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Tous les autres exemplaires originaux sont filmds en commengant par la premidre page qui comporte une empreinte d'impression ou d'illustration et en terminant par la dernidre page qui comporte uno telle empreinte. The last recorded frame on each microfiche shall contain the symbol — ^ (meaning "CON- TINUED "), or the symbol V (meaning "END"), whichever applies. Un des symboles suivants apparaitra sur la dernidre image de cheque microfiche, selon le cas: le symbole — ► signifie "A SUIVRE", le symbole V signifie "FIN". Maps, plates, charts, etc., may be filmed at different reduction ratios. Those too large to be entirely included in one exposure are filmed beginning in the upper left hand corner, left to right and top to bottom, as many frames as required. The following diagrams illustrate the method: Les cartes, planches, tableaux, etc., peuvent dtre filmds d des taux de reduction diffdrents. Lorsque le document est trop grand pour §tre reproduit en un seul clichd, il est filmd d partir de Tangle supdrieur gauche, de gauche d droite, et de haut en bas, en prenant le nombre d'images ndcessaire. Les diagrammes suivants illustrent la mdthode. 1 2 3 1 2 3 4 5 6 32X /, \ PROGRESSIVE SCHOOL SERIES. Academic Arithmetic, BEING PART I-OIR OF THE PROGRESSIVE SCHOOL SERIES OF ARITHMETICS. BY W. T. Kennedy, Principal of County Academy, Halifax, AND PETER O'HEARN, Principal of St. Patrick^; Boys' High School, Halifax. T. C. ALLEN & CO.. HALIFAX, NOVA SCOTIA. vl. Entered according to Act of the Parliament of Canada, in the j'ear 1898, by T. C. Allks &, Co., in the Department of Agriculture (Copyright Branch). PREFACE. h This part, No. IV, is intended, with the parts already published, to complete our Arithmetic. The parts I, II, and III, previously issued, cover the Common School Course up to and including- Grade VIII. Part IV has, besides new subjects, exercises oi a more difficult character than those of I, II, and III, and is designed for use in Academic and High Schools. The four parts are so g^raded and correlated that they make a complete, continuous text book on Arithmetic for Primary and Secondary Schools without overlapping- each other, an advantage to teachers, pupils and parents which will be apparent. The problems, we believe, will be found to be thorough- ly practical. Puzzling questions, long and difficult exercises and subjects that do not afford a mental training have been avoided. We think that a knowledge of principles can be more readilv acquired by working many comparatively easy exercises than by the use of those of a perplexing nature. It is hoped that the treatment of the subjects through- out this part, and especially those not usually found in school arithmetics, will commend itself to teachers. W. T. KENNEDY. P. OHEARN. CONTENTS. i!^ PAGE. Complex Fractions, --- 7-10 Continued Fractions, --------- 9 Recurring" Decimals, --_ 11-19 Cube Root, 20-26 Simple Scales, 29-33 Tables of Metric Weights .and Measures, ----- 33-34 Denominate Numbers in Irregular or Compound Scales, - 35-37 Tables of Weights and Measures, ----- -37-40 Values of Foreig-n Monetary Units in Canudian Currency, - 41 Table of Specific Gravities, -------42 Denominate Numbers, -------- 42-47 Longitude and Time, --------- 47-49 Thermometers, --------- 50-52 Percentag-es, - - 53-54 Taxes, Commission, Insurance, ------ 55-56 Custom House Business, -------- 57-58 Interest, 59-60 Partial Payments, - 61-62 Bank Discount, --------- 63 True Discount and Present Worth, ------ 63-65 Stocks, Bonds, Brokerage, - 66-70 Exchange, 71-77 Average of Payments and Accounts, ----- 78-83 Work Problems, 84-85 Clock Problems, - - 86-87 Ratio, 87-88 Proportion, 89-91 Series Arithmetical Progression and Geometrical Progression, 92-98 Compound Interest and Annuities, ----- 99-103 Problems in Square Root, --...--- 104 Measurement of Areas, etc., .--...- 105-110 Problems in Specific Gravity, ...... m Examination Papers, ........ 112-133 Answers, -•-.-..... 184-148 EXA Sim[ 1. 2. 3. 4. 6. 6. ; 7. i 8. 1 9. 10. 11. 12. 13. 14. COMPLEX FRACTIONS. Example 1. Simplity ^i^ - T).; ^ 4J x 1 ;\ - ^\ of (J - J*)- 8^^-52-11x1,^, -.\ of (J -;i) _ Q 1 11 — 1 U "~ 'J 1 I A of (;:-;;) i;i 11 1 1 '-'^ :; 4 — 4 1 _ n _ •' — 1 7 _ 5 _ 1 +7^)- 4 tt 17. 3-(l»- ' +?oflO;-Jof2i»5) 18. 87-(3|-Jj^). 19. (4-83)x(,Vof2;-^4;)x'J. 20. (i+.^)x(i+;)-<''-".)>< F> ^* 5 9 J 7 » 1 4 » li _1_ JL 1 7 0' 4 8 ♦ a • 3.V. 200Jr, g^ To express a Pure Recurring Decimal as a common fraction. Example 2. Express .6 as a common fraction. .6 = .666 . . . Ten times the decimal . =6.66 .. . Once the decimal =«= .66 . . . 9 times the decimal = 6 The decimal itself —^ = o a finite ^ = .225. ■ prime fac- ile decimal. te decimal number of rs 3 times .225, is 3. on, is also •tend must le common he number bits in the 3nly possi- of the re- equals 7. common on. RECURRING DECIMALS. 13 Example 3. Express .208 as a common fractloa .208 = .208208208 . 1000 times the decimal = 208.208208 . Once the decimal .208208 . 999 times the decimal = 208 The decimal itself _ 208 — 9 i>y To express a pure recurring decimal as a cofn?non frac- tion, write down the repetend for a numerator and as ynany nines as there are figures iti the repetend for a denominator. Any noug"hts occurring- in the repetend on the left of the first significant dig-it are of course omitted ; thus .00l8=5j§.j. EXERCISE 5. Reduce to common fractions in their lowest terms :— 1. .8, .45, .54. 2. 'r23G, .COsi, .b0088i. 3. 6.5445, 14.016, 100.059. 4. 1.0038, 18.01, 60.0003. . 5. 6.087, .0040, .0176. 6. 8.008, .100764, .999. To express a Mixed Recurring Decimal as a common fraction. • Example. 4. Express .654 as a common fraction. The decimal 1000 times the decimal 10 times the decimal 990 times the decimal The decimal itself .65454 . . . 054.54 . . . 6.54 . . . 648 rt 4 8 ^-. fl 5i- f 9 9(7 ^^ ~9 tfCT^ 14 RECURRING DECIMALS. Example 5. Express .4245 as a common fraction. The decimal =.424545 . . . 10,000 times the decimal =4245.45 . . . 100 times the decimal = 42.45 . . . 9900 times the decimal = 4203 The decimal itself — Al^il nr ±2 4 S -42 n of division can be easily per- formed without reducing' the recurring" decimal to a common fraction. CUBE ROOT. The Cube, or third power of a number, is the product of three factors, each of which is the number. The cube of 4 is 4 x 4 x 4, that is 64. The cube is denoted by a small 3 placed above the number and to the rig^ht, thus 4^ = 4x4x4 = 64. The number itself is the first power of the numbe •, thus 4 = 4'. Also 4- = 4 X 4 = 16, or second power of 4 ; 4* = 4x4x4x4 = 256, or fourth power of 4. 4 ^ = ? 3« = ? The Cube Root of a nuinber is one of the three equal factors whose product is that number. The cube root of 64 is 4 ; since 4x4x4 = 64. What is the cube root of 1 ? of 8 ? of 27 ? of 125 ? of 216 ? of 343 ? of 612 ? of 729 ? of 1000 ? J is called the radical or root sign. J 16 is read the square root of 16 ; ]^27 is read the cube root of 27. Read ^/ 64 ; J 243. A Perfect Cube is a number which has an integer for its cube root. Name some numbers that are perfect cubes. In extracting" the cube root, the number of digits in the root may be determined from the number of digits in the number whose cube root is to be extracted. This may be seen from the following examples : — 13 = 1. 103 = 1000. 1003 = 1,000,000. 1,0003 = 1,000,000,000. 93 = 729. 993 = 907,299. 9993=997,002,999. From these examples it may be seen : 1. That a root of one place may have from one to three places in the cube (1 to 729). CUBE ROUT. 21 2. Tliat the .'uklition of one place to the root adds three places to the cube (1,000 to 907/21)1)). 3. The cube root of a number of three dij^its or less is a number of one dii^it ( 1 to D) ; o( a number of not more than six places or not less than four is a number of two dig"its (10 to 90) ; of a number of not more than nine dij^-its or not less than seven is a number of three dij^its (100 to 999), etc. Tf wo point off a mimbor into periods of three fitriires each, be^ifiiiiiinj^ at the rij^-ht, the sum of the numhor of full pi'i lods and the one full or partial jieriod i>n the left will indicate the number of periods in the cuhe root. ^ Thus, 30 I 897 ] 845 has three fig^ures in its cube root. Also, 3')4 I 894 I 912 «' ♦« " '« «' «« To find the cube of G4 : 643 = (60 + 4)« = (GO + 4) (GO + 4) (60 + 4). By Multiplication : 60 + 4 GO 4- 4 60x4 + 4- 602+60x4 60^ + 2x60x4 + 42 60 + 4 602 X 4 + 2x60x42 +4^ 60^ + 2 x602x4 + 60x 4- 603 + (37602 X 4^ ^ (3 ^ 60 X 42)T43 = (64)3 That is, the cube of any number of two digits is equal to the cube of the tens, plus three times the square of the tens multiplied by the units, plus three times the tens multiplied by the square of the units, plus the cube of the units. Find by this method the cubes of 25, 34, 46 and other numbers, and verify by actual multiplication. J !(■ 22 CUBE ROOT. EXTRACTION OF THE CUBE ROOT, Example 12. Kind the cube root of 262144: 6 18 4 10800 786 11586 262 1 144 216 46144 46144 ^262144 = 64 1. Divide the number into two periods by a vertical Ijne separating- three figures on the right. 2. Take the nearest perfect cube not greater than 262, which is 216, and set it under the 2()2, and phice its cube root, which is 6, in a line with 262144. 3. Substract 216 from 262, and annex the second period, 144, to the remainder. 4. Place three times the first figure of the root, 18, to the extreme left, in a line with the remainder, and three time the square of the first figure of the root, lOH, with two noughts annexed, just on the left of the remainder and in a line with it. 5. Divide the remainder, 46144, by 10800 and set the quotient, 4, (the second figure of the root) midway between 18 and 10800. Read 18 4 as 184 ; multiply this by the 4 ; place the result, 786, under 10800 and add them ; this gives 11536 ; multiply this last by 4 and put the result 46144 under the first remainder. As it is equal to the remainder the operation is complete and the required root is 64. Note. — Sometimes when dividing the remainder we get a number (5) which when we carry on the operation, we find to be too large. We must then try a smaller number. By examining Example 12 it will be observed : 1. That 216 (216000) =60^ 2. That 10800 =(8x602) 8. That 736 =(3 x 60 + 4)4 = (3 x 60 x 4) + 42 CUBE ROOT. 23 let the ween he 4 ; this Iresult the root ^e get we iber. f 4. That 11536 (11 -fill) = (3x00-) -+- (3x00x4) + 4- 6. That 11536 X 4( = 46144) =(3 x60'-)4 : (3 x 60 x 4)4 + 4- X 4 = (3 X 60- X 4) 4- (3 x 60 x 4-) + 4'> 6. That 216000 + 46141 (-262144) = 60'* + (3 X 60- X 4) + (3 x 60 x 4 -) + 4'' EXERCISE 11. Find the cube roots of: — 1. 12167 2. 24889 3. 42875 15 4. 103H23 7. 132651 10. 704969 5. 97336 8. 3007(}3 11. H04357 6. 1331 9. 421875 12. 456538 PL E 13. Find the cube root of 182284263 : 5 182 284 263 125 6 7500 936^ 8436 \ 86J 67284 50616 7 940800 11809 6668263 952609 6668268 + 4= 168 f 182254263 = 507 1. Separate the number into three periods and set down the nearest perfect cube not j^^reater than 182, which is 125, and set down its cube root, 6, as before. Subtract as before and annex the next period, 284. 2. Set down 3 times 5 to the extreme left and 3 times the square of 5, with two noughts annexed, a little to the left of the remainder. 3. Divide 57284 by 7500 which gfives 7. As 7 will be found to be too larg-e try 6 for the second figure of the root, placing- it mid- way between the 16 and 7500. 4. Read 15 6 as 156 and multiply it by the 6, which gives 936 ; add this to the 7500 and multiply the result by the second figure (6) of the root. Subtract the product 50616, and annex the next period. 5. Set down three times 56, which is 168, and three times the square of 66 which is 9408. N.B. — This last result can be obtained by setting the square of 24 CUBE ROOT. ^J^. 6 (the last figure of the root obtained) under the last complete divisor and adding- the three numbers connected by the bracket. (The ex- planation of this will make an interesting- and not difficult exercise.) 6. Annex two noughts to the 9408, which g-ives 940800. Use this for a trial divisor, which g-ives 7. Read 168 7 as 1687 and multiply it by 7, which gives 11809. Add ; and multiply the result by 7 and place the product under last remainder. Example 14. Find the cube root of 130323843 : 15 150 7 7500 750000 10549 760549 130 I 323 I 843 125 5323843 6323843 In this case the first trial divisor, 7500, is so large that must be the next figure of the root since if any larger number be used, when the true divisor is found it will be too great. The next period, 843, is brought down. dO is multiplied by 3 and the result 150 is set down under the 15. Three times the square of 50 with two noughts an- nexed is also set down in a line with the 150 for a new trial divisor. The remainder of the process is the same as before, the answer being 507. Example 15. Find the cube root of 673373097125 : 8 24 261 6 2628 6 w 673 373 097 512 1 125 19200 1729] 161373 20929 146503 49) 2270700 156961 14870097 2286396 \ 13718376 36J 230212800 131425 1151721125 230344225 1151721125 i CUBE ROOT, EXERCISE 12. Find the cube root of : 1. 2048383 2. 16194277 3. 29503629 4. 67917312 5. 214921799 6. 224755712 7. 270840023 8. 5255:)7913 9. 746142643 10. 721734273 11. 955671625 12. T22615327232 13. 62712728317 14. 1076890625 < 15. 10460353203 CUBE ROOT OF DECIMALS. Since .1« = .001, .Ol^ = .000064, etc., it follows that the tht' ""k^ ^'""!f' ^^ "?' P^^"'^ '^ '^ ^^^''"^-I -f three pfaces £es e'tc ' '"""^ "' ^"" P^^^"^ ^^ ^ decimalTsix ^6'«r^, w;i^« ^/,e cube root of a decimal is to be extracted the decimal must be of the third, sixth or ninth place etc Example 16. Find the cube root of .02 to 3 places : .020 I 000 I 000 .008 12000 6 2 7 81 1200 469 ] 1669 [ 49 j 214700 81 1 215511 11683 Answer, .271 + . 317000 215511 101489 Since the root is to be extracted to three places 02 is made a decimal of nine places. This is separated Lo penods of three plac^es, each counting to the^i^4t f om num^e'rs!''"'^'"'^'' ""* '''' P'°'"'' '^ "^^ ^^"^^ ^^ ^" whole 26 CUBE ROOT. When a number is not a perfect cube its root can be extracted to any number of decimal places by adding periods, of three noughts each, to the remainders. The cube root of a common fraction may be extracted by ex- tracting the cube root of the numerator and denominator or by re- ducing- it to a decimal and then extracting- the root. Note particu- larly that when the cube root of any number that is a decimal, or that has a decimal as part of it, is to be extracted that the decimal places must be three or some multiple of three. When the decimal is a recurring one the repeating figures must be annexed instead of noughts. ' EXERCISE 13. Find to three places the cube roots of : — 1. 3 2. .3 3. .03 A 64 *K 343 S 1000 2744 6. 7. 8. 9. 10. 11. 2 9 128 446 900 .12345678 1.234567 12. 7 13. 14. 15. 49.296 .0001 00.8 A 16. 171.9 "V EXAMINATION PAPER. 36 X- e- u- 3r al al o( 27 EXAMINATION PAPER No. 1. ■ .. ^ ^- ^ "^a" had a lot of eg-g-s. He sold \ of them at 10 cents a dozen, i at 12 cents and 12 dozen at' 15 centT He hfs elgsT' "'' "'"' "^" ^P"^^^'^' ^^-^ ^'^ '-t-t " r 2. What is the square root of /^, correct to 4 places ? 3. Find the value of 4 ^-005. 4. J is 3i% of what number? 5. Multiply jTo 19s.ll|d. by 14^^. EXAMINATION PAPER No. 2. 1. Simplify _^L>< •J^^_.£lii__ 6-| of (5 -.ox 2)* 3. Find the square root of .01170724. lon^'^2 Tl^l '" '^H ^'^^§^ht of a piece of granite 9 ft. 4 in.\ long- 2 ft. 3 in wide, and 1 ft. 3 in. thick .P (Sp. P-r. of \ ., granite IS 2.72). ^ ^ S'- "» .., weJ^h; my^^ '" ^^^ 'P^?'^"^ .^'^'^^'>' °^ ^ substance that weighs 10% more in air than in water ? EXAMINATION PAPER No. 3. 1. What per cent, of a pound avoir, is a pound troy ? 2. What percent, of an ounce avoir, is an ounce troy ? 3. Simplify 6.03S + . 0875 -4.00376. 4. What is the value of ;^. 4o x . 39 ^ . 540 ? 1R ff^\ ^^f J? ^^f. '^^^^ «^ papering the walls of a room 16 ft. long, lo ft. 4 in. wide and 9 ft. 6 in. high, with paper 20 in. wide, 7 yds. in a roll, at 18 cents a roll ? EXAMINATION PAPER No. 4. 200 ft. long and 120 ft. wide, the plank being 'U in. thick, ana 3 cents a foot .-^ © - » 28 EXAMINATION PAPER. -f- 2. A school room is 82 ft. long- and 24 ft. wide. What is the distance from a corner near the floor to the corner farthest away near the ceiling-, ceiling- 10 ft. high ? 3. What will it cost to carpet a room 14 ft. by 12 ft. 4 in. with carpet 27 in. wide and $1.75 a yard ? 4. One side of a triangular field is 220 chains. A line drawn from the opposite angle at right angles to this side measures 200 chains. How many acres in the field ? 5. Simplify S^.0i5 x 4. EXAMINATION PAPER No. 5. 1. How many cubic feet in a globe whose diameter is 24 inches ? 2. Find the cube root of fj to four places. 3. A cistern is 20 ft. long, 10 ft. wide and 8 ft. deep. What is the area of the bottom of a cubical vessel that will hold as much ? • 4. Write the value of (.9)^ using^ as few figures as possible. • • • • 5. Multiply .549 by 729 without reducing .549 to a common fraction. Prove your answer by reducing .549 to a common fraction and multiplying. EXAMINATION PAPER No. 6. 1. By selling wheat at 5s. 6d. a bushel I gained 37|% ; what per cent, should I have gained if I had sold at 6s. 6d.? 2. Simplify (3.63 x .082) + (3.06 -2.719). 3. Find the value of .125 or an acre. 4. Bought 60 lbs. of tea at 20 cents a lb., 100 lbs. at 25 cents and 40 lbs. at 30 cents. At how much per pound must I sell the mixture to gain $13 ? 5. From 7 fur., 39 rods, 5 yds., 2 ft., 7 in. take 1 mi. SCALES. 29 SCALES. as a to I.? at Ind SIMPLE OR REGULAR SCALES. If a large number of pencils which are heaped promiscuously in a room are to he counted and prepared for shipment, it may be done by tying- them up, 10 in a bundle, and bindinic the bundles into p.ick- ag"es of 10 bundles in each packagfe, and phicini^ the packag-es, 10 in a box, and packing the boxes in bales of lO boxes each. If on tying the pencils into bundles of 10 each, we have 1329 bundles and 7 pencils over. We get ^W^^ or 132 packag-es and bundles over, And "Yq-, or 13 boxes and ^ packai^^es over, And \%f or 1 bale and 3 boxes over; Or 1 bale^ 3 boxes, 2 packages, 9 bundles, 7 pencils. Now, as 10 pencils make a bundle, 10 bundles 1 package, etc., and in the Common or Decimal system of Notation, 10 ones make 1 ten, 10 /^«5 make 1 hiitidred, etc., the 1 bale, 3 boxes, 'I packages, 9 bundles, 7 pencils, correspond to 1 ten-thousand, 3 thousands, 2 hundreds, 9 tens, 7 ones; or 13297 ones. But the pencils may be tied not only in systems ot 10, but in systems of 9, 8, 7, 6, 12 or any other number. Now, if they are tied in systems of 8, we have -''-i"-, or 1662 bundles and 1 pencil over. And 1662 bundles = ^8"-^, or 207 packagfes and 6 bundles over, And 207 packag^es = ^f^, or 25 boxes and 7 packages over, And 25 boxes — V , or 8 bales and 1 box over ; Or 3 bales, 1 box, 7 packages, 6 bundles, 1 pencil; that is 31761 in the eight or Octonary system. Ag-ain, if we take 13297 pencils as expressed in the decimal sys- tem, and collect them into a system of 7 each, we have ^^''p', or 1899 bundles and 4 pencils over. And 1899 bundles = ^^-2, or 271 packag-es and 2 bundles over, And 271 packag-es = ^^^, or 38 boxes and 5 packages over, 38 boxes = -y-, or 5 bales and 3 boxes over ; Or 5 bales, 3 boxes, 5 packag-es, 2 bundles, 4 pencils ; that is 53524 in the seven or Septenary system. This method is a universal one, and will be hereafter referred to as the Universal or Division rule. Ag-ain, reversingf the process: — 5 bales, 3 boxes —5x7 + 3, or 38 boxes, 38 boxes, 5 packages =38 x 7 + 5, or 271 packages, 271 packages, 2 bundles = 271 x 7 + 2, or 1899 bundles, 1899 bundles, 4 pencils = 1899 x 7 + 4, or 13297 pencils. HH 80 SCALES. Of thus, the two processes in the two usual arithmetical methods^ — the division and the multiplication methods respectively. 7 I 13297 penc ils. And : 5 ba. 3 bo. 5 pk. 2 bu. 4 pen. 7 I 1899 bundles + 4 pencils over. i_ 7 I 271packa^^ es + 2 bundles over. ^^ boxes. 7 I 38 boxes + 5 packag^es over. 5 bales + 3 boxes over. 271 packages. 7 Ans : 5 bales, 3 bo. 5 pk. 2 bu. 4 pen. 1899 bundles. 13297 pencils. Thus we see that in Notation there may be different Scales or ratios by which numbers increase or decrease. Of course the ordinarj' scale of notation for whole numbers is the decimal, but it is possible to express numbers in other scales. In any scale of notation the number of ones required to make one of the next higher order is called the Radix of the scale. When the radix does not vary, the scale is said to be RegfUlar or Simple but when the radix changes, as in the case of the table of sterling money, the scale is called Irregfular or Compound. In scales, other than the decimal, the radix is indicated by a small figure written below. Thus, 72468 means 7246 in the octonary scale. SOME REGULAR OR SIMPLE SCALES. Name. Radix. Name. Radix. Binary 2. Ternary 3. Quaternary 4. Quinary 5. Senary 6. Septenary 7. Octonary 8. Nonary 9. Denary or Decimal . . 10. Undenary 11. Duodenary or Duo- decimal 12. Vigesimal . . . Sexagesimal, 20. 60. SCALES. 81 It IS :o of )e y NOTATION.-REGULAR SCALE. When numbers are expressed in any uniform scale, it is necessary to employ as many characters as there are ones in the radix of the scale, and one of these must be 0. In expressing- numbers in scales hijjher than the decimal we use a single symbol such as t for 10, e for 11, etc. As the terms tens, hundreds^ etc. , are ahvays used in reference to the decimal scale, numbers in other scales must be read by naming" the number of units in each order. Thus 243 in the quinary scale is read thus : Quinary scale, 2 ones of the third order, 4 ones of second, and 3 ofthejirst. Example. — Write the numbers from 1 to 12 in the quaternary scale. Since 4 ones of any order are equal to 1 of the next hig^her, the characters we use are 1, 2, 3, 0, and the numbers from 1 to 12 would be set down 1, 2, 3, 10, 11, 12, 13, 20, 21, 22, 23, 30. In the common number 9 for instance there are 2 fours and 1 over. So 9 in the decimal scale is written 21 in the quaternary. Example. — Write 13297 of the decimal scale in the quinary. By dividing" by 5 we get 2659 ones of the second order and 2 of the first order remain- ing". The second time we divide by 5 we get 531 ones of the third order, and 4 ones of 5 I 106 and 1 over the second order over. The last time we 21and lover ^I'vide we get 4 ones of the sixth order and 1 of the fifth order remaining. 5 I 13297 5 I 2 659 and 2 ove r 5 I 531 and 4 over 5| 4and lover So the num- ber in the quinary scale is 411142. Example. — Chang-e 134052 from the senary scale to the decimal, and also 201210 from the ternary. Use the multiplication method, which is the more convenient when the reduction is to the decimal scale. 134052 6 senary. 201210 ternary. 3 9 6 6 8 58 19 6 3 348 59 6 a 2093 178 6 3 12560 decimal. 634 decimal. p 82 SCALES. Example. — Solve the same two problems by the division or universal method. 10 I 134052 senary scale to the decimal. 10"| 5452 and over 10 I 325lind 6 over 10 I 20 and 5 over 10 I 1 and 2 over and 1 over. Ans. 12560. 10 I 201210 ternary scale to the decimal. 10 I 1222 and~4j)ver liTl 12and^over and 5 over. Ans. 534. i i EXERCISE 14. 1. Express in the octonary scale 5706, 7134, 6544. 2. 8. 4. 5. 6. 7. 8. 9. 10. (( (( (( quinary scale 4321, 3042, 34012. nonary scale 7238, 77106, 88134. undenary scale 5137, 8369, 75903. duodecimal scale 13748, 78645, 70863. senary scale 25034, 111205, 34025li decimal scale 420316, I234567. " 71483o, 715428, lOlOlo. " 9t34„, 58S27o, 210102,. 37054^, 3819ei2, 268tei2. <( In arithmetical processes in any scale it is only necessary to re- member what the particular scale is. All numbers equal to, or g-reater than the radix, are divided by the radix, and the remainder set down. Numbers in different scales must be reduced to the same scale before any arithmetical operation is attempted. In multiplying or dividing numbers not in the decimal scale, time will probably be saved by reducing them first to the decimal scale, and then back again to the required scale after the operation has been performed. EXERCISE 15. 4253e, 12345^, 40231^, 52430;. 587278, 80062o, 5374^, 381449. 4322 6, 12340 6, 432 lo^, 33442 5. 58471^1, 89t44ii, 77692ii, 13729^. Multiply 77263 by 543,. 70843, by 6082a. 1. Add 2. (( 3. (( 4. (( 5. Mult 6. = [ = 1 meter-' =39.870H finches. = ldekameter'"'") =10 meters. = 1 nektometer*""" = 100 tt = 1 kilometer"^'"' =iooo „ = 1 myriameter '•^"»' = IOU(>'0 n MEASURES OF SURFACE. 100 square millimeters (i'""') - i ^^ 100 ^ „ centimeters - ^ "^ Centimeter *"■ ' 100 II decimeters 100 II meters 100 II dekameters 100 „ hektometers each LToHhTchT/l^ilfl^^^r'ned '^ "''"'"' ^ f"^*"" ^ »<1-™ each side 100". long is called f hekt'^r'! "' ' ^"^ ^ "'"""■'' ''^^'"^ 100 centars ''^») = 1 ar (»•> 100 ars = 1 hektar Ha.) decimeter *'*'^"" metep'i™- dekameter '*i^"" hektometer "'""" kilometer '•*'"'" MEASURES OF VOLUME. (CU. (.-IB.) 1000 cubic millimeferv: «="• mm.) 1 !_• 1000 „ centSs = J ="b.c centimeter 1000 /^''ynieiers = 1 „ decimeter""''"'! , " 'i^'^^eters = I „ metep'™-^' dekaster(i>ot.J, '^aectster' J, and ten cubic meters is called a 84 U SCALES. MEASURES OF CAPACITY. 10 milliliters <"'^' = 1 centiliter "=' > 10 centiliters = 1 deciliter '**' > 10 deciliters = 1 liter <" 10 liters = 1 dekaliter '»' > 10 dekaliters = 1 hektoliter <»'» 10 hektoliters =; 1 kiloliter "" ' = l"'" MEASURES OF WEIGHT. 10 7nilligrams •'"'•'' = 1 centiiTfufn ^'-'^ ' 10 centigrams 10 decigrams 10 grams 10 dekag^rams 10 hektograms 1 decigram "^^' 1 gram'^' =15.482848 grains. 1 dekag-ram "^'•'^ 1 hektog"ram '""' 1 kilogrram ""'■'^ 1 metric ton '"^^^ . 1000 kilograms EQUIVALENTS IN OTHER MEASURES. tneter = 39.87 + in., or 1.093G + yd. kilometer = .62137 + mi., or about § of a mile, square meter = 1.196 + sq. yd., or about lOJ sq. ft. ar hektar cubic meter liter hektoliter gram kilogram metric ton yard quart lb. 119.6 + sq. yd., or about 4 sq. rd. 2.471 + acres. 1.308 4- cu. yd., or about 35J cu. ft. .8804 + imperial quarts, or about If pints. 2.751+ bushels. 15.432 + grains. 2.2046 + lb., or about 21 lb. av. 2204.6 + lb. or about 1 ton, 200 lb. .9144™ + . 1 mile = 1.60935^^'". 1.186^ + . 1 bushel =.8636"'+. .4686^^. 1 ster = .2759 cord. The Associated Chambers of Commerce of the British Empire at their convention heid in London, June, 1896, at which thirty-two Canadian dele- gates were present, passed the following resolution: Whereas, The British system of weights and measures, which vary constantly in every part of the British Empire, is a source of constant annoy- ance, loss of time, and a formidable obstacle to local, imperial and foreign trade; and whereas the metric system has now been universally recognized as the most perfect decimal system, and it is generally adopted by nations of both continents, with the exception of the British Empire and United States of America, therefore, be it Resolved, That the metric system of weights and measures be adopted without further delay by the several governments of the British Empire, the yard being extended to the tneter, the quart to the liter, aiid the two pound weight to the kilogram. l SCALES. 85 DENOMINATE NUMBERS IN IRREGULAR OR COMPOUND SCALES. When the scale chaiiK-es with each order of ri.irures in the num- ber, tlie scale is called an Irregular or Compound Scale. Thus in the case of the pencils of which we had been speaking, instead i)f havuij,- the same number of pencils in each bundle as there are ot bundles m each packaj^-e, etc., we miKht use ilitVcrcnt numbers for each denomination. For instance, we miKht put 4 pencils in each bundle, 12 iHuulles in each package and 20 i>acka^;cs in each box. In that case the 13297 pencils would g-ive : ^3p% or 3324 bundles and 1 pencil over, And 3324 bundles = ■''J5S or 277 packages, 1Q k'^"'^ ^y^ packages Vr,", or 13 boxes, 17 packages over; that is Id boxes, 17 packages, bundles and 1 pencil. In the same way, if we take 13207 farthings (sterling money) and put them up in 4s or penny bundles, wi- liave '"i'''' or 3324 pence and 1 farthing- over, ' And 3324 pence- ^\l\ or 277 shillings, • .^d^'lFl^!^'']'"^'''"''-''^'' °'' ^'^ pounds and 17 shillings over; that IS ^13, 17s. Od. 1 tar. Or thus by the universal method: 4 I 13297 far. 12 |J{324^., 1 far. 20 I 277s. ^13 IT^Od. 1 far. And reversing the process by the multiplication method: £, S. D. 13 20 o77s. 12 17 3324d. 4 l.i297 far. Or by the Universal Method : £. S. D. 10 I 13 17 Oi 10 I 1 7 10 I u 8^ and 7 over. "9^ and 9 over. 10 1 3>^ and 2 over. 10 i)^ and 3 over. and 1 over. Ans. 13297 farthings. Again, if we take 850617 drams, the lowest denomination of 86 SCALES. common (avoirdupois) wcig-ht, and put them up in 16-dram or ounct packages, we get '*K8^^, or 63163 ounces and 9 drams over. And 631G3 ounces put up in 16-ounce or pound packag'es, g'ive *V«"» o** 3322 pounds and 11 ounces over, And 3322 pounds put up in 100-pound or hundredweight boxes, give YoV» o^ 33 hundredweight and 22 pounds over. And 33 hundredweight put up in 20-hundredweight or /on boxes, give II, or 1 ton and 13 hundredweight; that is, 850617 dr. = 1 ton, 13 cwt. 22 lb. 11 oz. 9 dr. Or thus, ascending- to higher denominations {universal method) ; 16 I 850617 d r. 16 I 63163 oz. 9 dr. 100 I 3322 lb . 11 oz. 20 I 33 cwt. 2 2 lb. Tt. 13 cwt, 22 lb. 11 oz. 9 dr. And thus, descendingtolowerdenominations{n\u\i\plicixi\on method): 1 t. 13 cwt. 22 1b. 11 oz. 9 dr. _20_ 33 cwt. _iqo 3322 lb. 16 19943 3322 53163 oz. 16 318987 53163 850617 dr. Or by the Universal Method : — ton. cwt. ib. oz. dr^ 10 I 1 13 22 11 9 10 3 32 4 6 arid 7 over. 10 33 3 10 and 1 over. 10 3 5 2 a' id 6 over. 10 5 5 and over. 8 and 5 over. Ans. 850617 drams. Here in the ordinary operation given above of reducing 850617 dr. to tons, etc., we proceeded as in the simple scale, beginning with the radix of the lowest order, 16 drams =1 ounce, and following with SCALES. 87 the other radices, 16 ox. = 1 lb. ; 100 lb. - 1 cwt. ; 20 cwt. - 1 ton. We thus showed that 860617 drams in the simple or re>fxilar decimal scale are equal to 1 ton, 13 cwt. 22 lb. 11 oz. U dr. in the ''Com- pound" Avoirdupois scale. Hut we have also shown, by the exercises done by the universal method, that the same division rule will change the notation from pounds to farthings, from tons to drams, or from one scale lo another. In this universal method of rr^/«<7/t»«, as will be seiMi from the examples, we divide the number in the given system by the radices of the required system^ in order, beginning unth the radix of the lowest order, and the remainders in similar order to ill be the number in the req u ircd syste m . BuL the more convenient method, when reducing from a com- pound to a simple scale, is the ordinary one of multiplication. In this, as has been observed in reducing 1 ton, 13 cwt., etc., to dr«ims, we began by multiplying the highest order in the given number by the radices, in order, beginning 7vit It the radix of the highest order, and adding in each lower order in its proper place. ' ii ' TABLES OF DENOMINATE NUMBERS IN THE IRREGULAR OR COMPOUND SCALES. MEASURES OF LENGTH. 12 inches (in.) = 8 feet 6jf yards = 40 rods = 8 furlongs = 1 mi. = 320 rd. = 1700 yd. = 5280 ft. 1 foot (ft.) 1 yard (yd.) 1 rod (rd.) 1 furlong" (fur.) 1 mile (mi.) = 63,360 in. A hand = 4 in. A span — \) in. A common cubit =1S in. A sacred cubit = 21.888 in. A fathom — i) ft. A inot, a nautical mile, or a geographic mile = 6086.1 ft., or ^V of n^js o^ the circumference of the earth. The standard yard is fixed by dividing a pendulum which vibrates seconds in a vacuum at 62° F. in the latitude of London and at the level of the sea, into 391,393 equal parts, and taking 360,000 of these parts for the yard. SURFACE MEASURES. 144 square inches (sq. in.) 9 " feet 301 << yards 160 ** rods 640 acres 1 square foot (sq. ft.) 1 " yard (sq. yd.) 1 *♦ rod (sq. rd.) 1 acre (A.) 1 square mile (sq. mi.' 1 sq. mi. =640 A. = 102,400 sq. rd. = 3,097,600 sq. yd. = 27,878,400 sq. ft. =4,014,489,600 sq. in. 88 SCALES. Land surveyors use a chain 4 rods or 66 ft. long, divided into 100 finks. A /m/& = 7.92 in. 10 sq. chains = 1 acre. 80 chains = 1 mile. An engineer's chain is 100 ft. long- and consists of 100 links. Shingling, roofing, etc., are commonly estimated by the square. Each side of the square is 10 ft. long and equal to 100 sq. ft. A square piece of land measuring 70 yds. on each side contains very nearly one acre. MEASURES OF VOLUME. 1728 cubic inches = 1 cubic foot (cu. ft.) 27 *' feet =1 '' yard (cu. yd.) A cord of wood or stone is a pile 8 ft. long, 4 ft. wide, and 4 ft. high. A perch of stone or masonry is 16^ ft. long, 1^ ft. thick, and 1 ft. high, and contains 24f cu. ft. A cubic yard of earth is consider- ed a load. MEASURES OF CAPACITY. 2 pints (pt.) = 1 quart (qt.) 4 quarts = 1 gallon (gal.) 2 gallons = 1 peck (pk.) 4 pecks = 1 bushel (bu.) The legal or "Imperial" bushel of Canada contains 8 imperial g-allons, and 1 imperial gallon contains 277.274 cubic inches, or 10 lb. of distilled water, temperature 62° F., the barometer standini^;- at 30 inches. Our Imperial bushel equals 2218.192 cu. in., and the Win- chester bushel used in the United States equals 2150.42 cu. in. The "Weights and Measures" Act of the Canadian Parliament of 1873 fixed the number of pounds to the bushel of each article named below as follows : — Oats 84 lb. Flax Seed. . .50 lb. Beans 60 lb. Barley 48 .• Corn 56 ., Peas 60 n Buckwheat . . 48 " Rye 56 «i CloverSeed 60 ti Timothy Seed 48 n Wheat 60 n Potatoes. . .60 ir MEASURES OF WEIGHT. AVOIR!)UP0IS WEIGHT. 16 arams (dr.) = 16 ounces = 100 pounds =- 20 hundred weight = A long ton 1 ounce (oz.) 1 pound (lb.) 1 hundred weight (cwt.) 1 ton (T.) 2240 lbs. 32000 oz. = 512000 dr.) (1 T. = 20 cwt. = 2000 lbs. = Everything except precious metals, jewels and medicines is weighed by avoirdupois weight . C^- ') . ' SCALES. 39 TROY OR JEWELLERS* WEIGHT. 24 grains (gr.) = 1 pennyvveig-ht (dwt.) 20 penny weig-hts = 1 ounce (oz.) 12 ounces = 1 pound (lb.) In weighing- diamonds 1 carat = 3 J. troy grains. APOTHECARIES* OR DRUGGISTS' WEIGHT. 20 grains (gr.) = 1 scruple (sc. or I)) 3 scruples = 1 dram (dr. or o) 8 drams =^ 1 ounce (oz. or 3) 12 ounces = 1 pound (lb.) The grain, ounce, and pound are identical in weight with the troy grain, ounce, and pound. 1 lb. avoirdupois = 7000 gr. 1 lb. | troy and \ .57^0 _ ° (apothecaries./ — •"""gr. I02. «« =437igr. ioz. " =480 gr. 144 lb. av. = 175 lbs. ap. or troy. Drugs are bought and sold by avoirdupois weight althoug-h com- pounded by apothecaries'. MEASURE OF TIME. 60 seconds (sec.) = 1 minute (min.) 60 minutes 24 hours 7 days 365 366 (i (( = 1 hour (hr.) = 1 day (da.) = 1 week (wk.) = 1 common year. = 1 leap ♦* Centennial years exactly divisible by 400, and other years ex- actly divisible by 4, are leap years. ^ «»«»«»- The civil day begins and ends at midnight. The earth revolves around the sun in 365 da., 5 hr., 48 mIn. CIRCULAR OR ANGO^^AR MEASURE. 60 seconds (") = 1 minute (') 60 minutes = 1 degree (°) 860 degrees = 1 circumference. «^/"f'* "*^^^° *^ '^'^"^^ * ''^;?--^^fl«^/tf, and an arc of 90° is called :li ' i '^1' iU SCALES. MISCELLANEOUS TABLE. 12 units =z 1 dozen. 12 dozen = 1 gross. 12 gross = 1 great gross. 20 units = 1 score. 196 lb. of flour = 1 barrel. 200 lb. pork or beef = 1 barrel. 100 lb. dried fish = 1 quintal, 24 sheets of paper = 1 quire. 20 quires = 1 ream. 2 reams = 1 bundle. 6 bundles = 1 bale. BOOKS. ' in 2 leaves is a folio. in 4 II a quarto or 4to. in 8 an octavo or 8vo. A book formed in 12 a 12mo. of sheets folded in 16 a 16mo. in 18 an 18mo. in 24 .. a 24mo. in 32 a 32mo. ENGLISH MONET. 4 farthings (far.) = 1 penny (d.) 12 pence = 1 shilling (s.) 20 shillings == 1 pound (£.) The Canadian gold coins are the British Sovereig^n, worth $4.86§, and the British half-sovereign. The Canadian silver coins are the 50-cent piece, the 26-cent piece, the 10-cent piece and the 6-cent piece. Our 1-cent piece, made of bronze, is one inch in diameter, and 100 cents weigh one pound. The gold coins of the United States are Eagle ($10), Double Eagle, Half Eagle, Quarter Eagle and Dollar. The British gold coin is ^^ pure metal and ^ alloy. The gold and silver coinage of the U. S. is ^^ pure. In Canada and Great Britain the silver coin is -JJ^ silver and ^ copper. Gold and silver alloyed as in the coinage is called standard. f f SCALES. 41 Standard gfold is ^J pure, or 22 carats fine. When pure g-old is alloyed with 6 parts of copper or other metal in 24 parts, it is then only 18 carats fine and is known as jewellers' g'old. Gold and silver before being- coined is called bullion^ and after being coined specie. The relations between Canadian money and the moneys of other countries vary slightly from time to time, but the following" table is substantially correct and can be used as a working basis in ordinary arithmetical operations : — Country. Arg-entine Rep. . Austria Belgium Bolivia Brazil Chili Cuba Denmark Ecuador Egypt France German Empire. Great Britain. . . Greece India Italy Monetary Unit. Peso Florin Franc I Boliviano Milreis Peto Peseta ,Crown Sucre Pound Franc* Mark Pound Sterling Drachma .... Rupee Lira Yen Japan . . Mexico Dollar Netherlands Florin Nor'yand Sweden Crown. Peru Sol .... Portugal Milreis Russia Rouble Spain Peseta. Switzerland Franc . , Turkey Piastre United States . . , . I Dollar. Venezuela. . j Bolivar rGold.. (Silver. Divisions of Units. 100 centavos- lOOkreutzers 100 c«,ntimes- 100 centavos: 1000 reis 100 centavos: 100 centimos 100 ore 100 centavos 100 piastres ■ 100 centimes loo pfennig - 20 shillings : 100 lepta 16 annas - lOOcentesimi 100 sens = 100 100 100 100 1000 100 100 100 40 100 100 centavos = cents ore centavos : reis copecks ; centimos centimes : paras cents centimos : peso. florin. franc. boliviano. milreis. peso. peseta. crown. Sucre. pound. franc. mark. pound. drachma rupee. lira. yen. dollar. florin. crown. sol. milreis. rouble. peseta. franc. piastre. dollar. bolivar. *T\vQ franc is a silver coin, 1 centimeter in diameter, and weighs 1 gram. 42 DENOMINATE NUMBERS. The specific gravity of some common substances is as follow : — Air Aluminum Brass (averag"e) . . . . Brick (common) . . . . " (pressed) Carbon, charcoal . . . g^raphite . . . diamond Coal, soft (average). " hard '* Copper Cork Deal (average) Ether . Glass, crown " ffint Gold Granite Ice Iron, cast wrought . .001292 .2.56 .7.611 .2. .2.4 .1.7 .2.3 3.6 .1.25 .1.5 , 8.9 ,. .29 ,. .66 .. .715 .2.5 .3.33 19.3 <( 7.i^ 7.77 Ivory , Lead Marble (average) Mercury Milk Nickel Oak Olive Oil Pine, dry Platinum Porcelain Quartz Salt Silver , Sulphur Sulphuric Acid. . . Tin Turpentine . Water, sea, Zinc , 1.91 ,11.3 , 2.73 13.6 , 1.032 . 8.8 . .84 , .915 . .4 ,21.5 , 2.38 , 2.65 . 2.13 10.5 , 2.5 , 1.841 . 7.3 . .869 , 1.026 , 7.1 DENOMINATE NUMBERS. EXERCISE 16. Reduce to higher denominations : 1. 69439 farthings. 6. 70301 pints. 2. 718096 grains Troy. 7. 6103876 seconds. 3. " " Apothecaries'. 8. 789247°^. 4. 8069644 drams. 9. 246078 cu. in. 5. 6876103 inches. 10. 197815 sq. in. Reduce to lowest denomination : 11. £45 166. lOJd. 16. 4.0371^^. 12. 24y.l27da.l8hr.44min. 17. 29 mi. 139 rd. 12 ft. 13. 4.0371''". 18. 201b. lOoz. 6dwt. 22gr. 14. 29 cwt. 87 lb. 15 oz. 19. 201b. lOoz. 2dr. 2scr. 6gr. 15. 18 bu. 8 pk. 7 qt. 20. 2 A. 68 sq. rd. 80 sq. yd. I DENOMINATE NUMBERS. 43 Multiply the sum of 18 mi. 15 rd, 4 yd. 2 ft., H7 mi. ^ rd., 1 mi. 500 yd. 10 in. by 39. EXERCISE 17. 1. From the sum of 3 T. 17 cwt. 29 lb., 18 cvvt. 4 lb. 9 oz., 78 lb. 17 oz. 9 dr., 3 T. 12 cwt. 51 lb. H oz. 11 dr.- take 4 T. 92 lb. 8 dr. 2. Ml 6 fur. 32 rd. 3. Divide the difference of 19 sq. mi. 4S0 A. 58 sq. rd. 6 sq. yd. 75 sq. in. and 5 sq. mi. 511 A. 27 sq. vd. 4 sq. ft. 88 sq. in. by 25. 4. Express in farthings £f^G lis. 3Jd.; and in pounds, etc., 23083 farthings. 5. Express in Troy weight 2 cwt. 45 lb. ; and also 1 ton 7 cwt. 14 lb. 6. Express in avoirdupois weight 1 lb. 4 oz. -S dwt. 3 gr.; and also 388 lb. 3 oz. 7 dwt. 12 gr. 7. When the sum of ;^241 19s. 72d. is divided into G7 equal parts what is the amount of each part ? 8. How many allotments of 83 sq. rd. 7 sq. ft. 81 sq. in. can be made from a piece of ground of 56 A. 7 sq. rd.^ and what will remain ? 9. Divide 17 T. 15 cwt. 53 lb. 2 oz. 3 dr. into 73 equal parts. 10. Multiply 8 mi. 3 fur. 11 rd. 1 yd. 8 in. by |. 11. In 65296108 sq. in. how many acres, etc..'' 12. How many minutes from 14.20 o'clock, June 24th, 1896, to 8.40 o'clock, January 3rd, 1901 ? 13. Two persons start at the same time from places 120 mi. apart, and trav^el towards each other. After one travels f and the other f of the distance, how far are they apart ? 14. The moon makes the circuit of the earth in 29 da. 12h. 44 min. 3 sec, and the earth revolves around the sun in 365 da. 5 h. 48 min. 49.7 sec. How many revolutions does the moon make while the earth makes one ? 15. The aggregate weight of 123 loads of hay is 57 T. 19 cwt. 42 lb. 14 oz. vVb^t is the average weight of a load ? ' i. I I. ii DENOMINATE NUMBERS. 16. How many '-ilver coins, each weighings 41 2|^ gr., can be coined from a bar of silver weighing 8 lb. 4 oz. av.? 17. When a steamer is going 20 knots an hour and a freight train 23 miles an hour, which is going the faster^ and how much ? 18. In water 43 fathoms deep, how many feet and inches will have to be added to a line 168J cubits long so that it may reach the bottom ? 19. Divide 12 lb. 11 oz. 7 dr. 19 gr. by 11, and 21 lb. 2 oz. 6 gr. by 17, and add the results. 20. Beginning on Friday, March 1st, a grocer sold during the month 1 T. 9 cwt. 18 lb. 8 oz. of butter. What was his average daily sale ? [For Explanation see Part I 11.^ Examples i-6.\ ... , ^ EXERCISE 18. ^ , 1. Express o cwtl 2 qr.' 14 lb. in tons. 9a 2. Express .778125 T. in lower denominations. 3. Reduce 13*> sq. rd. 167 sq. ft. 72 sq. in. to the decimal of an acre. 4. What is the value of .5555 of 1 lb. ap.? 5. What fraction of an ounce is x-Vg^ ^^ ■'■ cwt.? 6. What is the value of |- of | of £\ 16s 8Jd.? 7. What is the sum of .7 rd. +.625 yd. + .713 ft. + .91 in. 8. Express 2 da. 4 hr. as a decimal of 3 wk. 3 da. 9. What will 3 T. 6 cwt. 27 lb. of coal cost at $4.75 a ton ? 10. If a grocer's scales give only 15|- oz. for a pound, of how much money does he defraud his customers in the sale of 6 bbls. of sugar, each weighing 276 lb., at 5 cents a pound ? 1 1 . Express in inches the sum of 1 knot, 7 fathoms, 8 chains, 60 links, 12 cubits and 6 hands. 12. Find, in pounds av., the total weight of 2 bbls. of flour, 3 bbls. of beef, 27 bushels of wheat, 15 bushels of hi of .7£ i DENOMINATE NUMBERS. 45 ■J barley, 12 bushels of oats, 170 pounds of gold and 5 pounds of silver. 13. A stationer bought 6 reams of paper at $3.25 a ream, and sold | of it at 25 cents a quire and the remainder in sheets at the rate of 10 cents per half dozen. What did he gain ? 14. Add J of an acre, | of a sq. rod, f of a sq. yd., and -l^ of a sq. ft. 15. Add together .7956 of a week, 1.562 of a day, and .79 of an hour, and express the sum in days. \For Explanation see Part III., Examples y-i2.\ EXERCISE 19. 1. Express 178 yd. 2 ft. 5 in. in meters. 2. Express 391.008™ in yards, feet and inches. 3. In 273.4 yd. how many meters ? 4. Reduce 606.8262'" to yards. 5. In 55 mi. 5 fur. 15 rd. 5 yd. 2.28 in. how many kilo- meters ? 6. Express 831.978 quarts in liters. 7. Express 454.4^ in gallons. 8. Express 113.73^ in bushels, etc. 9. What is the cost of digging a cellar 8,4™ long, 6.5™ wide and 2.5™ deep, at $1.50 per cubic meter ? 10. A cask capable of holding a metric ton of water is filled with oil. What is the oil worth at 25 cents per gallon ? I 11. When a piece of matting 2 yards square costs $1|( iKhat is the cost per square meter ? 12. Express 25.748^^ in avoirdupois weight. 13. In 1 lb. 14 oz. 2.5 dr., how many grams ? 14. In 3 lb. 2 oz. 11 dwt. 14.4 gr., how many grams .^ 15. Reduce 15''' to tons, etc., avoirdupois weight. 16. In 18.5184 grains how many milligrams ? 17. What does a hektoliter of barley weigh ? I QOJ GC>"tt^O*>u^^^ * • '^■ ^k; 46 DENOMINATK MMBKRS. 18. What is «,^ained by buyin^^ 2000"* of doth at $1 per meter and selling- it at $1 per yard ? 19. Express 1051''' in Troy weight. 20. Find the width of a cistern 2.5"' long- and 1'" deep, which holds 80H1.4 quarts. Example 19. Express ^^18 12s. 6d. in dollars. ^18 12s. 6d. =;^18.625 Then since ^1 = $4.8C)2 .-. ;^18.625 = $1,862 X 18.625 = $90.64. Example 20. Change $453.21 to sterling" money. Since $4.86| = ;^1 :, $153.21 =^-^^i|^L^^=;^93 2s. 6d. Example 21. Express 2000 piastres in francs. 2000 piastres = $ .044 x 2000 = $88. And since $.193 = 1 franc 1 franc V. .-. $1 = $88 .1!»3 1 franc x 88 .193 EXERCISE 20. Express : 1. £27 6s. 8d. in dollars. 2. $852.60 in sterling- money. 3. $119 in German money. 4. 756.60 marks in dollars. 5. 837|- francs in drachmas. 6. 95 pfennig" in cents. 7. 8654.25 marks in dollars, 8. $7500 in Austrian money. 9. ;£486 13s. 4d. in marks. 455.96 francs. LONGITUDE AND TIME. 47 per •ep, V. 10. 8860 liras in Canadian money. 11. 3800 francs in United States money. 12. 3860 drachmas in dollars. 13. $386 in French, Belgian and Swiss money. 14. $13 10 in Danish, Swedish and Norweg^ian money. 15. 25 cents in pfennig" and in kreutzers. 16. $1 in rupees and annas. *< 17. £^ 10s. in roubles and copecks. 18. 1000 francs in sterling* money. 19. 1000 rupees in marks and pfennig-. 20. 50 cents in ore, and in reis (Portugal). :s. 4> LONGITUDE AND TIME. The earth makes a complete revolution on it^ axis every 24 hours, althoug^h to our senses it appears that the sun makes a circuit of the earth. The 360° thus apparently traversed by the sun in 24 hours is just 15" for each hour, 15' for each minute and 15" for each second. From this it is evident that when the difference in longitude of two places is known the difference in time can be calculated, and vice versa. Longihide is the number of degrees, etc. , which a place is east or west of a given meridian. In most countries long-itude is reckoned from the meridian which passes throug-h Greenwich, London. Places distant from each other 15° of longitude differ 1 hour in time. V II II 4 minutes in time. r II II 4 seconds Example 22. What is the difference in time between two places whose difference in longitude is 48° 25' 35" ? 15 I 48° 25' 35" Since the number of hours in a day ~ ; ' is iV the number of dej^-rees the sun 3 hr. 13 mm. 42J sec. travels in that time, therefore ^ of the difference of longitude expressed in deg^rees, minutes, and seconds is equal to the difference of time in hours, etc. 48 LONGITUDE AND TIME. Example 23. The difference in time between London and New York is 4 hrs. 55 mins. 87j; sees. What is their difference in longitude ? 4 hr r.«; 73' 00 min. 37':^ sec. Since there are 15 times as many o!e- 15 j^rces, minutes and seconds of long^itude ~ ■; ^ ,,; " as there are hours, minutes and seconds ^^ ^^ of time, we simply multiply the hours, etc., by 15 and write it down deg-rees, etc. \^ EXERCISE 21. 1. Halifax is G3° 35' 30" west longitude. When it is noon at Greenwich what is the true time in Halifax ? 2. Toronto is 79" 21' west longitude. What is the difference in the true time of Toronto and Halifax ? 3. St. John is 66° 3' 30" west longitude, Boston 71" 8' .0" and St. John's 52° 43'. What is the difference in true time between St. John and each of the other cities ? 4. What is the difference in time between Charlotte- town, which is 63° 7' west longitude and Chicago which is 87" 38' west. 5. The difference in the true time of two places is 1 hr. 22 min. 20 sec. What is the difference in longitude ? 6. Two persons observed a particular star to be hidden by the moon, one seeing it at 2H o'clock, and the other at 23|. What was the approximate difference in their longitude? 7. The standard time adopted in Nova Scotia is the true time of the 60th meridian. Is it faster or slower than Halifax true time, and how much ? 8. A gentleman whose watch was set to Halifax true time arrived in Montreal and was informed that his watch was 39 min. 2 sec. faster than Montreal local or true time. What is the longitude of Montreal ? 9. St. Petersburg is 80' 20' east longitude. When it is 10 o'clock at St. John (66° 3' 30" west), what is the time in St. Petersburg ? 10. If, on the occasion of a Dominion election, the polls open at 8 o'clock and close at 17, true time, how :: LONGITUDE AND TIME. 49 i lonj^il lan at Vancouver, B.C., much sooner \ 60° 12' 9' west 22' 24" west longitude ? 11. If a telegraph message is sent at noon without any loss of time from London to Washington 11" 1', at what time (true time) is it received ? 12. What is the difference in time between Cape of Good Hope (IN' 2LI' E.) and Quebec 71' UV 45' ? 13. A man travelling along the equator found when he stopped that his watch was 1 hr. Hr> min. slow. Did he travel east or west and how many miles ? 14. What is the difference in time of two places whose difference of longitude is 85^ 12' 15"? 15. Two places are 48° 24' 80", longitude, apart. What is the difference in their true time ? 16. The difference in the true time of two cities is 2 hr. 15' 27'. What number of degrees, etc., is one farther east than the other ? EXAMINATION PAPER No. 7. 1. What o'clock is it when the time from noon is ^\ of the time to midnight? 2. A man walks a certain distance and then rides back in 3 hrs., 25 mins. He could ride both ways in 2^\ hours. How long would it take him to walk both ways ? 3. Reduce the difference between ;^. 427088 and .2845 of £Q 17s. 6d. to the decimal of £5. 4. A river 5"" deep and 96"^ wide flows 3. 6*^^ per hour. What v/eight of water does it carry to the sea in 20 minutes ? 5. What is the value of a bar of aluminum 6™ long, 1dm ^ide and 5*^°^ thick at 65 cents r»er pound ? 60 THERMOMETERS. THERMOMETERS. There arc two thermometers in common use, — the Fahrenheit and the Centig-rade. The latter, only, is used for scientific purposes. In preparinjif his thermometer, about 1720, Fahrenheit, of Amsterdam, by a mixture of ice and common salt, pro- duced what he thought to be the greatest cold that could be produced by mechanical means and this he called zero. He divided the difference in temperature between this point and the point at which water boils into 212 equal part; degrees. By this scale he found that water freezes at ^<- above zero. In the Centigrade thermometer, first introduced by Celsius, of Sweden, about 1742, the freezing point of water is taken as 0", or zero, and the boiling poin*: as 100''. Thus, the difference between the freezing and boiling points of water as indicated by the Fahrenheit thermom- eter is the difference between 212° and 32% or 1H0% while the same difference as indicated by the Centigrade scale is 100°; hence 100° Centigrade are equal to 180° Fahrenheit, or 5" C. =9° F. Temperature below zero on either thermometer is iiiu. • cated by the minus sign. Thus 40* below zero is -- 40° C, or - 40° F. It is believed that 273 C. is the point which marks the absence of all heat. I SI Example 24. mometer ? 100' -82' = 68°. 68°x-i- = 37 70 i-. How is 100° F. indicated by a C. ther- Since in the F. thermometer the freezing" point of water is 32° above zero, we must subtract 32° from 100° in order to ascertain the number of Ans degrees 100° F. is above the freez- ing point of water. Then since 100" C.= I80° F. each of the 68' F. = ^C. Therefore, 68° F. = 68° x ^ or 37J° C. . THERMOMETERS. 51 ■ Example 2^. How would a temperature of 75* C. read on a I^'ahreiiheit thermometer? 7u°C. X » = 13r>' F 18r/ + a2°-l()7'"F. Since r C =r F., 75' C. = 136' F. Rut as this is 13.5' V. above the freezing point of ivater^ wt* must adil Wi io indi- cate the number ol degrees above zero. Example 20. What is - 50°C. on the Fahrenheit scale ? 56" C. ^- 100.8' F. Hut as the 50' C is below the freezing- pi>i(U nt" water, the 100. H" F is below the same point. And as zero or 0' F. is 32" he- low the freezing point of water, the ditlerenee between 100. H' aiul 'i'T is the distance below zero on the F. scale. SG** C. X 5 = 100.8° F. 100.8° -li2° = Gh.H^ F. As this is below zero, it is marked -08.8° V, Example 27. Whatsis 12^ F. on the C. thermometer? 32° F. -12° F. =20° F.= number o'i decrees below freezing* point of water. 20° F. X ^- = 11J° C. = number of deg^rees below freezing" point of water. Hence, 12° F. = - 11J° on a Centigrade thermometer. All exercises of converting readings of one scale into readings of the other can be done equally well, perhaps with greater facility, by referring to the boiling point of water instead of to zero. Thus : — Example 28. How is 100° F. indicated on a C. scale? As water boils at 212° F., 100* F. is 112° below the boiling point of water. 112° F. X A = 625° C. This is 62J° C. below the boil- ing point of water. As water boils at 100° C, G2^° below this point is 37J* C. Ans. Example 29. What is - 5G° C. on a Fahrenheit ther- mometer ? - 56° is 156° C. below the boiling point of water. 156* C. X 1 = 280.8° F. below the boiling point of water. , \ 52 THERMOMETERS. And as water boils at 212" F. above zero, 280.8° F. be- low that point must be 280.8' -212° or 68.8' F. below zero, or- 68.8° F. EXERCISE 22. 1. How would 65' C. read on a Fahrenheit ther- mometer ? 2. What temperature Centigrade does 86* F. repre- sent ? 3. The temperature of a room is 20' C. What would a Fahrenheit thermometer indicate ? 4. Alcohol boils at 173° F. What is this on a Centi- grade scale ? 5. Tin melts at 230' F. and lead at 334' F. At what degree Centigrade do they melt ? 6. How would temperatures of 40', 50°, 60°, 70°, 80', 90', 100° Centigrade be indicated by a Fahrenheit ther- mometer ? 7. Express 113°, 131°, 140°, 176°, 203° Fahrenheit in the Centigrade scale. 8. 75° C. is how many degrees hotter than 149° F.? 9. How does 25° C. read on a Fahrenheit thermo- meter ? 1 0. What is 40° F. on a centigrade thermometer ? 14. How do 46.4' F., 125.6° F., 145.4' F., and 195.8' F. read on a centigrade thermometer ? 12. Mercury freezes at - 39° C. At what degree F. does it freeze ? 13. Silver inelts at about 1000° F., copper at about 1090° F., gold at about 1250° F., and wrought iron at about 1650" F. How are their respective melting points indicated on the C. scale ? 14. On January 80th, 1898, in one town in Nova Scotia the mercury in a C. thermometer stood 25° below zero, and in another town the mercury in a F. thermometer stood 18' below. Which was the colder ? PERCENTAGES. EXAMINATION PAPER No. 8. 58 2. What does a fruit-;;eller Pain bv selUncr nf o 3. A train 88 yards lono- laL-o^ n i man walking by th'e side'°o"f thf acV^iru.t.Ite'':;^ ,„:n, 45.foot ladder placed between two bui!din,r« K: 1' tuMtnl/a^a"";^-- ^' ^ '^^'^'" °^ ^« f" How 5. Divide .777777777 by .63. and multiply the quotient by the product of .09 and 10. PERCENTAGES. (For Explanation see Part III., Examples 2,-25). EXERCISE 23. \^%o7l^%m\ """ ""^"■""^ ''''*"^" ^1% °f $*800 and then^«;^/"r.^'^''° '''''i '^""^ ^^'•'h $«00 sold 20°/ of it Oolong/ which'if 12". «7\";X °l'f^ *'',^'''= ^™ ^^ '" °f in the mixture ? '^^ ^' '^''°'^- ""^ "^"^ P°""ds 1 .b.t-p.To'rtC iTl^- ^v.^rr^Hi^y ^-^^'- -^° -"^ - 5. Vhat per cent, is made bv sellinp- rMf h o^ ^ u price per yard as it cost per meter ?^ ^^ ^^* "^'"^ 6. f is what per cent, of f .^ 7. I is what per cent, less than J ? 8. \ is what per cent, greater than | ? 5U PERCENTAGES. ( !!|l '^ hi' 9, A piece of cloth, on being- sponged, shrank 10% in length and 10% in breadth. What per cent, did it shrink ? 10. What per cent, of 30 is 20 ? What per cent, of a cube, each edge of which is 30 inches long, is a cube each edge of which is 20 inches long ? 1 1 . The female population of a town of 62000 is what per cent, of the whole, when the 5434: electors are 22% of the males ? 12. Two women buy pears at a cent each. One sells them at 4 for 6 cents, and the other at 3 for 4 cents. What per cent, does the second gain more than the first ? 13. If I sell |- of an article for what f of it cost me, what is my gain per cent.? 14. What per cent, of what he grinds doe^ a miller receive who takes as his toll 4 quarts out of each bushel ? And the flour which the miller keeps is what per cent, of the flour which the farmer receives ? 15. Which is better, and how much, on a purchase of $2340 worth of goods, a wholesale discount of 25% with a cash discount of 10%, or a single discount of 33^% ? 16. A druggist buys acid at 40 cents a gallon, and adds water so that when he sells it at 30 cents per gallon he gains 40%. What is the per cent, of water in each gallon ? \1» A sold a horse which cost him $200 to -5 at a gain of 10% ; B sold him to C at a loss of 10% ; C sold him to Z> at a gain of 10% ; andZ> sold him to ^ at a loss of 10%. If the horse is still worth $200 what did A gain ? 18. A gentleman bought two cottages and paid $1500 for one, and $2250 for the other. He sold each for the same sum, gaining the same per cent, on one as he lost on the other. What did he gain or lose on the transaction ? 19. A property was sold at 71% below cost. Had it been sold 10% higher it would have been at a gain of $56. What was the cost ? 20. A wholesale merchant sold to a retail dealer at a profit of 22|^%. The retail merchant failed and compro- mised with his creditors at 55 cents on the dollar. What per cent, did the wholesale dealer lose ? TAXES. COMMISSION. INSURANCE* 55, f^ TAXES. COMMISSION. INSURANCE. {For Explanation see Part III., Examples 26-jo). ■ EXERCISE 24. on property 1. What tax is paid by a man rated ; valued at $H65 1 ? 2. What is the amount, at $ . -^U per cent. , of a farmer's tax whose real estate is estimated at $4SG0, and personal property at $1175 ? ' 3. What per cent, of the area o.' Eng-land is the area of Nova Scotia, if England is 212 niHes square and Nova Scotia 147? 4. In 1897-8 property in Halifax was taxed, for all pur- poses, $1.63%. What were the taxes oi a man whose real estate was valued at $7450, and personal property at $1840? 5. At what rate is property assessed when an estate valued at $7860 pays a tax -^f $98.25 ? 6. What does an agent receive on sales aggregating $1275, on a commission of If % ? 7. An attorney collected 70% of a debt of $6400, and charged G^% commission. How much did the creditor lose ? . ' 8. A merchant sent his agent $1680, with instruc- tions to invest in goods, deducting his own cornmission of 1^/^. What is the amount of the commission ? 9. A fruit-growjr sent a commission merchant in St. John 400 bbl. of apples which were sold at an average price of $1.35 per barrel, and the proceeds, less the com- mission, Invested in goods. If the commission for selling is 6% and for buying ^J ,, what does the merchant make ? 10. Sent a commission merchant' $2472 to be invested in flour, his own commission being 3%. How many bbls. of flour, at $4.80 per bbl., can he purchase? 1 l^'ln the last question, after paying 8 J cents per bbl. freight, what must be charged per bbl. for the flour so as to gain 8J% ? 56 TAXES. COMMISSION. INSURANCE. ^ 1 2. What will it cost to insure a house worth $6000 for f of its value at a premium of 1|% ? 13. The premium on a building insured at f of its value at 1 2% is $105. What is the value of the building ? 14. A factory valued at $16000, and its machinery, which cost $8400, are both insured. The building is in- sured at 4 of its value at 1J%, and the machinery at f of its value at 2|% What is the annual premium paid ? 15. What sum of money must I send my agent so that after deducting his commission of 2^% he ma}'^ send me goods worth $4290 ? 16. For what sum must a building worth $4290 be insured at 2J% premium so as to cover both the value of the building and the cost of insurance ? 17. A building which cost $2000 was fraudulently in- sured at 8%, so that when it was burned the owner gained $425. What was the amount of the policy ? 18. For what sum must a schooner which was sold at auction for $5214 be insured at 2% so that in case it is lost the purchaser will lose exactly $1000 ? 19. A man whose property is assessed -|% pays $20.40. If his real estate is worth $2500, what is the value of his personal property? 20. What per cent, profit is made by a merchant who buys at 20% and TtV (cash discount) from list prices, and sells at 10% and 3% from list prices ? \ CUSTOMS HOUSE BUSINESS. CUSTOMS HOUSE BUSINESS. «7 The revenue necessary to defray the expenses of the Dominion Government is largely derived from taxes im- posed upon goods imported from foreign countries. These taxes are known as Customs Duties. Taxes on articles manufactured within the country are called Excise TaxeS. Duties are collected at seaport and other towns called POPtS of Entry. These Ports of Entry are furnished with customs houses where government officers collect theduties. Duties are either Specific or Ad Valorem. Specific duties are assessed on imported goods with reference to their quantity or weight and not to their value. Ad valorem duties are percentage taxes on the cost of goods in the country from which they are brought. Im- porters are required to submit to the customs officers an invoice of the goods, showing the cost of each article in the country where it was purchased. EXERCISE 25. 1. What is the duty on a dozen books, weighing 18 lb., at 6 cents per Ib..^ 2. What is the duty, at 18%, on 12G50 lb. of cordage, invoiced at 15 cents per lb.? 3. What is the duty on 12 dozen watches at $124 per doz. at 25%, and 20 jewel cases at $1.85 each at 80% ? 4. What is the rate of duty when $94.50 is paid on goods invoiced at $1260.? 5. What is the duty on 75 kegs of prunes of an average weight of 104 lb., at 1| cents per lb.; allowing a reduction of 7% for tare, that is for the weight of kegs, etc.? 6. What duty at 30% ad valorem must be paid at St. John on silk imported from England and invoiced at ;^872 2s. 6d.? 7. A merchant in Halifax made an importation of goods from the United States invoiced at $9865. On r. ' 58 CUSTOMS HOUSE BUSINESS. ■'-1 i:. goods invoiced at $2150 the duty was 17J% ; on goods amounting to $3720 the duty was 22^% ; goods costing $2612 were fre*^ of duty ; and on the balance the duty was What was the whole amount of the duties ? o' 8. When the duty on cut tobacco is 45 cents per lb. and i2J% advaloreniy what must be paid at the Charlotte- town customs house on 60 casks, each weighing 112 lb., tare 6%, which cost in Virginia 15 cents per lb.? 9. At 80% ad valorem what is the duty — Canadian currency — en an importation of 100 doz. kid gloves invoiced at 75 francs per doz.? 10, What is the duty at 6 cents per lb. and 25% ad valorem on woollens from England weighing 840 lb., tare 8%, and invoiced at £im 17s. 6d.? EXAMINATION PAPER No. 9. 1. What per cent, of the letters are vowels in the sentence, / ought, therefore I can ? 2* A gentleman whose house was destroyed by fire received from the underwriters $2945, which was ^ of the amount of the policy. The insurance was f of the value of the house and the premium paid was $66.24. What was the loss ? 3. What direct discount is equal to a trade discount of 174% and 8% ? 4. A number is increased by 4% of itself, and the number obtained is increased by 18% of itself. If the last number is 767, what was the first ? 5. The taxable property in a school section is $88600, and there are 18 persons who pay a poll tax of $2 each. When it is necessary to raise $640 for school purposes, what must a man pay whose real estate is valued at $920 and his personal property at $276 ? : ■ INTEREST. 69 INTEREST. EXERCISE 26. {For Explanations see Part III., Examples Ji-j6). Find the interest of : 1. $650 for 3 years at 6%. 2. $875.60 for 2^ years at 7%. 3. $892.85 for '^l years at 51%. 4. $1250 for 1 vear 9 mos. at 61°/. 5. $342.50 from June 3, 1892, to Dec. 24, 1H94, at 6%. 6. $88.70 from Jan. 10, 1896, to Nov. 30, 1896, at 5%. 7. $2968.80 for 2 years 155 days at 4 /o* 8. $1234.50 from Oct. 3, 1897, to March 15, 1890. at 5%. Find the compound interest of : 9. $350 for 3 years at 6%. 10. $426.75 for 4 years at 5%. 11. $2445.(>0for3yearsat6%. 1 2. $750 for 11. years at 6% interest payable half-yearly. 13. $450 for 2 years at 5%, interest payable half-yearly. 14. $200 for 1 year 9 mos. at 4%, interest payable quarterly. '' Find the amount of : 15. $194.60 for 5 years at 41%. 16. $349 for 2 years at 0% compound interest. 17. $4000 from Feb. 4, 1890, to Nov. 2, LS93, at G%. 18. $420 from May 16, 1890, to Jan. 7, 1898, at 6/, compound interest. n '' 19. $2000 from Sept. 18, 1893, to Feb. 11, 1S96, at 5 , compound interest. '^ 20. $500 from May 1, 1898, to July 15, 1899, interest compounded quarterly at 8%. Find the rate per cent, when : 21. $387 amounts to $468.27 in 3 years. 22. The interest of $187.60 for 81 years is $24.0a 60 INTEREST. 23. The interest of $3500 for 11 months is $152.40. In what time will : 24. $825 produce $67.81 interest, at 6% ? 25. $706.60 produce $80.87 interest, at 6J% ? 26. $875.45 amount to $959.06, at 6% ? What principal will give : 27. $1618.75 interest in SJ- years at 5% ? i^ 28. An amount of $1052.84 in 2 years 7 mos., at 6J%? 29. $28.10 interest, at 5^, from April 1, 1894, to Jan. 18, 1895 ? 30. An amount of $538.68 from Sept. 11, 1895, to Dec. 25, 1896 at 6; EXAMINATION PAPER No. 10. 1. A steamer from London to St. John traversed 10|- degrees daily. What was the length of time from noon one day to noon the next? 2. An agent who collected a debt on a commission of lf%, sent his principal $1247.12. What was the amount of the debt ? 3. From a lot of land 30 rods square were sold 300 square rods. What is the value of 58J% of the remainder at $46 per acre ? 4. A and B have respectively 7% more and 5% less money than C, and the three together have $1751.60. What sum has C ? 5. If a gallon of paint costing $1.50 will paint 150 square feet of board fence, what will the paint cost for a fence 6 ft. high, enclosing in four-sided form of least perimeter, a field containing 100 square rods ? INTEREST. 61 ? 1. 2 a f t PARTIAL PAYMENTS. Example 80. $4000. Toronto, June 1, 1892. Two years after date I promise to pay William Smith, or order, four thousand dollars, for value received, with interest at 7 per cent. Richard Powell. On this note were the following- endorsements : Sept. 15, 1892, $450; Dec. 15, 1892, $50; March 1, 1893, $500; Jan. 1, 1894, $1000. Whatwas payable when note became dueon June 4, '94? Interest on $4000 from June 1, '92, to Sept. 15, '92,— 106 days = $81.31 Interest on $3550 from Sept. 15, '92, to Dec. 15, '92,-91 days= 61.95 Interest on $8500 from Dec. 15, '92, to Mar. 1, '93,-76 days- 51.01 Interest on $8000 from Mar. 1, '98, to Jan. 1, '94,— 306 days = 176.06 Interest on $2000 from Jan. 1, '94, to June 4, '94,— 154 days = 59.07 Amount of interest - - - - $429.40 Balance due = $2000 + $429.40 = $2429.40. ^^==--= Note. — During- the first 106 days Richard Powell owes $4000 and pays interest accordingly ; during- the next 91 days he owes and pays interest on $3550, and so on to June 4, '94, he continues to pay interest on the amount of his indebtedness. The whole interest with the balance of principal unpaid amounts to $2429.40. As the note anticipates and the law allows only simple interest, it is obviously unjust to compound the interest every time a payment is made. Such a method would operate ag-ainst the creditor, in this example, to the extent of $26.10. The method may be stated as follows : Find the interest of the principal to the time of the first payment; the interest of the balance to the time of the next payment, and so on to the time of settlement. The balance of the principal unpaid, together with the several amounts of interest computed, is the amount of the debt. 62 INTEREST. i This method is known as Merchants' Rule. It is used by all bankers and is lej^al in Canada. It is sometimes expressed and ap- plied as follows : Find the amount of the principal for the entire time ; find the amount of each paytnent from the time that it was made to the time of settlement ; and from the first amount subtract the sum of the amounts of the several payments. EXERCISE 26a. $400. St. John, Jan. 1, 1891. 1. One year afterdate, for value received, I promise to pay Donald Elsdon, or order, four hundred dollars with, interest at 7%. Jackson Grant. Indorsements :— March 16, 1891, $200 ; July 1, 1891, $100. What was due at maturity ? 2. F'ind the balance due April 15, 1896, on a note of $2o0.60, given July 7, 1895, interest at 7%, on which the following payments have been made : — Sept. 20, 1895, $80 ; Jan. 1, 1890, $50; March 13, 1896, $50. 3. A note of $500 was given Jan. 1, 1896. Indorsements :— Jan. 20, '90, $100 ; Feb. 10, '96, $50; Feb. 25, '96, $100; March 1, 1890, $160. What was due April 4, 1896, interest 6%? 4. A note of $900, dated Sept. 1, 1895, with the fol- lowing indorsements :— Oct. 18, '95, $150; Dec. 22, '95, $200; March 15, '96, $300. Interest being at 7%, what was due on the note July 19, 1896 ? 5. The sum of $725.25 Nvas loaned on March 15, 1896, at 6%. Payments were rnade as follows : — April 3, $170 ; May 20, $245.30 ; June 17, $87.50. How much was due Sept. 5, 1896 ? " ^WV DISCOUNT. BANK DISCOUNT. [For Explanations see Pari III., Examples J/'-jS). EXERCISE 27. Find the bank discount and proceeds oi^ tlie following Face. 1. $850 2. $750 3. $500 4. $75 6. $1260 6. $5430 7. $5430 8. $485 9. $485 10. $485 Datk. Feb. 5, '91. Sept. 4, '90. Mar. 17, '95. July 8, '90. May 14, '97. Feb. 10, '95. Feb. 10, '95. Nov. 28, '97. Nov. 28, '97. Nov. 30, '97. TlMH. ,^ ^^'"'-^ DlSCOl N'TKn. 2 mo. Feb. 23, '91. 3 mo. Oct. 2, '96. 90 da. May 22, '95. 90 da. Aug. 4, '90. 00 da. June 25, '90. 3 mo. April 9, '95. 90 da. April 9, '95. 90 da. Jan. 3, '98. 3 mo. Jan. 3, '98. 3 mo. Jan. 3, '98. Ratf 4 o / / J o/ /o 5 % «■!% % O ' /o o/ /o o/ /o o/ /o % 7 5 5 5 TRUE DISCOUNT AND PRESENT WORTH. $100. Truro, June 10, '97. Twelve months after date, for value received, I pro- mise to pay James Smith, or order, the sum of one hundred six dollars, without interest. John Brown. What will this note be worth to James Smith on June 10, '98? Just $100, for on that day John Brown will pay him that amount. But what is the value of the note on June 10, 97, — the day on which it was given? It is worth such a sum as, being put at interest, will amount to $106 on June 10, '98. If money earns 6%, what sum will amount to $100 in 1 year? $100. What then is the pre- sent Worth of $106 due a year hence when money is worth 6% ? $100. And what is the true discount, or the sum which would be deducted from such a debt, if it were paid twelve months before it was due ? $6. 64 DISCOUNT. !i It is evident then that the Present Worth of any sum of money due at some future time is the sum which, put at Miterest for the g^iven time and rate, will amount to the ^iven debt. And True Discount is the difference between a debt and its present worth ; or the interest on the present worth for the j^iven time. ExAMPLK 31. Find the Present Worth and True Dis- count of $912.57, due 2 years hence, interest 4%. Amt. of $100 for 2 years at 1% = $108 Hence the Present Worth of $108 = 8100 $1 =«'"" And < ( (( (( It < i <( $942.57 = lOH $100x9 42.57 lOS $S72.75 True Discount = $942.57 - $872.75 = $G9.82. Example 32. A debt of $6G5.60 contracted on Feb. 11, to be paid in 6 months, without interest, was paid on March 1.8. Find the Discount, money being worth (>%. The debt becomes due on Aug. 11. Time from March 18 to Aug. 11. = 146 days. Amt. of $100 for 14G days at 0% - $102.40 HencetheTrue Discount of $102.40= S2.40 $1 = ^^"^ And t( ( ( (( (( $G65.60 = 102.40 $2.40xGG5.G0 102.40 Present Worth = $665.60 - $15.60 = $650. EXERCISE 28. Find the Present Worth and True Discount of : 1. $825 due in 2 years at 5%. 2. $1062 due in 3 years at 6%. 3. $352.30 due in S years at 4%. 4. $1518.55 due in 3* years at 7%. 5. $10868.75 due in S\ years at 5%. 6. $90.88 due in 4^ yJars at 5^%. = $15.60 DISCOI'NT. 68 7. A piece of land purchased tor $000 on July 1, 'OM, to be paid at the end of 2 years, but p.iitl on Marcli 1, '01, money beinj^* worth (I . 8. $10010.81 due on June H, liut paid on Mav 1, inonev worth G%. 9. S-'T^^.^O due in 8 years at 5' , when money ean be loaned at 6/^ compound interest. (Here the amt. of $100 for given time is $115. 7C). 10. $701.28 due in 1 .V years, when money can be loaned at G , interest compounded half-yearly. 11. What is the ditference between the true and bank discount of $2000 at 5^^ for 4 months, days of ^race not reckoned. 12. Whether is it cheaper to buy a horse for $2 10 cash or for $220 on a credit of 10 months, money worth i)/^. 13. I am offered a house for $2S8S, to be paid at the end of 18 months. If money can be hired at 5 /, what is that equivalent to in cash ? 14. The interest of any sum of money for 1 year at 6%, is J^y^ of that sum, and the true discount is -^i].. On what sum is the interest $4 greater than the discount ? 15. On what sum does the true discount for mos. at 6% amount to $2-1 ? EXAMINATION PAPER No. 11. 1. In what time will a sum of money double itself at a 6|-%. simple interest ? 2. What sum will amount to $085 in 4 J years at G%, simple interest ? 3. If you place a 3-month note for $100 with a bank when money is worth 8%, what sum do you receive, with- out allowing" for days of ^race ? What fraction of the money you receive for use do you pay lo the bank ? What fraction would you require to pay in the case of a private loan at the same rate ? 66 STOCKS. r.ONDS. nKOKERAGE. 4. A merchant, who used as liis private mark the word " P/^ccau/w/^ ." n\iirkcd a web of silk, " Cost price — p. ta., *' Selliiii;- price - r. ea." Afler iLrivinijf a customer a reduc- tion ot" 25 cents per yard, what per cent, of profit does he make ? 6. At an examination there were 7 candidates at the a^^e of 18, 4 at 11, 11 at 15, 5 at 16 and D at 17. What was the averai^e ag"e ? STOCKS. BONDS. BROKERAGE. When two or more individuals enter into an ag^reement for the purpose of carryinj;" on a commercial or other enterprise, the association is called a Partnership, and each individual a Partner. If, however, they secure a charter fr^m the g^overn- ment, defining the objects, powers and limitations of the Conipany, and org-anize by the election of such ofiicers as a president, secretary, treasurer and board of directors, the association is called a Joint StOCk Company, *' Ono of the special advantag-es of a charter is that it commonly limits each sJockhoklor's liability to the amount of the face value of his shares, whereas in an unincorporated company or firm, each member is liable for all debts of the company." The money required for carrying* on the enterprise is divided into shares of a definite value. Each shareholder receives a ccyfijicntc showing the number of shares he is entitled to, their par value, and the amount lie has paid on them. The value of the shares of stock named in the certifi- cate is called Par Value, When the business of the Company is very profitable, the market value of the stock is high or abovepar, or at a premium; when the business is unprofitable the shares are low or below par, or at a discount. When a $100-share seUs at $100 it isat/>rtr,- when it sells for $112 it is at 12% premium ; and when it sells for $96 it is at a discount of 4%. The balance of the gross earnings of a Company, left after paying expenses, is called Net Profit. STOCKS. HONDS. HKOKERAGE. 67 I Profit distributed amon<^ the shareholders is called Dividend. Divk'tMuis are ilocluroil annually, scMni-anmiallv or i|uartorly. Dividends and Assessments are always expressed as percentni^es of the par value. Wlien the Government of the Dominion, or of any province, or the municipal t^overnment of any county, city or town, or when the directors of any incorporated com- pany wish to raise funds, bonds are prepared and sold. The bonds are secured by the property of those who issue them, and bear a fixed rate of interest payable annually, semi-annually or quarterly. A Bond is a written obliij^ation under seal securing;- the payment of a sum oi money at or before a specified time. The bonds oi' a Company are soi iiivd by a niorii;-a^e on its assets. The most common loiin of bonds are government bonds, railroad bonds, eity bonds, etc. Government bonds are of various kinds, and they are briefiy described by abbreviations for rate of interest, date of payment, etc. Thus: U. S. 4's 1006, rej;"., means Unitetl States rej^-istered bonds, bearing- 4 ^^ interest, payable iti liK)*!. Dominion 8s, n-'JO, '87, means Dominion government bonds, bearing" interest at 3%, issued in 1887, and payable any time between ") ;uui 'JO ye.irs which the govern- ment chooses. British C>nso/s, "consolidated ammities," are per- petual annuities bearing- interest at 8%, . Bonds are sometimes called StOCk, because both are boug"ht and sold in the same way. Hut Bonds pay a reg-ular interest at a fixed rate, whereas the income from Stocks, depending" on the profits made from time to time, is variable. A person whose business is the purchase and sale o( Stocks and Bonds, is a Broker. Mis cotnmission is called Brokerag*e, and is always reckoned upon the par value oi' the stocks or bonds. How does a broker's I'ommission diifer '"nim that of'a i-i>mmission merchant ? ExAMPLK B5. ' What is the cost of oO rail-road shares at 91, brokerai^-e 1 J% ? Cost of Tshare - $94 + $1! - $i)r).nO ' ' "60 shares " = $1)5. 50 x GO - $073. 68 STOCKS. BONDS. BROKERAGE. Example 36. How much stock at a premium of 87, brokerag^e f %, can be purchased for $1734 .^ Cost of $100 stock = $108 + $^ = $1085. Since $108f will buy $100 stock . 31 .. ., $100 (( (( 108# Example 37. What sum must be invested in Dominion bonds, 5's at 108, to secure an annual income of $1500.^ Since $5 is received by investing- $108 . .1 „ .. $108 < ( $1500 (( (( $108x1500 = $32400. Example 38. What is the income from an invest- ment of $1898 in 3;% stock at 87[, brokerag-e -^%. Since $87J buvs $100 stock ■ SI •'■ ^'"^ ( < H7J $1898 ( ( $100 X 1898 87^ =2162.93 + Then SIX of $2162.93 = $75.60 income. Example 39. At what rate must 6% bonds be pur- chased to yield annually 0% of the investment ? Since it is desired to realize 5% of the investment, and the stock yields an income of $6 per share, therefore $6 must be 5)J of the price of a share. Thus : Since 5% of price of share = $6 " 5 $6x 100 / o 100 (< /o " =-iOO-=5120. STOCKS. BONDS. BROKERAGE. ca EXERCISE 29. 1. What is the cost of 50 shares Nicarag-ua Canal stock at oOJ, brokerage ^% ? 2.^ Find the value of 150 shares sugar refinen stock ^^ l^u% premium, brokerage ] %. 3. What must be paid for 220 shares electric railway stock at 94}, brokerage f ? 4. How much stock at 95, brokerag-e \^ can be bought for $955 ? "^ ' 5. What amount of Suez Canal stock at 181], broker- age 1|, can be bought for $7810 ? 6. How many shares C.P.R. stock 1072, brokera^-e 1 can be bought for $5400 ? ^ ** 7. What annual income is derived from $22500 in- vested in Dominion 5's at 105 ? 8. A broker sold 600 shares Portland Gas Co, stock at 61, and invested in City of St. John bonds at 109. Find amount of brokerage, the rate for selling- beiuir ^""^ and for buying f%. ^ * -" 9. What is the yearly income when $18350 is invested m 6}% bonds at 127, brokerage 1 J ? 10. What is gained by investing $10000 in British Consols at 103}, and selling immediately at 105, brokerage in each transaction |% ? 11. Which is the better investment 4?.% bonds at 92, brokerage IJ, or 5} bonds at lOG, brokerage IJ ? 41 51. Which is the larg^er fractionTj... or r^l; ? 12. Which will yield the better income, 87 bonds at 124, brokerage 2{%, or GJ- bonds at 110, brokerage 1|% ? 13. What must I pay for 5% stock that the invest- ment may yield 8% ? 14. At what rate must I buy Midland Railway stock paymg 6%, to receive an income of 7^% ? ' 15. A man who bought bank stock at lOSJ received $275 when a 5% dividend was declared by the bank. How much money did he invest ? 70 STOCKS. HONDS. RROKERAC.E. 16. In the I'nited States when i^old is quoted at 102J, what is the currency value of $H700 of i^^old ? 17. Wliat sum must be invested in U. S. 4's at 1'21}, brokerai^e at J , to secure an annual income of $900 ? 18. How much must I pay for Bank of Montreal stock which pa}s a di\idend of 11%, so that I may make 5% on my in\estment ? 19. Which investment will gfive me the better income, 5^-% stocks at 9i), or 6' ' stocks at 10(j ? 20. A man sells ?»?,^ stocks at 92, and invests in 4^% stocks at par. By what fraction is his income increased ? 21. Bv investinijf in Halifax city 5*% bonds I made (J.l '. At what rate did I buy the bonds ? 22. A i;as company declares a dividend of 15^%. What will a man receive who owns 28 $50-shares ? 23. To secure an annual revenue of $900, what capital must be invested in 5% bonds at 80 ? 24. A man boui^^ht 120 shares of suj^ar refinery stock y at 44, and after holding" it a year sold at a premium of 4%. What did he gain ? 25. A cotton factory company, whose capital stock was $30000, declared a semi-annual dividend of 4' ;, and 1 passed $804.50 to a reserve fund. What were the net earnings ? EXAMINATION PAPER No. 12. 1. How many gallons ot water must be added to 80 gallons of acid 75^;^ pure, so that it maybe only 00% pure ? 2. WHien money is worth 5%, how much more money can you hire for a year from a priv^ate individual for $50, than from a bank ? 8. When wheat is selling at $1 per bushel, what does a man gain or lose who buys HOOOO bu. by the Canadian bushel and sells at the same rate by the United States bushel ? 4. A building worth $3000 was insured for ;1 of its Value at 21%. The company, not wishing- to carry the entire EXCHANGE. 7 1 risk, reinsured J of the amount of the poHcv with anotlier company at 2|%. What amount, of the premium paid by the owner does the first company retain, and what will be the company's actual loss if the buildings is destroyed before the end of one year ? 5. When 35 lb. of te.i which cost 21 cents per lb. is mixed with 29 lb. which cost HO cents, what must be the selling price so that a profit of 20^^, may be realized ? EXCHANGE. Exchang'e is a method of making- payments in distant places without transmitting" money. Till' 1)11.'- iiiess is carri'jcl oil chiefly by banks whith iliaij^i'a small per c«MUai4e for each ttiinsaction. A Draft or Bill of Exchang'e is a written request or order upon one person or bank to pay a certain sum to another person or to his order, at a specified time. The Drawer or Maker is the person or bank whose name is si^iied to the draft. The Drawee is the person or bank to whom the order is addressed and on whom it is drawn. The Payee is the person to whom the money is to be paid. The Buyer or Remitter is the person who purchases the bill. In some cases he is himself the payee. It is j^-ciicrally saforaiulmorccoiivcniciit lorporsotis ^-oiiis^ abroad to lake with thetn one or more bills of exchange on foreign banks, payable to themselves, than to carr^' lartfe sums in bank notes or s^-okl. Wlien a Drawee accepts a draft he writes 'the word " accepted," with his name, and the date, across the face of the draft. He thus assumes the obligation of payinj,'' it at maturity. When the Payee writes his name on the back of a bill or draft he is said to endorse the bill, and it is then payable to the bearer. The payee may, however, endorse it to some particular individual, who in turn may endorse it either in blank or to some other person. Endorsers become separately responsible for the amount of the bill in case the 1 72 DOMESTIC OR INLAND EXCHANGE. drawee fails to meet the payment. A draft made payable to bearer is oi course transferable without endorsement. A Demand Draft is a draft payable on presentation. In an "'At Sigfht " draft three days ci i^race are allowed. In other respects it is the same as a Demand draft. A Time Draft is a draft payable at a specified time after date, or after sig-ht. In C.'inaila three days o^ j^^^race are allowed on all other than Demand drafts. \ bk ! DOMESTIC OR INLAND EXCHANGE. Domestic or Inland Exchangre treats of remittances made from one place to another in the same country. DoniL'stic exchanj^e is generally at par, and the cost of a draft above its face value is usually only a small charg-e made by the bank for its trouble. Hut it sometimes happens, for instance, that \'anco>iver banks have not sufficient money on deposit in Halifax to meet the drafts they lire making- upon Halifax. As it may then be necessary to go to the trouble and expense of forwarding- money to Halifax by express, drafts on Halifax are likely to be for the time at a premium. Or, again, iflarg^e sums paid in at Winnipeg- are drawn at St. John ; in other words, if many drafts on St. John are purch.'ised at Winnipeg, the St. John bankers will have surplus funds at Winnipeg- on which they are receiving- no interest. To save the expense of bringing- the money by express from Winnipeg- they may sell drafts on Winnipeg- at a discount. The premium or discount will never be g-reater than the cost of sending- the money by express from the one place to the other. This premium or discount is called the COUrse Of eXChangfe and depends largely on the balance Of trade. If Canada sells to Britain more than Britain sells to Canada, then British merchants will be making- payments to Canadian merchants in money. There will be more Canatlian drafts purchased in Britain than British drafts pur- chased in Canada. So the British merchants will find that exchange with Canada is at a premium. The same may happen between Winnipeg- and Halifax when Manitoba sells to Nova Scotia more wheat than is equivalent to the quantity of fish, apples or other Nova Scotian export which they buy. f t » i DOMESTIC OR INLAND EXCHANGE. 73 h I Form of Draft. St. John, July 20, 1897. $372.45. At sight, pay to the order of Lawrence, Thome & Co., three hundred seventy-two ^Vo-^^ollars, value received, and charg-e to the account of Harvey Elliott. To Jones & Co., Bankers, Toronto. Example 40. What will a sight draft on Quebec for $1200 cost, at i% premium .^ $1200 + 1% of $1200 = $1201.50. FlXAMPLE 41. $5^<^- Charlottetown, June 12, 1897. Thirty days after siy^ht, pay to G. U. Hay or bearer, five hundred eig-hty dollars, value received, and charge the same to the account of J. D. Seaman. To Harris & Wood, Bankers, St John. What is the cost of the above draft, exchange being- y% premium and money C% ? $580 + 1% of $580 = $581.45. $581.45 -$3.15, bank discount on $580 for 33 days = $578.30. If this draft were payable at si^-'it its cost would be $581.45.. But as the banker in St. John will have the use of the $580 for 33 days before he pays it, he allows bank discount on the face of the draft for that time. Hence the draft will cost $581.45 less $3.15. or $578.30. ' EXERCISE 30. What is the cost of a draft for : 1. $850 Q\\ Montreal at iV% premium.? 2. $1560 on New Westminster at j% discount ? 3. $872.50 on Toronto at 1% premium ? 4. $210.40 on Fredericton at |% premium ? 74 FOREIGN KXLIIANGE. 5. $4800 on Winnipci,'- at l/'] discount ? 6. $G40 on San I'Vancisco at -^ ^ premium ? 7. What is the cost of a O/i-tlay draft on Boston for $35GO, exchang^e 2% premium and money ' ? 8. What must be paid for a ]0-day draft for Si 000, exchang-e -j'jy% discount, money 6% ? 9. What must be paid for a draft of S7;">0, at 80 days, exchange being at J% premium and interest at G% ? 10. How large a sigiit draft on St. Louis, at 1.'% premium can be purchased for STiOOO ? 11. What is the value of a sight draft on Ottawa which cost $2000, exchange -^/^ premium ? 12. A commission merchant sold a consignment o( fruit in Halifax for $l)()0, and after deducting his commis- sion of 4%, he purchased a sig-Jit draft on New York at J% premium. What was the face of the draft ? FOREIGN EXCHANGE. Foreigrn Exchangee deals with drafts drawn in one country and payable in another. Foreign bills are generally, though not always, issued in sets of three, called respectively the //V.?/, second and third of exchange. They are transmitted by different routes, or on different days in order to niinimize the danger of accident or miscarriage. The one reaching its destina- tion first is paid at sig-ht or after the time specified, and the others become void. Exchange for £^25. Halifax, Aug. 20, 1808. At sight of this first of exchange (second and third of same date and tenor unpaid), pay to Murray, Fyshe & Co., four hundred twenty-five pounds and charge the same, as per advice, to WniTEWAV Bros. Baker & Sons, Bankers, London. ' i I'OKHUiX KXCHANtlE. 75 The Parliament of Canada fixed the A/r va/ue of i pound ster n«'- -if «^o ,^r. c i i 1 1 i> . ^ . ^(uuc or a of a noind ^n/. • \r ^ • ^^''^ ^'^^^ />//r/W6' value ot a pound (that is, the actual value oi the -old in one pound steHu.^) is $l.S(;^,or{)' ; above the old par .he - / u* valuo I,, V ,,., V '. ,■ ■ ^ "''^."■"i- pivMiium .m tlie oUI n,ir At ConfiHloration Iho mnv par value was fixoil -u f»i-^ „,. • \Nas leckoncd as fquivalonl to ^l Canadian. ^ The force of habit is ilhisfrateil hv the cmof-ifi.,n ,.r c:, r In the follouinjr examples .-ind in Exercise 20 we assume that a system of exchanf^e, on the basis of the va ues j,nve„ on pa^^e 41, exists between Canada and each ot the coimtnes named. En^lill^^dl^fr fr^"^ T'' ""^ P-^ '" ^--da fbr an ling-lish dralt ot /w50, exchani,'-e 10] premium ? IVice of ^,1 = — ^^ X ■* (( y 100 '' /:7^50 = ^^ Ml- 750 9 ^ 100 ^ 2^ -$3675. Example 43. How larg-e a bill of exchang-e on Edin- 76 FOREIGN EXCHANGE. burg-h can be bou^'-ht for $3200, when sterling' exchange Is quoted at $4. 88 J ? Since $1,881 will buy /:i $1 «< i< £^ 4.885 /a X 8200 ^ $3200 " '^ - j^^g— =^655 Is. SJd. Examplp: 44. What must be paid for a draft on Paris for 5000 francs, exchange being 3.1^/ premium? Price of 1 franc at par =$.198 t( (( (( "1 •' 5000 " '* 3.V premium-$.193x 1081 100 ( ( t ( $.193x207x5000 200 $998.78. Example 45. What will a draft on Marseilles for 5000 francs cost, when a franc is worth $.198. Cost of 1 franc = $.198 " '^ 5000 " =$.198x5000 = $990. Example 46. Find the cost of a draft on Berlin for 4000 marks, exchange 1}% discount. Cost of 1 mark at par =$.238 (( ( ( (( (( 1 4000 (( (< '' 1 {discount = $.238 x- 98f ( ( (( ( ( 100 $.238x895x400 400 = $940.10 Example 47. Find the face of a draft on Antwerp which can be purchased for $362.75 when the mark is quoted at $.243. Face of draft purchased for $.248 = 1 mark. 1 mark ii $1 ^ (( <{ (< u (( <( (( (( '^ $362.75 = .243 1 mark X 362.75 .243 = 1492.80 marks. ' < , FOREIGN EXCHANCIE. 'n ExAMF'LE 48. What will it cost to scnJ 2()(H) roubles to St. Petersburg,''!! when the rouble is quoted at $.r»<)2? Cost of 1 rouble -$.502. " '* 2000 roubles -$.502x2000 = $1121. EXERCISE 31. 1. What must be paid iu I lalifax for a bill oi' oxchang"e on London for J^fi^)^ IDs., when sterlinj^' exchange is quoted at $4.87] ? 2. Wliat is the cost of a bill of exchaiii^-e on Dublin when sterling" exchange isH^ /' premium ? 3. What Is the cost in St. John of a sight draft on London for ;^'JJ12 15s 5d., when exchang^e is $1.W7 ? 4. How much must be paid for a bill o( exchange on Paris for IJ5U0 francs, when 1 franc costs $.li)5? 5. What sum will purchase a 7000-franc bill o( ex- change on Paris at 5% premium ? 6. What will a sterling bill for ;^'317 9s. cost in Char- lottetown when exchange is $1.90' ? 7. Find the cost of a bill of exchange on Cieneva for 7250 francs at a premium of 10 \ 8. A gentleman visiting Europe wishes to carry with him a bill of exchange on a bank in Napks for 1000 lira. What will it cost him in Halifax, exchange being 0% premium ? 9. Find the price of a sight draft on Hamburg for 8540 marks, when the mark is quoted at $.284. 10. How many dollars are equal tO;^'l sterling, when $10 in United States gold coin, which is ^■\j pure, weigh 258 grains, and 1869 sovereigns, J J pure, weigh 40 lbs.? 78 AVKKAf.K Ol' PAVMKNTS AND OF ACCOUNTS. 'I f AVERAGE OF PAYMENTS AND OF ACCOUNTS. Averagfing" Accounts is llie process of fuulini;;- .-i time at wliich several sums due at difTerent times, and not bear- ing- interest, can be paid witbout loss to debtor or creditor. There are two cases, viz.: I. When the terms of credit bej^-in at the same time ; and II. When the terms of credit be^'-in at different times. Example 17. A. M. Brown & Co., sold to James McGrej^'or a bill of goods on the followinijf te-ms : $IK)0 cash, SIOO due in 1 month, $iioO due in 2 months, and $17'"> due in 1 months. At what time mii-ht the indebted- ness be discharged by one cash payment ? $800 for mo. = $1 for mo. $100 for 1 mo. = $1 for 400 mo. $250 for 2 mo. = $1 for 500 mo. $175 for 4 mo. =$1 for 70()jtio^ $1125 IGOcTmoT Now since $1 for 1000 mo. = $1125 for i'J "^l! i"-- Or 1 mo. 12 -f days ; the time required is 1 mo. 18 da. It is evident that $400 for 1 nio. is the same as $1 for 400 mo. Also that $2r)0for 2 mo. is equal to $1 tor 500 mo., and $175 lor 4 mo. = $1 tor 700 mo. Hence the credit of the whole debt is equal to the credit of $1 for 1000 mo. ExAMPi.H 50. Brown & Webb sold to Irwin & Sons bills of goods as follows : June 5, '90, $420 due in 3 months ; July 25^ 'i)0, $2H0 due in 8 months ; and Aug. 30, '90, $350 due in 2 months. Find the average time of payment. $420 due Sept. 5, $280 due Oct. 25, $350 due Oct. 30. Reckoning days from Sept. 5, — earliest date on which money is due. $420 X = mo. $280x50 = 14000 mo. $850x55 = 19250 mo. $1050 88250 mo. 33250 mo. -^ 1050 = 31 + da. 32 days after Sept. 5 is Oct. 7. AVKKAl-.K or I'AVMKNTS A\I> OP ACCOUNTS. EXERCISE 32. 79 When could the folknvin- noii-intcrcst boarin.^ debts be paid at one time uiilunil loss : "^ $80(Ki;!'';'Vnu,.uhI. '""■■''■ ''"""'"^' '" ^ ■"-"-. and / IuIvTh-'^'ViV-"', '^'^i-' ' ', *'"'" ^'"" J'""' 1«; «i900 due July Ih ; {t>12/o due Aul''. 1. for M^i ?'■"■"'' !• " *";" ''f iSltOO for (iO days ; April U, $100 » O. Jan. 2o, $8/)0 on 4 months: Feb 1 ;"> *nnn ^„ q moiUhs^; March 20, $.00 on i months '^Hfio^^S on 7. A man owes $1000 to be paid at the end of 9 months without interest. How muJh would he require to n.:^t^r' -^^ -onths to extend the time o^^^^:;;:;:^ 8. Henry Jones owes Peter Grant for ^^oods as follows: May 3, bill o( $200 on 8 mo. credit. May 24, '' - $uo .. 4 „^^,^ ., June 20, " •' $820 '' 2 mo. '' July 15, '' '' $400 •* 4 mo. (( On July 80, Peter Grant ^ot Henry Jones' note for the Tv had"JhT' ^^P'T'"' "''",^ """'■ ^^'- ^^""'- 'he same sStiar'^'hatdid htrii:'?" "'^ " "^^ "'""' "'• ^°- 80 AVERAGE OF PAYMENTS AND OF ACCOUNTS. Example 51. In the books of Frank Calder & Sons is the account g-iven below. What would be the date of an interest-bearing" note given to settle the account ? Dr. Bash. Bell. Cr. 18^J7. 1897. Aug-. 8 Sept. 10 Oct. 12 1 1 1 Julv 1 July 10 Aug. 3 To Mdse. .... " " 30 da.. " '' 2 mo. $450 300 550 00 i (iO i 00 i Rv Cash '"' Draft " " 10 da. $200 350 400 00 00 00 !;! 1 I < Dr. Taking Oct 25tii as the Focal Day. Cr. Due. July 1. Aug. 9 Oct. 3 Days. Items., Product. Due. JDajs. Items. Product. 116 77 22 450 300 550^ "1300' 950 Balance 350 52200 23100 12100 H7400 31350 1 Aug. 8.. 78 200 15000 Sept. 10. 1 45 350 15750 Oct. 25.. ■100 95(1 :! 1:^50 50050 56050 4-350— 160 + days, average time of interest. Oct. 25 - 161 days --May 17, date of note given by Mr. Bell. Explanation : — Adding 3 days of grace to the date ot maturity of the 10-day draft we get Oct. 25 as the latest date, and this we fix as our focal date. On Oct. 25, leaving the credit side oi the ace. out, Basil Bell owes Frank Calder tV Sons $1300, with interest on $1 for 87400 days. On the same day, leaving out the debtor .vo 00 i Feb. 20 400 0.) Ap. 25 (JOO 00 May 1 May 20 June 2 Dr. Taktn'. June 17 as Focal Day. Cr. Due. Days. ! Items, Product, Due. Days. ! Items. Product. Ap. 10 Jan. 21 Ap. 2 Ap. 14 June 17| 1 1 68 147 76 64 220 600 370 400 600 14960 73500 28120 25600 Mar. 12 Feb. 26 Feb. 20 Ap. 25 May 1 May 20 June 2 97 111 117 53 47 28 15 500 435 100 380 520 150 200 2285 2090 48500 48285 11700 20140 24440 4200 3000 2090 : 142180 lo02(>5 142180 195 i8(tHr) 18085 da. -195 = 92 + days, interest, dated Mafch 16, 1898. ^ ' ^'"'^ '^^"^ ^ "«^^' ^^'^''^^^ 82 AVERAGE OF PAYMENTS AND OF ACCOUNTS. Example 53, When, in equity, should the balance of the following account be pavd ? I Dr. John C. Chambers. 1898. May 5 June 7 June21 To Mdse " 2 mo. "30 da. 50 140 150 00 00 00 1898. May 15 June 10 June 30 Cr. Mdse 25 00 Draft 10 da. . 100 00 Cash 100 00 Selecting Aug. 7 as Foial Day. Due. Davs. Items, 1 Product. Due. Days. Items. Product. May 5 Au^^ 7 July 21 94 17 50 140 150 1 4700 2550 1 7250 i 1 May 15 June 23 June 84 45 38 25 100 100 225 2100 4500 3800 340 225 115 10400 7250 3150 3150 da. -^ 115 = 27 ^ days. Aug. 7-^28 days = Sept. 4. In this example the debit side of the ace. shows that J. C. Chambers owes $340 and the interest on $1 for 7250 days. The credit side shows that there is owing- to him $225 and the interest on $1 for 10400 days. Subtracting we find that he owes $115 less the interest on $i for J J SO days. This latter is equal to the interest on $115 for 28 days. Htrnce he is not required to pay the $115 until 28 days after Aug. 7. AVERAGE OF PAYMENTS AND OF ACCOUNTS. EXERCISE 33. 83 1. Wlun should interest begin on the balance of the following account ? Anderson Ro'. ks in Acc. with James McKenzie. Dr. Cr. 1898. Feb. 20 Mar. 19 Ap. 10 May 4 ToMdse. Imo " Draft 30cla " Cash •' Mdse.3Uda 380 175 120 650 loo loo 00 00 1898. Mar.?/) May 3 Mavis Juno 18 iJuly 10 BvCash. 100 00 *' Mdse 2nK-» 90 00 " Mdse. 3mo 125 00 " Cash. • . . . 540 00 " Cash. 140 00 2. A man bought on Sept. 14, '97, $100 worth of goods, at <) months' credit. On Nov. 25 he paid $115, and on Dec. 10 he paid $90. When, in equity, should he pay the balance ? 3. On July H, John Stewart gave John R. Fitzp;itrick mdse. worth $HG5 on HO da. ; on Sept. 2, goods worth $800 ; on Sent. 12 a :-U)-day draft for $180 ; and on Oct. I mdse. worth $*J50 on 1 mo. Fitzpatrick gave Stewart on Sept. 24, cash $200; Oct. 1, cash $100 '; and Dec. 1 a 10-da. draft for $525. When, in equity, should Fitzpatrick pay the balance ? 4. A man sold a farm for $2400. He received $'iOO cash, and was to receive $200 each month till all was paid. The purchaser, however, made no other payment until the end of 5 months when he paid $1000, and at the end of 7 months he paid $H00. When could the balance be paid without loss to either party ? 5. Alfred Dickie, on Jan. 1, 1H9G, gave Richmond Logan mdse. worth $500 on 1 mo.; on Jan. 20, mdse. $850 on 3 mo.; I^'eb. 15, mdse. $15(jO on 2 mo.; and April 8, mdse. $2500 on 4 mo. On Feb. 3, Logan gave Dickie cash $500; on Feb. 2H, cash $200; and on May 16, a 80-day draft for $1200. When could the balance be equit- ably paid in one sum ? »,'., mh Ul '■ 84 WORK PROBLEMS. WORK PROBLEMS. Example 54. A can do a piece of work in 8 days, woricing" lo hours a day, and B can do it in 6 days, work- ing* 12 hours a day; in how many days of 9 hours each can they together do the same work ? A does Jy and B ^.^ in 1 hour. Working together A & B do g^ + A ^^ tA ^" ^ hour A & B do A& B do 1.20 in 7^2^- hours y^- hours -r 9 = 4 j*y days of 9 hours each. Example 55. A cistern can be tilled by a pipe in 8J hours, and emptied by another in 18j hours ; if both pipes be opened in what time will the empty cistern be filled ? First pipe fills Second pipe empties 1 or 3 25 in 1 h. 1 or 3 in 1 h. 13.\ 40 Working together they fill ^\ - -^ or . l^j in 1 hour o Jo in I of an hour |S^in-|>^or22;hours i( t( i( <( (( (( EXERCISE 34. 1. A can do apiece of work in 12 days, B in 15 days and C in 20 days ; what fraction of the work can they do together in 3 days ? 2. A cistern has three pipes, two of which can fill it in 5 and G hours respectively, while the third can empty it in 4 hours ; if the three be opened when the cistern is empty how long will it take to fill it ? 3. A can do a piece of work in 20 days and B in 25 days. A works at it for 15 days ; in what time can B finish it ? WORK PROBLEMS. 85 4. A can do a piece of work in 20 days and B in 30 days. A works at it tor 5 days and then stops work. B then works at it for 15 days and stops work. C afterwards finishes it in 5 days. In how many days could C do the work alone ? 5. A does y^j of a piece of work in 14 days. He is then joined by B and they finish the work in 2 days. How long" would B take to finish the work by himself? 6. A can do a piece of work in 10 days, A and C in 7 days, and A and B in days ; in how many days can B and C, working" together, do it ? 7. A can do j\ of a piece of work in 5 days, B | in 4 days aiui C [;- in 10 days ; how long will it take them all working" together ? 8. A can do a piece of work in 10 days, working" 8 hours a day. B can do the same work in 9 days working" 12 hours a day. They work together and finish it in 6 days. How many hours a day do they work ? 9. A certain sum of money pays the wages of two men for lOo days. It would pay the wages of one of them for IS^i days. For how many days would it pay the wages of the other ? 10. A can do 2^ times and B 1^ times as much work as C in a day. A and C work for 10 days on a job which they could finish in 12 days ; A is taken off and B put on in his place J how many days does it take to do the whole work? y^'' 86 CLOCK PROBLEMS. CLOCK PROBLEMS. Example 56. At what time between two and three o'clock are the hands of a watch tog^ether ? The circle of the face of the watch is divided into GO minute divisions. At two o'clock the minute hand is at twelve, and is 10 minute-divisions behind the hour hand, which is at two. As the minute hand travels 12 times as fast as the hour hand it gains 1 1 minutes on the hour hand in every J 2 minutes. 11 minute-divisions are jrJiined in 12 minutes 1 10 (( (( k i in 1 ii 1 i ii ii (( IS are " in^ Vi'''. = 101? "lin Ans. 10}" min. past 2. EXERCISE 35. Find the time the hands of a clock are tog"ether between the hours of 1. Sand 4. 2. 6 and 7. 3. 5 and G. 4, 9 and 10. 5. 8 and 9. G. 10 and 11. Example 57. At what times between 4 and 5 are the hands of a clock at rig-ht angles ? To be at right angles the hands must be 1'") minute- divisions apart. As the hour hand is at 4 when the minute hand is at 12, the latter is 20 minute-divisions behind. When it gains 5 minute-divisions it will be at right angles. 11 minute-divisions are gained in 12 minutes 1 6 t( a it m 11 division is divisions are Ans. 5^^ minutes after 4. ii it in -fj- ''orSj-Ymin. It is evident that after the minute hand has passed the hour hand it will be again at right angles. It will then have gained 35 minite-divisions on the hour hand. This worked out as above gives 88 fj- min. past 4 as the second answer. RATIO. 87 At what time are the hands of a clock at right angles between 7. 6 and 7. 8. 7 and 8. 9. 11 and 12. NOTK. -When the hands of a clock are in a straig^ht line they are 30 minute-tiivisions apart. At what time are the hands of a clock in a straight line between 10. 1 and 2. 11. 5 and G. 12. 8 and 9. RATIO. Ratio is the relation o( magnitude in which one num- ber stands to another. The ratio of one number to another is briefly expressed by arranging them as dividend and divisor, or as numerator and denominator of a fraction. The sign of ratio is (:), which is a modification of ^, the sign of division. Thus the ratio of 4 to 9 is expressed 4:9, 4-^9, or J. Itj^ value is J. 9:4 is also a ratio. It is read •' the ratio of 9 to 4." Its value is ;* or 2^. I'he first Term of a ratio is called the Antecedent, and the second Term is called the Consequent. A ratio can exist between concrete numbers only when they are of the same denomination ; for example 8 lb. : 17 lb. equals 8 : 17. A Simple Ratio is the ratio between two terms ; as, 4: 12. Two or more simple ratios are compounded by multi- plying the antecedents together for a new antecedent and the consequents for a new consequent. 8:6] 4 : 5 = GO : 240. 5 : 8) The terms of a ratio may be multiplied or divided by the same number without chang^ing its value ; thus GO: 240 = 120 : 480 (multiplying by 2), or = 30 : 120 (dividing by 2). 88 RATIO. EXERCISE 36. What is the value of the ratios : 1. 8: 12; 9:4; 8 : 15 ? 16; 8:8; 30 : 4 ? 49; 49: 7 ; 18: 72? 2. 9 3. 7 4. 6 71 • 2' 5 » ;j 2 V 5. J f; 8J:10; HJ : ITJ ? 6. If the consequent is 16 and the value of the ratio is 4, what is the antecedent ? 7. If the antecedent is 14^ and the ratio 3, what is the consequent ? 8. Compound the ratios in each of the first five ques- tions of this exercise, and reduce to their simplest form. EXAMINATION PAPER No. 13. 1. Simplify 4.03 + 1.62 + 6.018. 2. Find the cube root of 49.296. 3. A note when drawn at two months, is discounted at 6%. The proceeds are $500. What is the face of the note ? 4. What is the difference between 1 A. 160 sq. rd. 80 sq. yd. 2 sq. ft. 72 sq. in. and 2 A. 5. Bought 60 lb. of tea at 80c., 90 at 40c. and 50 at 50c. a lb. At how much a lb. must I sell the mixture to gain 88 J%? EXAMINATION PAPER No. 14. 1. A man left f of his money to his elder son, and I of it to his brother and the remainder to his daughter. The elder son received $2000 more than his twister. How much did each receive ? 2. What is the change in income when $10,000 in 6% stock is sold out at 90|, brokerage |^%, and the proceeds lent on mortgage at 6^% ? PROPORTION. 89 3. Wlien 4 per cent, bonds sell for 1251, brokerage being" ]f%, vvhiit rate o( interest does an investor get for his money ? 4. Simplify .4(i x .9-7-.H7S. 5. Find the G. C. D. of 817684051 and 31388084951. d e .t kV % s PROPORTION. A Proportion is a statement of equality between ratios, which are expressed in different terms. Thus, ihe ratios 5:15 and 8 : being" each equal to I form a proportion. A proportion may be indicated in the following ways : Thus, 4 is to H as 12 is to 21 mav be written : 4 : 8 : : 12 : 24, or 4^8 = 12-f24, or 1 y The four points ( : :) represent the extremities of the lines ( = ) used as the sign of equality. Kv ery proportion must consist o( at least four terms ; for it involves at least two equal ratios. The first and third terms of a proportion are called the Antecedents (the antecedents of the ratios). The second and fourth terms are called the Conse- quents (the consequents of the ratios). When numbers form a proportion each ratio is called a couplet, and each term a proportional. Thus in the proportion, 5 : 10 : : 8 : KJ, there are two couplets and four terms or proportionals. In a proportion of four terms the first and last terms are called the Extremes and the second and third the Means. The product of the extremes is equal to the product of the means. Thus, in the proportion 4 : 8 : : 10 : 20, 4 x 20 = 8 x 10. Therefore i<^ any three terms of a proportion be given the fourth may be found. 00 PROPORTION. EXERCISE 37. Example 68. Find the missing term in the proportion : 8 : IG : : ( ) : 40. Since 8 x 40 = 820, the product of the extremes, 10 x by the missing term = 820, the product of the means. Therefore 820 -^ 16 = the missings term = 20. Find the missing^ term in each of the following- : 1. 4: 8: : ( ) : 82 ; 10 : 15 : : ( ) : 60. 2. 8 : ( ) : : 18 : 86 ; 2| : ( ) : : 5 : 15. 3. ( ) : 18 : : 20 : 10 ; ( ) : 7A : : J : f . 4. 5: 12: : 16: ( ) ; 14 : 20 : : 1 J : ( ). 5. $6 : $18 : : 9 yd. : ( ) ; ;^4 : 10s. Gd. : : 1 cwt : ( ). 6. 8 lb. 12 oz. : 11 lb. 4 oz. : : $3.50 : ( ). 7. 1 a. 80 rods : 88 rods : : £5 10s. : ( ). 8. § : .05 : : .05 : ( ). LI Example 59. If 5 lb. of butter cost $1.20, what will 16 lbs. cost? It is evident that the ratio 6 lb. : 15 lb. =the ratio, $1.20 the price of 5 lb. : the price of 15 lb. Therefore we have the proportion : 6 : 15 : : $1.20 : ( ). $1.20 X 15 Or, - 5 = $3.60 Ans. Example 60. If 5 lb. of butter cost $1.20, how much can be bought for $8.60. $1.20: $8.60: : 5 lb. : ( ). Or, 120 : 360 : : 5 ( ). 860x5 1800 = 15 120 120 Ans. 16 lb. PROPORTION. 01 EXERCISE 38. 1. If a man can travel 5 miles in 20 hours, how far could he travel in 10 days walking 10 hours a day ? 2. If 19 meters of tweed cost $57.57, how many yards can be bought for $1G0.5G ? (A meter - 89.^7 in. ) ' 3. 8 acres 100 rods of pasture were sold for $100. How many acres could be bought for $a0H0? 4. I bought drugs at $16 a lb. avoirdupois. At how much a dram apothecaries' weight must I sell to gain HO ? 5. If 120 men consume 40 bbls. o{ flour in a certain time, how many men will consume 1000 bbls. in the same time ? 6. Paid $10.50 for 11 lb. 4 oz. of tea ; what should I pay for 8 lb. 12 oz.? 7. An insolvent fails for $7000, and his assets amount to $H000. What does a creditor receive to whom he owes $450? 8. If 2 lb. of sugar cost 25 cents, and 8 lb. of sugar are worth 5 lb. of coffee, what will 50 lb. of coffee cost ? 9. A grocer has a false balance by which 1 lb. will weigh but 12 oz. He sells a barrel of sugar for $2s. How much more than the real value does the customer pay ? 10. A dealer in selling cheese gives only 14],'. oz. for a pound. A customer, who buys $80 worth pays how much more than he ought ? 11. A has land that he values at $50 an acre, and H land at $50 an acre. If in trade A gets $50 an acre what should B get to do as well ? 12. If B men can do /'.^ of a piece of work in 5 days of 12 hours each, how many men will it take to do J ot the work in 6 davs of 8 hours each ? IMAGE EVALUATION TEST TARGET (MT-3) 1.0 I.I ^ tii III 2.0 111^ IF 1.8 1.25 1.4 1.6 = — ^ M 6" ► V ^ %V' ."/? Photographic Sciences Corporation t3 WEST MAIN STREET WEBSTER, N.Y. 14580 (716) 872-4503 , and dividing- the result by 2. Example 64. Find the sum of the progression 3, 8, 13, 18, etc., to 20 terms. The last term (20th) = 3 + (19 x 6) - 98 „, (3 + 98) X 20 The sum = -^ = 1010. z EXERCISE 40. 1. Sum the series 3, 6, 9, 12, etc., to 10 terms. 2. 3. 4. 6. (( i< ( ( < ( ( < <( (( (( (( 60, 54, 48, etc., to 12 terms. 2, 2f , 3i, ^ to 21 terms. 100, 96f , 93J, 90 to 31 terms. |, 1^, 1, etc., to 50 terms. 6- Find the sum of the integers, beginning with one and end'iig with 100. 7. A body falls, approximately, 16 feet in the first second, and in each succeeding second 32 feet more than in the preceding one. How far will it fall in 20 seconds^ and what distance will it fall in the last second ? 8. A man sold 90 sheep, receiving" 10 cents for the first, 20 cents for the second, getting" ten cents more for each sheep, than for the preceding one sold. What did he get for the lot ? GEOMETRICAL PROGRESSION, 95 GEOMETRICAL PROGRESSION. A Geometrical Progression is a series of numbers which increase or decrease by a constant multiplier, called the Common Ratio. Thus : 1, 3, 9, 27, etc., is an ascending geometrical progression in which the common ratio is 8 ; And 64, 32, 16, 8, etc., is a descending geometrical progression in which the common ratio is ^. In an ascending geometrical progression the common ratio is more than unity ; in a descending geometrical progression the common ratio is less than unity. The numbers which compose the progression are called its Terms. In a geometrical progression there are five elements to be considered, any three of which being given, the other two may be found. They are the first tenn^ the last terrn^ the common ratioy the number of term^^ and the sum oj the ternns. The first and last terms are called the extremes and the intermediate terms the mean. To find any Term of a Geometrical Progression : Example 65. Find the sixth term of the geometrical progression 2, 6, 18, etc. Ratios = 6 -^ 2 = 3 Dividing the second term, 6, 6thterm = 2x8^ by the preceding term, 2, the = 2 X 243 common ratio is found to be 3. = 486 Ans, Since the second term is equal to the product of the first term and the first power of the ratio (2 x 3), and the third term is equal to the product of the first term and the second power of the ratio (2 x 3'"^), etc.; Therefore the sixth term will be equal to the product of the first term and the fifth power of the ratio, '"^ x 3^). To find any term of a geometrical progression ^ m,ultiply the first term by that power of the ratio indicated by the num- ber of terms less i. 96 GEOMETRICAL PROGRESSION. EXERCISE 41. 1. Find the 8th and 9th terms of the geometrical pro- gression 2, 4, 8, 16, etc. . 2. Find the 8th and 10th terms of the geometrical pro- g-ression 1, 4, 16, 64, etc, . 3. Find the 12th and 16th terms of the series J, \, 4. Find the 6th and 11th terms of the series 100, 60, 25, 12J, etc. . 6. The first term of a geometrical progression is f and the common ratio 1 J. Find 6th, 8th and 10th terms. 6. The first term of a series is 1, and the common ratio is ^ 5 what is the 8th term ? 7. The stx^h term of a geometrical progression is 3888, and the ratio 6 ; find the first term. 8. The last term of a geometrical series is 60J, the ratio f , and the number of terms 7 ; what is the first term ? ^ i To find the Sum of a Geometrical Progression : Example 66. Find the sum of the geometrical progres- sion 2, 8, 32, 128, 612. 4 times the sum = 8 + 32 + 128 + 612 + 2043 (1) 1 time the sum = 2 + 8 + 32 + 128 + 512 (2) By subtracting (2) from (1) we have 3 times the sum = 2048 - 2 1 time the sum ^= 2048 - 2 3 = 682 Now 2048 = 512 x 4 = the greater extreme multiplied by the ratio 2 = less extreme 3 = the ratio minus 1. Therefore to find the sum of n geometrical series^ multi- ply the greater extreme by the ratio, subtract the less extreme from, the product^ and divide the result by the ratio m^inus one. GEOMETRICAL PROGRESSION. 97 When the series is a descending one, write it in the reverse order, thus making the ratio more than one. Note. — By writing- a descending- series in the reverse order it becomes an ascending series. The ratio is then the reciprocal of the ratio of the descending series. Thus : o4, 32, 16, 8, 4 whose ratio is ^, becomes 4, 8, IG, 32, G4 whose ratio is 2. EXERCISE 42. 1. Find the sum of the series 8, 12, 48, etc., to 6 terms. Cth term = 8x46= 3Q72 (Art. 8072 X 4 - 8 4-1 = 4095 2. Find the sum of the geometrical series 162, 54, 18, etc., to 5 terms. 6th term = 162 X (J)* = 2. By writing in the reverse order the series becomes 2, 6. . . . 162 whose ratio is 8 (reciprocal of ^) 162x8-2 Sum 8-1 = 242 3. Find the sum of the progression 1,4, 16, etc., to 10 terms. 4. Find the sum of the progression f , f , y\ to 8 terms. 5. " " '* ** '* 120, 60, 80, etc., to 7 terms. 6. Find the sum of the progression 1, 2, 4, etc., to 11 terms. 7. Find the sum of the progression 20, 10, 5, etc., to 10 terms. 8. Find the sum of the progression J, y\, |J, etc., to 6 terms. 9. If a boy saves one cent on Monday, 2 cents on Tuesday, 4 cents on Wednesday, and so on for 7 days, how much will he save in the week ? 'I 98 GEOMETRICAL PROGRESSION. 10. A man in business doubled his capital every 6 years, and began with a $1000. How much has he at the end of 25 years ? 11. A sum of money is to be paid in 10 instalments, the first of which is $512, the second $256, the third $128 and so on. What is the sum ? An Infinite Series is a descending- series of an infinite number of terms. Thus: 1, J, ^, -^j, etc., is an infinite series, the last term of which is infinitely small and is regarded as 0. Example 67. Find the sum of the infinite series 9, 6, 4, etc. This is a descending series the last term of which is and the ratio |. By reversing the series we have 0, ... 4, 6, 9, with the ratio f , 9x3-0 Then 3 ' ., =27. EXERCISE 43. Sum to infinity the following series : 1. 12, 6, 3, etc. 2* 2> T> 8» ^^^' 3. 12, 4, 1^, etc. 4. 500, 100, 20, etc. 6. 768, 192, 48, etc. 6. 6, 1, J, etc. 7. 100, 10, j\, etc. 9. .5, .05, .005, etc. 19. .36. COMPOUND INTEREST. ANNUITIES. 99 18 be te is 6, IS ve COMPOUND INTEREST. Problems in compound interest may be solved by means of the principles of geometrical progression. Thus : Suppose $100 be put out at compound interest at 5%. The amount at the end of one year = $100 x 1.05. The amount at the end of two years = $100 x 1.05 x 1.05 or $100 X 1.05". The amount at the end of three years = $100 x 1.05 x 1.05 X 1.05 or $100 x 1.05» ; and so on. To find the amount of a sum of money at eompoiind interest for any number of years, multipty the principal by that power of the sum of i and the rate, whose exponent is equal to the number of years. Example GH. Find the amount of $250 for 8 years at 6% compound interest. $250 X (1.05)^ = $289.406250 This question makes a or, $289.41. geometrical progression of four terms, the first term being 250 and the common ratio 1.05. It is required to find the fourth term. EXERCISE 44. Find the amount and compound interest of : 1. $440 for 8 years at 6%. 2. $1000 for 4 years at 8%. 3. $690 for 2 years at 5%. 4. $S00 for 6 years @ 4%. 5. $2100 for 5 years @ 2%. 6. $1800 for 2 years (^ U%. %\ ANNUITIES. An Annuity is a sum of money payable yearly, to con- tinue a number of years, for life, or forever. The term is also applied to money payable at any regular times. 100 ANNUITIES. The Amount or Final Value of an annuity is the sum of all the payments, together with the interest on each pay- fnent from the time it becomes due until the annuity ceases. The Present Worth of an annuity is that sum of money which, at the specified rate of interest, will amount to the final value. ANNUITIES OF SIMPLE INTEREST. Example 69. Find the amount of an annuity of $800 for 6 years at 6%, simple interest. The first payment draws interest for four years ; the second for 8 years ; etc. Now the amount of the first payment at the end of the four years during which it draws interest, is $800 x 1.24 or $872. The amount of the second payment at the end of the three years during which it draws interest, is $800 x 1.18. The amount of the last payment is $300. This makes an arithmetical progression of 5 terms, the first term of which is $372, the last term $300, and the number of terms 5. Therefore the sum of the terms (amount or final value of the annuity) is ($372 + $300) X 5 ^ jgQQ A Example 70. Find the present worth of an annuity of $400 for 6 years at 5% simple interest. The amount of the annuity is $2700. Amount of $1 in 6 years at 5% = $1.3. Principal to produce $1.3 = $1 $1 $1 =• (( « (( (I (( " $2700 = 1.3 $1 X 2700 iTs = $2076.92. ANNUITIES. 101 Example 71. What annuity to continue for 4 years, at 6% simple interest, can be purchased for $2180 ? An annuity of $1 to continue for 4 years, at 6% simple interest, will amount to (^^'^^-^^^)-^ or $4.86. The amount of $2180 for 4 years, at 6% is $2708.20. $4.36 is the amount or final valiu* of an annuity of $1 $1 Of] t ( <« It it tt 2703.20 41 t( (i (< 4.36 $1 X 2703.20 ■ 473b cr$630 EXERCISE 44a. 1 . What are the present worth and final value of an annuity of $500 for 5 years, at 6% simple interest ? 2. What are the amount and present worth of an annuity of $600 for 5 years, at 4% simple interest ? 3. Find the amount and present worth of an annuity of $500 for 8 years, at 4.\'/^ simple interest. 4. Find the amount and present worth of an annuity of $800 for 8 years, at 6%. 5. What annuity to continue for 8 years, at 8% simple interest can be purchased for $2088 ? 6. Find the amount and present worth of an annuity of $720 for 7 years, at 82% simple interest. 7. What annuity to continue for 6 years, at 6% simple interest can be bought for $2070 ? 8. What is the annuity v/hose amount for 6 years, at 6% simple interest is $8450? 9. What annuity to continue for 9 years, at 4% simple interest, can be purchased for $1044 ? 10. $11220 is invested in an annuity to continue 11 years, at 6^% simple interest. What is the yearly pay- ment? il r 102 I . I ANNUITIES. ANNUITIES AT COMPOUND INTEREST. Example 72. Find the amount of an annuity for $100 for 6 years at 4% compound interest. The fifth or last payment is $400 and draws no interest. The fourth or second hist payment is $10u and draws interest for 1 year and therefore amounts to $400 x 1.04. The third payment amounts to $400 x 1.04 x 1.04 since it draws interest for two years, and so on. The last payment may be reg'arded as the first term of an ascending- geometrical progression of 6 terms whose common ratio is 1.04 and the sum of whose terms is equal to the amount of the annuity. The fifth term (greater extreme) = $400 x (1.04)*. ^ , ($400 X (1.04)*) x 1.04 -$400 Sum of the terms = • t-tt* — 7 1.04-1. $400x(1.04)S-$400 .04 = $2166.63. Ans. Example 7tS. Find the present worth of an annuity of $400 for 6 years, at 4% compound interest. The amount or final value of the annuity (see answer to preceding example) = $2166.53. Amount of $1 for 5 years at 4% = (1.04)5 = 1.2166629024. Present worth of $1.2166629024 = $! $1 i< i< " << $1 " '* $2166.68 1.2166629024 $1x2166.63 1.2166629024 $1780.73. ' ^ i ANNUITIES. 103 Example 74. What annuity, to continue for \ years, at 6% compound interest, can be purchased tor $(500? Amount or final value of an annuity of $1 to continue for 4 years, at 6% compound interest is (ll^^l'jl.^ or $4.81012;',. .06 Amount of $G00 for 4 years, at 5/, compound interest, is $600 X (1.05)* or $729.30875. $4.810125 is the amount of an annuity of $1 $1 (< $1 $729.80375 <' If <( (( <( (( <( 4.810125 $1x729.80375 4.310125" = $169.21. 1!!XERCISE 45. 1. What are the amount and present worth of an annuity of $400 for 8 yen s at 5% compound interest ? 2. What is the final value of an annuity of $300 for 4 years, at 6% compound interest; and what sum will pur- chase such an annuity ? 3. What annuity to continue for 2 years, at 6% com- pound interest, can be purchased for $5150 ? 4. What annuity to continue for 3 years, at 4% com- pound interest, can be bought for $3902 ? 5. The final value of an annuity for 5 years, at 6% compound interest is $2818.55 ; what is the annuity ? '< 104 SQUARE ROOT. II. SQUARE ROOT. (Part III., Articles 67, 58). EXERCISE 46. 1 . An army of 220900 men is drawn up in the form of a solid square. How m^ny men are there on each side ? 2. The sides about the right angle of a right angled triangle are 72 and 96 feet respectively. What is the length of the hypotenuse ? 3. A piece of land in the form of a square contains 10 acres. What is its perimeter ? 4. The hypotenuse of a right angled triangle is 125 yards and one of the sides is 100 yards. What is the length of the other ? 5. The end of a tree broken 39 feet from the top struck the ground 15 feet from the root. How many feet of the tree remained standing ? 6. A ladder 40 feet long being placed at a certain point in the street will reach a window 83 feet from the ground on one side and a window 21 feet from the ground on the other. What is the width of the street ? 7. A room is 40 feet long, 30 feet wide, and 15 feet high. What is the longest distance thi-t can be measured in a straight line in the room ? 8. The hypotenuse of a right angled triangle is 41", and one of the sides is .9"'. How many hektometers in the other side ? 9. The sides of a right angled triangle are 18™ and 33*^™, respectively. Find the hypotenuse in dekameters. 10. Two vessels sail from the same place at the same time. One sails north for two days at the rate of 5 miles an hour and the other sails east for three days at the rate of 6 miles an hour. How far are they then apart ? ■1 MEASUREMENT OF AREAS. 106 n the form of each side ? right angled What is the e contains 10 iangle is 125 What is the the top struck ny feet of the certain point m the ground rround on the , and 15 feet be measured iangle is 41", ektometers in e are 18" and iekameters. :e at the same rate of 5 miles ys at the rate ipart ? MEASUREMENT OF AREAS. (Part III., Articles 60, 62, 63). EXERCISE 47. 1. How many acres in a square field whose side is 24 rods ? 2. What will it cost to paint the walls of a room 10 feet 9 inches long, 9 feet wide and 10 feet high at 25 cents a square yard ? 3. The floor of a room 16 feet long and 12 feet wide is to be painted in alternate squares of black and white, the side of every square being 8 inches. If the white squares cost a cent each and the black 1 ^ cents each, what will be the total cost ? 4. A field contains 4 acres and is 70 yards long. What is its width ? 5. What is the cost of carpeting a room 18 feet 3 inches long and 16 feet 4 inches wide with carpet 27 inches wide at $2.70 a yard ? 6. How much must be paid for laying asphalt on ^ of a mile of sidewalk at 75 cents a square yard, the sidewalk being 10 feet wide ? 7. A contractor received $8000 for building a piece of road 800 yards long and 60 feet wide. How much did he receive a square yard ? 8. A square yard of a floor costs $2.80 and the whole floor costs $114. What is its length if the breadth is 5\ yards ? 9. How many suits can be made out of 60 yards of tweed, 64 inches wide, each suit requiring 7 yards, 27 inches wide ? 10. What will it cost to paper the walls of a room 19 feet 8 inches by 17 feet 9 inches and 12 feet high, with paper 18 inches wide, 7 yards in a roll, at 63 cents a roll ? 11. What will it cost to paint a house 54 feet deep, and 24 feet 6 inches, fronting the street, at 20 cents a square yard, the walls being rectangular and 24 feet high ? r 106 MEASUREMENT OF AREAS. 12. What must be paid for the plastering- of a wall 50.5 yards long and 5.25 yards high, at $2.10 a square yard? 13. There are 12 rooms in a school building, each room being 82 feet long, 24 feet wide and 12 feet high. What will it cost to paint the ceilings and walls of the whole building at 18 cents a square yard ? 14. One piece of ground is 200 yards square ^ and an- other is 200 square yards. What is the difference in price at 45 cents a square foot ? 15. What is the area of a triangle whose base is 40 feet and altitude 25 feet ? 16. One side of a triangular field is 12 chains, 76 links, and the perpendicular on it from the opposite angle is 9 chains, 43 links. How many acres in it ? 17. Find the area of a parallelogram, one of its sides being 40 inches and the shortest distance to the opposite side being 80 inches. 18. The area of a parallelogram is 1^00 square yards, and the perpendicular distance between two opposite sides is 20 yards. What is the length of each of those sides ? 19. It costs ;^12 to carpet a room 4 yards wide at \ of a pound a square yard ; what is the length of the room? 20. What is the area of a trapezium whose parallel sides are 85 and 45 feet, respectively, and altitude 25 feet ? To find the area of a triangle when the three sides are given. Example 75. Find the area of a triangle whose sides are 42, 45 and 39 j'ards. 42 + 45 + 89 -68 68(68 - 42) (63 - 45) (63 - 89) = 571586 ^571686 = 756 Ans. 756 sq. yds. From half the sum of the three sides subtract each side separate- ly ; multiply half the sum and the three remainders tog-ether ; the square root of the product is the area. MEASUREMENT OF THE CIRCLE. EXERCISE 48. 107 1. What is the area of a triangfle whose sides are 60, SO and 90 feet ? 2. How many square yards in a triangular field whose sides are 126, 247 and 296 yards ? 3. What is the area of an isosceles triangle whose base is 20 feet, and each of its equal sides 15 feet ? 4. The sides of a field in the form of an equilateral triangle are each 70 rods. How many acres in it. 5. The sides of a triangular field are 1200, 1800 and 2400 links respectively. Find the area in acres and rods. MEASUREMENT OF THE CIRCLE. (Part HI., Articles 74, 75). EXERCISE 49. 1. The diameter of a wheel is 26 inches. What is its circumference ? 2. What is the circumference of a circle whose radius is 5 feet ? 3. What is the length of a tire that will fit a wheel whose radius is 8 feet 10| inches ? 4. The circumference of a circle is 15708 feet. What is its diameter ? 5. What is the area of a circle whose diameter is 60 feet ? 6. The radius of a circle is 17 feet G inches. Find its area. 7. The distance from the centre of a circular pond to the bank is 225 links. How many acres in it ? 8. The circumference of a circle is 200 feet. Find its area. 9. The circumference of a circular field is 50 chains. What is its area ? 108 MEASUREMENT OF LUMBER. 10. The diameter of a circular garden is 40 rods. What is the area of a circular space 10 feet wide and just within its border ? 1 1 . The diameters of two circles having- the same centre are 20 and 32 feet. Find the area of the circular space (the annulus) inclosed between the two circumfer- ences. 12. A circular path 10 feet wide is laid at a distance of 40 feet from a certain statue in a public garden. What is its cost at 50 cents a square yard ? :l I MEASUREMENT OF LUMBER. (Art. 76, Part III.) EXERCISE 50. 1. How many superficial feet in a board 36 ft. long, 1^ ft. wide and 1^ in. thick ? 2. How many feet are there in nine joists, which are 15 ft. long, 5 in. wide, and 3 in. thick ? 3. Find the number of (board) feet of lumber required to floor a dock 100 feet long and 40 feet wide, the planks being 2^ inches thick. If the board is tapering, take half the sum of its ends for the width. 4. What is the number of feet in a tapering piece of plank, 20 feet long, 24 inches wide at one end, and 16 inches wide at the other, the board being 2 in. thick ? 5. How many feet in 3 beams 24 ft. long, 10 in. thick, whose width tapers from 18 to 16 inches ? 6. How many board feet in a cubical block of wood whose edge is 2 feet 9 inches ? MEASUREMENT OF SOLIDITY OR VOLUME. 109 6 MEASUREMENT OF SOLIDITY OR VOLUME, (Articles 78-80, Part III.) EXERCISE 51. 1. Find the volume of a cube whose edge is 3J- inches, and also find the area of its entire surface. 2. What is the weight of a cubical piece of granite whose edge is 2 feet ? 3. How many pounds avoirdupois in a piece of marble 2"^ long, .75™ wide and 8^"" thick ? 4. How many kilograms in a block of ice 1 ft. long, 10 in. broad and 9 in. thick ? 5. What will it cost to make an excavation 21 feet long, 18 feet broad and 10 feet deep at $1.30 a cubic yard ? 6. The walls of the foundation of a house are 50 feet long, and 30 feet wide, measured on the outside. If they are 7 feet high and 9 inches thick, what will they cost at 30 cents a cubic foot ? 7. What will be the cost of a pile of wood 35^ feet long, G^ feet high, and i feet wide at $3,84 a cord ? 8. How many granite blocks, each 8 in. long, 2J in. w^ide, and 2 in. thick, will be required to build a wall 18 ft. long, 3 ift. high, and 11 in. thick ? 9. How many bushels of grain can be-put into a bin S feet long, 3 feet wide and 4 feet deep ? 10. How many tons of coal can be put into a bin 10 feet long, 5 feet 3 inches wide, and 6 feet 4 in. deep^ a ton taking up 38 cubic feet of space ? 11. A bin, which is 10 ft. long, and 3 ft. 6 in. wide, holds 100 bu. How deep is it ? 1 2. A vessel 5 feet long, 4 feet wide, and 3 feet deep IS filled with wheat. What is the weight of the wheat and what does it cost @ $1.00 a bushel ? 110 MEASUREMENT OF SOLIDITY OR VOLUME. (See Part III., Articles 81-90). 'i . t ii'' I'fii EXERCISE 52. 1. What is the volume of a cylinder whose altitude is 7 feet, and the diameter of base 6 feet ? 2. The volume of a cylinder is 2412.7488 cu. ft., and its altitude is 12 feet. Find the radius of the base. 3. A cylindrical vessel is 28 feet hig-h, and measures 5 feet across the base. How many cubic feet are in it .'* 4. The diameter of the base of a cone is 1 ft. 6 in., and the altitude 15 feet. What is its solidity.'^ 5. The base of a marble pyramid is in the form of an equilateral triangle, each of whose sides is 3 feet. The altitude of the pyramid is 9 feet. What is its value at $2.50 a cubic foot ? 6. Find the length of each side of the base of a square pyramid whose altitude is 21 feet and volume 847 cu. ft 7. What is the volume of a square pyramid, the area of whose base is 36"', and whose height is three times the diagonal of the base ? 8. The diameter of a sphere is 25 inches. How many" cubic inches are in it ? 9. What is the weight of the hydrogen that will fill a spherical balloon, whose diameter is 3'", the weight of a liter of hydrogen being .09^? 10. What is the weight of a cast-iron ball whose diameter is 12 inches? 11. The diameter of a hollow sphere is 6J decimeters. How many liters will it hold ? 1 2. The altitude of a cone is 20 inches, and the diameter of its base is 10 inches. The upper part is sawed off in a plane parallel to the base and 10 inches from it. How many cubic inches in each part? (The lower part is called the Frustum of the ccne). SPECIFIC GRAVITY, 111 r a. V d SPECIFIC GRAVITY. EXERCISE 53. (For Explanation See Part III., Examples 68, 69). 1. Find the specific gravity of a stone which weig^hs 21 lb. in air and 12.26 in water. 2. What is the specific gravity of a piece of marble which weighs 46.41 lb. in air and 17 lb. less in water ? 3. If a substance weig-hs 26% less in water than in air, what is its specific g-ravity ? 4. What is the specific gravity of a substance which weighed 25% more in air than it did in water ? 5. What is the weight of a block of ordinary granite 1.6™ long, 7.5'*'" wide and 4.6'^'^ thick ? 6. If a bar of lead is 7"^ long, 1*^~ wide and 4'=" thick, what is its weight ? 7. A bar of iron weighing 7.6 lb. in air and 6.5 in water is fastened to a piece of wood weighing 5 lb. in air. Together they weigh 3.6875 lb. in water. What is the specific gravity of the wood ? 8. An empty glass bottle weighing 6.66 oz. is filled with olive oil weighing 7.32 oz. What is the specific gravity of the bottle of oil ? 9. A body weighs 7.56 grams in air, 6.17 grams in water, and 6.35 in another liquid. Find from these data the specific gra^'ity of the body in question, and also of the other liquid. 10. A block of wood floats with -^.j of its bulk out of the water. Find its specific gravity. 11. A pebble weighs 20 grams in air; immersed in water it weighs 16 grams ; immersed in another liquid it weighs 17 grams. What is the specific gravity of the latter liquid? 12. A piece of wood weighs 7 lb. in air, and a piece of iron 7.8 lb. in air and 6-7 lb. in water. The wood and iron together weigh 6.3 lb. in water. Show that the specific gravity of the wood is |. 112 EXAMINATION PAPERS. , ■ i n . 13. A solid, soluble in water, but not in alcohol, weighs 846 grains in air and 210 grains in alcohol. Find the specific gravity of the solid, that of alcohol being .85. 14. Find the specific gravity of a piece ^^ wood from the following data : Weight of wood in air =25.35 lb. • ** ** a metal sinker =11 lb. " ** wood and sinker in water = 5.1 lb. Specific gravity of metal sinker =8.95. EXAMINATION PAPERS. 35. 1 . What will eight hundred seventy-five thousandths of a ton cost, at $S.75 per ton ? .05 2. Simplify 09^ oy 3. My purse and money are worth 12s. 8d., the money being worth 7 times the purse. How much is the purse worth. 4. If f of a yard of satin lining is worth f of a yard of silk lining, and silk lining is worth $f per yard, how many yards of satin lining will $20 buy ? 5. Find the cube root of f to 3 decimal places. 1. Simplify- 3i + p ^2" 16. .09 2. i owned |^ of a ship and sold f of my share for $45000. What fraction of the whole ship had I left and what is its value at the same rate ? 3. One insurance company offered to insure my house ^t 3i%» 3.nd another oflfered to insure it at 2|^%. By accepting the latter offer I saved $10. How much insur- ance did I effect ? 4. Multiply £1 17s. 6Jd. by 17f EXAMINATION PAPERS. 118 5. I sent $4120 to my ag-ent in Ontario to buy flour, after deducting" his commission at 8/^. How many barrels did he buy, flour being $5 a barrel. 17. 1. A man sold J of his land and then }^ of the re- mainder. He afterwards sold J of what he had then left and there still remained 16 acres which he sold at $22.50 per acre. What was the whole farm worth at the same rate ? • 2. What is the interest of $150.90 from Dec. 7, 1895 to March 8, 1896 @ 6% ? 3. If .05 of an acre is worth $1000, what would be the cost of a lot 34 feet by 100 at the same rate ? 4. Find the value of 6.45 x 8.25 x 21 - 6.47 x 8.25 x 20. 5. What is the depth of a cubical cistern that will hold 10000 standard gallons ? 18, 1. A boy had a basket of apples. He gave •; of the whole to his brother and took 12 for himself. He gave what was left to his sister who found she had as manv as the two others. How many were in the basket ? 2. How many acres are there in a field 66.5 rods long, and 24.6 rods wide ? 3. $400.00. St. John, Jan. 1, 1898. Three months afterdate, for \alue received, I promise to pay James Hunter, or order, four hundred dollars. Thomas Marks. Discounted Jan. 8, at 6"'. Find proceeds. 4. What is 107^ C. on the Fahrenheit thermometer ? 5. Change — 31^ F. to the centigrade reading. 19. 1. 3 men, 4 women or 5 boys can do a piece of work in 2^- days. How long will it take a man, a woman, and a boy, working" together to do it ? I! 114 EXAMINATION PAPERS. it 2. What must be the face of a note drawn May 4, at 8 months, so that if discounted on that date, at G%, the proceeds will be $;iUO? 3. What is the amount of $300 for 4 years at 5%, compound interest? 4. A sum of money amounts to $H27 in H years at B%, simple interest. What would it amount to at compound interest ? 5. In what time will $640.80 amount to $7G0.90 at 6%, simple interest? 20. 1. At what time between twelve and one o'clovk will the hour and minute hands be in a straij^ht line ? 2. Divide $3600 among- A, B, and C, in the ratio of 5, 6, and 7. 3. A, B and C enter into partnership. A puts in $600 for 8 months, B $800 for 10 months, and C $1000 for 12 months. They g"ain $400. What is the dividend of each ? (See Art. 44, Part III.) 4. $200.00. Charlottetown, May 8, 1898. Two months after date, for value received, I promise to pay Thomas Bancroft, or order, two hundred dollars, with interest at 6%. George Crane. Discounted May 9, at 7%. Find proceeds and bank discount. 5. At what rate, simple interest, will a sum of money treble itself in 40 years ? 21. 1. The L. C. M. of 391 and another number is 12121, and the H. C. F. is 28. What is the other number? 2. If 3 horses are worth 10 cows, and 5 cows are worth 60 sheep, how many sheep are equal in value to 4 horses ? 1^ 3 3. From the sum of -f and tj take the difference of their quotient and product. 4. Reduce .692307 to a common fraction. EXAMINATION PAPKRS. 11, 5. A victualler buys a carcass of beef wei^liin*;' 5 cwt. 91 lbs. at 7] cents per lb. He sells oue-third of it at 'IHc. per kilog^ram, one-third at 22c. per kil. and one third at 10c. per kil. What was his g'ain ? 22. 1. Find the sum of all the proper fractions, in their lowest terms, that can be formed havinj^" only one fij^ure in the numerator and one in the denominator. 2. What is the cube of '^^ ' '• ? 21)/, 3. What is the cube root of ? 1401 4. If 12 workmen, in 12 days, working- 12 hours a day, can make up 75 yds. of cloth, 'l yd. wide, into articles of clothing", how many yards, 1 yd. wide, can be made up into similar articles, by 10 men, working H hours a day, for 9 days ? 5. A man who has a 2-acre field, twice as long as it is broad, can sell the whole at 10 cents per square foot, or he can sell it in lots of 100 ft. by 30 ft. for $380 each. Which is more advantageous, and how much ? If he sells in building lots what fraction of the land will remain un- sold ? 23. 1. Find the difference between 5 of J of 6 shillings and .0875 off of ;^1. 2. What would it cost to dig" a ditch 40 rods long, 3 feet wide, and 4i feet deep, at 18 cents a cubic yard? 3. A merchant bought a bbl. of apples for $3.20. The bbl. contained 400 apples, and two out of every 20 were decayed. At what rate per dozen must he sell the re- mainder so as to gain 25% on his outlay? 4. Of five legatees };he first receives .3 of the bequest, the second .3, the third .03, the fourth .03 and the fifth the remainder amounting to $36400. How much does each of the others receive ? ( UG EXAMINATION PAPERS. 5. How mucli money must be put at interest at 0%, simple interest, to yield in interest $H5L()0 in 2 years 6 months ? 24. 1. What is tlie weij^^ht of a t^ram in j^rains ? I low Was the weii,'"ht of the j^Tani lixed ? •-) 1 :. t V'lnd tile value of $5.07 x r- ■ '.--tt x .', i ' ■ <^t :; ;' + i 2. Divide .0()821 bv .OOOOOOM, and from 8.00823 take 1.009010G. 3. Find the difTerence between 8 miles and 2 mi. 7 fur. 39 rd. 5 yd, 2 ft 10 in. 4. When the income tax is Od. in the ;^, a inan pays £15 7s. Gd ; what is his income? 5. A sum of money at simple interest has in 4.V years amounted to $735, the rate of interest beini;- 5 ; what was the sum at first and in how many years more will it amount to $1140? 25. 1. There are in a li^ht house three revolving lig-hts. One revolves in 2G0 seconds, another in 195 seconds, and the third in 390 seconds. They are all in line with a small island out at sea at 1 1 o'clc">ck. When will they be again in line ? 2. A map is drawn on a scale of 10 miles to an inch, and a township is represented on it by a sc]uare whose side is half an inch. How many acres in the township ? 3. At what price must a bookseller mark a book which cost $1.20 so that he may allow a discount of 10% to a student and still make 20% ? 4. A square court-yard is bordered by a grand drive 10 yards wide, and the drive covers 4000 square yards. How many square yards in the enclosed grass-plot ? 5. A commission merchant in Montreal sells 15460 lb. of wool, at 24 J cents a pound. If his commission is 4% and exchange ^ premium, how large a draft can he buy to send to his consignor in Charlottetown ? EXAMINATION PAPI-RS. Il7 26. 1. A 8 per cent, stock is selling,'- at HG. If a man who has $10000 to invest delays until it rises to H7, what change is made in his income ? 2. A man who invested in 8% consols, received 4% on his investment after deduclinj;' an income tax oi' Is. Id. in the pound. What did he pay for the stock ? 3. Which is the more advantat^eous, to buy Hour at $5 a bbl. on montlis credit, or at $4.H7! cash, money bein^ worth 7 '{ ? 4. A and B working- toi^ether, can earn a sum of money in Hi days. B can earn it, working,'- alone, in 20 days. In how many da}s can A earn it, if working- alone ? 6. A merchant buys 1000 lb. of tea at 25 cents a nound. He pays 20% duty and $27. HO for freight and other expenses. At how much a lb. must he sell the article to '4Q 1 °/ gain 83J 27. 1. A man buys coal by the long ton and sells it at the same rate by the common ton. What per cent, does he g^ain ? 2. A square field, one side of which is 200 yards, con- tains a circular pond one hundred yards across. How much dry land is there in the field ? 3. If 1000 shingles cover 100 sq. ft., how many will it take to cover a barn 60 ft. long, 40 ft. wide, corner 21 ft. high, gable 8G ft. high, and rafter 25 ft. long ? . 4. A note of hand for $80, drawn May 1st, 1895, has the following endorsements: — June 10, '1)G, $20; Aug. 5th, '97, $15. What remains to be paid May 1st, 1H9H, interest being 6% ? 5. What must be the depth of a cylindrical vessel, whose diameter (inside) is 8 ft., so that it may contain just 10 bushels ? 28. 1. A, B, and C start from the same place at the same time, going in the same direction around an island whose circumference is ^ mile. They travel at the rate of 3, 4, 118 EXAMINATION PAPERS. and 4 J miles respectively. In what time will they be together again, and how far will each have travelled ? 2. A g-rocer gives 22 lb. of sugar for $1 cash, and 20 lb. on 1 year's credit. What is the rate of discount ? 3. What sum must be invested in U. S. 41%. bonds at 102, so as to provide an annual income of $1620 ? 4. Find the cost of excavating a cellar 6 ft. deep for a house 27 ft. by 31, at 20 cents a load for the first foot, 24 cents per load for the second, 28 cents a load for the third, etc.? 5. The cellar mentioned in the last question is walled 1 ft. above the ground with a wall 1 .! ft. thick, at $14 a perch. What is the mason's bill ? 29. 1. How many cubic inches in the largest ball that can be cut out of a cubical block whose edge is 6 inches ? 2. A man bought an article at 20% below the retail price. If he had paid a dollar more for it, it would still be 60 cents below the retail price. What was the retail price ? 3. Bought 160 lb. of tea at 20 cents a lb., 140 at 30 cents, and 100 at 40 cents. At how much a lb. must I sell the mixture to gain 20% ? 4. What principal will yield $60 interest in 219 days at 74% ? 30. 1 . Find the suiii of the series 2, 6, 18, etc. , to 8 terms ? 2. How many strokes does a clock which strikes the hours, strike in 13 weeks ? 3. A grocer mixes 10 lbs. of tea at 40 cents per lb., 20 at 45 cents, and 80 at 50 cents, and sells the mixture at 70 cents per lb. How much does he gain per cent.? 4. If a hatter sells hats at $1.25 each and loses 25 per cent., what per cent, would he lose by selling them at $1.60 each ? 5. What rate of interest does a person receive who invests in 4J per cent, stock at 90 ? EXAMINATION PAPERS. 119 will they be avelled ? cash, and 20 scount ? li%.bonds at ylo ? 6 ft. deep for the first foot, load for the tion is walled lick, at $14 a t ball that can ) inches ? low the retail would still be e retail price ? lb., 140 at 30 a lb. must I St in 219 days :., to8 terms ? ich strikes the cents per lb., he mixture at jr cent.? d loses 25 per illing them at n receive who 31. 1. The distance from Halifax to St. John by I.C.R. is 275 miles. A train leaves each place at 7 o'clock, the Halifax train running 25 miles an hour and the St. John train 30 miles, (a) When and where will they meet ? (b) When will they be 40 miles apart? (c) If the St. John train leaves at 11.30, where will it meet the Halifax train ? 2. Find the area of a circular bicycle track wh'ch measures 8 laps to the mile, measured on the smaller cir- cumference, the track being 20 feet wide. 3. A man invested $2400 in a publishing business, and at the end of 2^ years he withdrew $2940, being invest- ment and profits. What annual rate of interest did his investment pay ? 4. If a ton of coal occupies 40 cubic feet, what will it cost to fill a bin 12 ft. long, 6 ft. wide and 5 ft. deep, wher coal is worth $3.90 a ton ? 5. Iodine melts at 107° C, and boils at 178° C. At what temperature as indicated on a Fahrenheit thermome- ter does iodine melt and boil ? 32. 1. A merchant vessel starts at 8 o'clock Monday from a point 45'^ north latitude, and sails due south at an aver- age rate of 8 miles an hour. At 9 o'clock on Tuesday a cruiser starts in pursuit at the rate of 15 miles an hour. When and in what lati'^ude will the vessel be overtaken. 2. Why do we subtract one hour from the time we use throughout the province to get railway time ? 3. How many turns of a bicycle wheel 28 inches in diameter are made in going 10 miles ? 4. A pond whose area is I acre is covered with ice IS inches thick. If 1 cubic foot of water weighs G2.V lb., and sp. gravity of ice is .93, find the number of tons of ice on the pond. 5. A grocer buys a bbl. of sugar (gross weight 300 lb,, tare 12 lb.) at 4| ce^ts per lb. He sells J of k at the 120 EXAMINATION PAPERS. V llfnl 1 :! rate of 18 lb. for $1. How much must he get per lb. for the rest so as to g^ain 20% by the transaction ? 33. 6 + 1. Simplify —T+m- 2. Divide .0628 by .00002, and find the sum of 2.208 + .6588+4.028570. 3. If 4 of an estate be worth $7520, what is the value of f of the estate. 4. A can do a piece of work in 27 days, and B in 15 days ; A works alone at it for 12 days, B then works 5 days, and then C finishes the work in 4 days. In what time could C have done the work by himself. 5. $1500. St. John, Jan. 1, 1897. One year after date, for value received, I promise to pay John Smith, or order, fifteen hundred dollars, with interest, at 6%. James Jones. The followini^ payments were made and endorsed on this note : March 16,'$100 ; June 18, $100 ; Sept. 1, $200. What was due Jan. 1, 1898, interest at 7% ? 34. 1. A person sells 3% stock at 96 and invests the pro- ceeds in 5% stock at par. How much per cent, is his income increased ? 2. A owes B $900, of which $800 are due in 4 months, $400 in () months, and $200 in 9 months. What is the equated tim^^ for the payment of the whole amount ? 3. What is the compound interest of $2000, for 2 years, at 4 per cent., payable quarterly? 4. What readinyf on a Fahrenheit thermometer cor- responds to 170° C? 5. Find the sum of 32 terms of the series 1, IJ, 2, etc. EXAMINATION PAPERS. 121 35. 1. A father said to his son, who was 12 years old, "My ag-e is equal to ^ of the sum of your age and mine, with your age added." How old was 'the father ? 2. A man paid $54 for insurance on a house, at SJ per cent., and found that if the house was destroyed by fire he .would recover its value, the premium paid and $5 besides. What was the value of the property ? 3. What is the difference between the simple and compound interest on ;^'180O for 3 years at 6 per cent ? 4. A farmer sends his agent 415 barrels of potatoes, which the latter sells at $1.52 a barrel, charging 2-h% com- mission. He invests the net proceeds in cloth at*'$1.95 a yard, charging 3J% commission. How many yards did the agent send the farmer ? 5. Find the sum of the terms of the progression |, y|, etc., to 63 terms. _7_ 12» 36. 1. If 4 men, 5 women, boys or 8 girls can do a piece of work in 47 days, how long will it take 2 men, 4 women, 5 boys and 8 girls working together to do it ? 2. The discount on a sum of money at 5% is $108, and the interest on the same sum at the same ""rate and time is $125.28. Find the sum and the time. 3. 60 yards of carpet 27 inches wide are bought to cover a room 23 ft. 6 in. by 18 ft. The carpet cost 4s. 6d. per yd. , and the remnant was sold at 3s. 4d. per yd. What was the cost of carpeting the room ? 4. A bill due 4 mos. hence is discounted at 77 (true discount), and $1267 is received for it. What is its face value ? (Days of grace not reckoned). 5. Assuming a sovereign to be worth $4.86f in Canada and 25 francs 2 centimes in France, what wdi a traveller lose who changes £15 at the rate of 25 francs for ^'1 ? 122 EXAMINATION PAPERS. 37. 1. I lent a friend $1750, which he kept for 16 months. How long- must he lend me $700 to return the favor ? 2. I can sell my house for $4000 cash, or for $5000 payable in two years. If I accept the latter offer and receive its present worth in cash, at 10% discount, how much better off shall I be than if I accept the former offer ? 3. Sum the progression IJ, 2|, 4, etc., to 10 terms. 4. How much stock at 105 J, brokerage J%, can be purchased for $4781.25 ? What is the brokerage ? 5. How many gallons in a rectangular cistern 16 ft. long, 10 ft. 6 in, broad, and 8 ft. 4 in. deep ? 38. 1. Find the cube root of 5 to four decimal places. 2. If 30 men can dig a trench 108 ft. long, 8 ft. 9 in. wide, and 9 ft. deep, in 10| days of 6f hours each, how many days of eight hours each will it take 24 men to dig a trench 96 ft. long, 12 ft. 10 in, wide, and 12 ft. deep? 3. What must be the face of a note made on Jan. 9, at 3 months, in order that, if discounted at once, at 6%, the proceeds will be $200 ? 4. A merchant asked for goods 12|^% more than they cost him, but was obliged to take 12^ less than he asked. What per cent, of the cost did he lose ? 5. Three men bought a grindstone, 4 feet in diameter, paying equal sums. The first ground off his share ; the second an equal share, and likewise the third. If one- fourth of the grindstone was left, what was the thickness ground by each ? 39. 1. Ether boils at 95° F., and mercury boils at 662° F. Reduce these temperatures to the centigrade scale. 2. A merchant wishes to borrow $96.91 on a bill made on July 6th, for three months. What must be the face of the bill, interest being reckoned at 8|^% ? EXAMINATION PAPERS. 128 3. What sum must a man invest in Dominion 6's at 101 in order to have a clear income of $1775.50, after pay- ing- an income tax of If cents on the doUar on all over $400 ? 4. A steamer going- easterly in 0" latitude at the rate of 14 knots an hour, meets a ship in longitude 53° 25' west, and 10 hours later meets another ship. In what longitude did the steamer meet the second vessel ? 5. Calculate the specific gravity of alcohol from the following data : Weight of flask empty =14.3?j6 grams. *• ** '' filled with water =29.654 ** ** '* ** '' alcohol = 26.741 '* 40. 1 . The population of a city increases j^^ each year, and Its present population is 34560. What was ics population two years ago ? 2. A railroad runs through an estate for 18 miles, occupying a space 83 yds. wide, valued at $5.67 per acre. The owner receives compensation in land worth 12J cents per sq. rod. How many acres must he receive ? 3. An agent received a. consignment of wheat which he sold at a commission of 4 per cent., and invested the proceeds in sugar, less a commission of 5 per cent. His whole commission amounted to $107|. What was the value of the sugar bought ? 4. What must be the marked price of goods which cost $6 that the merchant may take off 10% and still make 25% profit ? 5. A cubic foot of copper weighs 550 lb., and tin 462 lb. What is the weight of a cubic foot of a mixture of 5 parts copper and 3 parts tin ? 41. 1. What is the diameter of a circular field which con- tains 20 acres ? 2. When a florin is worth 19f cents, what must be 124 EXAMINATION PAPERS. i paid for a draft on Vienna which will pay a debt in that city of 80J florins ? 3. When the days are of equal length, and it is noon in London (1st meridian), on what meridian is it then sun- rise ? sunset ? midnight ? 4. A box is made of plank 3 J in. thick. Its dimensions on the outside are 4 ft. 9 in. by 8 ft. 7 in., and its height is 2 ft. 11 in. How many square ft. did it require to make the box and how many cubic feet will it hold ? 5. A man who has a garden 100 ft. long and 80 ft. wide wishes to enclose it with a ditch, to be dug outside, 4 ft. wide. How deep must the ditch be dug so that the earth taken from it and placed on the garden may raise the surface 1 foot ? 42. 1. A hired a house for 1 year at $800 ; at the end of 4 months he takes in B, and at the end of 8 months he takes in C. What rent, in equity, must each pay at the end of the year? 2. What fraction of an acre in a triangular field whose sides are 7H, 84, and 90 yards ? 3. 20% of I of a number is what % of f of it ? 4. The highest common factor of two numbers is 23, and their least common multiple is 483. What is their product ? 5. How many liters in 10 imperial gallons ? 43. 1. A has 40 yards start of B. If B runs 7 yards for A's 5, how far must he travel to overtake A ? 2. Find the missing term in the proportion, 17 : ( ) : : 19 : 95. 3. The sum of the sides of a triangle is 162 feet and they are in the rates of 18, 20, 21. Find the area of the •Triangle. 4. A broker charged me 1J% for purchasing some banL notes that were selling at 25% discount, commission being charged on money invested. Three of the notes of $10 each and one of $50 became worthless. I sold the 'if ■ ' I J EXAMINATION PAPERS. 125 remainder at par and gained $520. What was tlie face value of the notes purchased ? 5. A farmer bequeathed ;^1200 to his tliree sons, leaving- V to his eldest son, I to the next and ]- tc the third son. How much should each receive in order that the intention of the testator may be carried out ? (Work by Proportion). 44. 1. When it was 1 o'clock a.m., on Jan. 1st, 1808, in Bang-or, Me., 68^ 47' west, what was the time at New Mexico 99° 5' west ? 2. If a credit of 3 mos. be allowed each item in the following" account, when might the whole be equitably settled by one payment ? Smith Fraser. 1898. • To David Singer. Jan. 8, To 30 bbls. fish, (a $7.00 $420 Feb. 4, *' 90 vds. carpet, '* 1.50 - - - - Mar. 22, *' 300 bbls flour, " 6.00 - - - - - 135 1800 3. For what sum must a note be drawn on July 3, at 3 months, so that discounted immediately at 7% it may produce $501.69? 4. If 3 per cent, stock be quoted at 90, how much must I invest in it so as to have an annual income of $2000, after payings a one per cent, income tax ? 5. If the hands of a clock indicate 3 o'clock when the proper time is one minute to three, and 4 o'clock when tlie proper time is half a minute past 4, what is the proper time when the hands of this clock coincide between 4 and 5 ? • 45. 1. What is the gain per cent, when 75% of the selling price is i less than the cost? 2. y^TT of A's money equals * of B's, and both together have $851. How much has each ? 126 EXAMINATION PAPERS. 3. What can a man afford to pay for stock which yields a dividend of 12%, so that he may realize at least 8% on his investment ? 4. What is the larg'est number of trees that can be set in a garden 120 yards square, so that the trees shall be at least 10 yds. apart and not less than 6 yds. from the fence by which the garden is enclosed ? 5. A British shilling is 37 parts pure silver and B parts alloy, and Q6 shillings weigh 1 lb. The United States dollar weighs 412.5 grains, and is 9 parts silver and 1 part alloy. How many shillings is a U. S. dollar worth ? 46. 1. Find the compound interest of $500 for 15 years at 5%, the amount of $1 for the same time and at the same rate being $2.0789. 2. What is the amount or final value of an annuity of $300 for 5 years, at 4%, simple interest ? 3. What sum will purchase an annuity of $240 for 3 years at 5%, compound interest ? What is the final value ? 4. A merchant dilutes a liquid, for which he paid 97J cents a gallon, with water, and sells the mixture at 73 cents a gallon. If he gain 20% on his outlay, how much water is there in every gallon sold ? 5. An artisan received $3.75 a day and his board for his labor and paid $1.25 a day for his board when he was 'die. At the end of 100 days he had saved $200. How many days did he work ? 47. 1. A man hired with a fa^-mer for $1.25 a day and his board, for every day he worked. On idle days he paid 50 cents for his board. At the end of 100 days he received $90. How many days did he work ? 2. If I purchase bank stock at 28 per cent, premium, and the bank pays a dividend of 9%, what interest do I receive on my investment ? 3. Which would be the more advantageous way for the Dominion to borrow money : on bonds bearing interest s EXAMINATION PAPERS. 127 uo.^"^^^' "^"^i ''^^^'"^" ""^ ^^' O'' o" bonds bearinsr interest at 8/^ and selling- at 101 ? '^ ri^'lv/^'^t'^^.'^^f''''"^ ^" ^''^^^^ ^O-*^- ^''' '^ "months is Xlo lOs., what IS the rate? 5. Find the cube root of .0001 to -4 places o( decimals. 48. fn ^'/^ ^^oT/' '''''^^' ^^ ^^"^^ '" '^ P^^'-t of his pasture, to contain 225 square rods, as a field on which to raise oats. Allowing himself $1.50 per day for his labor, fencing costs him 40 cents a rod. He marks out a field 25 rd long- and 9 rd. vvide. How much less would it cost to fence a square field of the same area ? 2. About how many miles must a man walk to ploug-h an acre, turning- a furrow of 9 inches? 3. What is the length of a rectang-ular field, 80 yds wide which contains 2 acres? Of a field 100 yds. wide which contains 5 acres. ' ^ ' /^"^^e feet of iron plate, -j- of an inch thick can be made from a cylindrical shaft 20 ft. long- and 4 inches in diameter? ^ 2. How long a trip up a river and back again to the starting point can be made in 6 hours by a small steamer which can travel 10 miles an hour with the current and 5 miles an hour against it ? Find also the velocity of the currentt -^ 3. A B and C were partners in business. A's capital was t of B's, and B's was | of C's. A's capital was in^he 128 EXAMINATION PAPERS. business 8 months, B's 9 months, and C's 10 months, and their net ^ain was $5348. What was the share of each ? 4. A man, whose horse is tethered by a halter to a stake in a grass field, uses a rope HO ft. long". He wishes the horse to have one-third of the fresh g-rass which the full length of the rope would give him in tiie forenoon, one-third in the afternoon and the remaining third at nig-lU. What length of rope must he allow the horse in tiie fore- noon and in the afternoon ? 5. Thomas Forbes g-ave Hugh Miller on March 10, 1898, mdse. worth $75; April 20, mdse. on 2 mo. $120; and May 80, on 3 mo. mdse. $180. Hugh miller paid Thos. Forbes on March 27, $G0 cash ; on April 15, a 10- day draft for $100, and on June 4, a 80-day draft for $50. When should the balance be paid ? 50. 1. The first term of a geometrical progression is 4 J, the last 4-J5, and the ratio J ; what is the sum of the series ? 2. A watchmaker sold 20 watches that cost him $8 a piece on the following" plan : He secured 20 persons who contributed $1 each, and at the end of a week one of them drew a watch by lot and retired. The remaining- 19 con- tributed $1 each and at the end of the second week one drew a watch by lot and retired, and so on until every person received a watch. What profit did the watchmaker reap. 3. A and B enter into partnership. A's capital is to B's as 5 to 8 ; at the end of 4 months A withdraws \ of his capital, and B | of his ; their gain, at the end of a year is $4000 ; how should this g"ain be divided ? 4. Stock bought at a discount of 80%, yields a dividend of 4% every six months. What is the annual rate per cent, on the investment ? 5. A cargfo was insured 4J%, to cover J of its value The premium was $122.50; what was the value of the cargo ? EXAMINATION PAPERS. 129 51. 1. If 7 men or 12 boys hoe a field of potatoes in days, working" 10 hours a day, in how many days, of 8 hours each, can 14 men andQ boys hoe a field five times as large ? 2. A grocer was selling cans of peaches so as to gain 20%. He increased the price to $1.50 pc* dozen cans and his profit rose to H3J%. At what rate per doz. was he selling them at first ? 3. A fruiterer buys equal quantities of apples at 2 for a cent and 8 for a cent, and mixes them. A customer comes and asks for 75 cents worth. How many can the dealer give him, selling at a gain of 25% ? 4. The stock in trade of two partners in a tea busi- ess consisted of 1020 chests of tea. B's share in the business being f of A's. Having agreed to dissolve partnership, B took 900 chests, A took 120 and received $1050 from B. Tea then rising 16§% in price, each sold his stock. After the sale, what fraction is A's property from this source, of B's ? 5. A square field is surrounded by a wall. The part immediately within the wall, all round the field, is covered with gravel, and is 13 ft. in width, and two straight g"ravel walks, IB ft. in width, join the middle parts of the opposite sides. The ungravelled part of the field contains 2^ acres. What is the length (in yards) of the bounding wall? 52. 1. The edges of a rectangular solid, whose content is 64 cubic feet, are in the ratio of 1 : 2 : 4. What are the lengths of the edges ? 2. A man hired a team to drive from Long Point Bridge to Port Hood, a distance of 15 miles, and back again, for $3, with tne privilege of taking in 1 or 2 persons at any place on the road. When 4 miles from Long Point Bridge, he took up the miller, and when 10 miles from Long Point Bridge he took up the postmaster. On his return he set them down at the points at which he took them up. How much, in equity, should each pay ? 180 EXAMINATION PAPERS. 3. At what times between 22 and 23 o'clock are the hands of a watch at rig-ht ang^les, the hour hand making* but one revolution of the dial plate in 24 hours, and the minute hand making" a revolution every hour? 4. A ten-foot pole casts a shadow of 15 feet. What is the height of a tree that casts a shadow of 80 feet ? 5. What sum will purchase an annuity of $500 for 7 years, at 0%, compound interest, the amount of $1 for 7 years at G percent., compound interest, being $1.50363. 53. 1. A railway train which runs at the rate of 20 miles an hour leaves a station 60 minutes before another train going in the same direction and travelling at the rate of 30 miles an hour. In what time will the second overtake the first ? 2. How many rolls of paper (the roll is usually 7 yds. long and 18 in. wide), must be purchased for a room 18 ft. long, 13 ft. wide and 10 ft. high ? There are 18 in. lost in ** matching" each strip with the next, but that is to be considered as off-set by the doors and windows. 3. At 90 cents per cubic yard what will it cost to build a brick and cement wall 72 ft. long, 7 ft. high, 3 ft. thick at the base and 18 in. at the top ? 4. A cubic foot of air weighs about .0808 lb. and a cubic foot of hydrogen about .0056 lb. A spherical balloon, the diameter of which is 20 ft., is filled with hydrogen gas. What weight will it support ? 5. A ship sails due north for { hours at the rate of 7 knots an hour, and then due ea?.t Tor 4 hours at the rate of 8 knots an hour. How many knots is she then from her starting" point. 54. 1. Divide $4941 among A, B and C, so that 3i% of A's share, 3f% of B's share, and 4|- of C's share may all be equal. 2. The excess of the present worth of a sum of money EXAMINATION PAPERS. 151 due in 1 year, interest 6%, over the present worth when interest is reckoned at G% is $0.60. Find the sum. 3. A, B and C enter into a partnership. A ird B put in $8475 ; A and C $3660 ; B and C $8H76. They gain $2062.60. What is each one's share in the profits: 4. A merchant boug-ht 200 meters of cloth in France at 16J trancs per meter, and after paying freight and duty amounting to 12J cents per yard, he sold it in Halifax at $4. 62 J per yd. What was his gain ? 6. If 4 men mow 15 acres in 6 days of 14 hours each, in how many days of 18 hours each can 7 men mow lOJ^ acres ? 55. 1. What per cent, is gained by buying stock at a dis- count of 20% selling it at a discount of 1U% ? 2. If the true discount on a bill due 8 mos. hence (days of grace not reckoned), at 7^ is $9?. 50, what is the amount of the bill ? 3. A man invested $28700 in U. S. 6's at 107J, the brokerage being i%. What will be his clear income after an income tax of 5% has been deducted ? 4. A bill of ;£'90 10s. is due in Edinburg, Scotland, by a school section in Canada for apparatus imported. What will the secretary of trustees pay for a bill of exchange to liquidate this debt when sterling exchange is quoted at 109f ? 5. A grain of gold can- be beaten into a leaf 8 in. square. How many of these leaves laid one upon another would make a block 1 in. thick, the weight of a cubic inch of gold being ^ ounces. u 56. 1. A man having invested $10000 at 4%, payable half-yearly, wishes to receive his interest in equal portions payable monthly and in advance. How much should he receive every month ? 2. A and B hired a horse and wagon for 30 days for 182 EXAMINATION PAPERS. r $30. A alone used it 10 days and B, 5 days. They used it together the remainder of the time. How much should each pay ? 3. A man invests a certain sum at 5% and twice the sum at 6%. His income from both investments amounts to $1700. How much has he invested at each rate ? 4. A man who owes a merchant a debt payable in 7 months agrees to pay one-half cash and three-eighths in 6 months. What time should he be allowed to pay the balance ? 5. The head of a fish is 12 inches long, the tail is as long as the head and half the body, and the body is as long as the head and tail together ; what is the length of the head and tail? 57. 1 . A merchant in Ontario sends flour to Prince Edward Island to be sold on commission at 5%. His agent is to invest the proceeds in herring after deducting his com- mission at 3%. How much will be invested in herring? 2. What is the sum of the numbers from 1 to 90 in- clusive ? 3. A lad bought a lot of oranges at 2^ cents a piece and twice as many at 2 cents a piece. He sold them all at 88 cents a dozen and made $4.20 profit. How many did he buy ? 4. A cubical box is made to just hold a globe. If the depth of the box is 15 inches, what is the solidity of the globe ? 5. Work by proportion : If 17 men in 24 days earn $612, in how many days will 68 men earn ^ that sum ? 58. 1. If the population of a country 'vhich is 450528 in 1891 increases annualiv 10 per cent, what would the popula- tion be in 1901 ? 2. A ladder 52 ft. long stands close against the side '^J^ ' EXAMINATION PAPERS. 188 of a building. How many feet must it be taken out at the bottom, that the top may be lowered 4 ft. 3. A man purchases goods as follows : June 4, 1897, $240.75; Aug. 9, 1897, $187.25; Aug. 29, 1897, $65.64; Sept. 4, 1897, $280.36; Nov. 12, 1897, $36. If all the goods are bought on a credit of 6 mos. , on what date may the bill be settled by paying the amount of the several purchases ? 4. A's money is 28% more than B's. What per cent, is B's less than A's ? 5. Find the difference in the incomes obtained by in- vesting $7700 in 2f per cent, consols at 96 J, or an equal sum in 4 per cent, government stock at 110. 59. 1. A man invests $1761.75 in 3 per cents at 97J, and sells out when they have risen to par. What sum does he clear, reckoning brokerage ^% both in buying and selling? 2. Extract the i^quare root of 1191078144. 3. Extract the cube root of 94818816. 4. What is the value in sterling money of 294.32 francs when the course of exchange is 24.90 francs per pound sterling? 5. The distance from X to Z is 170 miles. A train, travelling at the rate of 40 miles an hour leaves X at 10 o'clock, and at the same time another train leaves Z travelling at the rate of 50 miles an hour. If the latter train stopy from 11 tj 11.30 and then starts again at the same rate as before, when an J where will they meet ? 184 ANSWERS. ANSWERS. 5. 11. 18. 24. Exercise I.— (Page 7).- i. 6. 2. ion. 3, 2t;^. 4. 161?, mi 6. m- 7. ^^M. 8, 24HI. 9. 12^. 10. s^w 12. A. 13. ^lU' 14. 5^^ 15. ?i. 16. Vt' 17. H^ 19. A. 20. tVs. 21. liVs- 22. 25?^. 23. 16g|. ISA II' 'BIT' 25. 3^1 a. 26. 6. 27. 68^. 28. ItV 29. 2. 30. f Exercise II.— (Pag:e9).— Exercise III.-(Page lO).— l. 53U. 2. 29^^. 3. .093567. 4. 2. 5. .992424. 6. 366916§. 7. 2J. 8. 3.2034. 9. 37 mi. 4 fur. 1 ft. 6f in. 10. 13s. 2id. 11. 180.7 + . 12. 1.7142. 13. 11910.646 yd. 14. 436340.52 sq. in. 15. h Exercise IV.— (Page 12).— l. .6, .lA, .83. .714285, .6428571, .318. .0i42857, .14583, .1. 3.01, 200.012345679, .55681. Exercise V.— (Page 13).— i. |, ^j, ,\. 2. 2. 3. 4. 3. 5. 2. 5. 3. 6. 3. 8. 5. 9. «8« 9 a TJ > TTTTt rtrSTS' h\\ 6A5.^ 14^1". 10059 A 1 3S 18 fi 29 40 ^ Su5> 1 n 6. 8r« JJ7 32 1. 60 1 T3^7' 533> ■9Su5> i>()9' 'J' '='1J09> "SfO^P, Exercise VI.-(Page 14).-1. h, |f, ^,v »^, -S^j;^ .Onia Q 1913' 79 70 A TC5» OOOJ uuuj' "• IfiJoJj TF50U> ^^ffTT* *• :i4t' OQ 8 7 -t 1 2. 2.896178807069. 5. 17.8092502138. 2. 291.5524. OftS^SSJ* 17sJ8^o» 64{,V 6. 4^^^^, 4 Exercise VIL— (Page 17).— 1. 14.769587. 63.8198038274. 4. 52.526228203901471. 339.625268352. Exercise VIII.-(Pagel8).— l. 71.86193. 13.81824. 4. 44.789. 6. 500.916*. 6. 3.9046. 7. 1218.6. .61364073i. 9. 3451.386. 10. 0. Exercise IX.— (Page 19).— l. .03. 2. 9.928. 3. .082. 4. 1.8. 889.185, 6. 778. i48. 7. 760730.518. 8. 81.79i. 34998.4199003 + . 10. 2.297. > i ANSWERS, 136 ExePCiseX.— (Page 19).— 1. 1.31034 + . 2.55.69. S. 4.8731707. 4. 7.72. 5. 2. 6. l.i. 7. 1890.303. 8. .00013. 9. .06. 10. .lO'J. Exercise XL— (Page 23).— l. 23. 2. 29. 3. 35. 4. 47. 6. 46. 6. 11. 7. 51. 8. 67. 9. 75. 10. 89. 11. 93. 12. 77. Exercise XII.~(Pag:e 25).— l. 127. 2. 253. 8. 309. 4. 408. 5. 599. 6. 608. 7. 647. 8. 807. 9. 907. 10. 897. 11. 985. 1 2. 4968. 13. 3973. 14. 1025. 16. 2187. Exercise XIII.-(Pag:e 26).- 1. 1.442. 2. .669. 3. .310. 4. ^ or .571. 5. f or .714. 6. .480. 7. 4.973. 8. 7.640. 9. 9.654. 10. .497. 11. 1.072. 12. 1.912. 13. 3.666. 14. .046. 16. 3.932. 16. 6.561. Examination Paper No. I.— (Pag^e 27) l. $3.59 + . 2. .2070. 3. .2828. 4. 25H. 5. j{;85 ils. lOj^d. Examination Paper No. IL— (Page 27).— 1. $rfU. 2. 900 rd. 3. .1082. 4. 4444.65 lb. 5. 11. Examination Paper No. III.— (Page 27).— 1. 82^%. 2. 109^%. 3. 2.12220766. 4. 6s. 8d. 6. $3.06, Examination Paper No. IV.- (Page 27).— 1. $1800. 2. 41.2 ft. 3. $44,765. 4. 2200 acres. 6. 2.55 + . Examination Paper No. V.— (Page 28).— 1. 4.I888 cu. ft. 2. .8796. 3. 136.8 sq. ft. 4. 1. Examination Paper No. VI.— (Page 28).— 1. 62i%. 2. .6459368. 3. 20 sq. rd. 4. 31c. 6. 1 ft. 1 in. ExerciSO XIV.— (Page 32).-l. 13112, 16786, 14620. 2. 114241, 44132, 2042022. 3. 9832, 126683, 143806. 4. 3950, 6319, 62033. 6. 7e58, 39619, 85013. 6. 311522, 2214501, 11143^123. 7. 2766, 22875. 8. 47056, 29638, 21. 9. 13226, 39148, 578. 10. 15916, 76296, 58118. ExereiseXV.—(Page 32) -1.154143,. 2.204428,. 8.204414,. 4. 2t7715ii. 6. 6372702,. 6. 473126656,. 7. 5123417^. 8. 4124302.. 9. 2146ee9,,. 10. 83226^. Exercise XVI.-(Page 42).— 1. ;^?2 6s. 7id. 2. 124 lbs. 80Z. 16grs. 8. 124 lbs. 8 oz. 16 grs. 4. 6 t. 19 cwt. 90 lbs. 6 oz. 8 dr. 6. 108 mi. 4 fur. 3 rd. 4 yd. 8 in. 6 . 1098 bv . 1 pk. 1 gal. 2 qrt. 1 pt. 7. 8 wk. 3 da. 1 hr. 44 mm. 36 sec. 8. 7 Hm. 8 Dm. 9m. 2 dm. 4 cm. 7 mm. 9. 5 cu. yd. 7 cu. ft. 703cu.in. 106 ANSWERS. 10. 6 sq. rd. 1 sq. yd. 3 sq. ft. 67 sq. in. U. 44010 far. 12. 767904240 sec. 13. 403710 cm. 14. 752112 dr. 15. 1214 pt. 16. 40371000000 qcm. 17.1865106 m. 18. 120166 gr. 19. 120166 gr. 20.16250032 sq.m. Exercise XVIL— (Page 43.— l. 4 T. 7 cwt. 72 lb. 2 oz. 12 dr. 2. 4180 mi. 1 fur. 13 rd. 2 yd. 2 ft. 6 in. 3. 357 A. 27 sq. rd. 26 sq. yd. 8 sq. ft. 107^1. 4. 35101 ; ;^24 Os. lO^d. 6. 297 lb. 8 oz. 18 dwt. 8 gr.; 3298 lb. 3 oz. 3 dwt. 8 gr. 6. 1 lb. 2 oz.; 3 cwt. 19 lb. 8 oz. 7. 3;^; 12s. 2fd.; 14 farthings. 8. 1C8 ; No remainder. 9. 4 cwt. 87 lb. 7?f dr. 10. 1 mi. 7 fur. 23 rd. 3 yd. 2 in. 11. 10 A. 65 sq. rd. 16 sq. yd. 4 sq. ft. 136 sq. in. 1 2. 2379980 min. 13. 4: <' "^ fur. 9 rd. 2yd. 2 ft. 7f in. 14. 12|^VriW»- 15. 9 cwt. 421b. 10 oz. i HO. 17. The steamer by 98 yds. 18. 5 ft. 3 in. 19. 2 lb. 5 oz. ;i scr. 15^|^^ grs. 20. 1 cwt. 12 lb. 4 oz. Exercise XVIII.— (Page 44).— l. .282. 2. 15 cwt. 56 lb. 4 02. 3. .872595 + . 4. 6 oz. 5 dr. 19.C8 gr. 5. 5f. 6. £2 6s. 8,8yd. 7. 4 yd. 2 ft. 2.566 in. 8. .09027. 9. »15.74. 10. $4:Wr " ^ ,: - 1 1. 80516.4 in. 12. 3884. 13. $16.50. 14. 107 sq. rd. 2 sq. yd. 6 sq. ft. 51.6 sq. in. 15. 7.164116 days. I » Exercise XIX.— (Page 45). -l. 163.5™ 2. 427 yd. 1 ft. 10 in. 3. 250™ 4. 729.24 + . 5. 89.6Km. 6. 9451. 7. 100 gal. 8. 312 bu. 3 pk. 1 gal. 9. $204.70. 1 L $.299. 12. 56.764 lb. 13. 855g . 15. 16 T. 10 cwt. 69 lb. 16. 1.200*. 10. $55.02. . . . 14. 12008 17. 132.05 1b. 18. $187.20. 19. 2 lb. 9 oz. 15 dwt. 19.032 gr. 20. 1.4n». ^ Exercise XX.— (Page 46).— l. $133,022. 2. ;{;i75 3s. lOV^d. 3. 600 marks. 4. $180.0708. 5. 837^ drachmas. 6. $.2261. 7. $2059.7115. 8. 22321.428 florins. 9. 9951.44 marks. 10. $744.98. 11. $744.98. 12. $744.98. 13. 2000 francs in each case. 14. 6000 crowns in each case. 15. 105^^ pfennigs, 74^2 kreutzers. 16. 3 rupees, 1^ annas. 17. 49 roubles 20 copecks. 18. £S9 13s. l|d. 19. 1367 marks, 14 pfennigs. 20. 186.56 ore, 463 reis. ANSWERS. 137 2. 4. 7. 10. 12. 14. 1. 4. 4. 7. 11. 14. 2. 5. 4. 10. 15. 19. 3. 7. 12. 17. 4. 9. 2. 4. 9. 14. Exercise XXL— (Page 48).— l. 45 min. 38 sec. past 7 o'clock. 1 hr. 3 min. 2 sec. 3. 20 min.; 63 min. 22 sec. 1 hr. 38 min. 4 sec. 5. 20° 36'. 6. 26° 16'. 14 min. 22 sec. faster. 8. 73° 21'. 9. 25 min. 34 sec. past 16 o'clock. 4 hr. 12 min. 41 sec. 11. 51 min. 56 sec. past 6 o'clock. 5 hr. 68 min. 51 sec. 13. East ; 1425 geographical miles. 2 hr 20 min. 49 sec. 15. 3 hr. 13 min. 38f sec. 16. 33° 61' 45*^ Examination Paper No. VII.— (Page 49).— 24 min. past 17 o'clock. 2. 4 li. 2J m. 3. .237083. 576000 T. 5. $110.05. Exercise XXII.— (Page 52).— l. 149°F. 2. 30°C. 3. 68'F. 78rC. 5. IIOX, 167rC 6. 104°, 122°, 140°, 158°, 176°, 194°, 212°F 45°, 55°, 60°, 80°, 9o^C. 8. 10°C. or 18°F. 9. 77°F. 10. M'C. 8°, 52°, 63°, 9rC. 12. - 38J°F. 13. 537r, 5875°, 676|°, 898f °C. Equal. Examination Paper No. VIII.— (Page 53).— i. I0924i^j. 483Vff kreutzers, or about 16 cents. 3. 33^ miles. 4. 73.3 ft. 1.2345679. Exercise XXIII.— (Page 53).— i. $o. 2. $2100. 3. 160. 21i|%. 6. 9^^% gain. 6. 88|%. 7. lli%. 8. 12^%. 9. 19%. 66r/o, 29Jf%. 11. 52i%. 12. 8|%. 13. 2fO/^. 14. 12^°^, 14f%. The latter by $19.50. 16. 46^%. 17. $3.98. 18. $150. $3200. 20. $32|. Exercise XXIV.— (Page 55).— l. $54.09. 2. $50.69. 36x»7% nearly. 4. $143.28. 5. $1.25%. 6. $22.31. $2211.20. 8. $24.83. 9. $44.78. 10. 500 bbls. H. $5.46. $67.50. 13. $9000. 14. $289.40. 15. $4397.25. 16. $4400. $2600. 18. $4300. 19. $900. 20. 17^f%. Exercise XXV.— (Page 67).— 1. $1.08. 2, $341.55. 3. $380.10 7i%. 6. $90.68. 6. $1273.30. 7. $1628.15. 8. $2493.75. $434.26. 10. $969.75. Examination Paper No. IX.— (Page 68).— i. 47/5 /o» $2029.67. 8. 24^15%. 4. 625. 5. $15,603. Exercise XXVL— (Page 59).-l. $117. 2. $163.23. 3. $169.60. $136.71. 6. $52,685. 6. $3.95. 7. $287.93. 8. $89.28 + . $66,855. 10. $91.97. 11. $467.16. 12. $69.55. 13. $46,715. $14.43. 15. $238 38. 16. $392.14. 17. $4898.19. 138 ANSWERS. 18. $490.22. 19. $2249.10. 20. $550,112. 21.7%. 22.6%. 23. 42%. 24. 1 yr. 135 days. 25. If years. 26. 1 yr. 216 days. 27. $9250. 28. $927.10. 29. $550. SO. $500. Examination Paper No. X.— (Page 60).— 1. 24 hr. 42 miii. 2. $1264.50. 3. $92. 4. $580. 5. $33. Exercise XXVIa.— (Page 62).— l. |113.31. 2. $79.67. 3. $104.59. 4. $282.86. 5. $233.36. Exercise XXVII.-(Page 63).-l. $6.16, $843.86 2. $5.42, $744.58. 3. $1.85, $498.15. 4. $.74, $74.26. 5. $5.07, $1254.93. 6. $30.35, $5399.65. 7. $31.24, $5398.76. 8. $3.79, $481.21. 9. $3.92, $481.08. 10. $3.92, $481.08. Exercise XXVIII.— (Page 64).— l. $750, $75. 2. $900, $162. 3. $315, $37.80. 4. $1255, $263.55. 5. $9250, »1618.75. 6. $72.85, $18.03 7, $925, $74. 8. $10,000, $49.31. 9. $500, $78.82. 10. $699.44; $64.84. 11. $.64. 12. The latter. 13 ^2(i40, 14. $1680. 15. $824. Examination Paper No. XL— (Page 65).— l. is^v yr. 2. $500. 8. $98 ; ,V ; ^k 4. 20%. 5. 15 yr. 1 mo. 20 da. Exercise XXIX.-(Page 69).— l. $2531.25. 2. $16912.50. 8. $207.90. 4. $1000. 5. $4000. 6. 50 shares. 7. $1071.43. 8. $573.94. 9. $892.51. 10. No gain or loss. 11. The latter. 12. The former. 13. 62^%. 14. $80. 15. $5967.50. 16. $3788.80. 17. $27309.38. 18. $220. 19. The latter. 20. A'?- 21. 76^. 22. $217. 23. $14400. 24. $6936. 25. $2214.50 Examination Paper No. XII.— (Page 70).— l. 20gai. 2. $50. 8. $946.47 gain. 4. $17.60, $482.50. 5. 30j=V cents a lb. Exercise XXX.— (Page 73).— l. $860.85. 2. $1556.10. 8. $873.69. 4. $210.93. & 4853.93. 6. $642.40. 7. $3540.94. 8. $997.22. 9. $760.62. 10. $4926.11. 11. $1992.63. 12. $917.0L Exercise XXXI.-(Pagre 77).— 1. $1727.30. 2. $4.82} for each ;^1. 3. $1523.19. 4. $682.60. 5. $1418.65. 6. $1657.09. 7. $1639.17. 8. $204.68. 9. $1998.36. 10. 14.8665. Exercise XXXIL— (Page 79).— 1. 2 mos. 18 da. 2. 41 da. 8. Si mo. 4. June 29th. 5. Aug. 1901. 6. June 6th. 7. $375. 8. $1068.04 ; equated time Sept 23. ANSWERS. 189 0. 3. ). Exercise XXXIIL— (Page 83).-l. Nov. 29th, 1897. 2. July 6th, 1898. 3. March 11th of the same year. 4. 6^ mos. 6. June loih, 1896. Exercise XXXIV.-{Page 84).-l. |. 2. 8f hr. 3. 6i da. 4. 20. 5. 3 da. 6. 92^. 7. 4^V da. 8. 7^. 9. 24^. 10. 13. Exercise XXXV.-(Pag-e S^y—i. le^^r min. past 3. 2. S2^ min. past 6. 3. 27-fV min. past 5. 4. 49 ^^ min. past 9. 6. 43^7^ min. past 8. 6. 54y^ min. past 8. Exercise XXXVI.-(Page88).-l. «; I; i. 2. A; §; 9. 3. ■f;7;i. 4. ^;A;t. 5. 2;J;f|. 6.64. 7. 4|, 8. 3:10; 27:2; 1:4; 3 : 10 ; 11:35. Examination Paper No. XIII.— (Page 88) — i. ii.0665483. 2. 3.6. 3. $505.30 (term of discount being 2yVmo.) 4. 30 sq. rd. 2 sq. ft. 72 sq. in. 5. 52|c. Examination Paper No. XIV.— (Page 88) — 1. $6000, $5000, $4000, respectively. 2. $104.3125 loss. 5. 3/^04. 4. 1.226. 6. 317. Exercise XXXVII.-(Page 90).-l. 16;40. 2. 6;6^ 3. 36; 15. 4. 38i, IH. 5. 27 yd.; 13^ lb. 6. $10.50. 1, £l 2s. 8. ^i^. Exercise XXXVIII.— (Page 9i).-l. 25 mi. 2. I66.2+ yd. 3. 32A. 4. 24f Jc. 5. 4800. 6. $3.50. 7. $192,855. 8. $10. 9. $7. 10. $2.46. 11. $62.72. 12. 4. Exercise XXXIX. -(Page 93).— l. 58. 2. 15. 3. 23. 4. Ci. 5. 46. 6. I 7. 17. 8. .3. Exercise XL.-(Page 94)._1. 165. 2. 324. 3. 168. 4. 1550. 5. 237 J. 6.5060. 7. 6400 ft.; 624 ft. 8. $409.50. Exercise XLI. -(Page 96). -l. 256 ; 512. 2. 16384 ; 262144. 3. 256; 4096. 4. 3i ; ^y,. 5. 1^; 2ffa ; 4|nf 6. „W 7. i. 8. 34H. Exercise XLII.-(Page 97).-3. 349525. 4. HH' 6. 238^. 6, 2047. 7. 39P4. 8. 2i|«f. 9. $1.27. 10. $31000. H. $102a Exercise XLIIL— (Page 98). -l. 24. 2. l. 3. 18. 4. 625. 5. 1024. 6. 7i. 7. 111*. 8. IJ. 9. I 10. A- 140 ANSWERS. h 1 Exercise LXIV.~(Page 99).— 1. $524.05 ; $84.06. 2. $1125.508 ; $126,608. 3. $760,726 ; $70,725. 4. $1012.255 ; $212,255. 5. $2649.79 ; $249.79. 6. $1864.405 ; $54,406. Exercise XLIVa.— (Page 101).— 1. $2163.86; $2800. 2. $3240; $2700. 3. $1667.60 ; $1381.06. 4. $7520 ; $6371.42+, 5. $292,886. 6. $5607; $4441.188 + . 7. $408. 8. $500. 9. $136. 10. $1284. Exercise XLV.— (Page 103).— 1. $1261 ; $1089.30. 2. $1312.38; $1039.53. 3. $2809. 4. $1406.08. 5. $600. Exercise XLVL— (Page 104).— l. 470. 2. 120 ft. 3. 160 rods. 4. 76 yd. 6. 36. 6. 66.64 ft. 7. 52,2 ft. 8. .4099H°'. 9. 1.83»°'. 10. 494. 18 + mi. Exercise XLVIL— (Page 105).— l. 3.6A. 2. $10.97. 3. $5.40. 4. 276f rods. 5. $120,866. 6. $2750. 7. 60c. 8. 7.755 yd. 9. 17 suits and J yd. over. 10. $17.96. H. $83.73. 12. $566.76. 13. $506.88. 14. $161190. 15. 500 sq. ft. 16. 6.01634. 17. 8J sq. ft. 1 8. 60 yd. 19. 16 yd. 20. 1000 sq. ft Exercise XLVIII.— (Page 107).— 1. 2352.5 + sq. ft. 2. 15327.98 sq. yd. 3. 111.8 sq. ft. 4. 13.26 A. 5. 10 A. 73.12 + rods. Exercise XLIX.— (Page 107). -1. 81.6816 in. 2. 31.416 ft. 3. 24 ft. 4. 168 in. 4. 5000 ft. 5. 2827.44 sq. ft. 6. 962.116 sq. ft. 7. lA. 94.46 rd. 8. 3183.1 sq. ft. 9. 19 A. 143 + rd. 10. 75.006 sq. rd. 11. 490.089 sq. ft. 12. $157.08. Exercise L.— (Page 108).— 1. 81. 2. 168f . 3. 10000. 4. 66|. 5. 1020. 6.249^\. Exercise LI.— (Page 109).— 1. 34.328 cu. in.; 63.375 sq. in. 2,13601b. 3. 722.2 + lb. 4^ 16.478^^. 5. $182. 6. $247.27^. 7. $27.69. 8. 1944. 9. 74.78+ bus. 10. 14. 11. 3 ft. 8 + in. 12. IT. 8cwt. 4.41b.; $46.74. Exercise LII.- (Page llO).— 1. 197.9208 cu. ft. 2. 8 ft. 3. 549.78 cu. ft. 4. 8.83576 cu. ft. 5. $29.23. 6.11ft. 7. 305.4672 cu. m. 8. 8181.25 cu. in. 9. 1272.3488. 10. 233.98 + lb. 11. 122.786+ liters. 12. Conical part, 65.46 cu. in ;. frustum, 468.16 cu. in. Exercise LIII.-(Page 111).-1. 2.4. 2. 2.73. 3. 4. 4. 5. 6. 1468.8Kt. 6. 316.4KC. 7. .64. 8. 1.398. 9. 3.172. 10. ,*,. 1 1. .6. 13. 2.1625. 14. .844. ANSWERS. Ul EXAMINATION PAPERS. 15. (Pag-e 112).—!. $3.28 + . 2. .04. 3. Is. 7d. 4. 24 yd. 6. .822. •" 16. (Page 112).-!. 5«J,. 2. A, $15000. 3. $4000. 4. £32 14s. 7Hd. 6. 800. 17. (Page 113).-!. $1215. 2. $6.81. 3. 1661.066. 4. 49.9126. 6. lift. 8J in. 18. (Pag-e 113).— 1. 96. 2. 8.686875. 3. $394.35. 4. 224.6°?. 5. -35C. 19. (Page 113).—!. 3jV da. 2. $304.76. 3. $437.58. 4. $327.82. 6. 3 yr. 4G days. 20. (Page 114). 1. 32,\min. pastl2. 2. A. $1000. B. $1200, C. $1400. 3. A. $77.42, B. $120.03. C. $193.55. 4. $199.GG, $2.44. 5. 5%. 21. (Page 114).-1. 713. 2.100. 3. f|. 4. HSI^S- 5. $16.21. 22. (Page 115).-1. 13^. 2. U- 3. |. 4. 23/^ h. Take 100 feet as the depth of a hot and measured on the breadth of the field. The latter is more advantageous by $1168. Fraction left over is s%%. 23. (Page 115). 1. 21fd. 2. $59.40. 3. 13J cents. 4. $40,000, $36,000, 4,000, 3,G00 respectively. 5. $23G4. 24. (Page 116).-!. $1.76. 2. 210700 ; 1.9992194. 5. 1 ft. 4 in. 4.' ;^615. ■ 5. »C00 ; 13^ yr. 25. (Pagell6).—1. 13 min. pastil. 2.16000. 3.01.60. 4. 8100. 5. 03631.65. 26. (Page 117).—!. $4.01. 2. 70. 3. The former by $.04 + . 4. 15. 6. $.437. 27. (Page 117).— 1. 12%. 2. 32146 sq. yd. 3. 68800. 4. $56.47. 6. 1 ft. 9.8 in. 28. (Page 117).— 1. 1 hr.; 3, 4, and 4^ miles respectively. 2. 9tV%. 3. $38720. 4. $55.80. 5. $653^. 29. (Page 118).-!. 113.0976. 2. $7.50. 3. 34ic. 4. $1333i. 30. (Page 118).-!. 6560. 2. 14196. 3. 50%. 4. 4%. 5. 6%. 31. (Page 119).— 1. (A) In 5 hr.; 125 mi. from Halifax. (B) In 4^ hr.; also in 5.\ hr. (C) SS^t: mi. from St. John. 2. 14457f sq. ft. 3.9%. 4. $35.10. 6. 224.6^.. 352.4-F 142 ANSWERS. 32. (Page 119).—!. 34 J min. past 13 on Wednesday; 87* 51?'N. S. 7202.9. 4. 949.47. 6. oj cents (nearly). 33. (Paj,'el20).-1. 12 J2f. 2.3115.6.88626204872789. 8. $7833i. 4. 18 da. 5. $867.92. 34. (Paj-e 120).—!. 60%. 2. 6 mo. 3. $166.71. 4. 348. S'F. 6. 280. 35. (Page 121).—!. 52 yr. 2. $1381. 3. i;i9 16s. 6.912d. 4. 304. 5. 299J. 36. (Page 121).-!, 16 da. 2. $783, 3^ yr. 8. Ans. /lo 2s. 21d. 4. $1296.56^. 5. $.0579. 37. (Page 122).—!. 40 mos. 2. $132.23. 3. 201JU. 4. $4500; $53.75. 6. 8724.9. 38. (Page 122).-1. 1.7099. 2. 19f. 3. $203,105. 4. 1^%. 5. 1st, .26795 ft.; 2nd, .31784 ft.; 3rd, .41421ft. 39. (Page 122).— 1. 3o°C., 350°C. 2. $99.00. 3. $30300. 4. 49° 41' W. 6. .8099. 40. (Page 123).— 1. 30375. 2. 61.236. 3. $1142f. 4. $8.33^. 5. 517 1b. 41. (Page 123).—!. 63.8 rods. 2. $158. 8. The 90th meridian west, the 90th east and the 180th respectively, 4. 70^j sq. ft.; 29i cu. ft. (The box has a lid). 5. 5\^ ft. 42. (Page 124). -1. A. $183i, B. $S3i, C. $33^. 2. Uh 8. 12^%. 4, 11109. 6. 45.4 + liters. 43. (Page 124).—!. 140 yd. 2. 85. 3. 1134 sq. ft. 4. $2600, 6. i;553Ji, ;£;369^, ;{;276H respectively. 44. (Page 125).— 1. 58 min. 48 sec. past 10 o'clock, Dec. 81, '97. 2. June 6th. 3. $511. 4. $60606^^. 6. 22^ min. past 4. 45. (Page 125).- 1. 6|%. 2. A. $437, B. $414. 3.150. 4. 152. 5. 4.6, nearly. 46. (Page 126).— 1. $539.45. 2. $1620. 3. $668.68 ; $756.60. 4. f gallon. 5. 65. 47. (Page 126).—!. 80. 2. 7^%. 3. The former. 4. 4%. 5. .0464. 48. (Page 127).— 1. $3.20. 2. 11 miles. 3. 121 yd. ; 242 yd. 4. Sli cents. 5. 27,', ft.; 1^|^ sec. 49. (Page 127).— 1. 83.776 sq. ft. 2. 20 mi.; 2\ mi. per hour. 3. A.$896, B. $1512, C. $2940. 4. 17.82 ft., 24.49 ft. 5. Sept. 26» It ^« ANSWERS 148 50. (Pa&e 128).~1. 7,Vir. 2. $50. 3. A. $1714f , B. $2285f. 4. 1H%. 5. $4480. 51. (P'l&e 129).—!. 15 da. 2. $1.35. 3. 144. 4. g. 5. 492 yd. 52. (P^ge 129).-!. 2, 4, and 8 feet. 2. $1.7aj, $.935, $.33J. 3. 10?J and 41. » 5 min. past 22. 4. 53 J ft. 6. $2791.19. 53. (Pag-e 130).—!. 1 hr. 40 min. after it starts. 2. 19^3. 3. $37.80. 4. 315 lb. nearly. 5. 38.27. 54. (P'is:e 130).— 1. A. $1822.50 ; B. $1701 ; C. $1417.50. 2. $o5.65. 3. A. $609,375. B. $693.75, C. $759,375. 4. $356.99. 5.4. 55. (Page 131).- I2i%. 2. $2047.50. 3. $1520. 4. $141.44. 5. 307200. 56. (Page 131). -1. $32J5f. 2. A. $17.50, B. $12.50. 3. $10,000 and $20,000 respectively. 4. 3 yr. 2 mo. 5. 48 inches. 57. (Page 132).-!. VVir of value of flour. 2. 4095. 3, 720. 4. 1707.15 cu. in. 5. li da. 58. (Page 132).-!. 1168521. 2. 20 ft. 3. Feb. 1. 4. 21|%. 5. The latter $G0 more. 59. (Page 133).-1. $36 2. 34512. 3. 456. 4. ;{;il 16s. 4||d. 6. 10 min. past 12 ; 86J miles from X.