons : 28 qt. ; 87 qt. ; 56 qt. ; 8(i qt. ; 107 qt. ; 200 qt. 11. Find the number of scliool weeks in 20 school (lays. In 35 da. In 100 da. In 200 da. 12. If a man works G days a week, in how many weeks will he work 96 da.? 150 da.? 192 da.? 222 da.? 264 da.? 13. Reduce to weeks: 84 da.; 98 da.; 126 da.; 175 da. ; 240 da. ; 365 da. 14. Reduce to pecks: 40 qt.; 58 qt.; 72 qt.; 125 qt. ; 197 qt. ; 215 qt. ; 288 qt. 15. Reduce to square yards : 81 sq. ft.; 108 sq. ft. ; 164 sq, ft.; 200 sq. ft.; 243 sq. ft. 16. Reduce to dimes: 80^; 120^; 93^; 168/; 248/: 290/; 312/; 415/. 17. Reduce to meters: 40 dm.; 70 dm.; 82 dm.; 94 dm.; 99 dm.: 100 dm. 18. IIoAv many yards in 1 rod ? Measure off 1 rd. ill the schoolroom. Measure this distance afjain with a miit half a yard in length. Count the number of half-yards in 1 rd. How many? 19. What is the number of rods in 33 half-yd.? 66 half-yd.? 88 half-yd.? 20. Divide by 11: 75; 92; 132; 189; 210; 265; 331; 484. .1 9 i.ESSON 15 41 21. Reduce to feet: 72 in.; 89 in.; 11 T in • 109 in.; 254 in.; 288 in. 22. How many dozen are tliere in 60 units ^ 93 units ? 384 units ? 23. State liow you reduce a quantity expressed in pints to quarts. Quarts to gallons. (Quarts to pecks. Indies to feet. Feet to yards. S(iuare feet to square yards. Days to weeks. Cents to dimes. Lesson 15 1. What must 5 lb. be multiplied by to get 40 lb. ? 40 lb. of butter is packed into pails each containincr 5 lb. How many ? '^ 2. 4 is one factor of 3G, what is the other ? If 4 yd. of ribbon cost 3G^, what is the price per yd.? 3. A pail contains 9 qt. of water. How many quarts will be left in the pail after 8 pt. are taken out ? 4. What will a dozen oranges cost at the rate of 3 for 10^? 5. How many lemons can I buy for 50 ^ at tlie rate of 4 for 10^? 6. A man travels 18 mi. at the rate of G mi. an hr., and returns immediately at the rate of 9 mi. an hr. How many hours is he gone ? 7. A man drove to the city in 3 hr. at the rate of 8 mi. an hr., remained 2 hr. to transact business, and returned at the rate of G mi. an hr. How lono- was he ^'one ? ^ ^•ri I. :i;i ;:;i i J 1 1 il M I V JvJ: 1 If! t II: I ■: ■It :' I' 42 MENTAL ARITHMETIC 8. A dairyman traded 4 horses worth $ 90 apiece for cows worth i30 apiece. How many cows did he get ? 9. If 3 lb. of rice cost 24 /, what will 2 lb. cost ? 10. I select the cheaper of two rugs, one of which costs f 48 and the other $66. With the difference in price, how many chairs can I buy at $ 6 apiece ? 11. A room is 10 yd. 2 ft. long. How many steps will a boy take in walking the length of the room if he steps 2 ft. each time ? 12. A room is 12 ft. wide. How many strips of carpet each 2 ft. wide are needed to carpet it ? How many yards of carpet are needed if the room is 15 ft. long ? 13. A room is 15 ft. wide and 18 ft. long. • How many yards of carpet are needed to carpet it, if the carpet is 3 ft. wide, the carpet running lengthwise ? 14. A room is 20 ft. wide and 24 ft. long. How many yards of carpet are needed to carpet it, if the carpet is 2 ft. wide, the carpet running lengthwise ? 15. If two 5-lb. pails of butter cost 11.05, what will 20 lb. cost ? 16. If 6 bu. of corn cost 1 3, what will 30 bu. cost ? 17. What is the cost of 12 boxes of berries at 3 for 25^? 18. A farmer's wife sold to a grocer 8 lb. of butter at 18 ^ a lb. and took in exchange raisins at 12 ^ a lb. How many pounds of raisins did she get ? LESSON 16 43 i; 19. A man divided 3 lb. 8 oz. of candy equally among 4 children. How many ounces did each get ? 20. Some workmen work 9 hr. a day at 24 j^ an hour. They demand the same daily wages for an 8-hr. day. How much would that be per hour? 21. What is the cost of 5 qt. 1 pt. of milk at 24^ a gal.? 22. To how many customers does a milkman sell 8 gal. 1 qt. of milk if he sells 3 pt. to each customer ? 23. I bought 7 chickens giving for them a $2 bill and receiving in change 25^. What did they cost apiece ? 24. A grocer buys 6 baskets of peaches at 30^ a basket. He uses two baskets and sells the remainder for what all cost him. Find the selling' price of each basket. 25. Find the number of strips of paper in a room 20 ft. long and 18 ft. wide, the paper being 2 ft. wide. Lesson 16 1. Find the weight of an iron bar 5 yd. 1 ft. long of which 2 ft. weigh 13 lb. 2. A block of stone is 10 in. long, 8 in. wide, and 6 in. thick. Find its weight if 5 cu. in. weigh lib. ««."''' >••.'.' :;!,t i:ii^ I'M 1,1 -• Pi 'n, ill I f ' i I 44 M I-: NTA L A urn IM ETIC n if .■)'r ..i 6* 'U' t\ ! ffl f! 'I ! 54 MENTA L AF 11 IIM ETIC 15. Ai'tlmr can ride around a two-mile block in 12 niin. and Cecil in 10 niin. If they start together, when will they be together again ? How many times will each go round the block in the time ? 16. What is the least sum of money with which I can buy sheep at ${j apiece or cows at $2ij ? 17. Find the least sum of money with which you can buy an exact number of a dozen lemons or bananas at 20^ and 15 P a dozen respectively. 18. What is the least number which, when divided by 6 or 7, leaves 3 as remainder in each case ? 19. Find the least number which, when divided by 4, 0, or 8, leaves 2 as remainder in each case. 20. The fore and hind wheels of a carriage are 9 ft. and 12 ft. in circumference. Find the least number of revolutions of each that will bring them to the same relative position as at starting. 21. Find the L. C. M. of the denominators of the following fractions : -1. 1 3 1 . 2 1. 3 _8_ . '" " 3^9' y 2 1 ' _5_ _1_ 12' 18* 2' 3 ' 3' 4 ' 3' 6 ' 5' 3 ' 22. What is the least number of parts into which the primary unit 1 yd. must be divided in order that both l yd. and | yd. may be expressed by a whole number of parts ? How many parts in each case ? 23. As in example 22, give the least number of parts into which the primary unit must be divided LESSON 19 55 and the number of parts equal to the given quanti- ties in the following : i yd., ^ yd. ; ^ ft., i ft. ; | lb., ■i lb. ; f gal., 1 gal. ; f bu., | bu. ; ^ 1, ,1i^ |. 24. Solve as in example 23 : | da., i da. ; i yr., } yr.; I- wk., I wk.; | hr., | hr.; f min., f niin ; I mo., I mo. 25. Solve as in example 23 : i -i. ^ - i 2 5 . 1 1 . o Q r ...... 2' 3' 6 2' 3' 6 ' 1 1 1 . 2 3 5 2' t' 5 » 3' I^' 6 ' 2 2' 6' r ' 1' 12' i* '•<■ ■I 'll^ . iBi li I II Ik 1 ■■ 1 ! ! • ■ S 1 ■1 ! 3 |! H ! '' li '! <1 'ii I! i; ! m 1 1 i\ FRACTIONS Lesson 20 1. If on measuring a cortain unknown quantity witli a unit 1 ft. in length I count 5 units in the ([uantity, then I know that the quantity is 5 ft. Similarly, if I measure another quantity witli a unit 2 ft. in length and count 4 units in the quantity, I know that it is 4 (2 ft.). If I measure a third quan- tity with a unit |- ft. in length and count 3 units in the quantity, then I know that it is 3 times the unit ^ ft. or I ft. Thus when a measured quantity is deiioted by the expression | ft., tlie measuring unit is I ft. and tlie number of units in the quantity is 3. Let actual work he done in measurhuf imknoivn quantities with fractional i(nits. 2. 1 measurcMl a quantity with the unit J ft. and counted ;") units. What was the (j^uantity ? 3. Name the quantities wliicli, wlien measured by 60 11 111 I r>, l; 1 V vj Liii; ji u 111 » ;«. 1 li nil 11 l/ilLUll . Unit iV ft. i mi. Ni'MBEU 4 t 3 Unit 1 hr 1 gid. t bu. NUMBKR % 4 t A IIJ- If iT. 13 9 ! ilb. 5 1 doz. i I LESSON 20 57 4. I measured the quantity of milk in a pitcher with a measure containing J qt., and in so doing tilled it 5 times. Find the quantity of milk in the pitcher. 5. I measured a distance and found it equal to I ft. What unit did I use and how often did 1 measure ? 6. In each of the following quantities name the measuring unit and give the num])er of units that measure the quantity : | ft.; | yd.; | gal.; | sq. yd.; Ji A.; i%cd.; ^g sq. mi.; -^^ T. 7. What is the measuring unit for the quantity I ft.? Into how many parts do we think of tlie primary unit, 1 ft., as being divided to give the measuring unit J ft.? 8. Name the })rimary units indicated by the fol- lowing quantities and state into how many parts we think of tliem as being divided in order to give the me; curing units: | ft.; >i^-^\; {^^ lb.; | sq. yd.; -^Q vn. yd.: ^\ A.; || yr. ; -^\- mo. Name tlie measuring unit in each case. 9. Fhid 1 of nO; I of [) lb.; ] of 16 mi.; J of It) hr.; i(«f ^8 T.; -| of :5T A. 10. How do you find l of a quantity? (Divide tlic (piaiitily bv 2.) Jb)\v do you lind J of a (pian- '-> • 4- 5- (5 • ^- 8- 9- 10- IT- 12 • 11. Find the fractions, given in example 10, of these quantities: -t-jO; 1^4 mi.; 45 yr.; GO A.; 42^. f « itj .k\ 'I i 5S MENTAL AKITIIMETIC H * r ! i 12. Find I of i?12; | of 20^; | of 25^; f of 24 lu-. ; I of 28 da. 13. IIow do you find | of a quantity? (Divide the quantity by 3 and multiply by 2.) How do you find I of a quantity? |? |? |? How do you lind any fraction of a quantity ? 14. Find 5- of 18 ; ^ of 49; I of 72: 1 of 54; A of 80 ; j\ of 88 12 of 48. 10 15. Can you find a fraction of a quantity by mul- tiplying it by the numerator and dividing the prod- uct by the denominator ? Find in two ways the value of I of -^12 ; f of ^8 ; | of ^18 ; f of $14. 16. By the method given in example 15, find the value of the following : -| of 7 ; | of 11 ; -f of 12 ; I of 10 , f of 15 ; I of 20 ; I of 13 ; ^^^ of 23. 17. If I cut a piece of string 48 in. long into halves, a ^hat will be the length of each piece ? If into thirds? Into fourths? Fifths? Sixths? Sevenths? Eighths ? In examples 18 to 28 give the statements of bal- a.n(^e. 18. A boy had 30 words in his spelling lesson and -'led correctly | of them. How many did he spell c( »rrectly ? 19. A man is 32 yr. old, and his wife | as old. AVliat is her acre? 20. James rode 28 mi. on his bicycle in 2 J lir., and Harry -| as far. How far did Harry ride? LESSON 21 59 21. During tlie montli of April a commercial trav- eller was away from home f of the time. How many days was he away ? 22. A merchant bought cloth for 60 ^ a yard and sold it for I of the cost. Find the selling price. 23. A man earns 120 a week, and his son | as much. Find the sum of their earnings. 24. A man whose salary is $30 a week pays I of it for rent and | of it to meet otlier expenses. How much does he spend a week and how much does he save ? 25. A man who earns 1 32 a week spends i of it for meat and | of the remainder for groceries. How much does he spend each week for meat and grocer- ies. 26. A horse which cost $dS was sold for f of the cost price. Find the loss on selling. 27. A house is sold for .1j>2400, and | of that sum is paid down. How much remains unpaid ? 28. What actual coins and how many of each kind are equal, respectively, to |, -/^, and ^\ of 1 1 ? Ml ;i '- ti Lesson 21 1. Find|of812; |of.fl2; lof-i^l2; fof^fl2; |of$12; |of-^12; iof812; I ofai2. 2. Which is greater | or | or |- or | of 1 12? Of any quantity ? il ! m 1 • ■U l ■ il l1 ; .1 1 60 MENTAL AIUTILMETIC IJ- = ao == !;o 'O' tiikiiig these rnictions of -i^GO. 4. What is I of a quantity equal to ? How can you write a fraction that is equal to 1 ? 5. How many hours are there in J da.? f da.? fda.? I da.? 6. What is meant by saying that J = | = g = 4 y 7 1^ 2 _ n _ 4 _ r> p 9 "99 ? ~ il ~ 16 ~ "! ~ r AVhat is the relation between the denominator of a fraction and the numerator in order that a fraction may be equal to ^ ? 8. How many inches 'are there in J yd.? | yd.? I yd.? -^.j yd.? Name ten fractions each equal to ^. 9. Name fractions equal to |^ ; -J ; J ; | ; ^; J ; 10 ' 11 ' 12* 10. What change can be made in the numerator and denominator of such a fraction as | without alter- ing its value ? 11. I yd. = ? in. | yd. = ? in. -| yd. = ? in. J'^- yd. = ? in. -|| yd. = ? in. | J yd. = ? in. Name live fractions each equal to |. 12. What chanofe can be made in tlie numerator and denominator of such a fraction as |^ Avithout altering its value ? 13. By what process is | changed to -^^ or j| ? By what process is ^ or |^ clianged to J? LESSON 21 Gl I 14. I aa. = ? I,,.. I ^|.^^ ^ , J Name livo fnictioiis eaeli equal to ?. 15. ShOAV that 2 == 4 J', _ 8 _ 12 1 /^ T t,e number of „,i„ute« in ead. o/lhcso" fraction ot' ai hour. 16. Name five fractions equal to each of tlie fol- i"^-"'g ■■ h h h f f. t, ■i''o-. 1^. When we U.SC the expression J y.l., we think ot 1 yd. as divided mto how many ],arts ? \VI,en we ehnge |- y.l to ,% yd., we think of each of these paits -s divided into liow many j-arts ? 18. Why can you multiply both tenas of a frac- oH^ieteZT ""'"'"• ^""'""^ ^"""«""'«' *■'" -1- 19. 1/d. is equal to how many halves of a yanl v How ma,y thirds of a yard V Foa,th« ? i^ft],, . bixtlis r 20. 1 f, is equal to how many fourths of a foot '> - tt. is eqnl to how many thirds of a yard ? 21 Exj,,,ss f 1 as 4ths of a dollar; .*.'", as 4tl,s: C yd. as Sils ; 2 ft. as 12ths ; 8 gal. as 4ths. In each case nme the measuring unit of the quantitv expressed asn fraction. Name this unit also as an actual unit ii common use. loff' ^T ^ ''^- "^ ™'' of '-^ ^^«ek; 6 vr. as 12ths ; da.as 24ths ; 5 min. as UOths ; pk. as 8ths ; 11 bu. a 4ths. :i ; I i'l 62 MENTA L A U ITI I M ETIC 23. In each example in tlie preceding question name tlie direct measuring unit as an actual unit of measure in common use. 24. Wliat is I of 24 equal to ? If 1 yd. is dividel into 24 equal i»cir^s, into Low many parts is | yl. divided ? | yd. = J^ yd. 25. lleduce | lb. to 1 2tlis ; g lb. to Klths ; j qt. t()21sts; I da. to SOths ; -^ftoloths; | lir. to Sahs. 26. Express as fractions witli 100 as denominator : 113 12 3 4 1_ t_ 12 15 111 2' '4' V b' b' 5' 5' 10' 10^ 10' 3' 3' G' 6' "T' 8' "9* fli V' lirt til If ^ I Lesson 22 1. Reduce the following to tlieir lowest terms: 6 ft • 4 vd • -8- ...\ . 1^(1;, . 12 ],!• . 15 loi 12 v)]^ 8 ^^'> 6 J*-^"' 12 ^'^^•5 20 '^•' IG ^ •' 35 ^^^^^ 2^7 J^*^' 2. Reduce to their lo\vest terms : -^^^ |., |^, ||, IJ, ||. IIow do you reduce a fraction to ts lowest terms ? 3. Reduce to their lowest terms : 3 6 ' 3 3 5 1' 42 '5 4' 78 87' 2 4 C 3 99' 8i' 4. From a rope 30 yd. long 1 cut of 20 yd. for a clotlies-line. What part of the who'3 piece was cut off ? 5. Express as a fraction of 2 doz. : 4, G, 3, 8, 10, 9, 12, 15, 2, 30, 36. 6. Express as a fraction of a peck 1 qt., 4 qt., 2 qt., 6 qt., 8 qt., 9 (|t. 12 qt., 14 qt. LESSON 22 68 I. 7. Express as a fraction of an hour: 20 niin., 15 min., 40 niin., CO min., 45 min., 80 min., 90 niin. 8. l^lxpress as a fraction of a Imsliel : 8 qt., 24 qt., 28 qt., 16 qt., nCj (j^t., 48 qt., 80 qt. 9. Express as a fraction of 2 ft. G in.: 10 in., G in., 9 in., 15 in., 20 in., 24 in., 30 in., 3G in., 45 in. 10. What part of a mile is 220 yd.? 880 yd.? 440 yd.? 170 yd.? 100 yd.? 480 yd.? What part of an acre is 484 sq. yd.? 121 sq. yd.? 242 sq. yd.? 908 sq. yd.? G05 sq. yd.? 11. If IS-!'. 16. I ])Jii(l 12f' for l)criies, 14^ for pcjis, Jiiid lOp for he'iius. VV^liiit j)ai't of whole sum was paid for ca(;li ? 17. At G4^ a yd., liow iiuicli cloth can I buy for 48^? 18. At 20^ a (luirc, what part of a (juiro of papor can I buy for If))^? Ifow many sheets? 19. A man walks a certain distance in G hr. AVluit part of it does he walk in 2 lir.? in 3 hr.? In [ hr.? Inlhr.? In .] hr.? In -J hr.? 20. I had *30 and si)ent J of it for an overcoat and i of it for a bicvle suit. Wliat was tlie cost of both? This is what part of ^ 30 ? 21. The smaller of two numbers is 10 and tlieir sum 40. Their difference is what part of their sum ? 22. On ^ of a held containini:!^ 12 A. I sowed oats ; on I of the remainder wheat. What part of the lield did I sow in wheat ? 23. A pipe will fill a tank containing' 300 ufal. in 10 hr.; a second pipe will fill it in 5 hr. How many gallons Avill enter the tank in 1 hr. if both pi2)es are open? This is what part of the capacity of the tank? 24. A can do a piece of work measured by 20 units in 4 hr., and B in 5 hr. How many units of work will they do in 1 hr., workino- together? This is what part of the Avhole work? 'S 'i! I LESSON 23 65 25. A sum of money is divided between A and li so that A gets twice as much as I J. What part of the money docs eaeli get? Jf the money is ^m, wliat does each get? 26. T Imuglit 2 hu. of plums put up in 4 boxes of erpial size, eacli of which was divided into two equal j»ai'ts. What part of a bushel did each part contain? What part of half a bushel ? Lesson 23 1. 1240 was divided between A and B, so that 15 received three times as much money as A. How many units reju-cscnt the entire sum? A receives what part ? B what part ? How much in each case ? 2. -it^l80 Avas divided among A, 15, and C. T5 re- ceived twice as much as A, and C as much as A and 15 together. What part of the money did each receive ? How many dollars? 3. In one corner of an oblojig 8 in. by "" ' . is drawn an oblong 3 in. by 2 in. 11ie small oblonir occupies what part of the area of the large one? 4. At a recent election 150 votes w^ei-e cast in favor of the successful candidate and 120 for his op[)onent. The majority was what part uf the whole lunuber of V(jtes? A 8-in. square is what part of a G-in. square? 5. 6. A 8-in. cube is what part of a G-in. cube ? \;^ U : :i F I G(^ MENTAL AllITII.MKTlC r r ft ^m V I ■ Si •( 7. A Lookciisc, G ft. high and 4 ft. wide, is put at one end of a room 12 ft. wide and 9 ft. high. What part of the wall space does it occu[)y? 8. A rug consists of a centre piece 4 ft. square and a horder 2 ft. wide. Whjit part of the whole rug is the centre piece? The border is what part? 9. A tablecloth 10 ft. long is folded under 2 ft. The part folded uiider is what fraction of what is needed for a tablecloth? 10. How often will a line in. long measure a line 18 in. long? 3 is often called the ratio of 18 in. to (J in. 11. What part of a line 18 in. is equal to a line 6 in. long ? J is often called the ratio of G in. to 18 in. 12. What is the ratio of 20 in. to 5 in.? Of 5 in. to 20 in.? 13. What is the ratio of G A. to 24 A.? If 24 A. of land cost 1720, what will G A. cost at the same rate ? 14. If 20 bu. of wheat cost 816, what will 4 bu. cost at the same rate ? 15. The ratio of the value of a purse to the money in it is J. If there is !i>12 in the purse, what is the value of the purse ? 16. Give the multiplication table of 4. Name numbers of which 4 is the ratio. Of which ^ is the ratio. LESSON 23 G7 17. rJivo nunilH!rs of which tlio ratio is ,% J ; G, J ; 71-1-1. 1.1.1 - 'V' 8' U' lU' 11' 12* 18. Whiit is iho ratio of the cost of 5 yd. of rib- bon to that of If) yd.? If 15 yd. of ribbon cost .ii<1.88, Avliat will o yd. cost at the same rate? 19. Wliat is tlie ratio of 3 in. to 4 in.? Of 4 in. to 3 in.? Of 3 in. to 8 in.? Of 8 in. to 3 in.? 20. What is tlie ratio of a line 3 2-in. lonrr to a line 4 2-in. hmo'? Of .,, line 3 3-in. lon4' Og, -g, -tg, Jg, Ig, ig, 14:y, ri-^f-g^ IO3, 1--^, I 1 g, O^. 15. Aniojig how many children can you divide 1 8| equally, if each cliild receives ^4? 16. How many boxes of candy, each containing half a pound, can be put up from SI lb. 7 LESSON 24 fji 17. How many weeks will 3| T. of coal last a furnace that burns on the average J T. a week? 18. How often is 5 contained in 19 and with what remainder? If you measure a line \^- ft. long with a foot rule, how often could you measure an exact number of feet ? The remainder would be what i)art of a foot ? 19. The expression ^r} ft. denotes a quantity made up of 19 units each equal to i ft. The expression 3| ft. denotes the same quayitity divided into 3 parts, each equal to the primary unit 1 ft., and the re- maining part divided into 4 parts, each equal to the direct measuring unit of the fraction ; viz. J^ ft. 20. Have you a clearer idea of the value of a quan- tity when expressed as ^^- ft. or as 3| ft. ? 21. Express the following quantities in terms of the primary units and the direct measuring units : f ft., -2^ yd., -sf mi., ioo lb., i^9. T., ^|4, ,^4^, || a., ^4'- A. 22. Name the two units of measurement in each quantity in example 21. Reduce to i|}|;egm's find proper fractions : 23 ^i M !5S U 6t 8 6 ^^' 11' 9 ' 6 ' 12' "9 ' ~i~' 24 liI-0. 2_0 10 2 5 25 5 ^'' 3 ' 9 ' T ' "2"' "3"' 3"* 25 J n P<'> 121 J 0.1 9 § •> (J ' 9 ' 12 •' 9"* » .; «,•* ♦ : f , i: 'TT »',« < i 1 1 I'^'i 4\ 'i A » ' * ' t' I' i 72 MENTAL ARITHMETIC Lesson 25 1. Find the sum of; 2 in. iind 3 in.; | ft. and I ft. ; I pk. and | pk. ; -| sq. yd. and | sq. yd. ; I wk. and | wk. llow can you add -| and -|? 2. Can you add \ ft. and -J ft. as you added the (piantities in example 1? Why not? 3. To what unit must you reduce 2 gal. before you can add it to 3 qt.? 5 ft. before you can add it to G in.? 4. To what must you reduce l before you can add 1 ? it to 1? ? f of 12 = ? I of 12 5. I of 12 = ? -|of 12 I of 12 = ? I of 12 = ? 6. Keduce the following fractions to 12ths : |, -J, fi l^ |i Ji 6* ilo^^' ^^ you find tl'^ numerator when you reduce | to 12ths? (Take |- ol 12.) 7. Reduce to lOtlis: J, |, J, |, |, |. 8. Reduce to loths ; J, |, |, |, |, |-. 9. Reduce to IGtlis 10. Reduce to 18ths 11. Reduce to 20ths 12. 13 16 3 1 2' 8' 4' 8^ 4' 8* 15 2 2 17 3^ (J' 9^ 3' G' "9* 1 3 11 4 12 1 _3_ 11 1_9. 21 2' ^2' 6' ^6' 5' 10' ^2' ^10* Reduce to 24tlis : |, f , { V, 3, |, 1 1, 11, li 1^, 1 ^. 13. Reduce to 30ths : J, IJ, |, |, 1|, f, -^%, -if. 14. What is the least connnon denominator of the fractions |, |, | ? Reduce them to equivjdent frac- tions having" this least common den'^iinator for their common denominator. LESSON 25 i •) Reduce to equivalent fnictions iiavln.i^ for ilicii- common denominator tlie least common denominator of the given fractions : 11 16. 17. I 1 1 _1. . ii' 3' G' 12 ' 112 5 4 2' 3'' G' 9* 1 _3 _ 5' 10' 3' (5 ' 1 2 5.31153 '3' 6' '8' 1 1 1 3.312 3 4' 7' 2' 4 ' 5' 6"' 15' 10* 18. 2V, 11 1-3 5 . 5 13 n 5 -2' ^3' -^4' 6 ' 8' ^4' 16' 12- 19. A piece of land containino- | A. is divided into lots each containing i A. Find the number of lots and their selling price at .f'SOO each. 20. Pepper is sold in bottles each containing i lb. How many of these bottles Avill contain 2| lb.? 21. IIow many tins each containing | lb. of mus- tard will contain all together 2i lb.? 22. A charity organization divided 5 bbl. of flour among several families, giving each family 1 bbl. How many families received help ? 23. I spent 1 1 in buying collars at 8 ^V each. How many collars? 24. A grocer sold 31- bbl. of flour in ^ bbl. sacks at $ 2 each. Find the total selling price. 25. A girl works a problem in arithmetic on the average in /^^ hr. How many v/ill she work at this rate in 1.1 hr.? t.'l ■ I it J H **1 >■?, ADDITION AND SUBTEACTION OF FRACTIONS ll u Lesson 26 1. What is the sum of | and J ? | and J ? ^ and ^ ? I and I? ^ and ^? 2. What is the sum of ^ and | ? ^ and |- ? ^- and ^ ? ^ and J ? J and ^? 3. A man whose lot contained -|- A. bought an adjoining piece that contained ^ A. How much did he then have ? 4. A can do | of a piece of work in an hour, and B ^ of it. How much can they do in one hour work- ing together? 5. Find the difference between J and I ; ^ and ^ ; I and i ; I and 1 ; -} and -i- ; ^ and 3^2 ; i '^^^cl i. 6. Bought I bbl. of flour. How much was left after using J bbl.? 7. What is the difference between ^ of my age and I of it? li this is 2 yr., how old am I? If 6 mo., how old am I? 74 LKSSON 26 75 8. What is the sum of i and I ? I aii b 3 ^^^^ 6 • f and I? 2 jind -3-V 5 '•'^^'-^ 10 • land ji? } and I ? f andi? 9. Find the difference between the fractions in example 8. 10. If 1- of the cost of a hat was 20 ^, wliat did the hat cost? 11. If I of the vahie of a liouse is ^1^^1200, what is J of its value? Wliat is f of its value? What is the house worth? 12. If I of the size of a farm is 75 A., what is J of its size ? How many acres in the farm ? 13. If I of a boy's weight is 45 lb., how much does he Aveigh ? What is your divisor liere ? Your multiplier? With the fractions |, |, j\, -f, respec- tively, what would your divisors and multipliers be ? 14. The sum of i and J of a boy's weight is what part of it? If this is 60 lb., how much does the boy weigh ? 15. A man invested ^ of his money in business and i of it in real estate. What part of Ids money did he invest in both ? If this Wcas ^ 14,000, liow mucli money had he? 16. 1-3 = ? 1-|=? l-f=? 1-|=? 9 1 _ A — 9 9 1 _ r, _ V 9 _ r. 9 — • -- 9 r«ti «i ti 2 = ? 8 — - ^ — ^ ■ ii I' i 1 .i" Iti 11* I) \ t- i h... " r*' ,. !'lv, H :' . i* I • 7G MENTAL AlUTIIMETIC 17. A tailor cut off | of a piece of cloth to make a suit of clothes. If G 3^1. are left in the piece, how many did it contain at first? 18. A cistern was J full of water, and during a rain filled up ^ more. The water was then 2 ft. from the top. How deep is the cistern? 19. 'J'he trunk of a tree was cut into three parts. The first contained | of the length, the second -J, and the third part was 20 ft. long. How long was the trunk ? 20. A merchant sold cloth at 45^ a yd., gaining ^ of the cost. The selling price was what fraction of the cost ? What was the cost per yard ? In examples 21-27 give the statements of halance. 21. A merchant sold apples at GO ^ a bushel, gain- ing I of the cost. 1 ind the cost. 22. A boy lost ^ of his marbles and has 30 left. How many had he at first? 23. A man divides his lot into two parts, one of which, containing | of the entire lot, he keeps for himself. The other part, which has a frontage of 100 ft., he sells. What is the frontage of the entire lot? 24. Two boys ran a race, and when the winner had reached the goal, the second boy, who was 10 yd. behind, had run -^^ of the whole distance. How long was the course? LI-:SSON 27 77 25. Three boys ran a race, and when tlie winntT reached the goal the second boy liad vna J of tlu; whole distance, and the tliird boy, who was 2 yd. behind him, had run J of the whole distance. How louGf was the course? 26. The torrid zone extends 2o.V° on each side of tlie equator. How many degrees wide is it? 27. How many degrees from the north pole to the equator ? The frigid zone extends 23,V° from the north pole, and the torrid zone 2o|° north from the equator? Find the number of degrees in the north temperate zone. Lesson 27 1. What is the sum of 3 and 2? Of 31 and 21? 2. Find the sum of the followincr : Of I and i ? 2| and 31 4| and 2f 3f and 41- 2| and 1 1 21- and 1| 4 1 and T| 35 and 1 4| and 5f 2i and 3I 4| and 1| 21 and 34 { and 2| IJow do you add two fractions ? 3. Find the difference between tlie following : 2 1 and 11- 3f and 2^ 2| and 2| 35 and 11 5 1 and 4| ^ and 1| M IJ 78 MENTAL AUITIIMETIC It1, I ''1 m Itf :•■. '11 - ill I r ' ,■ ■ '\ Pi . '" '.H !! it I ill I 3 _ 2 ^ ■^ n _ 2 ? 2 3 3 ~" • A _ 3 3 4 = ' n -I ='! -'r. 2 = n -1 ='^ = I + ' C| _ 2 _ y 3 41—3 ^2 4 — * 51 - V = ? 2 • 72 _ 43 '3 ^4 = . 8| _ 45 = ? 5. How do you find the difference between two sucli fractions as 8 2 and 4^ ? o 6. Find the difference between the following : and 4? 1 9 12 and C)/^ 36| and 2i| 41 and2| ^ andSfl 6|^ and 5 J 31- and 2^3^ 12 s and 8 1 7. Find tlie sum of the following : 1 1 2' 3' J 1 2' 1' 1 2 2' 3' 4 1 1 "6 12 1 X 3 91 2' ■*' "^2 --2' ^3' 2-1 .'^)i 83 *-2^ •>-3^ O^ _7_ 12 9' 12' 6 42 1\ ^5' ^4' 6^ 8. 1 + 1 5 1 3 3 i 1 '6 _ 1 - ¥- _ 1 - 3 ~ _ 1 _ ¥- 9 3_ I 1 _ 10 ^ 4 ~ 2|- + -ti - 4i H - H + ^ n + 2j - 2iV 4f - 2J - 1-1 ? ? ? ? 9. A boy who had i4| earned 1 2| more. How many dollars did he then have ? 10. A dress was made from two remnants, one containing 4| yd., and the other 8f yd. How many yards did the dress contain? :'l.1 .•if LESSON 27 w iO 11. Two brothers gathered hickory iints. Tlie ohler gathered 2|j bu., and the younger 21 bu. How many bushels (Ud tli(;y gather together ? 12. A hidy is 48^ yr. ohl, and Iier husband is 5] yr. ohler. Find liis age. 13. A boy traveled ;jl)| mi. »n his bicycle and tlie next day 3()| mi. Hon' much farther did he travel the first day than the second ? 14. What fraction subtracted from the sum of i and I will leave 1 for reii aiiider? 15. A farmer divided a portion of his farm among his three sons, giving to the two oldest ^^ each, and to the youngest J of it. What part of his farm did he divide. If this was 1G5 A., how large was his farm ? What was each son's share ? 16. Two boys contributed 1 and |, respectively, toward the expenses of an outing. The third boy, who contributed the remainder, put in 75 ^. What did the outing cost and what did each contribute ? 17. A number is larger by 50 than | of it. Find the number. 18. A man paid a debt in four payments. At first he paid |-, then J, then | of it, and finally $500. Find the original debt and the amount of each pay- ment. 19. My furnace burns 12| T. of coal during the winter, and my neighbor's 8| T. How much more does it cost me for coal than my neighbor, if coal is f 6 a ton ? ' ■•« »*; h\ %- '^ r^%k IMAGE EVALUATION TEST TARGET (MT-S) <- .^r^ f,%^. i 1.0 1.1 ll£|2£ |2.5 ■ 50 ■^™ H^B u lii '•25 1 '-^ •* < 6" ► V2 ^ 7] ^^J>\^ :^:> "♦V^ /^ 7 Hiotographic Sciences Corporation 23 WEST MAIN STREET WEBSTER, N.Y. MS80 (716)872-4503 %^ \ \ V ». 6^ '^^^ ^ 4^0 6^ MHli 80 MENT \L AK ITI I M I:TIC u m 20. A bicyclist travelling at the rate of 8 mi. an hour goes around a quadrilateral section of country of wliicli the sides are G| mi., 5| mi., 3| mi., and 6|- mi. How long is he gojie if he is delayed a quar- ter of an hour on the road ? 21. If a certain number is increased by -| of itself the sum will equal 75. Find the number. 22. At the rate of 8 mi. an hour how far will a boy ride in two trips, one lasting 3j lir., and the other 4| hr., if he stops ^ hr. during each trip? 23. The difference in weight between two parcels is 8 J 11)., and the lighter parcel weighs 12-}| lb. What is the weight of the heavier ? Of both ? 24. Divide 10| lb. of sugar into two parcels, one of wluch shall be 1} lb. heavier than the other. 25. What fraction must be added to the sum of ^ and -|- that their sum may be | ? MULTIPLICATION AND DIVISION OF FRACTIONS >'m Lesson 28 1. Find the cost of 10 yd. of clotli at $ | a yd. Name tli3 number^ the imiU and the quantity, 2. Find the cost of | yd. of silk at 88 a yd. Name the number^ tlie unit^ and the quantity. 3. In taking f of a nnmber is it better to divide by 4 and then multiply by 8, or to multiply first by 3 and then divide by 4 ? Test in these questions : I of 20 = ? I of 7 =? f of 32 = ? Find the cost of : 4. f of 15 = ? I of 13 = ? I of 48 = ? 10 lb. sut^ar at 5 J ^ a lb. 6 bu. potatoes at 85 a bu. 5 baskets melons at %-^-^ a basket. 5. ^- doz. 5-lb. jars butter at % 1-^^ each. 1 yd. cloth at 8 2 a yd. 2 doz. yd. towelling at 8 | a yd. Find the weight of : 6. 8 yd. carpet at 1\ lb. a yd.; l^lb. ; l^^g lb.; 1_3_ 11) . 16 11) f W 82 MENTAL AllITlIMETIC il 1^ 7. 8 double rolls of wall paper ut 1| lb. each ; 6 double rolls ; 12 ; 7 ; 9 ; 10. 8. 3 spoons at | oz. each ; 5 at | oz. each ; | doz. at I oz. each ; | doz. at 1| oz. each. 9. Half a dozen teaspoons, each 4J in. long are placed end to end ; how long a line do they make ? How long if each is 5^ in. long? 5| in.? 10. How many ounces of pure silver in a dozen tablespoons weighing 1| oz. each, |J fine ? 11. Reduce to inches : f ft. ; 1|- ft. ; 2J ft. ; 3| ft. v" IL. I -q It. f J.V.- It. 12. Reduce to yards : 2 rd.; 4 rd. ; G rd.; 5 rd. 9 rd. ; 10 rd. ; 12 rd. 13. Reduce to feet : 2 rd. ; 6 rd. ; 3 rd. ; 10 rd. 12 rd. 14. Reduce to quarts : 1| gal. ; | gal. ; 1| gal. 2 J gal.; 3i\gal.; U/^ gal. 15. Reduce to square feet : | sq. yd. ; ^ sq. yd. I sq. yd. ; 21 sq. yd. 16. Reduce to ounces avoirdupois : J lb. ; | lb. I lb. ; 2\ lb. ; 51 lb. 17. What is tlie area of a table 3 ft. long and 2|- ft. wide ? 18. Find the areas of the floors of the following rooms : Length Width Length Width 8 yd. 6fyd. 10 yd. 7f yd. 9 yd. 7fyd. 12 yd. 8f yd. 7 yd. 5Jyd. 9 yd. 6J yd. ■ LESSON 29 6 83 19. Find the areas of the walls of the followinir rooms ; Perimeter Height Perimeter Height GO ft. 8 ft. 6 in. 80 ft. 10 ft. 4 in. 80 ft. 9 ft. 9 in. 90 ft. 9 ft. 4 in. 20. The base of a triangle is 12 ft. and the alti- tude 6 ft. 6 in. Find its area. (See p. 35, ex. 2.) 21. Reduce 2| and 4| to improper fractions. iof4i = ? 8x|=? 2|x4i-=? 22. How do you find the product of two numbers like 21 and U ? 23. Find the value of : H X H Sfxlf If x9^ 2| X 41 24. Find the value of : 3fx5J 71 X 2-2-- • 3 ^ ""11 X 5 y 5. 6 l|x2i 181 X 3f 12 xlf 2f xCf 131 X 41 2f X 5^ 8f X 4f of fofl2 fof f of 41 of fof 24 f of 36 i of 15 f of 4f of 27 f of 61 of 8 25. If a man earns f 2| a day, how much will he earn in 3| da.? 26. Find the volume of a rectangular slab of gran- ite 4i ft. long, 2^ ft. wide, and 11 ft. thick. I I I t ?n I 11 i.^ ! 1^' •I 84 MENTAL AIUTIIMETIC 27. What is the weight in pounds and ounces of a s(liiare piece of floor oil cloth 1| yd. long, weighing 3 J lb. to the square yard ? 28. What is the weight of a mat 27 in. wide, 48 in. long, which weighs 2 J lb. to the square yard ? Lesson 30 1. If 1 bu. of corn is worth $5|, what is -| bu. worth ? 2. What will 5 J T. of coal cost at $5\ a ton ? 3. What will a 5-gal. can of kerosene cost at 12|^agal.? 4. How many square miles are there in a section of country 4 J mi. long and 2| mi. wide ? 5. A piece of park is in the form of a triangle. If its base is 7^ rd. and its altitude G| rd., what is its area? (See Lesson 13, example 2.) 6. A lot in the form of a trapezoid is 9| rd. wide in front and GJ rd. wide in the back. If its depth is 10|^ rd., what is its area? (See Lesson 13, ex- ample 15.) 7. How far will a person travel in 5| hr. at 5| mi. an hr. ? 8. How many yards in 1 rd.? How many square yards in 1 sq. rd. ? 9. At ^1 a yd., what part of a dollar will | yd. cost ? LESSON 30 85 10. A farmer can plough a iield in 18 hr. What part of it can he plough in 2 hr. ? In 3| hr. ? In 2 hr. 15 min. ? In 6 hr. 45 min. ? 11. If f of f of 1 of a board 20 ft. long is cut off, what length of board is left ? 12. How much greater is -j^ of 2| rd. than 1 yd. ? 13. Texas is 4f times as large as Illinois, and Illinois is 2-| as large as West Virginia. How many times as large as West Virginia is Texas? What part of Texas is as large as West Virginia? 14. If Michigan is lo as large as California and Oklahoma is f as large as Michigan, compare the area of Oklahoma with that of California. 15. If the population of the District of Columbia is I that of Rhode Island, how many people live in the District of Columbia for every 99 who live in Rhode Island ? 16. If the population of Utah is J that of Colorado, and that of Colorado } that of Michigan, how many people live in Utah for every 100 who live in Michi- gan ? How many people in Michigan for every 100 in Utah ? 17. The Mackenzie River is I as long as the Ama- zon, and the Rio Grande ^\ as long as the lAIac- kenzie. How long is the Rio Grande, the Amazon being 4000 mi. in length ? ' 1 1(1 1 I t L 86 MENTAL ARITHMETIC hi :* ♦ iigi 18. I bought a liouse and lot and paid | of the cost price in cash? If this was $3600, what was the cost price ? 19. A man sold his horse, gaining | of the cost l)rice. If his gain was $24, what did the horse cost liim? 20. A man owned a | interest in a factory and sold I of his share. What part of the factory did he sell and what jiart did he keep? 21. A man invested ^ of his money in a house. What part of his money was left ? If he invested ^ of this remainder in busincbs, what part of his money did lie invest in business ? If this was $2400, how much money had he at first? 22. A man spent ^^ of his money for a farm and I of the remainder for cattle. If he paid $1800 for the cattle, how much money had he at first ? 23. A merchant bought goods to the amount of $ 3(J0, but he got ^ off as a merchant, and then 2V off the remainder by paying cash. What did the goods cost him ? 24. A wholesale merchant sold goods amounting to $(J00, but he deducted -^ of the amount of the bill and then threw off -^^ of the remainder for cash. Find the net selling price. 25. A v/holesale merchant threw off ^ uf the amount of a bill and then ^ of the remainder, and the bill then amounted to $300. What was the original bill? LESSON 31 87 26. A piece of cloth vvlieii measured with a yard- stick that is I in. too long seems to be lOf yd. long. How many inches longer than that is it? How many yd., ft., and in. long is it ? Lesson 31 1. Into how many parts, each 2 ft. long, can you divide a line 6 ft. long? 6 ft. -^ 2 ft. = ? 2. Imagine a line 1 ft. long divided into parts each J ft. long. How many parts? 1 ft. -f- ^ ft. = ? 3. 1 ft. -^ 1 ft. = ? 1 ft. - 1 ft. = ? 4. lib. -l-lb. = ? 1 A. -lA. =? 1ft. 1ft. i ft. = I- ft. = IT. - 1 T. = 1 bu. -f- Jg^ bu. = 5. Into how many lots, each equal to J A., can you divide 1 A.? 6. A society distributed 1 T. of coal to several families, giving to each | T. How many families were there? 7. Into how many parts, each J ft. long, can you divide a line 1 ft. long? 4 ft. long ? 6 ft.? 6 ft. -ift. =? 8 yd. -lyd. =? 7 gal. -f- 1 gal. = ? 4 ft. - l ft. = ? 9 hr. H- 1 lir. = ? 5 bu. -^ -V bu. = ? ^- I I ii i ' ,& I !V II I r \i ii 'V >'|i ■111' ' it :i m \-. » 1 i I ji ''l .! ■" ' 88 MENTAL AUITIIMI:TIC 8. A five-p^jiUon eiiii of nuiplo syrup is einptiecl into jjii's, each containing | gal. Find the number of jars. Name tlie qnantity, the unit, and the number. 9. A girl, working problems in arithmetic, did 2 in every ] hr. How many did she do in 2 hr, ? 10. Into how many parts, each ^ ft. long, can you divide a line 6 ft. h ng? How many of these 18 I)arts make | ft.? How many parts, each equal to I ft., can you count in these 18 parts. 6 ft. -- 2 ft.=? 4 ft. -^ 2 ft. = ? 11. In the first example in question 10 by v/hat did you multiply 6 to get 18? By what did you divide 18 to get D? Is it correct to divide by 2 first, and then multiply by 3? Which gives the smaller numbers? 12. How do you divide a whole number by a fraction ? 13. Find the quotient in each of the following : 6 ft. -^|ft. n : -f 1 8 ft. . 2 • 3 10 yd. • 1 y^^- i^8-5iii Or 2| times the rate = 27 mi. (2) The actual distance = 12 yd. - \^ in. (live the statements of balance for the following examplos in this lesson. 1. A bicyclist rode 27 mi. in 2 lir. 15 niiii. What WHS his rate in miles per hour? 2. if a yard measure is J in. too short, what is the actual distance between two points which is found by this measure to be 12 yd. ? 3. After pumping out 3G0 gal. of water from a cistern, | of the water still remained. How many gallons did the cistern contain at first? 4. After selling 400 bu. of wheat, a farmer had I of his crop left. How many busliels renuiincd unsold ? 5. What part of a mile is 440 yd.? 30 sec. is what part of 1 min. ? Of 1 hr. ? A train travels 440 yd. in 30 sec. This is at the rate of how many miles an hour ? 6. Railway fencing costs $5.50 a rod. What is the cost of 1 yd. of fencing ? Of 2 mi. ? 7. Find the number which is 35 more than the sum of 1 and J of it. Prov-e your answer correct. '1 «•♦*. {» i i! 92 MENTAL ARITHMETIC 1* i 8. If to a certain number its | and J be added, the sum will be 611 94 MENTAL ARITHMETIC 22. I sold a horse for 160 and thereby lost | of the cost price. What should I have sold it for to gain I of the cost ? 23. A boy bought 3 doz. eggs for his mother at 12j^ a doz., and with the remainder, which was | of the money she gave him, he bought oranges at 4 ^ each. How many oranges did he buy ? 24. Three farms join each other. The first con- tains I as much as the second, and the second | as much as the third. The third farm contains 120 A. How large is the first farm ? 25. I traded a -| interest in a grocery store for a I interest in a general store. If the grocery store was valued at 19000, what was the value of the general store? ii to at DECIMALS ♦ i A. Lesson 33 1. In the expression '^^6.37 the 6 is 6 what? The 3 is 3 what ? What part of a dollar ? The 7 is 7 what ? What part of a dollar ? The 37 is 37 what ? What part of a dollar ? 2. In the expression -f 2.19 what part of a dollar is the 1 ? The 9 ? The 19 ? 3. Read as dollars and as hundredths of a dollar: $4.29, 16.37, 12.15, $8.05, $8.99, $.76, $.26, $.06, $.09. 4. Read the following expressions : 3.45 T., 2.96 mi., 7.15 A., 6.43 yr., .37 da., .89 hr., .07 hr., .08 gal., 3.06 gal-., 5.01 bu., 3.18, .64, 3.50, .04, .02. Give the place value of each figure in the above example. 5. Read the following expressions : 3.146 lb., 4.293 T., 8.141 A., 5.16 A., 8.028 T., 3.006 da., .456 da., .75 hr., .005 bu., .003 gal., 4.568 mi., 3.333, 4.593, .063, .008, .06, .006. Give the place value of each figure in the above example. 0d liii I J ■ ( H ^M I m\i If ''i i ii'-.. I nr^i 9G MENTAL ARITHMETIC 6. Ueiid the foUowiiii^ expressions : .292, G.04, .6, .04, .045, .0457, .04572, .6, .00, .000, .0000, .0054, 94.8000, .0010, 7.24732, 3.0300, 12.47025, .42819, .00328, .8210, 3.1410, 2150.42, 39.371. 7. Reduce to common fractions in their lowest terms : .5, .0, .20, .25, .75, .80, .125, .375, .025, .875. 8. A farmer sold .8 of his flock of 325 slieep. How many did he sell ? Give tlie statements of balance for this and the following examples. 9. A farmer gave .375 of his farm of 200 A. to his son. How many acres did he keep ? 10. A merchant sold cloth at an advance of .125 of tlie cost and chained 21 a vd. What did a cus- tomer pay for 10 yd.? 11. Ileduce to common fractions in their lowest terms: .331 .r,Of, .lOf, .14?, .12|, .37|, .02^, .871, 111 ^)^)2 114 555 771 e«8 (;2 ai 08^ Commit these results to memory. 12. A man withdrew .44 J of the money he had in a hank to invest it in business. T'lere still remained in the bank '"^8500. How much did lie Avithdraw? 13. A merchant sent .00| of a piece of cloth, which originally contained 45 yd. and sold at 50^ a yd., to the remnant counter, reducing the price .5. What Avas the })iece marked to sell at? LESSON 33 97 113 122 3 41 51 4^ 4> 3^ 3' f)' f)' f)^ G' 8 11111 14. A inerclumt bought goods from a wholesiile house for which he received a bill for '^480 less .S^VJ and .Oi)| for cash. Find the cash amount of the bill. 15. .Gf)| of a piece of ribbon was sold to one cus- tomer and .!()? of it to another. The remainder sold for 80^ at 12^ a yard. Find the number of yards in the })iece at first. 16. Reduce to hundredths : G- 7' h h I' 8^ 9' 9' 9' IP 9' 9' 10' l"l' 12' 15' IG* Commit these results to memory. 17. What part of 1 yd. is 1 ft.? How many liun- dredths? What part of 1 ft. is 1 hi.? How many hundredths? 18. Express your answers to the following ques- tions in tenths or hundredths. What part of 1 gal. islqt.? Of 1 bu. is 2 pk.? Of 1 pk. is 1 qt.? Of 1 lb. avoir, is 1 oz.? Of 1 lb. troy is 1 oz.? Of .25 of 1 yr. is 1 mo.? Of 1 da. is 4 hr.? Of .lOf of 1 quire is .38| of a dozen sheets of paper? Of 1 rd. is 1 yd.? Of 1 sq. yd. is 1 sq. ft.? Of the side of a square containing 25 sq. in. is the side of a square containing 10 sq. in.? 19. Carpet bought at 5G ^ a yd. is sold for 70^ a yd. What is the gain as a fraction of the cost? How ni.ny hundredths? 20. A grocer bought tea at 00^ a lb., and sold it for 72^ a lb. How many hundredths of the cost did he gain ? II Mil I 4 I ! I( k I I 'I 11 ;f 'I « i I'M Mi! ;iio 98 MENTAL ARITHMETIC 21. Bought pencils at 30^ a dozen and sold them at 4 ^ each. What is the gain on Ccach dollar invested? 22. A laboring man earns $270 in 240 da. at 81.50 a day. How many days does he work out of every 100 da.? 23. A person invested .SSJ of his money in real estate and .44 J of it in business. He had 1 1200 left. Wliat was the amount of each investment? 24. A grocer sold .10| of his stock of eggs for 24^ a doz., .5 of it for 22^ a doz., and the remainder 18^ a doz. Find the average i)rice per doz. 25. The stock in a drygoods store was damaged .331 of its value by fire. If the stock was worth $ 9000, what was the loss of one of the partners Avho had a | interest in the business ? f f COMPOUND QUANTITIES 'I 'I Mm Lesson 34 In the following examples give tlie statements of balance : 1. Find the quantity measured by the number G and the unit 1 gal. 2 qt. 2. Mark off the space occupied by a cord. How many cords in a pile of cord wood 24 ft. long and 4 ft. high ? 24 ft. long and 8 ft. high? What is tlio cost of the latter pile at f 4 a cord ? 3. How many acres in 1 sq. mi. ? What i)art of a square mile is a farm of IGO A. ? If this farm is in the form of a square, what is the length of its side ? 4. How long will it take to walk around the farm in example 3 at the rate of 3 mi. an hr. ? 5. How might such expressions as the following arise by acts of measurement : 2 rd. 3 yd. ; 3 bu. 1 pk. ; 4 gal. 2 qt. 1 pt. ; 5 lb. G oz. ; 2 T. 400 lb. ; 3 yr. 2 mo. ; 4 hr. 2G min. , 3 doz. G ? 6. What part of 1 yd. is 2 ft. ? G in. ? 2 ft. 6 in. ? Find the weight of an iron bar 5 yd. 2 ft. G in. long if 1 yd. weighs 15 lb. 99 m I' ■ 100 MENTAL AKITIIMETIC y 'I N 7. How inaiiy pecks in 1 bii. ? A farmer sowed 5 pk. of oats on a field and raised IG bu. 1 pk. What is the yield per peek of seed ? per bushel ? 8. A train travels at the rate of | mi. in 30 sec. AVhat is its rate per hour ? 9. What is the value of £1 in dollars and cents ? A o'entleman in New York places c£100 to his credit in London. What will it cost him, exchange being (pioted at •ii<4.87 ? 10. A cubic foot of water weighs 1000 oz. How many pounds? 20 cu. ft. of water weigh what part of a ton ? 11. Apples are bought at $2 a bbl. of 2| bu., and sold at i)0 ^ a pk. AVhat is the gain per bbl. ? What would the gain be if 1 pk. in each barrel were spoiled? 12. How many square rods in 1 A. ? What is the value of a triangular piece of land whose base is 82 rd. and altitude 15 rd. at 860 an acre? (p. 35.) 13. How much ink will it take to fill the ink-wells in a room containing 48 single desks, if each well holds \ gill ? 14. IIow many pounds does 1 bu. of wheat weigh? What is the cost of 2 bu. 15 lb. of wheat at 84^ a bu. ? 15. IIow many square chains in 1 A. ? What is the selling price of a piece of land 20 ch. long and 16 ch. wide at # 90 an acre ? LKSSOX 34 101 16. A fruit dejiler l)()iio-lit ;i l.uslicl of jipplos for '^l.TiO, jind sold thoin at the rate of 8 For T)^. II,,nv inucii (lid he gain if there were i'40 ai)i)k's in the bushel ? 17. How many loads of earth were removed to make a eellar If) ft. long, 12 ft. wide, and G ft. deep ? 18. JIow many le^d peneils in a dozen boxes if eaeh ecjntains .] <''i'()ss ? 19. When apples are selling at 40 ^ a peek, wliat profit is made by the seller who buys them jit >i^l.2i) ii bushel ? 20. What will the lumber cost for a tight board fence 250 ft. long and 4 ft. high at .^2 per C ? 21. A square rug 2^ yd. long costs -^^5; what is the price per square yard ? 22. A bottle of sachet powder weighs (] oz. What is the least number of bottles that weigh an exact number of pounds ? 23. How many pounds of silver will make 240 teaspoons, |^ fine, if a dozen spoons weigh (3 oz. ? . 24. Each of the two sides of a roof is 50 ft. long und 24 ft. wide. What is the cost of iron rooling to cover this roof at ^^2.50 per square of 100 sq. ft.''? 25. How many 4-in. cubes can be cut from a block of wood 1 ft. 8 in. long, 1 ft. wide, and 8 in. thick, no allowance being made for Avaste ? 26. Find the weight in pounds of 12 doz. silver tablespoons w^eighing 20 oz. per doz. Of 12 yd. of carpet weighing 20 oz. each. •I ?; H. *l ii \\l I i ■41.. li! 102 M ICNT A L A K ITI I M ICTIC Lesson 35 1. Find tlio value of the qiijintity measured by tlie number 8 according as the unit is 2 ft. G in. or 2 bu. G pk. 2. What are tlie dimensions of a cord ? Find tlie value of a pile of cordwood 32 ft. long and G ft. liigh at $ 8.50 a cord. 3. What part of a rod is 1 ft. ? nj ft. ? 22 ft. ? The driving wheel of a locomotive, Avhich is 22 ft. in circumference, turns 1)0 tim(3S in 1 min. Through how many rods does it draw the train in 1 min. ? 4. What part of a mile is 40 rd. ? 120 rd. ? IGO rd. ? 280 rd. ? A train runs 120 rd. in 1 min. How many miles is that an hour ? 5. How many rods in 1 mi.? How many yards? feet ? A horse trotted 1 mi. in 2 min. 5G sec. Tak- ing his stride at 5 yd., how many times a second did his feet touch the ground ? 6. What part of an hour is 18 min.? 3G min.? -45 min.? 48 min.? 54 min.? A horse travels 24 mi. in 2 hr. 24 min. What is tlie rate per hour ? 7. Reduce J of ^ 48 to a decimal of 1 80. 8. How many shillings in £1? £15? £1 16s.? X3 4s.? £2 17s.? How many ounces of gold are worth £21 if .G oz. is worth £ 2 14s.? LKSSON 35 10;} are 9. TTow do you fiiul tlio perimeter of mi oblong iield ? How iiKiny rods of wire feiieiiicr will it take to enclose a field 40 rd. long and 25 rd. wide, there being 5 rows of wire ? 10. What will the wire cost at ^ 2 per spool of 100 lb., if wire fencing weighs 1 lb. to the rod? 11. What part of a yard is 18 in.? How many strips of wall paper 18 in. wide will be required to paper a room 20 ft. long and IG ft. wide, no allow- ance being made for waste in matchino' ? 12. If the room in the previous example is 12 ft. high, how many double rolls of paper will be needed for the room, and what will be its weight? (A double roll weighs 1| lb. and is 10 yd. long.) 13. State clearly and briefly how you solved each of the foregoing examples in this lesson. State how you will solve each of the following examples in this lesson. 14. What are the factors of 231 ? I buy molasses put up in a till dish 10^' by V^ by 2i- in. What part of a gallon do I get ? 15. Give the table of Troy weight. Find the weight in pounds of one dozen silver teaspoons weighing I oz. each. Find their cost at 50^ each if 3 initials are engraved on each spoon at a charcre of liy a letter. :^ 104 MKNTAi. AurriiMF/nc i 16. A luiiii pjiul !M;^ for 1 11 >. of teji, but on wi'1l»1i- iiiL»' ii found tluit it was .] oz. too light. Wluit should hu have paid for it? 17. What is 1 ft. board mcMisure? A lot 200 ft. louL,^ and 150 ft. widi3 is to bo surroundod by a (rloso hoard funco T) ft. hi<^di. What will the lund)or cost at -if 1:5 i)er M ? 18. A watch gains 3 sec. in 12 hr. In how many days will it gahi 5 niin.? 19. How many acres in 1 sq. mi.? ^ scj. mi.? I sq. mi.? How many scjuare miles in a strip of land 20 mi. long and | mi. wide? 20 mi. long and ^}^^ nu. Avide? How many acres in the last strip? 20. What part ofl mi. is 880 yd.? 88 yd.? 22 yd.? 176 yd.? A railway company pays ''ii<20 an avva for a strip of land 44 yd. wide and 20 mi. long. Find the entire cost. 21. Name the months that have 30 days ; 31 days. A note drawn on Aug. 27 is due Oct. 15. How many days has it to run? 22. A note due Deo. 8, 1898, was discounted Oct. 10. Find the number of days between the day of discount and the day on which the note is due. 23. Through how many degrees does the sun pass in 1 hr.? How do you change longitude expressed in time to longitude expressed in degrees? LKSSOX 36 10.- 24. A man tnivcls until liis wati-li is -J lir. { ,„i„. fast. Does lie travel east or west, and tluou-'li how many degrees? 25. How many square cliains in 1 A.? How maiiv links in a eliain? What is the value of land 24 eh. oO 1. hm<^ and 20 eh. wide at >i^i)0 an aere? 26. Two elocks are toirether at Monday noon ; one loses 10 see. and the otlier gains 20 see. a day. In how many days will one be 10 min. ahead of the other? On what day of tlie week? Lesson 36 1. How many pounds of ilour are there in a saek containing V bhl.? -J hhl.? 2. How many grains in 1 lb. Avoir.? In 1 11). Apothecaries'? ])o druggists use Avoirdupois or Apothecaries' Aveight in buying drugs? In selling drugs? 3. A druggist buys 1 lb. of quinine for .^4 and uses it to make 2 gr. quinine pills which he sells at 30^ a hundred. How much does he train? 4. Give the table of Apothecaries' li(pii(l meas- ure. How many minims, or drops, will iill a tea- spoon? A physician prescribes a 4 oz. bottle of medicine to be taken in teaspoonful doses three times a day. How many days Avill it last if a teaspoon holds 60 drops? lOG MENTAL AUlTIlMirnC illi :l ■ '!'••!« 5. What does 1 cu. ft. of water weigh? How many pounds do 8 cu. ft. of water weigh? What part of a ton ? A cylindrical column of water whose base is 4 sq. ft. and height 8 ft. weighs how many tons? 6. How many board feet in 8 scantlings 12 ft. by 4 in. by 3 in.? 7. Find the cost of 5 bu. 3 pk. 2 qt. of nuts at 11.28 a bu.? 8. How many hours of daylight are there on Aug. 18, the sun rising at 5.10 A.M.? 9. How many sheets in a quire ? Quires in a ream? A stationer pays $1.56 a ream for letter paper and sells it at 15^ a quire. How much does he gain on 1 ream ? This is what part of the cost ? 10. Show by measuring that the circumference of a circle (plate, etc.) is about 3^- times the diameter. How many feet will a bicycle having a 28-in. wheel go in making 12 revolutions? 11. How often does the unit 88 ft. measure the quantity 1 mi.? How many revolutions will a 28-in. bicycle wheel make in going 1 mi.? Also the pedal, the gear being 3 ? 12. What is the ratio of the weight of 1 lb. of brass to that of 1 lb. of gold ? How many pounds of gold weigh as much as 72 lb. of brass? LESSON 36 107 on 13. If 1-4 ytl. of cloth 30 in. wide will make a dress, how many yards of cloth 28 in. wide will he needed to make another dress of the same size ? 14. An ingrain rug 30 in. by 54 in. cost dO^. What was the price per square yard ? 15. A man rowed 4 miles down stream in 40 min., and back again up stream in 1 hr. Find his rate of rowing per hour down stream, up stream, in still water, and also the rate of the current. 16. If paint brushes that cost 4 ^ each cost | as much when bought by the dozen, what is the cost of 6 doz.? 17. The border of an oblong ^Moquette carpet is 223 in. wide. How much longer is the perimeter of the border on the outside than on the inside ? 18. In the centre of a square room 12 yd. on a side is a square rug covering | o^f the floor. Find the length of its side. 19. If 6 double rolls of wall paper 18 in. wide are needed to paper a room, how many double rolls of 22-in. paper would paper a room of the same wall area ? 20. What is the ratio of the side of a 6-in. square to tliat of a 9-in. square ? Of the perimeter ? Of the area ? 21. What is the ratio of the side of a square rug 1|- yd. long to that of a square rug IJ yd. long ? Of the perimeter ? Of the area ? 108 MENTAL ARITHMETIC 22. If the larger rug in example 21 costs -ii) 1.44, what will the smaller rug cost at the same rate per square yard ? 23. A stationer buys lead pencils at 90 ^ per half gross, and sells them at the rate of 2 for 5 ^. Find his gain on 1 gross. 24. How many quarts in 1 bu.? If a horse eats 1 pk. 3 qt. of oats in 1 da., in how many days will it eat 6 bu.? 25. A township 6 mi. square is divided into how many sections ? What is the area of one section in square miles ? In acres ? A farmer who owns \ of a section of land lias liow many acres in his farm ? If his farm is in the form of a square, what is the dis- tance around it? 26. 25 bu. of grain occupy 32 cu. ft. A bin 8 ft. by 4 ft. by 4 ft. contains 3200 lb. of grain. Find tlie weight of 1 bu. of this grain. What kind of grain is it ? 'W. w\ PERCENTAGE Lesson 37 1. A quantity divided into thirds is measured by 3 units. By liow many units is a quantity measured that is divided into fourths ? Into fifths ? Fortieths ? Sixtieths? Hundredths? 2. A quantity considered in percentac^^e is meas- ured by 100 units, | of tlie quantity by | of 100 or 25 units. Thus | of a quantity is 25 hundredths or 25 per cent (25%) of it. 3. What per cent of a quantity is I of it ? J ? -^q ? _1_9 .2? 3? 20 • 5 • ¥' 4. Whot is meant by saying that l = 50 % ? 1 = 25%? 1 = 20%? 5. Express the following fractions in per cent : rn\ 1 1 3 4.1 2 3 4 5 . _L _3_ _7_ .n_. .1_ 3 yaj p "2", 1' 1 ' 5^ 5' 5' 5' 5 ' 10^ 10' 10' 10 ' 2 0' 2 0' 9 11 . _1_ ^3_ 13 2 4 20' "2 ' 2 5' 2 5' 2 5' 2 5* 1 3 5 7.1 _2 4 1 8 //A 1 2 3 . 1 5 . 1 3 5 7 . 1 _ _ VO 3' 3' 3 ' 6' G ' 8' 8' 8' 8 ' 9' 9' 9' 9' 9' 9 ' 11 ' 1_ . _1_ . _1- 12 ' 15 ' 16* 6. What is the ratio of 1 qt. to 1 gal.? AVliat per cent is 1 qt. of 1 gal.? What is the ratio of 1 ft. to 1 yd.? What per cent is 1 ft. of 1 yd.? 109 110 MENTAL ARITHMETIC 7. What per cent is 1 pt. of 1 qt.? 1 dime of 11? 1 da. of 1 wk.? 1 in. of 1 ft.? 1 ft. of 1 fathom? 1 sq. ft. of 1 sq. yd.? 1 sq. eh. of 1 A.? 1 oz. of 1 lb. Avoir.? 1 oz. of 1 lb. Troy? 1 qt. of 1 pk.? 1 sh. of £1? 8. What per cent is 2 in. of 1 ft.? 1 yd. of 1 rd.? A 3-ft. square of 1 sq. yd.? A 2-ft. square of a 3-ft. square? 8 oz. of 2^ lb. Avoir.? 30 miu. of 2 hr.? 4 units of 2 doz.? 1 dm. of 1 m.? 1 qr. of 1 rm.? 77 cu. in. of 1 gal.? 1 rd. of 1 ch.? 9. What is the ratio of $3 to $4? What per cent is ^3 of $4? What is the ratio of $200 to $ 600 ? What per cent is $ 200 of $ 600 ? 10. What per cent of $2 is 50^ > Of 864 is 124? Of 840 is 14^? Of 48^ is 6^? Of $36 is 149? Of 1225 is $45? Of $2500 is 1400? Of 15.28 is 81.32? Of 85 is 1 2.45? Of 83 is 81.25? Of 81 is 59^? 11. Memorize the following table : 20 % - i 831 % _ 1 50 %-| 22f % = 2 9 40 % = 1 66f % - 1 621 ^, = 1 16f % - 1 6' 60 fc = 1 121 % _ J 75 %-| 8i% = 1 12 80 % = -t 25 %={ 871 % = f 6|- % - 1 15 100 % = 1 37| % = 1 iH % - * 6i % - 1 16 What is meant by saying that 6| % = -^q ? In the following examples give the statements of balance : 12. A grocer bought tea at 75/ a lb., and sold it at a gain of 15 / a lb. What is the ratio of the gain to the cost ? What per cent of the cost did he gain ? — 2 ~ 9 _ 1 ■" 6^ _ _1 ~ 12 _ _1 "" 1.^ _ _1 ~ 16^ LESSON 37 111 13. A merchant bought flour at -^6 a hbl., aucl sold it for $7.50 a bbl. Find his gain per cent. 14. Bought cloth at 80 ^ a yd., and sold it for 70 ^ a yd. Find the loss per cent. 15. At the end of the season the following articles were advertised in the Chicago pai)ers as marked down in price. Find in each case the rate per cent of reduction : 1. Tan shoes from 15 to $3.75. 2. Leather belts from 15 ^ to 5 ^. 3. Couches from $ 15 to $ 10. 4. Screen doors from $ 1 to 29 ^. 5. Window screens from 25 1 to 10 ^. 6. Rockers from 12.50 to $1.70. 7. Handkerchiefs from 15^ to 9^. 8. Sheeting unbleached from Gi^ to 3)^. 9. Bicycles from $25 to $16.50. 10. Boys' bicycles from $ 20 to $ 14.50. 11. Bicycle shoes from $ 2.50 to $ 1.25. 12. Shoes from $3 to $1.85. 13. Veils from $5 to $2.25. 14. Dress skirts from $ 8 to $ 5.75. 15. Bookcases from $ 12.50 to $ 7. 16. Towels from 15^ to 12^^. 17. Towels from 50 ^ to 35 ^. 18. Towels from $ 75 Ao 50 ^. '!• 112 MENTAL AUITILMETIC \i i;!l! m 10. Curtains from .f 2 1 to 'tl I 20. 8kirts from 40Ao 25^. 21. Blankets from ^ 5 to ^ 3|. 22. Comforters from ^ 4 to ^ 2.98. 23. Comforters from I S^ to % 2|-. 24. Laces from ^ 1 J to 25 ^. 25. Jackets from $ 20 to .1)7 J. Lesson 38 What fractions in tlieir lowest terms are equiva- lent to the followhig per cents ? 1. 1%, 2%, 4%, 5%, 20%, 25%, 50%, G0%, 100%? 2. 75 %, 45 %, 3G %, 90 %, 18 %, 32 %, b6 % ? 3. 121- ^^^ 621 %^ 871 %, 371 %, 33-1- ^^, 06|- % ? 4. l(3f %, 831%, 81%, 111%, 142%, G|%, GJ%? 5. 9-i\ %, 44| %, 5-1 %, 88-1 %, 27| % ? 6. 100 %, 120 %, 125 %, 225 %, 350 %, 475 % ? 7. A lot which cost f 600 Avas sold at a gain of 40 % . The gain is equal to what part of the cost ? What was the gain ? The selling price ? 8. Cloth which cost 42^ a yd. was sold at a loss of 1C)|%. The loss was what fraction of the cost? What was the loss on each yard? The selling price ? Wm. loss LESSON 38 113 9. Silk which cost 8 3. GO a j^d. was sold at an ad- vance of 33 J ^lo . Find the gain and the selling price per yd. 10. A farmer who had 3G0 sheep sold 22 ^ % of them. How many did he sell ? How many did lie keep ? 11. The followinGf articles were advertised in the papers subject to the discounts given. Find the actual selling price : Mauked riiiCE . % 6.00 1. Tan shoes . 2. Lace curtains 3. Capes . . 4. Shirts . . 5. Towels . . 6. Towelling . 7. Parlor suits 8. Linen skirts $ 2.00 % 7.50 % 1.25 I 0.50 % 0.12| 842.50 % 4.00 Discount 25 % 371 % 331 % CO % 30 % 40 % 50 % 1. Silks . 2. Tables . 3. Chairs . 4. Couches 5. Umbrellas 6. Suits Discount 141 % G2^% 12. Find the selling price of the following articles subject to the discounts given : Marked trice % 1.75 % 4.50 I 0.75 112.00 % 1.98 815.00 lGf% 20 % Hi % 10 % I', 114 MENTAL A llFl IIMETIC \m 13. A farmer sold his crop of wheat for 1890 for 'tl80. The next year his wheat crop was | greater, and the price of wheat ^-31% more than in 18i)(3. Find what he got for the crop of 1897. 14. What is the selling price of an article costing *ii^lG and sold at a loss of 12J % ? At a gain of 15. What part of 1 % is \ % ? i%? \%-'- l%? 'h1o->- 16. What is 1% of #1800? A%? i%? \1o'f Ifol -I % ? -I%? 17. If ^% is charged for sending money from New York to Boston, what is charged for sending $800? 1600? $300? $500? 18. If 1^ % commission is charged for selling stock, what commission is charged for selling $1600 stock ? $2400? $848? $3264? 19. What is the cost of insuring 600 bbl. of flour worth $5 per bbl., the cost of insurance being | % of the value of the flour ? 20. A merchant paid 60^ a yd. for cloth and sold it at an advance of 33^%. After selling all but 2 yd., he put the remainder on the remnant counter and marked it to sell at 50 % below the first selling price. Find what it was marked at. 21. A grocer bought potatoes at 45 ^ a bushel and sold them at a gain of 33|%. If he received $48 for the potatoes, how many bushels did he sell ? I i' ' '-I LESSON 39 115 22. A fruit (lealor paid 810 for peticlies at 20^ a basket, part of wliicli being spinled lie sold the re- mainder at an advance of GO %, receiving '812. Iluw many baskets were damaged? 23. A lawyer collected 8320 and charged i'y^/o for his services. How much did he retain and liow much did he pay over to his employer? What per cent is the amount paid over of sum collected ? 24. A bookseller paid 80^ each for some books. The expense of selling them was 5% of the cost, and the gain 1G| % of the whole outlay. Find their sell- ing price. 25. A grocer mixes two kinds of tea, which cost him 41^ and 60^ respectively, in equal quantities. What must be the selling price of the mixture to gain 20%? Lesson 39 1. A house worth 12500 rents for 1 300. What is the ratio of the rent to the value of the house ? For what per cent of its value does it rent ? 2. A town of 2400 population increases to 2700 in one year. What is the ratio of the increase to the original population ? Find the rate per cent of increase. 3. A drover bought sheep at 85 each ; their food cost 81.25 apiece. He then sold them at 87.50 each. Find his gain per cent. 110 MKNTAL AiuriiMiync .itl! t1 4. A man invested ^GOO in real estate and sold out at a ,i>iiin of t)')}^%. He re-invested the entire amount and ag'ain sold out at an advance of 2;")%. Find tlie last selling i)rice and the total gain. 5. A trader sold a liorse at an advance of 22|%, gauiing sl(3. This 222 r^^ [^ 222 r/^ ,,i ^y]^.^^ y wi^.^t fraction of the cost ? What was the cost of the horse? 6. A grocer made •ii<12 by selling flour at a gain of 102 c/^^ wiiiit did it cost him ? 7. A mercliant lost -"ii^lS hy selling cloth at a loss of 37.1 %. Find wliat he paid for it. 8. I was offered ^t^SOO for a lot, but sold it for ff82-l:, gaining 4% more tlian if I had accepted the first offer. Find wliat tlie lot cost me. 9. A mercliant sold tea at G4 ^ a lb., gaining 33 J %. The "fain is what fraction of the cost? The sellin^• price is what fraction of the cost ? How much a pound did the tea cost the merchant ? 10. A man sold a cow for ^oij, gaining 33J%. Find the cost. 11. What would have been tlie cost price of the cow given in example 10 if there had been a gain of 20 7^? Of 50%? 121%? 80%? 9Jj%? 11^%? 12. A grocer sold coffee at 2Sj^ a lb., thereby losing 12.J %. The loss is what fraction of the cost ? The selling price is what fraction of the cost? What was the cost? :\m LESSON 39 117 sold 13. A mcroliant sold clotli at 48^ u yard, tliureby losing 33i % . What did it cost him ? 14. Wliat would havx> been the cost price of the ch)th in example 13 if tliere Iiad been a loss of •>:>% '> ()f20%y 14f%y lli^y 331^. 7^.^^. 40%? (;o%? ^'^^ ^ 15. Kirgs sold at 12/ a doz. net a profit of 20% ; lind the cost price per th)z. 16. 15y selling elothes at Jif'12 a suit a nuM-cliant gains 20 %. Wliat would have been the actual gain, and what the gain per cent, if they liad been sohl for 112.50 a suit? 17. A merchant sells overcoats at >jf'21, thereby losing 12|%. What was the cost, and what per cent would he gain by selling them at -^28 apiece? 18. A wholesale merchant sold goods to a cus- tomer at a profit of 25%, but tlie buyer becomes baidvrupt and pays only 60/ on the doUar. AVhat per cent does the wholesale merchant lose on the sale? 19. A mercliant bought goc^ls for -tOOO. He sold I of them at a gain of 15 % and the remainder at a loss of 24%. Find his gain or loss on the Avhole transaction. 20. A grocer sold a chest of tea at 09/ a lb., thereby gaining 15%. If his entire gain was ^^0.75, how many jjounds of tea did the chest hold? 118 MKNTAL AIMTIIMKTIC 21. I sold a span of horsi's jit >ii<105 ejicli, jT^jiining l^>ij% oil one and losing lOj % on the other. Find the eost price of each horse and the gain or loss on both transactions. 22. A specnlator sokl two lots at $720 each, los- ing on one 20% f^iid gaining 20% on the other. Find the gain or loss on the whole transaction. 23. IIow many yards of ch)th at i3 a yd. mnst a merchant buy, so that by sellhig it at 12] % profit he gains -it' 24? 24. Give answers to the following : (rt) 1S|% of 48 = ? (6) *480 is IHI % of what number? (c?) .8% of 20 = ? ((?) 20 is .8 of what number? (tj) ^ is what per cent of |? 25. A grocer buys flour at the rate of 15.40 a bbl. At what price per 49-lb. sack must he sell it so as to gahi 111 ^^ y Lesson 40 1. A wholesale merchant sold a bill of goods, list price $240, discounts IGf % and 10%. Find the net amount of the bill. 2. What is the net amount of a bill of goods, the list price of which is $300, trade discounts 20% and 12|- % ? What would have been the net amount with trade discounts 12| % and 20 % ? LESSON 40 111) 3. What .single (liscount would give the same net amount for a bill of Jii300 as discounts of 20 rL a,ul 12^%? ^ 4. Find the net cash amount of a hill for ^-iOi), subject to discounts of 25%, 1(;| %, uiul 4% oif for cash. 5. What is the difference between the net amount of a bill of goods for ^800, trade discounts 25% and 15 %, or a single discount of 40 % ? 6. A merchant buys at 40% below the list price, and sells at 20% below the list price. Find his gain per cent. 7. An auctioneer sold goods on a commission of 2 % . What did he receive on a sale of '^ 1400 ? What per cent of the sales does his employer receive ? 8. A commission merchant received 2J % for selling goods. If his connnission amounted^to -^^75, what was the selling price? 9. A broker sold $ 6000 stock on a commission of J%. Find his commission. 10. A commission merchant sold 600 bbl. of flour at 14.50 on a commission of 3 %. Find the commis- sion and the amount remitted to his employer. 11. My agent received $12 for selling apples at 12.50 a barrel on a commission of 4%,. How numy barrels did he sell ? I I ( 120 MENTAL ARITHMETIC 12. I paid my insurance premium of $50 with J of my commission at 6% for selling land. For how much did the land sell ? 13. A young man was paid 6 % commission by a mercliant for collecting overdue accounts. If he earned !^ 31.80, how much did he collect? 14. A man insured his house for $ 1875 at 2 % premium. IIow much did he pay? 15. I insured my furniture for three years for 1600 at 1J% premium. What was the amount of the premium ? 16. A mill owner insured his lumber for $ 800, paying $25 premium. What rate of insurance did he pay ? 17. An insurance company received $22.50 for insuring a house for 75 % of its value at | % pre- mium. Find its value. 18. A farmer paid $ 6.25 for insuring his barn and contents at | % . For what amount did he insure ? 19. A merchant insured his stock for $400 for one year at |^ % . Six months after the policy was can- celled at the request of the insured. Find the amount of premium returned, the short rate for six months being |%. 20. I paid $ 27 for insuring my house at | of its value at 1 1 % . What is the value of the house ? LESSON 41 121 how *nm '^Y "•'"' "'^' '"'"''"^ '" o"" «'^"'l"i"v for *600 aiKl in another iov $m. It was dunm.-:.! 1,. the extent of * 30. For what aniount was each con. pany liable ? 22. My property is assessed for $ 7200. Wlnt t'lx must I pay, the rate of taxation being 1 1 % ? ^11,400, the collectors commission being f) % y 24. The rate of taxation being reduced from 19 mills to 17 mills on the dollar, my taxes are lowered * ». h or how much am 1 assessed ? Lesson 41 Find the interest of ; 1. 'f' 600 for 1 yr. at 5 % ; 4%; G%; S%, 2. 1400 for 1 yr. at 7 %; 3 %; II %; 10 %, 3. 1800 for 1 yr. at 4 %; q %. (ji ^^. 31 ^^^ 4. 1450 for lyr. at 4%; 5%; 8%; 0%!' 5. 1 250 for 1 yr. at 3 % ; 3^ % ; 5]- % ; 7| %. 6. 1.375 for 1 yr. at 4 %; 3 %; 5 %; 8 %^ 7. 6 % per annum is what i)er cent for (3 mo '^ 4 mo.? 8 mo.? 3 mo.? mo.? 10 mo.? 8. 8 % per annum is what per cent for 3 mo '^ 61110.? 4 mo.? 4| mo.? 9 mo.? Find the interest of : 3. #500 for 6 mo. at 8 %; 3 mo.; 9 mo.; 4 mo. 10. % 700 for 6 mo. at G % ; 10 mo. ; 4 mo. ; 9 mo. I!! if >1 122 MENTAL AUITIIMETIC 11. $ 350 for 6 mo. at 4 % ; 9 mo. ; 3 mo. ; 4 mo. 12. What part of a year is 2 mo. 12 da.? 1 mo. 15 da.? 9 mo. 18 da.? 6 mo. 12 da.? Find the interest of : 13. $ 640 for 2 mo. 12 da. at 5 % ; 7 mo. 6 da. 14. $ 325 for 6 mo. 12 da. at 11 % ; 9 mo. 18 da. 15. $ 400 for 3 mo. 15 da. at 6 % ; 10 mo. 15 da. 16. In each of the above questions in which you find the interest, how would you find the amount? Find the amount for the first part of each of these questions. 17. A merchant whose capital was f 15,000 was making a profit of 12| % per annum on his capital, but on account of ill health he quit the business, and loaned his money at 6 %. How much is his yearly income diminished ? 18. A man leaves unpaid a sum of money on which he pays 8 % interest until the interest equals ■^Q of the principal. Find the time. 19. A person borrows a sum of money for 9 mo. at 6 %, and at the end of the time repays, as princi- pal and interest, 1418. How much did he borrow? 20. Feb. 15, 1898, a young man deposited in the savings bank $ 450. July 15, 1898, he withdrew the principal and interest at 4 % per annum. What amount did he withdraw? LESSON 42 123 21. I drew a note for ^1^ 550, due in 6 mo., and pre- sented it at a bank for disconnt at 8 %. What was the discount? What were the proceeds of the note? 22. What is the bank discount on a note for $800 due in 71- mo. at 6 %? How much cash woukl the owner of tliis note receive for it ? Does the bank make 6 %, or more or less than G %, on this note ? 23. A note due in 3 mo. was presented at a bank, and when discounted at 8 % yiekled $ -A90. liequired the face of the note. 24. A wholesale merchant took a note due in 4 mo. in payment of certain goods. He had it dis- counted immediately at 9%, and received -^15 582. For what sum were the goods sold ? 25. A merchant sold goods for ^ 540, subject to trade discounts of I'SJ % and 1G|- %, and took in pay- ment a note due in 4 mo., which lie liad immediately discounted at 8 %. Find the cash value of the sale. Lesson 42 1. If I buy rugs at $ 4.90 and sell them for $ 5.60 each, what is my gain per cent ? 2. A retail merchant bought carpet at 11.55 list price, trade discount 20 %, and sold it at an advance of 25 % on cost. Find tlie selling price. 3. If 331 % is gained by seUing cloth at 'it? 1.00 a yd., find the gam or loss per cent if sold at 80^ per yd. 124 MENTAL AKITIIMETIC 4. A triidesmairs prices iii-e 40 % above cost. If he allows a customer 10% on his bill, what profit does he make ? 5. A person at the age of 25 insured his life in each of two companies for 84000, the premiums being at the rate of 2^ % and 21 % respectively. Find his annual payment. 6. A trader insured 300 bbl. of flour for ^ of its value at 2%, paying 825 premium. What did the flour cost him a barrel ? 7. A man whose property is assessed for 812,000 is taxed 15 mills on the dollar. Find his tax. 8. A town levied a tax to build a town hall at a cost of 84900. If the collector's fee is 2 %, find the tax. 9. I bought goods for 8400, list price. Is it to my advantage to accept trade discounts of 25 and 5 or of 20 and 10 ? 10. Goods are sold at a loss of 20%. By what percentage of itself must the selling price be advanced to yield a profit of 20 % ? 11. A man began business with a certain capital ; he gained 20 % the first year, which he added to his capital, and B7| % the second year, Avhich he also added to his cai)ital. If his capital at close of the second year was 80G00, what was his original capital? .S'm LKSSON 42 125 12. _ A ma.) 1.c.o-a„ l.usi.iess Avitl, a certain capital • he gained 10% the lirst year, wind, he added to lu^ capital ami 20% the second year, whicli he also atlded to las capital ; the thir,l year he lost 0,V % He then had flOOO more investe.l in the hnsin,ts than at first. Find the capital witli which he com'- menced business. 13. A grocer bouglit 60 lb. of tea and 90 lb. of coffee, the cost of the latter per lb. being 3U% that of the former. He sold the tea at a profit of 2.5 % and the coffee at a profit of Vq%. gaining on L of eth *'" ^"^■"'^' '""' '^""'"= l'"'-''^^ 14. A carpet dealer sold carpet at 1)8? a yd., o-ain- ing ICf %. Afterward he sold some of 'the same carpet to the amount of $28.80 and gained 142% What was the last selling price and how many y/trds did lie sell at that price ? 15. A merchant sold 40 yd. of silk at a gain of <5<2 %, and 4.5 yd., which cost him the same per yard at a gain of Of %. If his entire gain was >fSH, fln.l the cost price per yard. 16. A mercliant bouglit goods for a certain sum, and jnarked | of tliem at a profit of 2,)%, and tlie remainder at a profit of 15 %. If his entii'e gain is * 76.38, find what lie paid for tJie goods. 17. A stationer buys a gross of lead i^encils for 12.50. He sells U at the rate of 2 for 5^ and the remainder at 3^ each. Find his gain per cent. ■^ 126 MENTAL AKITHMETIC 1^ 18. A grocer purchases syrup in 50 gal. barrels at $ 12.50 a bbl. At what price per gallon must he sell it to make 10 % ? 19. I bought a house for $ 3000 and spent 33|- % of the cost in repairs. What must I rent it for a month to make a clear income of G % of the total cost, taxes and repairs amounting to $ 00 a year ? 20. A man having lost 6 J % of his capital is worth twice as much as another who has just gained 50 % on his capital. What was the first man's capital ? 21. If a furniture dealer buys chairs at a discount of 30 % from list price, and sells them at 9% below list price, find his gain per cent. 22. A certain sum of money amounts to $ 1236 in 6 mo., and to #1251: in 9 mo. What is the rate per cent ? 23. Bought 2000 bu. of wheat at 82^ a bu., pay- able in 4 mo. ; I sold it immediately for 80 ^ a bu. cash, and put the money at interest at 6 % . At the end of 1 mo. I paid for the wheat. Did I gain or lose by the transaction, and how much ? 24. A wholesale merchant bought 400 yd. of silk at $ 2.50 a yd., and sold it at once, receiving- in pay- ment a note for $1100 due in 00 da., which he at once discounted at a bank at 0%. Find his gain. '!i),'r LESSON 43 127 MISCELLANEOUS EXAMPLES Lesson 43 1. How many strips of paper 2 ft. wide are needed to cover the walls of a room whose length is 18 ft. and whose breadth is |- its length ? 2. A gardener sold 7 bu. of potatoes at 35^ a bushel, and with the proceeds he bought 3 yd. of ribbon at 15^ a yard and sugar at 5^ ii lb. How many pounds of sugar did he buy ? 3. A township is cut into farms of 80 A. Find the number of farms. 4. Two men separate, one driving north at the rate of 7 mi. an hr. and then returning at the same rate. The second walks south at the rate of 3 mi. an hr. How far apart will they be in 3 hr.? 5. The distance of a suburb of Chicago from the city is indicated in any one of the following ways : 11 mi., or 24: min., or ^5 for a monthly ticket. Which unit gives the most practical measurement of the distance of the suburb from the city ? 6. Harriet puts 68^ into the Fenny Savings at one time, and has stamps of the largest denomination possible placed on her card as a receipt. If 1 ^, h )^, 10^, and 25^ stamps are used for the purpose, liuw many of each kind are used ? 7. Dorothy, who has 7 lOi^ stamps, 3 5^ stamps, and 9 1^ stamps on her penny savings card, has it cashed. How much does she realize ? 128 MENTAL AllITIIMETIC 8. The annual mcnibersliip fee for adults in the Itiversitle Art League is 50^, and for eliildren 10^. What will a family of two adults and three ehildren pay in dues in 3 yr. if all are memhers ? 9. What is the total cost of a revolver weighing IJ lb., for wliich >i('9.50 was paid, if sent through the mail in a registered parcel, the buyer paying postage and the cost of registration ? 10. I bought two crates of currants at different times and made 28 glasses of jelly out of each. The jelly from the first crate, for which I paid 90 ^, cost a cent a glass more than that made from the second crate. Find what the second crate cost me. 11. La Grange is 13 mi. west of Chicago, and Niles 00 mi. east. How far apart are two places, one of which is 7 mi. east of La Grange, and the olher 8 mi. west of Niles ? 12. The sum of the ages of three brothers is 51 yr. Tlie youngest is 15 yr. old and the next 2 yr. older. Find the difference between the ages of the oldest and the youngest. 13. A lady goes to the city by train at the cost of 'f 3.10 for a 25-ride ticket and a 5^ street-car fare each way, using 4 25-ride tickets in one year. If she had gone by electric car it would have cost her 10 ^ each way. IIow mucli would she have saved in one year if she had gone by the electric car ? LESSON 43 121) 14. If in the previous question it takes 30 min. to go by train and street car and 1 lir. by electric car, how- many hours are saved in one year by going by ti'uin? 15. The sides of three boxes, each 4 ft. k^ng, 3 ft. wide, and 1 ft. G in. high, are covered with wall paper cut from the same double roll. IIow nnich is left over if there is no waste ? 16. Alice has 8 oranges, Caryl (>, and Marguerite 1. The oranges are divided equally amongst them, and iVIarguerite pays Alice and Caryl IG^. How much should each receive ? 17. When is a number divisible by 2 ? liy 4 ? 8 ? 5? 10? 18. A lady buys stockings at the rate of 3 pairs for $1 instead of Shj^ a pair. How many })airs of stockings will she buy before she saves the price of a single pair ? 19. The average age of 10 boys is 8 yr. Five boys join the group whose average age is 3 yr. below the average of the others. What is the average of the entire group ? 20. A barn is 32 ft. wide and its two gables are 12 ft. above the eaves. IIow long are the rafters if they project 1 ft. G in. over the sides ? 21. Two men start from the same point and w^alk, one south at the rate of 3 mi. an hr., and the other west at the rate of 4 mi. an hr. IIow far will they be apart in 3 hr. ? 130 MENTAL ARITHMETIC h»m 22. What is the volume of a cube whose surface oontains 90 si^. in. ? What is the surface, of a cube 8 times as large ? 23. A man walks across the diagonal of a lot 8 rd. long and 6 rd. wide 4 times a day, instead of walking around the corner on the sidewalk. IIow much time does he save in a year of 300 da., if he walks at the rate of 3 mi. an hour ? 24. A man spent | of his money and then earned a sum equal to J of what he had left. He then had ^ 00 less than he had at first. How much did he spend ? 25. A lady spent -| of her money for a pound of tea, and with the remainder she bought a dozen lemons at 3^ each. What did the tea cost per lb. ? 26. A train ran 2 mi. 80 rd. at the rate of 30 mi. an hour. How far would it have gone in 1^ times as long at the rate of 48 mi. an hour ? 27. Sugar worth 1 1. 55 is weighed in a false balance which gives only 15|^ oz. to the pound. What is the selling price of the sugar ? 28. A man agreed to dig a cellar for $ 12, but after doing J of the work he found that, owing to the nature of the soil, he could only work on the remainder 1^ as fast as he expected. What should he have charged for the entire work ? 29. Divide $ 30 between two men, giving the first $ more than | of what the second gets. LESSON 43 131 30. Wliat is the weight of 1 cii. ft. of water? What is meant by saying that the specilic gravity of mica is 3? What is tlio weight of 1 ca. t't. of mica ill air ? In water ? 31. The specific gravity of lead ore is 7.4. How many pounds will 1 cii. ft. of lead ore weigli in water? 32. Magnetite and liematite are two iron ores of specific gravity G./3 and 5.3 respectively. I low mucli more would 1 cu. ft. of magnetite weigli than 1 cu. ft. of hematite ? How much more woidd it weicrh in water ? 33. A gallon of water Aveighs about 8 J lb. Into a pail full of water a piece of mica, specific gravity 3, is carefully put and 1 gal. of water overflows. Find the weight of the piece of mica. 34. At the rate of how many feet per second must air be forced into a schoolroom containing 40 persons, through an opening whose area is 2 sq. ft., in order that each one may have 30 cu. ft. of fresh air per minute? 35. A vessel can sail up stream at the rate of 8 mi. jind down stream at the rate of 12 mi. an hour. How far can it go down stream and up in 5 hr.? 36. The parallel sides of a field in the form of a trapezoid are respectively 60 rd. and 100 rd., and its depth 20 rd. What is the field worth at ti^48.50 an acre? 132 MKXTAL AlHTIIMETIC 37. A rectuii<^ular l)l()ck of Ijind coiitjiiiiiiig 24 A. is 80 rd. on oiiu sick'. How lon^* will it take a boy to ri(Ui ai'omul the block on his bicycle at the rate of 1 lui. ill 5 mill.? 38. What is the weiij^lit of the wall oapcr required tor the walls of a square room 15 ft. 1 )iig and ft. liii;h, allowinij^ 12 sq. yd. for doors and windows? ( A double roll ot wall paper is 10 yd. lon^, 18 in. wide, and weighs 1| lb.) 39. A box contains 150 lii,dits of glass, each in. by 8 in. How many lights of glass, each 12 in. by 80 in., are there in a second box that c(mtains the same quantity as the first? 40. Lubricating oil for bicycles is soh^ 8-oz. cans at '11^1.54 a dozen and in 4-oz. cans at 't i-.iO a dozen. I>y what per cent is it cheaper to buy a dozen 8-oz. cans of oil than to buy the same quantity in 4-oz. cans ? 41. The cost of a box of glass G in. by 8 in., con- taining 150 lights, is 't'2.50, and the cost of a box 12 in. by 24 in., containing 25 lights, is tji'3.25. By Avhat per cent is the price of the same quantity of glass increased in ordering the larger instead of the smaller panes ? 42. A man loans | of his money at 6 % and the remainder at 4i %. If his income from the former is >S)250 more than from the latter, what is his capi- tal? LESSON 43 133 43. Which is cheaper, goods hoiiglit for 8200 sub- ject to discounts of 25 % luid 20 % or to discounts of 20% and 25%? 44. I bought a bill of goods for f§?240, discounts 20 % and 10§%. Find my gain per cent on selling them for $ i;35, if I lose -i^ 15 in bad debts. 45. A grower sold | of his potatoes for 10 % less than all cost 'dm and the remainder at 28% below cost. Find liis gain per cent on the entire lot. 46. I sold a lot of land, receiving for it 8 800 cash and a note for 1640 due in 6 mo., which was imme- diately discounted at a bank at G % . Find the cash value of the selling price. It'li 134 MENTAL ARITHMETIC Multiplication Table n It I Tw CO Three times Four times Five times 8ix times Seven times I is 2 1 i S 3 1 i s 4 1 is 5 1 is 6 1 1 s 7 2 " 4 2 ' ' 6 2 ' ' 8 2 ' ' 10 2 " 12 2 ' 14 3 " 6 3 ' ' 9 3 ' ' 12 3 ' ' 15 3 " 18 3 ' 21 4 " 8 4 ' ' 12 4 ' ' 16 4 ' ' 20 4 " 24 4 ' 28 T) " 10 5 ' ' 15 5 « ' 20 5 ' ' 25 5 " 30 5 ' 35 6 " 12 6 ' ' 18 6 ' ' 24 6 ' ' 30 6 " 30 6 ' 42 7 " 14 7 ' ' 21 7 ' ' 28 7 ' ' 35 7 " 42 7 " 49 8 " 16 8 ' ' 24 8 ' ' 32 8 ' ' 40 8 " 48 8 ' 56 " 18 9 ' ' 27 9 ' ' 36 9 ' ' 45 9 " 64 9 ' 63 10 " 20 10 ' ' 30 10 ' ' 40 10 ' ' 50 10 " 60 10 •' 70 11 " 22 11 ' ' 33 11 ' ' 44 11 ' ' 55 11 " m 11 ' 77 12 " 24 12 ' ' 36 12 ' ' 48 12 ' ' 60 12 " 72 12 " 84 Eij,'ht times Nine times Ten times Eleven tiuies Twelve times 1 is 8 1 i s 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 " 44 4 " 48 5 " 40 5 ' ' 45 5 ' ' 50 5 " 55 6 " 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 U " 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 v\ 1 times is 7 u 14 " 21 " 28 " 35 " 42 " 49 " 5G " G3 " 70 " 77 " 84 e ;iiiies is 12 u 24 u 36 L( 48 U 60 I 72 t 84 I 96 TABLES UNITS OF VALUE United States Money 10 mills (m.) = 1 cent (ct. or f) 10 cents = 1 dime (d.) 10 dimes = 1 dollar (^) 10 dollars = 1 eagle (E.) British or Sterling Money 4 farthings (far.) = 1 penny (d.) 12 pence = 1 shilling (.s.) 20 shilUngs = 1 pound Ci) 5 shillings = 1 crown 21 shillings = 1 guhiea UNITS OF WEICxHT Avoirdupois Weight 135 16 ounces (oz.) 100 pounds 20 hundredweight 1 pound Avoirdupois 1 ounce Avoirdupois 1 pound (lb.) 1 hundredweight (cwt.) 1 ton (T.) 7000 grains 437 1 grains Troy Weight 24 grains (gr.) = 1 pennyweight (pwt.) 20 pennyweights = 1 ounce (oz.) 12 ounces = 1 pound (lb.) 1 pound Troy = 5700 grains 1 ounce Troy = 480 grains 136 MENTAL ARITHMETIC AroTiiECAiiiKs' Weight 20 grains (gr.) = 1 scruple (3) 3 scruples = 1 dram (3) 8 drams = 1 ounce ( 5 ) 12 ounces = 1 pound (tb.) 1 pound Apothecaries' weight = 5760 grains 1 ounce Apothecaries' weight = 480 grains UNITS OF LENGTH Long Measure 12 inches (in.) 3 feet 5i yards or 16^ feet 320 rods 1 mi. = 320 rd. 1 foot (ft.) 1 yard (yd.) 1 rod (rd.) 1 mile (mi.) 17G0 yd. = 5280 ft. Ich. Surveyors' Linear Measure 100 links (l.)=l chain (ch.) 80 chains = 1 mile (mi.) = 4 rd. = 22 yd. = 66 ft. = 792 in. UNITS OF SURFACE OR SQUARE MEASURE Surface 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 A township is 6 mi. square, and contains 36 sq. mi. It is divided into 36 sections, each 1 mi. square. 1 1 TABLES 137 UNITS OF VOLUIME Cubic or Volume ^Ikasuue 1728 cubic inches (cu. in.)= 1 cubic foot (cu. ft.) 27 cubic feet (cu. ft.) = 1 cubic yard (cu. yd.) A cord is a pile 8 ft. long, 4 ft. wide, and 4 ft. higli. It con- tains 128 cu. ft. A cubic yard of earth is called a load. •^ UNITS OF CAPACITY Liquid jNIeasuue 4 gills (gi.) = 1 pint (pt.) 2 pints = 1 quart (qt.) 4 quarts = 1 gallon (gal.) 1 gallon contains 231 cu. in. E Dry Measure 2 pints (pt.) = 1 quart ((jt.) 8 quarts = 1 peck (pk.) 4 pecks = 1 bushel (bu.) 1 bushel contains 2150.42 cubic inches Is It is Apothecaries' Fluid Measure GO minims (n^) = 1 fluid dram (f 3) 8 fluid drams = 1 fluid ounce (f 5 ) 16 fluid ounces = 1 pint (O) 8 pints = 1 gallon (cong.) 1 minim is about equal to 1 drop 138 MENTAL AllITllMETlC UNITS OF TIME Measure of Time 60 seconds (sec.) = 1 minute (min.) 60 minutes = 1 hour (hr.) 24 hours = 1 day (da.) 7 days = 1 week (wk.) 365 days = 1 common year (yr.) 366 days = 1 leap year (1. yr.) 100 years = 1 century (C.) February has 28 or 29 days; April, June, September, and November, 30 days each; the other months, 31 days each. Angular Measure 60 seconds (") = 1 minute (') 60 minutes = 1 degree (°) 360 degrees = 1 circumference (c.) MISCELLANEOUS UNITS Numbers 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 Wheat, beans, clover seed, and potatoes each weigh 60 lb. to the bushel ; corn, rye, and coarse salt, 56 lb. ; barley, 48 lb. ; and oats, 32 lb. Public School Arithmetic BASED ON McLellan and Dewey's ••PSYCHOLOGY OF NUMBER" BY J. A. McLELLAN, A.M., LL.D., AND A. F. AMES, A.B. iimo. Strong Buckram. Price, 60 cents, net. This book, based upon sound psychological principles, stands for a needed reform in the methods of teaching Arithmetic. SPECIAL FEATURES. The treatment of the subject is in strict line with the idea of num- ber as measurement. This true idea of number running through the whole work estab- lishes the unity of the whole. Fractions are divested of their traditional difficulty by being placed in their true relation to integers. Great care has been taken in selecting and grading the examples. This treatment of the subject will prove a good preparation for algebra. THE MACMILLAN COMPANY, 66 FIFTH AVENUE, NEW YORK. COMMENTS. " 1 can see that it is an important contribution to the art of teaching numbers." — W. T. Hakkis, U.S. Commissioner, Bureau 0/ Education. " From a crireful examination it seems to me to have many advantages over the books on the subject now in use. Its wise omission of topics of no practical use, the clearness of its methods and problems, and its neat typography appeal to every teacher who has occasion to deplore the bulky and involved arithmetics in so many of our schools." — George Giluekt, Principal Chester Academy, Chester, Pa. "I heartily approve of the method of this book." — W. B. Smith, McDonogh School, McDonogh, Md. " The processes are explained logically and the subjects are arranged in their proper order in the course. The examples given are si,,..i .^s require thought and at the same time are not such as can be considered unfairly puzzling, nor are they too simple, but are ail such as might be learned from some of the preceding parts of the book." Fkeoeuick Dodi.ittle, Acting School Visitor and Clerk 0/ Committee on Education, 0/ Connecticut. " 'Ihis volume is a very successful attempt to give expression to the better teach- ing of psychology as to the growth and development of the idea of number. The plan of the work is thoroughly scientific, the methods are well presented anti the ex- amples well chosen. This little book should assist greatly in the reform in teaching arithmetic, now in progress." — Prof. Alfred I. De Lurv, University of Toronto. " This bonk contains many admirable features. I like especially the early intro- duction of decimal tperations." — CvRUS DucER, A.M., Superintendent Schools, Lebanon, Pa. " Naturally I am pleased with the extent to which the book bases the treatment of fundamental operations of fractions and ratio upon the dea of measure and of numbers as units of measurement. I am particularly struck with the fact that the pupil's attention is definitely called to some special quantity or whole which furnishes the object of attention, and within which, so to speak, the numerical processes take place; also with the clearness and conciseness of the method of treatment; the logi- cal order of the selection of topics; and the exclusion of useless and irrelevant mat- ter. The simplification of treatment due to sticking close to fundamental principles, must recommend the book to teachers and pupils who have been bewildered by the great number of topics treated in the ordin.iry arithmetic — topics which do not differ at all in their logical or arithmetical basis, but are simply different practical expressions of the same principle. I wish the book the succe.ss it deserves." — Pro- fessor Dewey, University of Chicago, " ' The Psychology of Number,' by Professors McLellan and Dewey, placed on a rational basis the methods to be pursued in the elementary treatment of number. This has now been followed by ' The Public School Arithmetic,' by Professors McLellan and A. F. Ames, in which these rational methods, set forth in the former work, are systematically and successfully presented. The special feature of this book consists in its treatment of number as the result of measurement. The authors have brought out very clearly the proper methods of dealing with the fundamental operations, with fractions, and with the commercial rules. The definitions are con- sistent and accurate — a feature not common in elementary texts of arithmetic. There is an excellent collection of well-graded examples. Few of the tricky prob- lems which have done so much to discredit arithmetic are to be found. The book consequently deserves speedily to win a place among recognized text-books." — Professor McKay, McMaster University. "One of those wise books that make school study more pleasant and effective than in the * Id days when routine study was the rule. . . . Both teachers and pupils will welcome this very valuable book." — Saturday Evening Gazette, Boston. THE MACMILLAN COMPANY, 66 FIFTH AVENUE, NEW YORK.