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Lorsque I ) document est trop grand pour dtre reproduit en un seui clich6. 11 est film6 A partir de I'angle sup^rieur gauche, de gauche A droite. et de haut en bas. en prenant le nombre d'images nicessaire. Les diagrammes suivants illustrent la mAthode. 1 2 3 32 X 1 2 3 4 5 6 A KEY TO THE SOLUTIONS OF PROBLEMS IN THE HIGH SCHOOL ARITHMETIC TORONTO : THE G. M. ROSE & SONS COMPANY. Limited. 1898. Entered according to Act of the Parliament of Canada, in the year one thousand eight hundred and ninety-eight, by Thb G. M. Rose & Sons Company. Limitbp, at the Department of Atjriculiure. This Key to the High Schoo/ Arithmetic has been prepared for the use of teachers. 16: the 9 -=1 ■rS 1( spao 11 m SOLUTIONS OF PROBLEMS IN THI HIGH SCHOOL ARITHMETIC. METEIO SYSTEM. 95 + 1000. • ^°°' '23406789^ 1000; 123486789.^ 1000 «• 8.66 X 1000 X 100 ; 6.632 X 1000; .2468 K 1000. le-MO^rKr "" """ -"■»««" •" "-6, 2.27, 1623, thin ster* " "■''■ ""* -"""««« «™ B67800, 13648.9, 9. 103 Km » 10300000 cm ^^'d ^'^f " f ^'^ ^^ « '^ ^ 2. -=103 Km4-64^1760=103OOnftA ^ = ?*™»J«8-j.64^ 1760 •^8 + 8^8^2^11) cm ^^'""^^^-^^^eo. (1030000 10. 66000 m in 3600 sec. , 66000 • ^Rm « "• In 1 min. the train goes 1 ^m i/SS."" P**" '^^ 'P^_.-. 1 8pace-60m^ Km- 1000 m. which ... -20 12. xne man's height = 5 ft '70.6.f39.37xl& cm. m m.- 70.6 ^39.37 SOLUTIONS OF PROBLEMS 18. 19. 20. 21. 22. 13. 29.6-^39.37xlOOO. 14 10 m - 1000 cm. In taking one-half of one part and two thirds of the other we have taken one-half the whole string and one-sixth of the other part : but one-half the whole htnng IS 500 cm. .-. one sixth of the other part is 100 cm. .-. the other part is 600 cm, and the one part is 400 cm. 15. No. minutes = 65 Km -r 80 m - 65000 m-^ 80 m -812 5 =s'°"^-^°» = 100cub.dm. ft. - ic! " ^ '^^^ *'"^- **'• =(39-37)» - 1728 cub. KA vT ^5a ^ o!J^' ^*- = 35.31 - 128 cord = &o. K?- I?^' t^^ ^ 3^ ^ irV cub. m = 24 8. 51 20 cub units cost $54. .-. 1 cub. nnit cost $2.70 = cost oM7s. .Mcub.unit=27s=27cub.m. .-. Hnear unitl 52. Vol.of first wall -15x1x3.4 cub. m^Bls- • each stere of wall contains 1000 bricks ; &c. " ' ^ BurfL^6cr:X' ""' '^' ^' *^« ^"^ ^« •* --*-- ••• -hoi. ^K^ Jl'' J^ ^T^ " ^^^ **^ ^**«' " 2i cub. m water - 2*a00^» cub. cm of water- 2f(100)»K-2*aOO^» 4. 1000 Tr.-li ""^ 8 water : 4^ TT of 1000 Eg-. otfiiUTIONS OF PROBLEMS thickness » .01 h- 80m. < 56. Vol.-(1.002)a8 = &c. 57. lcs=.01s = .01 cub. m. .01 -i- 80 X 1000000 microns - &c. 58. A vol. of 1 cub. m has a surface of | sq. m. .*. its depth must be f m = &c. 59. The vol. when the water is 1 d m deep=>4x 2.5x.l oub m "^ -4 5 J**^*"*"^ ^'^^ "^-4500 cu. dm, and .-.weighs 4500 Kg ??• ^It^^vi^ *^'®® "^'Sits as decimals ; the last six. ^^91. 2.679 lb. T. = 2.679 x 5760 grs = 2.679 x 5760 -=- 7000 Ih. 92. Sides of the triangle, in dm, are 13, 14, 15, .-. area- 84 8q. dm = .84sq. m = .84ca. 93. The trapezium -a rectangle of the same width 2 m long, . •. width = .375 m. 94. The height is 10 cm. .-. vol. » ("\« v a« s, lo «„ e_ -1155CU. cm = 1.165cu.dm-1.1551.^' -^^^^cn.cm A'SL'^^ height is 60 cm. .-. vol.-(40)«xY cu. om- 32000 cu. cm, and .-. its "^t, = 32000 x 11.4g = 364.8 Kg. •OLunoirs of problkmi xox 96. Radius = 35 cm. .-. vol. - * x V- x (36)» ou. em - 1 x 639000 ott. cm, and .-. its wt. -^ x 539000 x 7.3g = 1311.666 Kg. 97. Area = 1.44 Ha=: 144 a. 144 sq. Dm, .-. side- 12 Dm, .-. perimeter = i8 Dm-480 m = .48 Km. .-. time in hrs.- .48 Km -^ 5 Km = .096 - 5.76 min. 98. 5 Km per hour -600000 cm per 3600 sec. -Ac. 99. Vol. of hemisphere = | that of its circumscribing cylin- der: vol. of cone-^ &c. .•. cone must be twice as high as the sphere =» .56 x 2 m = &c. 100. 1 ch.- 66 ft. = 66 X 12-^39.37 m = Ac. 101. 250 ac- 1568160000 sq. in.- 1568160000 ~(39.37)« sq. m. = 156816 ^ (39.37)» Ha- &c. 102. 229 miles = 229 x 5280 x 12 in. - 229 x 5280 x 12 + 39.37 m - 229 X 5280 x 1 2 4- 39.37 - 1000 Km. 103. 1 mile- 5280 ft. = 5280x12 in. = 5280x12 -5- 39. 37 m. 104. 1 tonneau = 1000 Kg. = 1 cu. m. of water «= (39.37 >» cu. in. water-(39.37)»^1728 cu. ft. water- (39.37)*^- 1728 x 1000 oz. = (39.37)» -I- 1728 X 1000 -j- 32000 tons -Ac. 105. 1 Kg. on a sq. cm. - 2.2 lb. on (.3937)* sq. in. - 2.2 + (.3937)Mbonl8q. in. v / 1 106. •. 8 Km. = 5 miles ; but 8 Km. = 8000 m, and 6 milei -316800 in. .-. 1 m.- 31 6800 4- 8000 in. 107. 1 pt. = I gal = i -5- .22 1 - II 1 ; and .-. worth SM. franoa -^B^ X T^Jir » V^ X T^ X 240 pence = etc. 108. 1 Ha = 10000 sq. m - 1000000 sq. dm, and the depth is .1 dm. J .♦. the vol. = 1000000 x .1 cu. dm - 100000 1. 109. 1 tonneau - 1000 Kg. = the wt. of 1000 1. 110. 1 sq. yd- 1296 sq. in. = 1296 -j- (39.37079)* sq. m. or sq. yd.- 4840 X. 836 ca. = 4840 x". 836 -j- sq. m. - (4000000)* 4- V^ S3 ca. 1 acre — 4840 10000 Ha »,uath-rf^^^^ '-■ = ''''' lap« ♦»,- 64. yh = ^i/2^..707] + ; Ac. 66. ^=i»,l LL.„Ac 31 TV 17x2' • 6 _ 72 67. =^^2 IIvjoo_i,„ 3 2 3 6 ^lr5""»C. 107 71. In the text .0001 should be .00001. The expressions become respectively .00001369863 and .0000137 and Zd« ference is .000000001369863, which is 0001 of ti,! f 73. l-r3.14159 = &c. thus: *^® ^°""«'- 3.1415)100000 3813 94245 5755 3143 74. 1+2.302585 as above «. 4343 »o„, nearly than. 4342. 76. L.C.M. of numerators -90, .-. we have A^/ ^, and 18x5 243x5 -*e. 119x6 er 8. IK TH« BIOH SCHOOL ARITHMBTIO. 78. Vy— 4-1230769; v/17-4.123105 79. li/5=TVl/180-«fec.;f v/2-iy'8=io. 80. lm.-70-.64yd.-39.375in. .-.dif.-Ac. ♦k'^Ju-S't^T'?'!.^''*^^- ^^ ^°''™«^ ^y dividing the first by 5 • the third by dividing the second by 7 ; the fuSrth by dividing the third by 9; and so on ; hence we proceed thus : ^ i - .33333333 ; dividing this by 5 we have .066G6G66 ** .0095 .'380.. .00105820.. .00009620.. .000 0740.. .00000049 « « ({ 7 9 11 13 15 .410686 82. See pp. 69, 70. zo8 . ®t' .?®.u."? ^^!^ *®™^ *™ ^' -25, 09375 J the fourth is Self th! ^c! ' " ^^ ""^ '^' ^^"""'^ ' '^^ "***^' X of 85. These can be more readily calculated as follows :~ 1.25 I .09375 = .046875 = .007812 dif.-.u39063 = .019531 ' .002441 .017090 i = .008545 tV« . 000854 .0u;691 i = .003845 ,4-000320 .003626 .09375 .039063 .017090 .007691 .003526 1.411119 I ■OLUTIONS OF PROBLBUS 86. 1 + 1^2 1?F = .5, the next is I of this - .16666 the next is 4. of this- .041 (i6.. * one-fifth- .00833 one-sixth- .001388.. one-seventh - .0001 984 i 269 one eighth = Ac. 90. The dif-inW of one and ^.^^ of the other 109 nato; mustTi - jV^^. 'f *^° ""'"' °^ *^^ ^'•*^*^<'" ^^« denomi- iukwr must be -?Tr^ 01 its numerator or YrV" of 209 = n.,Aa jj 1.6. d.v* b, 49,^.^d 1, divide^y 49, MlX^Xi 7)-357 14285 .05102040 L5 7)1.''>5102040 ^to?l"'.f^. '™:„t* P'«« -"586487. The ,,u«, Leswhen^Astrrej!n>:!„"'hi%5l'3r. V"V°™ 'i 107. 32x25xlU.,728.100x3...70'00 ll " '"«"• eqi^f. fghrof w.lT±f .'^'"•'.^ one.ten.S'/^follows th.t fore oq^afvo ulr^Uter !f„r- '" """J"" " '" ^ "• ''•««- 109. A cubic foot weighs 1728.* 277.2 X 101b..- 997,Voz.; 110. £12 10s,^4a 2d. = 3000d.^80d._4.. • « ■n--„ ft. ... volume of rain -4840x9x4. onK ft I. and flt. R± /vol *- iv - 1 .. ., , ^^ io»i/ X IIq'tkJ^^ ^ !' . =""°' "• '^^w gives 170154 rr« 112. The tarm contains 184.48 .c.fthe remainde^r ac., Ac. pal. .36.976 •0LUTI0N8 OF PBAiBLim I zxx 114. If the work which one man can do in an hour be Uken as the unit, then the mowing of 15 acres will- 180 units of work; .••tomow 11 ac. will require 132 units, and as each man can do but 22 units in the given time, .-. six men will be needed. .*!i?' n 8*^* i '^'^'"S i ' o^ *h'« B got I, leaving i of 4 ; of this got », leaving i of | of ^ of the farm whi?h at tL price received, mu8t = f ac; .-. ^ ot j of ^ of the farm = | ac. ; on.\l®* '^^%l^ V"'^' ^c -1147 pt., and .-. contain 1147 x 1 V/''^; ^*'"'*^y. and •. measure 1 147 x 3000 in , *« ••15 m. working Ton ;^ ^****® * ^*y . •. 2100 H- 150 - Ac 139. 10quar=bbl.of8alt-2801b. .-. 1 quar. -281b Ac 142. d 243 107 80 14 h 2 7 10 7 11 4 7 11 49 7 42 17 143. See 138. V ^ »«• }li* ^ ^^^® **^ ''®'^^*' *'"'*^- P- 7. 8. 145. A fall in price of 40c a gal. on one^ixth rif A* ««.«. ■OLUnONS OF PBOBLEMS ifii^?oA*i" *"*^^ '^^ *^** *^^ ^°* ''^^^^ ^* ^» ^ respectively 160, 180, 163, the unit of price being as yet undetermined • but the cost to being $306 fixes the unit at $2, and .-. the lot cost A $320. 147. The value given is that of 780 per. .-. 1 ac. or 160 per. 18 worth /^ of this value; but previous value = |S of rrfir^8ri7s:6drrr ^''"^^^^ ^' ^"^- ^^-^^ ^^ 114 148. The unit of measurement in selling the coal is |4 of that used in buying it ; .-. the number of short tons is M of the number of long tons. * 149. A's rate : B's as 20 : 19 = B's : C's ; .-. A's : C's as 400 • 361 = 200 : 180i = a diflference of 19^ yd. in a 200 yd. race. * 150. 1 pound of thread makes 3f sq. yd. .-. 45 sq. yds. will require 12 pounds. 151. U.S. dol. contains T»^ of 412J grs. pure silver: Can. dol. contains |J of 360 grs. pure silver. . •. silver in Can. dol. is worth 85 X f^ of 360 -j-t^ of 41 2 J, cts. 152 Area of end of bar = yV of 1 sq. ft.-3.6 sq. in. •. edge of end of bar- v^ ^ 1.8973 in., and if this length is cut from the bar, the cube so cut oflF will weigh I45i^ x 560 lbs = 2.2136 lbs. nearly. ^ ^tftt- x oou ids. 153. 1 gal. = 4.534 litres. 1 pt. = | of 4.534 1., 10 fr.= f '^^,7 H^J.^^ = ^" X II X 240 d. r. 1 pt. is worih 1 M* x i§x||x240d. = 53.1d. ^ 154. Time = 14| hr. = Y hr. Distance = 863 x 3281 ~ 5280 milea Speed= 863 x 3281 ^ 5280-f- ^ miles per hr. 155. In 5 days 10 min. it loses 10 min., &c. 156. Whole selling price = $52. 10 + $6.75, and this -5- 536 = lie. = selling price per lb. 157. Enough water must be added to 63 gal. to make 90 gal. 158. Since the time is diminished in the ratio 3 : 2|, .-. the rate of speed must be increased in the ratio 2f : 3 = 11 : 12 = an increase of Jli^ . 159. If the unit of val. is one-fourth the val. of the house, then the nnniiA will Ko 1lr/^«•^Vl A a.».*J fi.~ i^f o i xu. x , -"^ TTirivii -x S)£iu Clio iOu u, niivL hUO IWO to- gether, 7 of these units. .-. the unit is $300. IN THK HIOH SCHOOL ARITOMSTIO. "5 *Ki?^*-/l?® quantity of hay eaten by a sheep in a day be Jn fin^h, 7*^?o^ ^^^ "^*^ ^^^^^'^g ^95.25, from which to find the cost of 112 units. f hin 1' Jf ^^"^ "" ^*' *™^^ °* P"""^ ««^^ ^'•^ ^orth a dollar, then 437^ grs. are worth 437.5 -r 25.8 -^ . 9, dollars = &c. 163. In efficiency these taps are as J : | : -L - 21 : u ; 30. «™ ?'''^^ ^""^ *T*^®f ^''^ ^ *^^ 3rd as 7 : 6. that is, out of 1^7. Lfl'J^r'^J^ ^y *?^ ^'•^^ *^« *^P« 6 gal. are' drawn th«^lf !"''«• T^ T?. ^^^' '^"^^^^ ^^ ^^« «i«tern. ... when d^wn ^''^^^*'y ^""'.'^^^ ^^^' *^« °^"«^ ^i" have been xT hr. = Ph^^ '"" *'°'^' *' ^°°^ ^' *° ^""P^y '* °"^^ = ^ of »^i ft« whT"'^ ?^ A f I of the price - an increase of .« 165. Divide the whole sum into 91 equal parts, and give them respectively 17. 20. 24 and 30 of these parts. ^ 167. 5 m. - 1 140 ft. - no. of seconds = 11-^^ hr. ... dist. gone by train = [1^ miles- &c. we]g\%?li\;TJf ^teV'g^ ''' '^' ^- ^^ ^-^^ -^^ -^ the'?alu^T|o^VI ^^^- *^^ ^^- «^ 128 gal. and ... gains 170. For one day's work A should receive ttIt and B ,4, ZI6 410 = 42:41 .-. B will win 171. B's rate is to C's as 420 by ^ of the distance run. 179 Total call: — ~^^^ *„,«,,. i9097'qnV ." ""'""^. P"pe = IdI^j./d; first sale brings SOLUTIONS or PROBLEMS ]l i 174. A and 6 together earn 9| s. a day ; A and C, 9 s. ; B and C, 8^ s. .*. if each works 2 days their earnings will aggregate 27 s. or 13^ s. in one day ; but A and B earn 9^ s. in a day .•. C earns 4 s.; &c. 175. L.c.m. of 914c. and 99|c.- $229041 ; &c. 176. lxUxi^=lir^i .•.gain = ^. 177. To build 12 ft. of the sidewalk will require 240 ft. of lumber, or 20 ft. lumber will build a ft. of the walk which .*. cost 34c. a ft. or $.34 x 1320 «&c. 178. Total area to be papered = 1331 J sq. ft. 179. Total length of ditch - 1 332^ ft. 180. The difference between the squares of two consecutive numbers is their sum, so that the numbers required are the whole nos. next less and next greater than one half of 691. 181. 7926 X -2^ ^360, miles = &c. 117 182. Had his speed been a third greater A would have run 1760 yds. while B ran 1738 yds.; .*. his real rate : B'srate as f of 1760 : 1738= 60 : 79. 183. 6 yr. = J of father's age - ^ of father's age = -^ 3" «" 8i«s J39.60 a year =osf.' •: g'^!:.tVof Lt^ "' °""- ••• -■""« p'- « i of M of 119 in u\ ^" "^"^^ ^"^'^ ^^ "^^^- ^^ lO^i h^- --d ••• loses 1< min. 204. ^405 lis. 4d..4.46- ^ft ifi- >«d ■ -«-•- Aa o ,- , and this ' AR ij ■ ' ' ^ ^^' > ^""^ -j-46 = 3s. iud.. iOD. Dmde each by 1,250,000. J' i ' .' SOLUTIONS or PBOBLlMg .206. 54 men in 13d. dig 1053 ft.; .-. 46 m. in lid. dig 759 ft., leaving 77 ft. to be dug by 8 boys in lid. or 7 ft. in one day. 207. In 35 hr. the first would fill the tank 20 times, the second 14 times and the two together 34 times, or once ia, 3 5 FT hr. 208. 365 X 400 + 97, there being 97 leap years. 211. If the cost of the first is the unit> the second cost U. . ^iif^^i' ^""^ *^® ^®"^**» ^i-a total cost of 8, making thi unit $3,000 ; (fee. » o 120 212. In 10 02. gold, nine-tenths fine, there are 9oi.. pure go d ; to reduce this pure gold to a fineness of three-fourths will require 3 oz. alloy. L ?^^^;. ^ quart = 69.3 cu. in., and .-. weighs .693x31 crs.; a half dime weighs 4 12. 5-;- 20, grs. .. &c. lit' yj^^^n *''® ^*^ ^""^ ^v ^^ *^e ^a"n» respectively. 215. 63460.6 X. 82 ^7000. ~» i~ j 216. It takes 18 min. to row the extra f m.; the other 30 mm. must be the time of resting, or the steam flows J m. in 30 rams. * sideiy/lr/ii;''*"'"*-*'"-' «.230» (total lengti. of 121 219. If they had gone on as they began they would have shared equally in the sum paid ; but the first increases his efficiency | for half the time ; the second | ; and the third ^ ; they are .-. now entitled to share in the ratio ItV * l^ ' UV-765:760:756. Or, thus : In efficiency the first istci be second as 9 : 8, the second : the third as 10 : 9 j .'.the three rank as 45 : 40 : 36; the times during which they work are as 17 : 19 : 21 ; .'.in work done they are as 45 x 17 : 40 x 19 • 36x21 = 765:760:756. 220. $4064.55 x^x^%x |^ = $4073.16. 221. In the firstcase 704 units of work = 5133^ yd. of road ; in the second case there are 19162J yd. of road ( = 2628 units of work) to be done, and since 36 units are done in a dav. • 73 dava. /{7fi. ^' ' ' 222. If the number is a square, each- prime factor in the number must occur an even number of times ; in 1600 the IW THE HIOH SCHOOL ARITHMBTia factor 2 occurs twice, 3 once, and 5 three times : ... another q and another 5 must be introduced. anotner 3 nr^lf* l^ * """^^ .^^iX'"^^ ^^°*°^ ""St occur 3 or 6 or 9 or^&c, times: m 14175 the factor 5 occurs twice, 7 once and 3 four t.'nes ; .-a 5, two 3's, and two 7's must be introduced 991^;^ f.^'^ ^' multiplied by 5 X 3 X 3 X 7 X 7. "''"^"^^^' mu^tVLroducTd. *'"' '' '""^ *"° '''' ''' '^ ^^^ ^- ^'s llf 7i \"; '' ^V of a cubic ft., or in cu. in. 64. ^ii». ine trams go rest'ectivelv -^ I ? a « nnrl is 3*8 -i hour, which equal ||0| I gi/^J SF^ gT \i^ '"l^"' ^' • 91 nAr V,./ o„^ wi L . *n»^^°^- ^he rates are me^urfment ""^ '' ^' ^"' "^^^^ ^^ °^^ - ^^^e "nit of 230. See 142. 122 2560 ac^ ^' "^ '^^'^^''^^ * ^^oct 2 miles sq. 232. 233. 234. 235. 30A 194 , -3^ x-^ -5-4840 = (fee. 5J,x4t?jx3^o^27 = &c. 5 yd. 2 ft. 6 in.H-1 mile = 5|^1 760 = &c ms.-20s. = 18l4-480 = &c ' 4 sq. m. = 240. The measure when the vd. is the imif I'o »?i *• ^reat as when the required unftV ustd ""* L InitZ.h" IS 5^ times as great as the yd. - 1 rod ^ 2I3 I'i'osr'.r ^.""^ f ^^'^^ = -^^'ft. - 1 link. 243. $1,085 X 3f ; 7 ft. 4 in. x 3.1416. 244. L.c.m. of 437| grs. and 480 grs. 123 246. Length of side of township = ^ in, = 5 miles, &c, ill hi SOLUTIONS OP PROBLEMS 246. L.c.m. of 3 pt, 10 pt, 12 pt, 40 pt. ^ 247. One sq. m. on the map = 64 sq. miles, .-. 1^^ x 1^ sq. in. ^^'^.t^^^^^^ ^^' "--- ^' -- ^^- - on the 'f ' ; 1 sq. in. on map = 250000 sq. ft. =&o. * sotn' ^2^'t"^i<^« = TV*c. = 484sq.yd. ... 1 sq. unit-4 sq. yd. .-. 1 linear unit- 2 yds. ' ^ umt-* ^ 251. Sun = 354936E = 10486.9 J, ... J- (354936 -10486.^ 253" ^\^Q^f«:«^=.5f^' •■• M-1.25^79.89 = &c. J03. -= 7925.648 X 11 1.454 = &c 254. 2000 X 2000 x 50 ^ 360 - 33000 = &c. 2o5. 320 units of work remain to be done • and PP?r weighs Vx X 550 lbs. = 300 lbs. » 4 u wei^cs y*^ X 462 iba. « 210 lbs. cu. '2-1- \i)' " ™» HioH sbneoL AsiTHMmia 17?8, giving the value of d * ^ ^^ *'"' "^•' ^*»ich » ,66E„g ao._4840x9x&?''^ " 10x42-144, a,, ft. tional weght = 4L3«ii ^f 070 T^ ^"7 ^^^1 cub. m., .-. addi- 279 Thl k ^^^5^5; *** ^'2 lbs. . &c. mer measurement, i." by™ ^ ™'''"*^ *" »? of its for- B, olU-Dt^St^ef ^3^^ "^ r ' *'J « = f «• ••• A is to in V# of the time tak^n bV the nitt T ''° " PJ'"" "^ *»''l' as the other, i„ ^ss ofle^.l' ||VofTo da^-^'r """"■ ^74. In value 1 part allov _ 1 1 *? ^^^ * <''^<'- - m value ^ot 15 G., Ac ' " ' '^^ °' * °^ '"^ "^ »!% 126 TOOOgrs-YiiesleBo lli i„f!V-°'u™^''; "-i' divMedby gal.; and this by 8 gal tte no of h" ,^ "^ "«• 8'™» ""« -"^ 279. Since 1397X727- n\f "*"'?• tori^h;:ryiStiLni:r/*.r;!r=i^^- •••" OZ. • i-ha.t-l . " ''■*'''^* * OZ. each to mnL-a«n il,_ rp f + i'Ax * of the whok £J^°* * *°™-^'^'>' 2 tons-^. f>F^ w ■ I'^i SOLUTIONS OF PROBLEMS 281. A convenient method is to wort fr«« ♦!.* i^m* . m each case, remembering that 9V S^C^^^'^P^*** -82. Divide 1cm of 1 13.002 and 89.604 by 113 002 ^283. L.c.n..of 16 ft. and 27^ ft. = 880 ft./9 mile«;880 ft. 284. If the first set do as much in 1 ^^n /i -o *i. -, ■, ;n 350, the» the , r.,t set will do J^ much fa i9d tf.r™' ^ 127 28fi* Jf*"^ ^r = ^VtV^^ of ^1075 . &c. we^\' nornd\raSlo'?n'if ^' ^'^^^ ^^ *^« «"* -i" allov 19 Tf^u *"°yiO> in the second the gold 108 the bej/ib" i« L" r«"inti: t^t^tr'a wrr Tsf do'e"h: ^;^ stiaaa '"' '^ *' """ "^'^ «2««' »<> -ten cu. in Vol v( s mrl 7fiJ >< \" X *> " 17 X 9 X 5 cu. in. = 315 lb 1 Ih 1 " ^5 "■"•. '''• ^ °"- i^i- '"^ood weighs J-« IqS tiks re ::• r^t^tt ^^^'^ ^'- - ^'- ••• -^^"^ J^L'et ;3n%ot^^^^ '• *^« <^epth 3J. ... no. 128 J92. The flint occupies the space of ?^ ^, of water. ^ttoti^^25P;''nf ; '^^^'}y the granite occupies the space ^7— ~°*^" pints; leavmc 8-(^,4. i \ loooo ^- 1*^ ^ water. * ViTt+xt; Wiir pmts of 293. Faat train goes | of the distance while the other should ow train „fto„]d niuke the journey in 3J hrs., but owing I ! li do W THl HIOH SCHOOL ARITHMETIC. in 9 sec. the train Soes Qfi ft ^Aat I^^' *** 10 sec. . . se«,„d pe^„ 8oes>^7d.tVs^rJ2r.„\? '-^ -• ">• ^hells. ... 1| sq. cub. cost ^«-%«We'.'s:^ij^eV6f HT 30lLt^'.f40t'cTb':'^:^^^^^^^^ ^ l'^ ^^ -^° -bits. ... but as they are to be ca^ed VfL ^^ T^* ?^ **»^ ^'^t ^^t, the same, or 8 dr? ^ *"°*' ^ ^^^ ^^^ cost will b^ 9(Uolo ^^® ^''^,* P**'^®^ -x)ntains 485640 crs • f >,« c. -i l97?«? ^'^•' u*^^ third = 32.452 Ib8.r32 452:700ft ''''• 298 IS' *^« «•/•/»• of these is 76 grs."^'^^ «"- 6 in., 112 in.; 2800 i in 22A hra °^' ''^- *^"^" ^«>°»« 129 each; .-. A's shaVof rt„K',i" ' „ ^'""™ '"^y received 3 300. T^iC? 6 K ;^k ^~ "J' ^' """^ his first sharr2I .«. oi wSS^Jd'^iLg J^V'ri^-'«^* -O height, the sq. units- 168 sq. yd '' ' ' ™' *"* "' floor-42 hr. In the second «Sttefrat,JI «'»''»'"» « 2S m. «, in 6 sec. = 76 m an Jn? . i ^f™ " '""e rate of 220 yd. m. an hr. " Mnd dlw^V'^ °' »«™»d down train is 40 rate of 16 n,! Th?, tTis*"" miSTs" d* w'.'^f^ "' *« »™in wiU re«h 1 ^tt it' l1™ Xti. aJ ^6'^^ i If •If ,' I 80LUTI0W9 OF PBOBLBMg sec. past 1 o'clock. If these two trains are on parallel tracks they will be abreast in another 1 2 sec ^^^ trenPh Z^^ ^/"'^ ^T""^ '"''"^ P*'* P«^ «"' ^^ ''' ^^e second trench being f as deep costs ^ of ^c =|4c. per cu. ft • total co8t = 6x 10 x3G0x^°c =$62 50. P«r ^u. it. .. second 3U*I^r^'■'^TK•^' '^'^'' ^^ P"" *^'^*-' ^'^^^ ^^ ^^e second, ddj per cent, of his income. .-. &o. X30 307. Every even number except 2 has for factors 2 and a number greater than 1, and .-. cannot be a prime number 308. The difference between the squares of any two con- secutive numbers = twice the less no. 1 1 and is .-. an^dd no Also since G^ - 5^ = 2 x 5 + 1 = 1 1 and 5^ - 4« - 2 x 4 +1-9 &c., It IS readily seen that th6 successive pairs of square num- bers produce the consecutive odd nos. '^ i "• Anf .!; ly ^'"''^ "'' ^''^'' J""- '^ * multiple of 2, the r.™ of any no of ev. n nos. must be a multiple of 2 and . •. ever. y> Any pair of odd nos. makes an even no. : .-. any no of I^irs of odd nos. will make an even no. The e may^aTs^ be lw«« •"'•' ^^tf^"'^^ expression for an even noT2n where n is any whole no. ; and for any odd no. 2n + 1 If • we take the sura of any no. of even nos. as 2m + 2n + 2d we get 2 (m + n + p) whioh, being divis.ble by 2, is an evenlT^r anevenno of odd nos. as 2a + 1, 2b/l ;'2c+l. 2d + l'; 2^ Ih^.^' "7" ^'""^ ^ (a + b) + 2; 2(c + d) + 2; 2(m + n) ■^ /qWv.Tu.*'!^''^'' '''''" '-^^^^^hole sum is even. ^ ^ (6) -Ihe third case is simply the addition of an odd no. to Ji'cSrio."' ^ ^ ^"^ "^"^ '"''' "^^'^ '^ ^^ "^"^ ^°- P^^"«^ 310. Since any two consecutive odd nos. differ bv 2 • ?7 Fh^A"^^ '^ u' ^"^ ^'^'' ^y 2 or by some multiple if 2 ': t.e., the difference between any two odd nos. is an even no Or thu8:2m+l-(2n+l, = 2m-2n = 2(m-n) = amuTtip?;of2 = an even no.^ So that if an odd no. is diviLd by anSd 1 and the quot. is odd the subtrahend must be odd and .-. the i^d tr;Sm.tls.^ *'^ ^'^°*- ^ -- *^^ -^^-^-^ ^ eve': iJh^'J^oZ^rS^\^.'^'^' "P. «f *^« f-«to/«> one-half of i_^ ^a „^ a«« tuaw ^ vaa contain no part of the odd no, IN THE HIGH SCHOOL ARITHMETIC. no. is'Znii'Zl;ZyZlZ "• '' ^'^^ ^^^^^^ ^' ^ -- = 2 /smnr^f"^' '''''' """^^ no. = even no. for. 2mV(2n + l) 4l%^^V2n:ri'2%t^ -nt^r^^^^^^^^ :;o?r ^jdd :l;:t^i:"e:r '^^^ ^ '""^''^ ^^^^^^^^ '^^ a^x'o.'^hr.uot; From (1) and (2) it appears that the quot. of an even no ViVtk"' 7n'" ^^""'^ ^^ '''^'' odd or even ''^ °^* ma. Ihe following should now be clear •— Odd) even (even odd) even (odd ^^^" odd odd even 34687 320648 873294 -r 277496 221'J968 11099840 1112231717fi mult haa been multiplied by 320000 + 640 + 8 = 320648 ' ^*^":L^: 62{''J^''"''1 thV6 SS^rX'^t qivision, make 62 for the complete remainder. 80LUTI058 Of PBOBLUfa 316. 10 times .7-7.777. . .t- .777! 9 times .7=: 7 Law Iht^lddiTtTrndTabr'T '^'' '^ *^« Commutative to Jower terms " ^^^' ^""^ ''' ^^"^^^ be reduced due to the last two didis '^'''/' J*^J> ^ > •• ^^7 remainder is divisible by 4 so ittte whoL tm^r^^^^^^^^ ^^ ^^^ ^ ed'fnit,''afdYf"tr J auanStr\- 1"' *'^^* ^*- ^« * ^-- unit 5 times. "" ^ ^*'^^ ''^**'^ ^^'^^^^ ^^^ derived 131 323. Every number must divide ov 3 exactlv n,. »,-*k mamder 1 or with rem 9 f h^f ;» "//^ exactly, or with re- end in 9 ; « i„ 2 or 8, the sq. end. in V J ."n l' ^J V^"""' S30. Three more terms must be taken. It -it! ■• •• ♦ Pb»m th. i..t term di^otly in order to ve4"t£ '^0^1^ IK THK BIUU SCHOOL ABITaillTIO. may be done thus:— 10)J_ 8).l 8 ) .0125 9 ). 0015625 9 ). 00017361 7 ) .0000 1 929 .00(»00276 2x^?4U9"^^n^*^^'r-^'^°^^^^^' ^^'^^ when divided by anytdrr^^aii^^Sri^^^^^^^^^ - an2\\ These two noa added together :. the sq. of the odd no d f Zt% ' ""^^ T ^"* *^^ ^^'"^ «W« - «q- rt. of the dif of the squares of these no8. = 8q. rt. of prod, of their sum and dif =sq. rt. of the sq of the odd no. =?he odd no. .'. th^ third side in all such cases is the odd no. itself. Ihis inay be used to form right angled triangles whose sides sor must end m same dicit. «uu u*ia «Jn^^; ■['' ^^d'*^«n.*^e units, tens, hundreds, etc., are added separately. In mult plication, the units, tens, e c.^n m^ tiphcand and multiplier are practically reatil as sepamte STe latt:."' '"" '"''' '°^"^^ ^^ "^'^^^P"^<^ by eachTrS m,?»f ^.^"L*^!.'*''''^''**^.^ ""^"'^ ^« *^a<^ the different orders Tel^, Itit^'^''^^^^' -^ ^^- ^^- -- added to ge" ^9J^his alao holds if the same number be subtracted SOLUTIONS or PROBLKMS 340. So far as this is true it will be exempUfied by a care- I i "»' inspection of the operation in each case. N f' A\^tl'i^7 ''?L'** ^^^ " ^^^ *^*» 1000, any no. of 4 foCnnnn''^*^*'' ^^^^?' •• ^^^^^ P''^"<'t i« less than lU.UOO.OOO and .-. cannot have more than 7 digits, and so on : li ii ^^'/*''^''*J ^' ?^ """^ consisting of m digits is less than 10- thL iV+n'' F^ "' *"'' *^''^. ^^"^ ••• ^^^'' Pr«d"°t i« less «r«^ iT+n • "lu-"-, **''"''* consist of more than m + n digits since 10-+- IS the least no. consisting of m + n+ 1 digits. ^ «n„ai ; .r^!?.®*^ *^® dividend as subdivided into units, each ZtlJTt Tf ir°^'-^-^r A^ number of these units is the quotient. If this dividend be multiplied by any no., say 3 we may consider the result in either of two ways : (1) ascon- tm^L 1 '*°^^?e- i "^*"^ ^^«^«' each unit being 3 times as large as in the former case, when evidently the divi- ^me"".^'^ '""^^P"^^ by 3 and the quotient rema^^s he tT,!ni/^ as.consisting of 3 times as many units as before, the unit remaining the same. If now we grJup these units in indtLr\^?."P"^" ^" ^*1"^^ "^ the multiplied diW «n,!!/i'^!t^'*'?^^*?*'^''^ the dividend left after all units equal to the divisor have been taken, then evidently this re- minder will be multiplied by whatever no. the dividend h^ ^d^^'J^"^ "^'^ *?P®*'' **'**"^ ^^^''^^^ above, in which, when great. ''"'*'''' "*""'' '^' 'l"^*^'"* ^^°^«« 3 times ^ inatottWnfTv^'*^^^ is a larger fraction of the denom. mator than of the numerator. .-. the denominator has been rZ?lor'^r*^'"*i?'^*" the numerator. .-. the vS of the fraction has been diminished. 346. Multiply any numbers, say 837 by 429, and then divide the product by the multipUcand; and compare the steps in the two operations. ^ ^« .liff' ^5^'^''*'^ *^*''®^*^ *°y '^o- ^'^d its sq. is either to oS' nof ^r^'" '^^ """'^ °"^- °' ^^'^ d^^'^^^ between two odd nos., and is .-. an even no. iJl\ v^r^^ T"'' T* ''°''**^ ^ " * '*«*or. .-. ite sq. con- tains 2 X 2, or 4 as a factor. ^ ; I 'i u IN THB HIGH SCHOOL ABITHKBTIC. nrilfn;f '^*^nnn'? * "»"J*»Pl»er as 1999, then the last partial product 18 1000 times the multiplicand, and the sum of all the other partial prods is 999 times the multiplicand, .-. the statement holds in this case, and .-. a forlioH when tie left hand digit IS greater than 1, or any of the other digits less 349. See 325. 350. The chance of errors occurring is always diminished clnVr'^^'l.*'" T?'5^ ^y °^«'« *^^^ one method, but cannot always be guarded against with certainty. 133 351. See Arith. pp. 42, 47. 352. See Arith. p. 66. it T?fi Joo/f'"!? *^.** 10-*« ^^ 358. See 316. nf ?h!'^^'T ^}^ <^J^^«^? i« 9 times the quotient, .-. the sum digit is a ^^"^ ^""^ ^"''^''^' "^ ^^ *^^« ^^« W. ••• *l»e last 360. See Arith. p. 56. 361. (99.9899995)=^ =aOO-.01 -.0000005)^ = 100^ + ;ootoo-oiV25rk-^r^^^ ^ (-^^^^^'^^^ - ^^^« - ^uotlenl d^ff^'-^^^^tl^e true quotient bV%00277 of the .?^^V? J*- » min. = 15a ft. in 15 min. = 1 mUe, mile = 352 ft, Iff 364 in ultimately reduces introducing or ir -^ •• • itself to the question whether itriking out two lectors, say 3 and 6, d 1 a t s •Ol-l^TiONi OK PROBLEM, 366. See 317 '^ <«the^o&^^^ ." « the terms ar^; • ^^^ "^^^ '^ot appjy ^ ?h« ^ *'*^'''* " e^^al of the fr^^^P;;^?^^ each oth^^ ^n^S^r^?* ^« S 368. If we ?al"^ ^'*^<^^o« cannot b^ealf."''^.*^"*^ ''atios, « 15 and 35 t« ^""^ ^^^ nos. that hav2 *° ^^^ ^Id. ^ and the sq. of ZT *^/* ^^^^^ ProduclV '"?• '"^'^•' «««h " true when/mstei^'';?"?- ^« ^^e other's InH .1?"^"°* 3- contain the faVor^^^- ^ - £^., ^tS Td^f 3^0. Every «a no "^^^^ **^«^ «q- must do. See si-.? .'?■ H-SKr-s» * w Jdbyatonoe *^o ternas. It tt u (S on is equal case since ' the terms uaJ ratios. • old. leas., such P of 3, 7 • product the same there are '6ir sqrs. sqrs. « 2 • *• does lo. See 6. 1, 6, • Nos. J of 5 ; luJtipIe trthan ; and of 9, ^ «.«., rem.; mad W THE HIGH SCHOOL ABITHMKTZC. % 7e first" ?htl;rfmrr"' *'"^ •• ' ^^0 + 600 + 40 6, the third, 4, so that J Hrsu^t 'of' Th ^'^^^^ ^^^^^'^ there would remain 7 + 6 + 4T7 K . .u^ separate divisions with ^rem. 6. . •. &c. ^ "^ ^ ^ ^"'^ *'»««« ^'ontain 9 twice products collected, and the Vrocp-^ ' ^""^ ^^^ P'^^'tial ^ul iplying 16 b/3 and b^i and « bvr^^^'^l^ ^^^'^^^^^^ uig the products so obtained* ^ ^ ^ *"^ ^^ i ^'^d collect- <>75. The$18.75 ia^fcia 7K i ., proximately ' .-. igfsS- 18 75 xVol '-'J ^^^^^"^^ x '06 ap- other products. ^ ^'"^ *nd similarly for the ^^^''-dthel^'t.t' tZlt fo 'f. ""^^^^ 3 from from each term and the fract^T!, >;1.^^- ^«^« ^ more «t down, nothing appeanW bSfcth. ° P'^"""' "^ ••• not , 379. The addition or "„btr».f T'''™ ^i^^nd,. factor or unit cannot can." tLat ?«V* ""> """"'P'^ »£ any ff™" 3's and four 3's are elevi '%"$?■■ '"■ «"' to disappea/ U quarto. Seven S-qte ^dlour 3 ol^"'* f""' * 'I'"'-^ "^ a82. Whether 6 x 1 - la ,• Pend "Pon the meaning atfaihl?*"!t' '"'"*'<"' or not will de- athereasomngisthat3e.i.they:erfl'Z,» »^^--6ata.p3wmo„3t6 thnes „ ««<^ .« ^ ^,, j«' .. « ---^n Luo cost is 3 *-««.r«.t;,ewou,dha™^a,the.,u«o. 18o. 1^ SOLUTIONS OP PROBLEMS 384. tV-tV of T where the form of the repeating period is determined by the 7. See arith. p. 59. 385. The civil year is 365d. or 366d. ; the solar year is the period of time in which the earth performs a revolution in its orbit round the sun= 365 days, 5 hrs., 48 min., 46 sec, mean solar time. See any good encyclopedia. 386. Silver coinage is legal tender to the amount of copper to the amount of 25c. 5 + 8 S87. 6 + 9 -13 = 28 _ 8 IIT 'ZT — S'^JYS 136 of an odd no. assumes the » 4(n2 + n) + 1 = a multiple of 388. See 345, 348, 372. 389. See 347, 369. The sq form (2>i+l)2„4na + 4n+l 4+1. £90. Hquare nos. must contain factors in pairs, if then 3 occui's as a factor it must occur twice, i.e., 9 is a factor. 391. The g.c.m, of two nos. contains all the factors common to these nos. ; any other cm. must be made up of a part only of these same factors. . •. g.cm. is a multiple of each cm. and since the g.c.m. is itself a cm. it must be the least com. multiple of all the com. measures. 392. 3 and 5 are cm. of 30 and 45, but neither is a factor of the other. 393. Each com. mult, of two nos. must contain every factor that these nos. contain ; thus each com. mult, of 20 and 15 must contain 2, 2, 5, in order to contain 20 and 3, 5, in order to contain 15. .-. in order to contain them both it must con- tain 2, 2, 5, 3. The l.c.m. contains these factors only, any other c m. contains these with additional factors, (either the same or different) and .-. is a multiple of the l.c.m. 394. 12 and 18 are com. mults. of 2 and 3, but neither is a mult, of the other. 395. By 2 if the last digit is 0, or divisible by 2; by 4 if the last two digits are O's, or form a no. divisible by 4 ; by 8, if the last three ; by 16, if the last four, &o. By 3 if the sum of the digits is divisible by 3 ; by 9 if the sum is divisible by 9 : by _ J — .^,„ vi.g.v 1.J V Tji t/ , Dj -a ii tiic xiujc two are Us or form a no. divisible bv 25 ; by 125 if the last three, &c By 6 if the no. is divisible hj 2 and by 3. "'I 4 IN THB HIGH SCHOOL AIIITHMETIC. are^^Ln^n^fi^J'^.i*'^"''''*!*^.^^^ '^' ""^^ "^ measurement unit t... to find Its measure. In the second the quan. and its measure are given to find the unit of measurement. 097. See arith. p. 42. 398. The dividend is 101 times the quot. .• if the auof hf, subtracted from the dividend the remainder wil be l(?Otmes the quot.; t... will end in two O's ; .-. the last two digits in dT vxdend and quot. must have been the same. See 336. + 2 6n + 3, 6n + 4, 6n + 5, i.e., must be divisible by 6 either without rem. or with rem. 1, 2. 3, 4 or 5. But 6n il a multt pie of 6; 6n + 2 and 6n + 4 are multiples of 2 : 6n + 3 of 3 7aa^ o^ ^'^^^ possible primes 6n + 1, 6n + 5 ' 400 Since the dividend is 9 times the quot., if we can sub- quot. If the quot. is multiplied by 10 the last digit in the pod IS either 0, or the first digit to the right of thVdec. pt m the quot. which is always the rem.; thus the full quot. in the aAi^- ■' *^® remainder after dividing by 9. ^•^ i'u 7? ^^'^ ®''®° P°'^^'' ^^ 10' ^100, 10000, Ac.) whendi- 380000, &c., when divided by 11 will leave the same rem. a^ /4 J8, &c. .-. the rem., after dividing 387426, is the sum of 38 yTIk f^l.'1't^^if ^?!'l.^*00. 26 = sum after dividing finnl ' 26-5 + 8 + 4 = 17, which contains 11 once, leaving I final rem 6. Similarly the rem. left by 5783742 = sum of rems. left by 5, 78, 37, 42 = 5 + 1 + 4 + 9, giving final rZ 8 137 nth^l ^^^J ^^^ """/fe ^^".^^^8 : if the sum of the 1st, 3d, 404. Divide 8 by 2, add 7, divide by 2, add 6 &c and it will be found that 8 has been divided^ 9' tTmes 'by 2 7 8 times i 6, 7 ti^es, &c. The last division is by 2» wWch gives .98046875 as the final result. ^ 405. By extractincr so. root, onb rnnh A:« 406. See arith. p. 6O/ " ' "* ~" 407. See 334. 408. See 364. li BOLUTI0N8 OF PROBLEMS timra„^°4it°''Tt" *'"?'!i *S O™*- ••■ dividend- 10 «*J line , ij. and units digits, viz 7 79 xkt,. mineinfiw«o« *k . ' ''^" ^e can now deter- 410. See 317. '^ a btl'and S'e f " Vis'r'lh^S '':J'''- *'^? ^'^^ ^- ^ -' - - Ac' But thif does nit hoMi^ fh^ieneri'^'' ''^ = t ^ or x r» of the nos. may have a com f«n^ the general case where two I^t. A, B, r^pTsenVthe hrle ^ol TheT A '"^ '\''^'^' factor X not contained in either B or O « / . ^ ""^^ ^'^^^ * in B but not in C. or factoTnTn C bu^' ott°^' "" T'^'^ r com. to B and C ■ thii« a ' ^"^^^^.^^^ B, and a factor and C-znpr The 1 c m^nf^Jif '^ ^^^^^^ly B = y m p r g.o.m.isrJndtheprod of k BanlV- ^^^^fp'; the the statement wiU |p;J^r °* *''^ "™- ^7 » ""^ truth of "iTw.m.^sV"''''''"''''"'^' Thede- ^'iJ:J^:Z^^lt<.^T!,^ *«-P<«ed or when 4 1 y. oee 014. ft«*?>. ^?-60 is the price of a bushel. Al sn M , ,- . fo.Du IS tile price of half a h»oV.^7'' V' ,"^^» uue-aair 01 i. the price o? Tgi. td 22jr^Vqu'^'" ' ''*^^' «* PROBLEMS ARISINO FROM BUSINESS TRANS. ACTIONS. PERCENTAGE. t;' ) :'■ 100% of the number- Z40 21. 10% of the number- 13. 130. 22. Number of boys =60% of number of girls = Ao of 60 - 36. .-. number of pupils - 60 + 36 = 96. 23. 1 1 2^% of av. attendance last term - 225. . •. 1 y of av attendance last term-2. ... av. attendance last term- X 100 i\to ""^ * ''"°'^'' " ^®^" •** *^^ ''"°'^' - 286 X Tix 26. 100-lli-88|. 88|% of a number = 710. .-.number -710x^x100 = 800. in^^' A loaf formerly cost 10c. It will now cost 1267 of 97 **ono/H •'• *.^*\*^«« "ay no'^ be bought for 50c. 27. 90^ of remainder after battle- 360 men. .-. remain- der after battle - 400 men. ... 80% of original r;gimenT! 400 men. .'. original regiment = 500 men 205/, of Ist year's earnings = 16660. .-. 106% of 1st year's earmngs = $6560 x J0|. - ^3360. ° ^ wif^f^^^ i5^' " T^^^ **^ ^ *"*• **• *»* ^**e'- ••• 2000 Ibe. - m§y ci 4? '"• ^!- ^* ^^'^'' 0.^,2 cu. ft. of water become. ^^Wo J* 32) cu. ft. of ice, or35| cu. ft. of ice. fx, A C: ^'^l^*' receives 30% of the debt, he loses 707 of is* V nf •■• ! ""Tr^'^ ""^ amountof loss. •.«., he reodres *^T/o Of amount of loss. 31. The increased value -|880a 141 *220% of cost -220% of $4000 I if. lllj r |K.«! SOLUliONS OP PROBLEMS 32 Ho saves 7J% of a jWs salary in I year. ... he save. 100% of a year's salary in ^.years, or 13} years. g^ = 1 20. . •. greater number = 240. 34 First number = 1 20% of second number • 1 oi o/ ^^^ « . 35^ Lh ;, r/"rH" ^^0' ^'^^ .'first number is 120 ftn?nni ^ """'^ °^ ''o^""^ becomes (] -001)3 units or 1 -00^' 003001 units, .-.increase = -3003001 7. "'"^s. or 1 003- 36. 33^%=^. A's money = 4 of B's°'monfiv . -R'c ==to^,^;smoney,==75%ofVmo„ey '°'^- '-^'^---7 was i3 ^. f h! ''^''^'' '"''?'^ ^ ^^•^^' ••• *he silver money La ® P^P®^ money was $9. "loney d8. A cu. metre = (39-37)3 cu. inches -61 09 ^■qr'r^L ^f ^/bT" = ''' ""• " = *«^5« «"•" 6rom?7Tl307+ y 3^x1 nLrr-orC'ls'37l?&,'''"" V"'^*^ number of girls is 62iy of ^K?'^ r"""'^ ''""''^'•- •"• "-e „ *2" •■*• ??°' W' °^ ■""* in *™ days, and B does 37' V of l2i'?^f r^"^'- ■^'""? »■»<"'■'' done issfl/oflik ' ■ /i *4fo/^ u ■^'"»'°' t" be done. *'° . ■. value of mine w^«l 20,000 '^ "' ™'"' °* "'"'« ""^ «3«'«<">- the hou^^l|'/or''rf *""•■«%,-'* 3 times the value of or^airorthe2!useU J20oT'''''^"''°' the house - J6000, of 3rk '^i°' \%T:fX^t1is^y-,}!'^^-m% Dart » f'l'^io/ «* Q J 1 P , ~ ^^3% of 3rd part .-. 2nd - 4W of¥rd narf ^?So •■- '",' T' + ^""^ P"» + ^.d part 2nd7ai = 486.'^ '"°- •'• ^--dp-rt-SeO. Istpart.goO, 44. J is 100% of i. .-. 1 is 2007 of i • >i8 40»/l of seilii"^tf '°'f """»« M • • ^ -3 B re"ile 967 or76.8%^of sellingprice ^^'^^ "^'^^ °' ^^/« ^^ ^^« P"*«^ IN THB HIOH SCHOOL ARITHMETIC. 46. 126 gal. 504 qts. 2gal.. l|qfc.«9| qt. ^.11.0/ the cost of 1240 era • r.n 1 ^ u ' ' ^ '^" S""^' ^^ g^ins ■HX i TRADE DISCOUNT. 1. 10% of $600 = 860;. ..he.paid$540. ii6 *™o'»nt of bill = $850 .-. 15% of amount of bill = $150 = discount . Z,?''iT^ ^'"°"''* ^^ ^^" w^ $200 : discount was iDS-ift • .'. rate of discount was 15% ^ • ^wcouni; was $dO : .*. marked price = 1 l.57|. 143 S! B»n^.,^o„?rf:•,;•,^■>^°'^°'_""8'^'''l marked Brics. £nce. But 207 off leavpa «no/ !ie^° - • f . ^/o °^ original Uno/J'^' -7 . /^ °^ f>"ginal price. Hence dif- w 1% of onginal price, or I| cents. J)ncec erenoe SOLUTIONS OP PROBLIMS 10. 95% of marked price = $7.60 . •. marked price = $8. 00 11 i»c , 1^3^% of cost = 18,00 :.'. cost = $6.00. ^Anp; f*^^®*^ price = 140% of cost : .-. selling price = 907 of = ${i °°«*= 126% of cost. .-. 26% of co8tife.60 : . .'to^t \^' one/ °! i?^ ''^®'' *<^ ^2.50 per dozen is 831.25. .-. net cost IS 90% of $31.25, or $28. 12^. 13. 66§% of marked price = 30 cents .*. marked price = 45 cents, houseil^lst '"' °' ''' ho«se = $4,000: .-. cost of 1st $5 W)0 ""^ ''''^^ ""^ ^""^ house = $4,000 : .-. cost of 2nd house = .-.cost of two houses was $8333 J : .-. loss was $3331. i.io; ^^^••^«d price-140% of cost. Cash price = 707 of U0% of cost = 98% of cost. .-. loss waa 2% of cost. ^° ^« 1 «i r},?? l^i*'^^^ P"°® o* 16 OS. equals a discount of A on 16 J. This 18 3^% ' 17. See solution of 16, i?'«n^^ onoTf^ cashprice = $3.42..-. usual cash price -|3.60. .-.90% of marked price =$3.60..-. marked price 19. The first reduced price is a certain fraction of the marked price. The cash price is the same fraction of the first reduced price. .-. the cash price is obtained by multiplying the marked price by the square of the fraction. .-. thesquafe of the fraction is j^, or the fraction is if. .-. rate of discount is TT or ^_^/o- 20. Nine gallons of mixture contain 1 gallon of water • he can throw off ^, or 1 1 »%. 144 21. The selling price of one article is U, or 4 of the list price of one article. . •. discount is ^, or 20% ll'Jo^of T'^ltrr^W^^ °^ ^°*- ••• "^^'^^ price lin is 161°/ •*• ^^%<^^ "parked price -116|% of cost. .-. 23. The merchant gives 35.28 in. for 90% of marked price peryard. .-. hewould give 36 in. for(90xi« ,„r/ of markiKl pnce per yard, that is, for 91|^% of mark^'price". He could give a discount of 8^%. ^ « wuia '%of coat net IN THB HIGH SCHOOL ARITHMBTIO. 25 lo^ of T(^'7h'- '9 -^d 10% off- 28% discount. 1 :\ A ***. 1^* reduced price -85% of original orice • martJ '^^ * ''^"'^^i^ price -i$3.60. .-. reduced price-»4 . marked price wus reduced by $1, or 20% nrirl'i!;^"^^'- 'J^^'..°'* tV^'*^' «^ ^^0." The first reduced price IS a certain fraction of the marked price. The second Tt"th- J"'^^''^'?'"^ ^'•"^'^^^ of *^« first educed pdce The third reduced price is the same fraction of the seconcl re- thrt^A •■' k" r^li"?P^i<^eiB obtained byruldplyC III f^actbl^' -> '^M"^ "' ^^^ ^'•'^^^^°^- - ^»^« «'b'e of the fraction = ^V17V .'. the fraction = ^V .-. rate of discount cosf ^:^^^^ ^' ^^^% °^ -^^ *^-^ -. for 120% of of L.^^:^L:i'|r^ ^' ^'^^ °^ -^^ ^'^^^ -. for 991% «,„^?* J^^.loss is equal to the discount on the amount of the S hvl^u- r-^^V^" ^«»*> *^^^ i«' ^h« l^-'^ ™ay be obtain ed by multiplying the cost by the square of the discount fr^. won. . •. the discount fraction = |, or 1 2i% or 257^^ Henoi%V ""^ ^l' T'-' •'• *^" discount fraction is l Qo KA ^®^°® *"o marked price was $120. * 45 I'nff w^^^l™ * ''^'^"'''' P"*" °^"<^- of t^o marked price : 45 cents waa the same per cent, of that marked price when tTZ^tVr^'^' ^^"^^' ^'^^^^-^ the SZu^toff 50 cents, or rate of discount was 10%. ... original price was »5.00, and selhng price $4.05. f ^« was pri^c^-«^'7r'''' °""«V'" for S1.75. ... 75% of marked pnce - f 1 . 7o. ... marked price = $2. 33 J. fin?/' Z,?^.''^ ^.^^'^^ Price = 4% of list price. .-. selling price = -¥otnt^"^' •••20%oflistpricell0cents. .-.Kprire -o^o^o l^Z^ ^%f ^'^^'^ ^^? ^f ^^% of ^'ost, that is. leaves Ttrirjnr ot cost. When a rate of discount is taken off there IZ'nfA- ^^^*^^.f^-<'tion of original price. When the same rate of discount is applied a second time, there rem.fns T h^L^""T^'^'^l'^'' ""^^^"^ P"^o by the square of ihe iraction. Hence the square of the fraction = ,Ask^ ■ <-h« £r«5Uoa-|^*. .-.rate^f discount- 22-5 + %.'"'^' ' ■OLUTIOirs OF PROBLEMS IliJl M PROFIT AND LOSS. drover for ^SW^.uu " ^f? sheep were sold by the urover tor SJJ04, or each sheep sold for $10. lOiS ream, 04nnT'* -^ "''H ^'''" ^ ^ ^ J% of $1000, oJ|l 1 25. 1 200 for'2of ol7;i,lr2%f "- '^"«'*- ••• ^'^ --^ -" ^ «a^- n J?; ? P*'"^ ^^^% «^ a^'ount A paid. .-. C paid 120°/ of ^^0/ /'°°""* A paid. ... C paid 138% of arunt A paid . . 38% of amount A paid-$I90. .-. A paid $500? ^ 12. That which costs 7 cents, is sold for 12 cents • t,- gams 5 cents on 7 cents, that is', 4. or 71?%. ' * ^" 11 r S^^^}^^ goods sell for cost of the gO^xls • t»-« - •••oo,t„fl6,t,.. be SScVor MJ"/ ° ''^''- ^'°°« «"" ^ '» q"- «ould of oyp^ir'-r^rAtt^r '""• - ^'"-^ "-^"-s 28 H sells 11 qts. for 18 times cost oil qt • „li. i „. gal. of whiskey, and ^^-gX^ «1"/ ."'^fj^ T- ™/¥ of amount of whiskey titt' °^ irr tori,Ji:jTii%.zzp ercti' - 18 7y ®^3^ ' "^ ^"*^ v^ cost ot 1 oz. .'. he gains ^^ or 33. SeeSolution of No. 30, page 144. 04. iSee Solution of No. 30, page 144. 1l:i ^: aoLunoire of pboblims lo«f • li?® **S^«,^1<^0 ^PP'es for 25 cents. Ten of these are ^L ?S^ ''1^' the remaining 90 for 54c. .-. on 25 cento she & 116%' ^""^ "^ ^^^ *'''^*' '^' S'^ 11« <^^^- ••• «^e Thfw^^®°?*''"?5''r"'*'' ^*'^*^ ^°^* ^'e^itt fraction of cost. Th! I^r^";j°i*^ ^^'"'^"^^ ^'^^"^'^ «« ^hat goods cost h^ The reteiler sold for same fraction of what gocSs cost him • 18-64 ''''• •'• ^'"^ ^'^ *^« fraction- *P^ -IJfl- ••• fraction is |.§, and rate of profit is 20%. 148 /. 1 ; i ~^*? *^, ^ '°" (^) °* o"«i'i'^l cost. B sold to C for (irJ;« of ongmal cost. U sold to D for (|^)3 of original cos? C's gain was { (|x)l (« x)^ | or ^Vir of original cost. A's • • i^v) of ongmal cost = $5 x -S^LOo ^^ 9 241 _ ^1 i ooi « * 40. The gain is 20% of cost.?^o;;ycoiof f JSh =8c 195^; nf % ,T*i'7o*?^ = ''^^^- •■• COStof l8tC0W = $54A. -S% 6fi2°/ f"'^ '^^ ^"^/rr^l^O.. .-. cost of these two thfe iwo-l4o '°'^. f 1**^^ * ^*^. cows = $120., .-. cost of vZft5nn ^ f • •'• ^^^ ^'^^ ^*« $330X; the selling price was ^300. . •. loss was $30X on $330 « ot 9 ^ a 0/ "*"» P"*'® 42. IIOJ/ of cost of 1 buS. of mixtSe = 50c^/r cost of 1 bnsh^f -xture- 45^0 On 1 bush, of oato ihe irtould nnVTf* u* •* ° ^OOjmsh. of oats the loss would be ^^ Onl bush, of corn the gain would be 5X0. .-. on (1ooo2rT{ ^hr^^S^""' «"" "''^^ ^ ^^-•- - bu^of co7i 43. 3 articles are sold for 4 times cost of 1 article • 1 .rticle ,8 so dfor | of cost of 1 article. ... gain isTor'ssiV increased capital = sum each invested + $100. invested + $100 =JLs nf au«, «„«k ; Txli invested - 9100. . •. sum each invested « |7l ». 8 .*. sum each sum each r T IH TH» HIOH SCHOOL ABITHMBTIO. 45. The mixture of (a + c) lbs. cost (a b + c d) cento. .-. the mixture cost i^±l^ cents n«.r Ih Ti,« • a + c ~ ^ ^°* -^"^ g*^» on each lb. - (e- "L^^i) cents -?l£±^_?-» b-cd . v ._ ; cents- _ ce^^^ .-. rate of a+c a + c 46. 115%of cost=$16.10. .-. cost-euoo. Selling at 1120 the gam would be |6. .-. rate of gain would be^i la^/y Qnt V?,«*^^* ''''^^ ^* **^« molasses was $88. 76 HesI^ nf 4?i^^? ^^^'-'n'^'" 2«3^ gals. He receives payment for 947 siiiTeifj^^^a^^:,^^^^^^^ 46.6+0. '° **8-'''' •'• lio must sell 1 gal. for h J w'fl^'t tr''^, ''"• 1»5% °'.«'"»1 «»'• n the oct uaa oeen yD / of actual cost, to gain 10°/ the a«»lHn» ««• J wou^d have been 110% of 95% of a^ctual cirthaMriOUy of actua cost. This is 81 less than former seLg price W of actual cost-$l. .-. actual cost = $200. ^ ^ * ' ^'^ 108V J n T*'^^^?"'* ^ 1 20% of cost. The selling price is 108% of cost .-.discount is 127 of coat on 7 9ft''/^S 1 .-. rate of discount is 10%. '^ "^ ""* °'*''*- 60. Jones' present capital - 120% of $9600 = $11520 .'. Smith's present capital is $11620. ' * COMMISSION. 149 1. 2%of$12000-$240. 2. 2i% of $1850 -$41.62*. of Ut^5"J'o.'" ^''''- ^^ ^«^* ^ ^^« -plover 97 J% 4. Agent sent employer 98A% of selling price • QSiV ^# selhng price o 1 bush. 1 65§c*/^.. selling^pST: eeic^'^ ^^ 6. Cargo sold for $540. .-. com. was 1*1% of $640ror $104 O. Sale arnnnnfAd *o ^'^ '^fio J. «0" *^A%f: z"^^* "*^ ♦*"#• .-. rate of com. wi 27 ^^^"" ■*■*'"=' «^ ♦^^OO' ^^' was $32. 7. See Solution of 6. i! ,! H SOLUTIONS OP PB0BLBM8 On a sale of $5200 the com. was the com. was $2. . •. rate of com. h% of ei25, or 62Jc. the com. on $100 was 8. The com. was $104. $1C4. .'. on a sale of 9. For every $100 received by consignor, agent retained Uh .'. on a sale amounting to $104^, the com. i M In a sale amountu.g to $10000, the com! is $4(10. ** ' IJJ. o/^ of amount returned to consignor was thp n^™ frt; •• amount of sale- $945. 11. The com. for selling 1 acre was .-. no. of acres sold = ^^'* _ 80 63^c.~''"- 12. The com. on $ 1 OOuO was $75. . «f. .-. rate was |%. 'I50 13. The com. is 2^% of $650, or $16.25. 14. Ihe agent must receive the price he pays for the horse $2^r$m ^« musVreceivfll'oT:' cam'J^! T"* '^''^'^®' ?^ ^°°'- °^ ^100 invested. .-. his Sl%r' ^"^""^ ^^^^^^^^ ^y ^-- ••• t- -- iB .Lot 16. The amount invested is ^-M- of sum oo«4. *^ .-. amount paid by agent f or goo^d.^^ oHlTlSt $17T5?|' 17. The com. was j^^. of $3570, or $70. ''^ $100 invested h^ com. was $31 .-. rate of com. = 3iy 19. He paid 95c. for book,' and charged 20c. com' At same rate, on 100c. the com. would be 20c. x^V 0T21I .-. rate of com. = 21,-^% ^ ft . or ^1^. 20. Agent invested |^o ^f $1224, or $1200, in pork. $4 75 buys 100 lbs. of pork. .-. $1 buys i2?lbs. of pork. 100 4.75 $1200 ^"^'475 "" ^^^^' ""' 25263^^^ lbs, of pork. 21. The price paid for the wheat was $9600. The agent's com. was U V of $9600 nr «i9n . * , ' "^"^ agents $9720. *»ow, or ^IM. .-. employer must remit 22. The agent sold the annles fnr .$1QR0 H- mm f^^ "^ was 2% of »1250. or* $25. "Tli-teft fofi^v^^J^^t 1» TH« BIOB SCBOOI, ARITHMCTtO. (Muding «,„.) ,1225. Hi, „„„. ,„ , •mount paid fo. ,„g„. ., y, ^^ ^ e - ' J^ of left for investment. .• his sfiPnn.i U or {518^8^. ... hia total com. was Wi ^ i^^^ ^>^m^- «i.5o-,«A. r,^^. ^'^ -u^«« lumber, and com. Oat of « 10. T'^*' *"'■ '-vestment in vested ip .„„be, jioo *'?'','''« «8^»? k-P' «1 and in. MOl • *^**> ^ 1000, or 31 1963 + ft. QR^it" ^® broker invests ^go ^^ dtionnn • 25 lbs. . .. he buys 25 x 10/ J? , onS ^on", P^""^' $1 b^ya •f i'^^' ^-^000, or 294 11 7U. lbs. 615610 for investment in cotfon anrf f ''• ^^T remained paidforcottonwas 100_ ^'';^^^ ^-o- The amount k ,. iOOl ^'^^ISeiO. .-.theno. of lbs. of cotton bought was $.|oo of 15io) + $.16, or 10329Uiax T 26. The agent evidently retainpd «i%n ^^^^ ^^^• Bale amounting to |1 2600 wTs i^^o ^ ^ ^^ °°"- ' '• ^'om- on >ng to $100 w^ $ll " ^fj^^^- •'• <'^n'- on sale amount- ^^ 27. The agent pays 6c for lb """Z ™ H%- the freight on the same aidhJ.i; "^^J) ^« P»y« ic for com. ... the total co^lVlIb'^ ^5 ^c^' ''' Z^^' ^" of lbs. bought is ^ ^ 120000. '"' '"^^' agent pays 46c. for 1 lb. of w ch ""' ''^ ''^^eOO. The of tea. Pays freight Ic. foVub. of S" '^h^.'^; ^ ^ ^b' 1 lb. of tea is 45^0. • ,k. u * * ^' *^*^^ i^«* ^^ lib. of tea is 4510 Z V^^ '''''''' "^'^^ .7847X11 ••*^^'^'^^r of lbs. bought is ■78471^. 83600 lib V It •1 ■r ; lOLUTIONS OF PROBLIMS II' Sy *«?*^* receives (2+2i)% of sales, or $54. $8000 *"" 8ale-e380. .-. amount of 8ale« «ei?L^!lt^"*?o? ^°'- ™ H% o* amount of sale. The sale = $40. . •. amount of sale = $5333|. 32. Com. for selling cotton = y^ of selling price of cotton. Com. for buying sugars _*|. of^^^of selling price of cotton. =2r§l§o-^of selling price of cotton. .-. two coms. amount to \Txsv + Tu^^xrh or ^g^g. of selling price of cotton. . •. JUL of sell- ing price ot cotton = $220. .-fcotton sold for $406o!^ r.»rA on ? *?1^,<' P^id ^2.80 for each cwt. of flour. He also paid $. 20 freight for each cwt. of flour. He retains also $.06 A com for each cwt. of flour. ^ .-. it costs $3.06 ^ to buy 1 cwY of flour^ But com on sale of apples is j^ of^mount of sale, aLt. «,r-"'l\H* ^, P"^°^«^« fl°^--- «>"»• on sale of apples, sufficient to buy 1 cwt. of flour, is ^ of $3.06A. .'. total (jm. when 1 cwt of flour is purchasedTw $.06^+^^ of 152 34. See Solution of 33. 36. The com. for buying tea was ^4^ of cost of tea. • net Kmn'T "*'' "* ?""^.™ ^ ^««^* o^ tea. The co^ for ^^ 2**"' ™ ^ ^^,^"^"« P"«« o* flour. ... the com. for selhng flour was x of net proceeds from sale of flour, thatis, was ^i^of f^J of cost of tea. .-. total com. was (^ 36. Agent's first com. =^^ of sellingprice. .-. second com. 18 oflOO-a . ... . :Oi -^QQ— of selling pnce. 100 + b "' ' 100 °^ seuing pnce. .-. total com. = ( JL + b(lOO-a) , . „. - .u ^^ io o(iou+b) ^ "^^ '^^'''^ P"^^= roSTb ""^ '^^^« P"°^ ia 5i' T\" 5'"'* T!!^'.!f oaJcuJated on $4060. The second com. Z i^^^ '*'' f ^*^V , 0/ ^^ 20 is the com. on $4060 + $3940, or $8000. .'.rate is Uy. * 3ft SL»« Mrvl,,*: r ^4° 40. The selling com. is calculated on $4040. .-. the Iir TBI HIOH SCHOOL ABITHMKTia 41. The aelliug com. is calculated on $1421 • *k n- ei3;2, or $32661 atinVlL raW * ^t^r ^^ ^'?^^t + .-. selling rate is 2%. ^ ' * ^^^^sting rate is l|%. 42. Selling com.^ia calculated on 9i17^± n • calculated on $1649 SelUn^^™ • *JL , ?"y'°g ^om. is com. on $1734 calculat «f '' ^^^ "^^ ^^^34, together $.734 = $T7.34* • »-$^V3^^^ ' % o^ $1734 + $1649 or^ ' "^% of risk : lum. Company III. receives ^Vr.tVinnnAn ^i^^ ""^ P''®"" ium. Divide loss in Zn^Z- "^4 ^^^^^^0, or $875 as prem- divide $10(r000 £ ratbof 5 fi"" 1^^""' °^ "^^ '^^^"^d. ... $100000, or $15151^ et. ' ''"'P^''^ ^- ^°^^« ^"^ «^ Sme. value °f fSX_ ^"T™,"" ''?f '?% »* f "f 8 premiums VA of tre~o??„^'i^tl $ st"'"™' /"*»' furniture = $1 2.60 x %Aoo . toi„ /u ~ "^^ 2-^<^- • "• value of 5 = $1901|| ^^^- ••vaI«eofhou8e = $12.60xV^oox ^'of ^afue^v^^sirUli^ '' ^ ^' ^*^"« ^^ ^--^ *^-t -. ^2 times value ^fveslel ?hTir T *^1 ^*^«o ^^ §% o^ I The two premiums- Ts ,'.tT V^^ °^ ^^^"« ^^ ^Wl of vessel £$128To!-^.rui;^^^^^^^^ - value the premium is $316^ ^(° " ^Ifol ,' •'• '''' ^^l^'^ ^^^"^^'^'^^ ei?. .-. rate is |«% ^' ' ' ^ '^^^^ "^surance the premium is su 'ed sIpa'rLtt IVto^^^^^^^^ ^^^^!! ^^ ^^ -e in- solution will be^Ls folbws ToofvirT.*'^^ premiums, the by $100 insurance riSOOnll 7i''*''^''^^> ^ ^'o^*"-*^ $18136-02. A^a^n ^901 1? 7?^"^ (of vessel) is covered by insurance. .^ $14000 vair/T^''^T^°>" ^^^ered by $100 ••• the vessd and car/oLui^^^^^ '^T'''"^ by$24120.60. premium paid willXT22r6 ^he prfli'tn$^^rsr^^^^ is the .'. value - Unnn "^^l^^^^^- • • f of value of flour - $3000 iioot+Vrrr- «6 tr:^, t ^".r ■'"'^ i".'. «^^' $10-15. '^ 'S'^'viD. ... i }ji)i^ ggiig f^j. 27. The premium is value of house « ~ 156 1% of I of value -$21. aoLvriosa of probleujb a "'.ft irjjsr "" *''" "" ""="■* ^« -»-^ or W7m ^""^hTrJ^suT™"^' "■" »3000-»2962.60, 31. I of value of vessel =*° of ^49015 _ «qq>ia ,, vwue, or | of value. Companr II. load 4. nM nf ™i * ? 1 vidu* ... net loss of ComC/l faa +11 "^ ' Tl*- .°7 °l vriue. And net lose of (C^y if.fa'^r^^T 11^ "! value. .-. MSL-jjil. r«- DtT^itt i '* ^W'hJ" «rV »' - W9000 ,^&. %• lis ?i;tTf.irid- ^"'"0? vaUe^fprenuun.^ ■•• owner, lost «4lo00x^r(?+at ^oHS??-'-*-" ti;'^^wte»^rr,roJ that i^ L nf ^""5' ^^ ^/^ ^^ t <>f I of ^^l"e of ^y hou8^ !4!l nf liy. 7^"^ °u '"y '^°"««- r^ifference of premiums k • Vtjwl? ^ t^ my house: diflFerence isgiven rqual toTl? . . value of my ^ouse is $6283^. ^ *^'^* 35. $i is premium on $100 insurancft • Aik i» - on $6000 insurance • I of v«i,»f ^* a ' * ^^^^ Premium or SlfiqftR ." ™ • • i oi value of flour was $6000 -«1 5 ^« 22: ••^*''»® of flour was $11970. * ' OD. The cost of the cattle was %! ftonn tj,« .•« 3?^^> .V * °l* ? "r*"^^ '■y »l»312Jf insurance. of?^i^T^'°'?i'<'y-t°f™'''«<'«^»'»e-.--^ount " P<««9y - TO o* vol"* of house, that is, }f of value of house. [* insnranoe, seived f of he received -82962-50, D. .-. the >i or $60. cmium on value, that t H% of I tf of|of ue, or I of or T% of •r AW of .'. value Qd j^^ of >use, that a is 1% of ny house, )miums is J to $12. premium 00 -$15, mce is to by $100 nee. . amount ^f house. IN THB HIGH SCHOOL ABlTHMETia TAXES. 1. 2. 3. 4. 157 2% of $5000 = $100. On $1 the tax is l|c.: .-. on $2500 the tax i« «q7 Kn On $1 the tax is 2c.: • on «1 9nnr; fu . /sfS'.SO. .„ On $1 the tax is 15 miik 0^2000 h ':^^^^- mills, or 130 ' * *^""^ <''»« tax is 30000 ta-fisto f ■^n?;, '"V' ^'^■"OO- ••• on »1 th, 7 ThA f n ' ' ^ ^ ^^ "'^"^ on the dollar received is $14400- $288 or $14112 '*' *°***"''^ $19b®r^'i^'JS^^^^ •••98% of tax = the tax is $20000 on an Z^ ^^ ^««e««nient or$1200000 or 16f mills. ' ^ ^ ^"^ assessment of $1 the tax is $^V. Pa^ l1^% ^f m t $H^0°^ ^^'''-^^^^' - ^«00. ... A $.016 x'^l'ooro^liT"'"' " ^''''- T^« *- on $1000 i. 1400 JrSr'!t^ for $1400. The tax on $1400 is $ 015 x ^^ 12: Thf tix i; $^?oo!i$i4S Tiu To'' - '^i 100 $lV600 • "''''' ^^'"^^ = ^25. . .. the income = $25 x ^=-^. .-.25% of the capital = 11^0^ ,. .^1^ $5714f 7 ^*'*' = 158 »10150. .-. assessment = 8607500 ' ^/o* assessment. - W95 pays no tax. ^r. L |f bette?^""" ""»* ^«'"'<' 80LDTI0NS OP PROBLEMS .'. on $1800 the tax is Man pays tax on $900. 17. On 8266000 the tax is $4250. $4250 X „ifoo^, or $30. 18. On $930000 tlio tax is $1.'3000 .-. oi$900 ihe tax is $14^«. nr l%r!n''^^ '' *''^.*'''' *"" ^^' •■• ^2* '■'' '^>« *'^'^ "" TTiV X 24, OA o . /■• ^y ^"*'°°'® "^"^ $1500 + $40(), or $1900. 20. Pmif h pays in tax $G6. Ho pays in ir.surunce $22.50 loses 11 teres t $300. .-. house costs him yearly $388 50 or monthly $32.37^. "v..cv, 21. 95% «.f tax = $.-»700. .-. tax = $G000. 22. See solution of 17 above. < DUTIES AND CUSTOMS. «oln'"'m/'?^f'' r>'' ^*''' *^^ ^°«k ^2.40, $.20 and 15% of f.; n?' ^^i '^' ^® P'^y*' ^^•^^- •'■ he charges me 125% of $2.9G, or $3.70. 159 w. Tho cost would be $2.G0. To make same rate of gain he would ^ har^ra 12o% of $2.G0, or $3.25. The amount li gain u ^ "T^.A '"^1 ^'^ ^^^^ ^h« «a™e amount of gain he would chargo $2.60 + $.74, or $ 5.34. 3. Th^ duty is 15% of $5, that is, $.75. 4. The specific duty is $30, the total cost is $390. .-. the in- voice price and the ad valorem duty = $360. .• 120°/ of in voice price = $360. .'. invoice price = $300. $25;2olT3'78!"'^ " ^'^^ "" ^^' """ ^^^•^^- •■• ^"*y = 15% «* 6. With duty the cost per gallon is 115% of 40c, or 46c. . .selhng price would be 125% of 46c., or 57ic. Selling price, would" be ?fc' ^^^^ ""^ ^^'" °' ^''- ••• ''^^'''^^ 7. The area of the section is (H x l)^ x V- sq. ft. • no of ^ 8. 1 sq. ft., board measure (i.e., 1 in. in thickness) contains Tjr c"- ft., or 1 cii. ft. contains 12 feet, board measure. .-. the stick contains (20 x 3 x 2 x 12) feet, board measure. .-. exi duty. .12 X 2 0x3x 2 X 12 1000 cport = $2.88. IW THB HIGH SCHOOL ARITHMETIC. 9. The specific duty on the tobacco was 30c v 50 or«lB .*. the ad valorem duty was $2 50 • 191°/ ^* i "'. ^ ^^^• -12.50 ; ... value = $20 ' * ^^^^^ ^^ ^*^"*' <"^^°i««) «ol?l ^*T^ ""^ *^^} T* *^ ^«aler«e304. . . cost to dealer- t ilsTl 'P'"!'" ^"*^ n" '^•^^- ••• ^ 15% o^ invoke pri.l -HpiaZf .Mnroice price = $162. 73+ « p"^^ $30?-^?27^82°:S'6'lV+.-'-^^^^^ ^^"^^ ^ -^--^'^ 12. The whole cost wm $4384, and cartage exDenses IIIIOO Myrf'g'.P/'T"""'^ valorem duty -«428*4! TtTty »^ 160 14. If the duty on cases had been also 207, the- total Hntv would have been 207 of «30 or «fi Ti»f Vk j®/*^*^*^ ^^^7 was 3^7 . 1 r.V /t' • ^ ^ « "^"^ ^^^ ^"*y on cases Ti r:n "^^ •• ■ ?^ o^.^nvo^ce price of case was $7.50- $6 or *l.o0. .'. invoice price of case was 1^10 ^^a " . v'^'.or of instruments was $20. ^ ' ^'"'^ •'• ''^^^^'^^ P"^« o«^^fl'^i^""'"^^L*'^ P°""^« <>^ »'«sins was 15000 • th« $16 and specific duty = $40. $2 = 8peeific duty on 1 doz • . $40 sp. duty on 20 doz. ^y »a i aoz. bale contains 75 ^yl ... the ^^.^w^'f ,.. .T -^V'^ $200==$54!'' .^totaTdutyisiyn' J"^^ = '^^^ ° ^V of = $97 -«fi9i 1 «1A 2 "V ^r^Tff- • • gam in saving of duty 1^!^ Vor%r*l¥j-. . 1^88 in value of wine is $20. .-. net Io88.$20-4i4^ = j5j|. SOLUTIONS OF PROBLBMS » I 19. Since each shade is worth twice as much as each roller. .•. 3 times invoice price of 1 roller- $3. .-. inroice price of 1 roller is $1, and invoice price of 1 shade is $2. The ad valorem duty on 1 roller • 30c., and on 1 shade = 30c. The total duty on 1 shade and 1 roller- $9.90 -^ 12 = 82^c. .-. the specific duty on 1 shade is 22|o. .•. each shade contains 4| f «« 32 ^e xu- -fti«n ? "'■««'"'« HU'Ji- . • Ut— nru'. oputw of this sum -$4«0 ., the sum received for stuck sold was $100500. Since 4/ stock was at par, $100500 is amount of stock bought. «inn r , """i^ receives from the broker $67^-$' or $67, for f Sinn /'!^*^,::''"l^®f*y^ the broker $99^ + $!, or $100 for $100 stock bought $100 of 3% stock retur'ned $3 income. This stock yielded $67. $67 invested in the 4% stock at $100 returned income $2.68. .-. $3-$2 68, or $.32 was loss of inl $150000^^'ck' ^"^^^ "''• ••• ^''' ^" ^^^ «^ --- - 27. By selling $4800 stock I gain $150. .-. by selling $100 s^k I gam $3f .-. I must seU $100 stock for $75 + $3^, or : Pi lii i I 'i SOLUTIONS OF PROBLEMS 28. $120 buys $100 stock, which yields half-yearly $4 • stock yields half-yearly ^-^, or ^\, of sum invested in Jt. 'The secqpd half-yearly dividen-l is equal to the first dividend, in- °''®^S ^ *^® S'""" ""^ ^""^^ divid. nd. .-. 1^ of first dividend «=^496. .-. first dividend was |480. OQ?/^" i?i?^ ^^°^°^^ ^^^^^ ^^ income. .-. $100 consols gives iko! TJ'o^^ ^"^°"'®- ••• 3J% of price paid for $100 consols "^ o A^ ®^ • •• price paid for $ 1 00 consols = $84. 30. -g^y of sum invested in 4% stock must be equal to ^^ of sum invested in 5% stock. .-. sum invested in 4% stock must be T^ of sum invested in 5% stock. .-. j* of sum invested in ¥ f^^'^'J = ^^^^^ + ^1 2000 = $21 000. . •. sum invested in 5°/ stock = $11052if. .-. $12000 - $11052j|, or $U7^\, must ^ transferred. * 31. The amount of the mortcage was $1,600,000. .-. the amount of shares (stock) was $2,400,000. The yearly inter- est on the mortgage is 6% of $1,600,000, or $96000. The yearly dividend to shareholders is 5% of $2,400,000, or $120,- 000. .-. the neb income must be $96000 -t- $120000, or $216- 000. .-.35% of yearly gross receipts = $216000. .-. yearly re- ceipts = $216000 x \%o. ,., average weekly receipts $216000 xWxiV = $11868.13. ^ 32. The amount of preference stock is $500,000, and the iZ^t^^ 2"" '* "^ ^% ""^ $500000, or $40000. The balance $55000 - $40000, or $15000, is dividend on $500000 ordinary stock ; hence on $100 ordinary stock the dividend is $3 • rate is 3%. 163 33. In the 6% stock $1 is the income derived from $128A -^ 6 f ' ^^iti invested. In the 5% stock $1 is the income derived from $99^^5, or $19f§ invested. .•. the latter stock is the better investment. 34. The net gain is 6% of $1,000,000, or $60,000. 24-7 of the whole stock is $25,000. But preference stock yields 87 • ■• 5 i % of preference stnck = $60000— $25,000 = $35000 • preference stock = $63o363y^y. 35. The dividend was 8% of $200,000, or $16,000. .-. 57 of increased stock = .f 1 6000. . •. increased stock = $320.00o! .-. new stock = $120,000. . 00/ $^5 invested in 3% stock yields $80. .-. $6000 invested m 3% stock yields $6400. m invested in 3J% stock gives of m IN THE HIGH SCHOOL ARITHMETIC. $34. incomfi • $^400 . . •""""• • -r- '--ted in 3i% »t«k gives »77J income. $6400 X 2 $100 invested in 5% stock gives income. in- total incomes: vested in 5% stock civea S!9iqi • 37 On $60 the gain is $21 : • on «1 nn .i, • .-. rate ff gain is 4J%. ^ • . on ?piOO the gain is 38. The stock soId°*at |90. • 1 1 910/ ^- , . . •. buying price = $80. Stock wV« kT*?. "^'"^ P"*^® = $90. come. $120 navs for «iinftT 7 i^f/?^ ^^^ yields $2760 in- .-. $6000'0 payXl'stttfclW the total income was^^esoS Th¥ir? ^''^'^' ^^^^^- •'• bought was $100,000 • ihl'u 1 ^^°^® amount of stock 41. $15000 stock pav'in. 50/ '^^'!'T.r'' ^^SO. from 6% stock waaV5Tr$6o^:Sl^«'^^ •'-«<'«"'« investedf in the 67 stock • S!«?n • ^?'«'«<'ome from $120 invested in the 67 stock .jfti^nnnV'i ToT^ *''""» ^^6200 42. $1779-$400, or^ISTQ in -n .-. 98|% of taxableinclri^ 379' "tVah,*'' '^"^^ '2 ^^%- .-. whole income = $1800 «>> ,'1 ; * ' ***r^e income = $1400. .^ $180^ « income from $41400 ''''''''' ^'^^ ^^ 5 invested. yearly faymentewnUniOOoI^^^^^ ^"'^ *"d the (1.06;« + $60 X (1.06;« + $60 1 1 05 .^^^^^^ >< (1.05) + $60 x 5384 TU *iQQVKoo7^*^*"'°"^t8to$1331 ■5< i o' «4: ...the 19. See'Solutionof 18. %?Vl! • ^^^^rT ^^^^' °^ ^^^O- ^he interest for year is $500 x -04, or $20. .-. the number of years is V*^, 22. See Solution of 21. ve^v'Jla interest is $100. .-. the interest on $100 for 1 year is $6. •. the time is 100-6, or 16f years. The amount theram^:^'m7''^"^^^^^- ••• ^^^^'^^ JoLts AT^^ ofi%:srrii^:^^^^^j:.j/- ^^ of money will double itself in 25 years ^ * ' ^ ""^ pritipI^i^'tfe^Kri^r'oVt^-^^^ ^ y-- ••• the will double itself ?n 33J ytrs"^"' "' ^^^ ^''''- ''' ^^^ '^ ' IL'int.l^^rf"^^''" ^ ^•^\'' T^ «'• 1^' «^' the «"°» lent. • • 07 ^'?*'':e«t for 3 years is ^% of the sum lenv. ^/. Ihe interest for 1 year is -05 of the print inaL • f»,o interest for 4 years is 2, or i, of the principal ^ ' ' ^' 28. (a) For 1 year the fraction is ^X, or ^^V .% for 4 years the fraction is ^ M For 1 year the fraction is ^, fraJtTon' s I ^ ^"7 *^ooT'^"" ^^ ^^ (^) For 1 ya^rX traction IS ,V- .'. for 292 days, or * year, the fraction 1. A. sufieT^if.t'/.!rJ:t^^.^« -merest for 1 year. .-. tYe •- •"^.-u-nroo ror 1^ years. dO. jj^ of the sum lent is the interest for 1 vear • fk> sum lent is the interest for 20 years. ^ * ' ^ SOLUTIONS OF PROBLEMS I liii- 31. 6|% of the sum lent is the interest for 1 year, im lent is thA in^aw^ct- *«- 100 _ '. the the sum of Slim lent is the interest for ^, or, 16 years. ""To^^.TJ".^^"^^^ i*«eJf i« 16 years. , */b.or the sum lent is the interest for 1 vP^r • ^u sum lent is the interest for 25 years tw^PA 7h ,^* is the interest for 50 vears • % hi I, ' ' * ^^^ ^""^ ^^""^ itself in 50 years. ' " *"* ""'"''^y ^^^^ ^^^^le to HXattlte^^duT^^^^ '" ^f '^- ?*^ '«°^ '^"-"-^ the value . 710/ ™/i% £the value ; that is, to 7^7 of U tL ; • ^^ °i "'t^''^ = ^240. . •. value = $3200. ^^° • t:\t iS''®' ^"""^ *^® interest -9% of the value - «720 • • If'r^^' .12 months = $720. .-. rent for 1 month-VeO 35. The interest on $51 1000 for 5 davs = iosn ? 7? ■ . estonftSllonof^.. s«K ^^"V"ioroaays = ^j«0. .*. the mter- esc on $011000 for 365 days, or 1 year-$280 x 73. .-. the interest onJlOOfor I year = §2|0>^.,i. .., ^ '^ ^^ i5%, and second rate is 4^ *-«... ursc rate is J "». xue interest on ^bOO for 3 years = int^rpaf «»» ftionn ?»* ZrLt'tTte-k or''°? «600 + »3600. or'a42M .t tZ ,ate= 2t " fi;;t*',S?e -Vif' '"' *""' '"' ' '"'' 38. The interest on $250, for 6 months = interest on ft! 9fi 'm\ \7. ,:- '^&- r^ra /I rHS rate = interest on $506^ for 1 vear afc W^r ' ^- f ™^ on $125 + «506i nr ««^i 1 * i ^ ^* ^**®- • *• interest interest on 41 n^' ? 1 ^' ^^'' ^P""^^ *<^ ^^^^ rate, is $25i. • interest on $100, for 1 year, at first rate = $4. .- first ratr ' 4%, and second rate = 37. ' ™tc = 39. Interest on $100° for 2 vears af 1°/ «q *,«« amounts to $108 in 2 years. ^ y^*"" ** 4% = $8. ... $100 40. Interest for 2 years at 4% = 8"/ of pnn.Jn-i . 1 aoo/ of principal = «540. .-. principal = $500. ^-^^'^" ' ' 1®% IN THK HIGH SCHOOL ARITHMETIC. intl'o/^}^^^^^ ^°^ ^ months at 5% = 2i7 of princical • 102|% of principal = $820. .-.principaUSGOO.^ ^ " ofpLrpai'i/r^ •••103% f? T^i ~* ^- •■•P'incipal = $1300. ^° for 91 Q ^'^ '''''''Tof ""^ *^^^' ^-01^ X SOOOO = $750. Interest ?033y^lf ^% - m ''. 6% "f P"-'ipal = ^% of princ "al $723 93'+°. depusited = $750. ... sum deposited = or st590^^^.'Si^«Sf"^^.' ^* ^% «°^""»<^« to 106% of $1500, 45 tL • ; ^^^^^ "i^ "^'"'^^ i« ^^« better o«er. nnioKnf 'i.T/'^ ""^ ^250 iu the timH = $25. .-. the interest $250 for tlf the'l" ''"^^ - lil^ ••• ^'''' '^ the aLott : half the W ' '''^'' ''• ^'^' '' *^^ ^^--' °^ ^2C- 1 for 46. Interest for 3| years at 8% = 28°/ of princioal • 1 98°/ of principal $320. ... principal . $250 ^ ^ ' ' ^^ sum J?oVeT-Tl6r ^'^^''^^^ "^ ^""^'P^^- •'• 111% <>* 48 T^^^ x~^l ,' •■• ^"™ l'<'rrowe«l = $149 «V. 48. The interest for 12 n.onths = 6°/ of princioll • th^ interest for 2 months -i°/ ,><■• ^ • • /°, P'\"''^P»^'- •• the anvnumhprof . I^ lu~- ^ ot principal. .-. the interest for pHed r^^P h 1 f ?i! '' '^T^ *° 1% ^^ the principal multi- Iq A T i:^! ^^^ """'^e'" of months. ^^ ^ " e45tni'$4l5^' fd^^^'^^ ^^^"^ ^^ 3 y-- Coffers I455 at rArflr/ "^ °^J y^^*"' ''^^^^ at end of 2 j-ears, and to $568/ $530« E?; 1 ^'7.r'^ ^* -^^ «f 3 yeiraam'ount 3 years *' C offSs fieS ^"^.^f^^.^ ^" ^'^ ^2047^, at end of y/ars. .-. AfXrlsTeSst' ''" '"^""^^ ^° ^'^^^ ^ ^ of Im^olTlVel^lfsV^;^ at 5% = $,260. Amount year at '5°/ \VqoQ ,^^^1^23. Amount of $1323 for 1 $1389 15 ^°^''^^^^-^^- ••• ^* ^^<1 «^ 3 years I must pay $llm^or&0 'w'^'^f ^'Tn^^o? *^"^^«^«« ^as 15% of v^onv, or ;jpjJDO, He loaned 90% of $1500( or tlViOO .'. yearly interest was 8% of $13500 or «ir»sn ^^13500. yearly income was $2250^1 $loSo, or $1 1 70 ''' ^°'' ^" 168 , 52. He sold the flour for 112Jl°/ of «9non ^- jgooKn rr, s/o tuosum. ..lU^i% of sum deposited = $2250. .-.sum n E&iJ |l III SOLUTIONS OF PliOBLBMS or'^j^rr?!'"'' +• ••■ "^ ''"^ '=f' »2260 - »1961.21 +, moM^lt^^/it^ ^ °""'"" "r "'™' W *2500. I„6 for 1 ^„ i. 4f y'sJn'^ i„:t' ":..Ya"e'.-:™^'"- ■• "^^''" 81000 ?;"l'f i»* j « itv is Spnf 4fV, ,, * '"^^- ^st IS feept. 1st. .-.dayof matur- dL^Ss $ 00 "7; f „- t>%Y Sept. 4th is 95 days. .-. - $12.60 == 1987.40. ^^'' ^ ^' ' "• P'^^eeds = $1000 169 before it/d„e^''".'i!:tunt=l2?6Vo%trr.'l'J» . •. proceeds = $2332. 10. "" ^ totj >< ^t = v-^*- 40- 5. Three months after Jan. 15th is Anril i»!fk . j * maturity i, Apr. 18th. From Feb."ut''S'ipr!'l8ti.tM days. .-. discount is $1250 x ^ x ^^ -flsifl qo j =«i233.08. "m^m-^''-'^ ■■■^"-^ 6. Four months after May 23rd is Sflnf 9qr^ . ^ * 3; 4b6:Sr *'''"•'' " '^ " ^-«106.32. .-. p* . Sr Lr^Mn^- %Jf ^i'- *"■' ^- ^™"» »»■=• 2«h. If " io •oiar. 4tin, io90. la 70 dn^jo . j: — ,-_.l a<».»^ « \. fdl.ae. .-. proceeds = 12737. 14. ^ '^""^Tuv^v^i- IN THE HIGH SCHOOL ARITHMETIC. S* Four months after Ap. 1st is A „«, 1 of .j - Q q;«««^u ^^^ ,^3^^~'''^-"^* •'• proceeds = 82 71 39 9. Since there are only 28 days inFebruarv IMft fv.? * would be nominally due on Feb ^sui t^ •^' lu ' ^^'^^""^ tice at banks THp rul r.f ?' •. V ^^^ ^^ *^® "S"al Prac- Dec. 1st. ?889, t^Mt^fd'llsrrsI d" """• ^J^' ^^^" 10. Six months after May 5th is Nov fi^^. .XT* r turxty is Nov. 8th. From June 4th to Not 8th is 7.7 !l'"** .-. discount = $1234.56 x t4^ x » » ^ -«^i s« '^ ''.^y^* $1202.70. '•''"^TTr(rXiriry = *J1.86. .-.proceeds- 170 11. Thirty days after Jan. 29th is Feb SftfK . a 4. * matunty is Mar. 3rd. From Jan 29 th to Mar 3rd I^^hI.^^ . '. discount = $400 x ^^ x a ^ - «9 «q i P °*y^- 12. Four months If^r l^rfd is Jun^JTf ' = ^f ^•^^• maturi.y is June f?th PnflV i *^"1® ^^^- •'• <^ate of .-. discoLtiSe 75 X Six «^' ~\%'or '*^ '^ ''^.'^y^- $564.49. •'^ ^ TXH7 X yg.y = $12.26. .-. proceeds =. of the ^ ^ jieol nltLr^:nVb:u^ fArZ 80 day^ ... discount= S486.05 x .^ x .0 _«« ,q^ * proceeds-»486 05-«6.39 = 1479 66 " JnrT-»6-39. .-. »2j^b ?ot''63'rAT=to^ x'^x ^'":-. -\^oVr™"r |i992.8l" •'• "" P'™<»«18-«2020.71-»27.90- «420b!i:2l;Vi^*"3"l '?,^7- 26th The i„.««,t on note is «l274.91."l^„r«rpf tf to Not^^rhTse'd "" .-. the di9count = $4274.91 x,S,x " _«sVi m I? ^"y^' oeed8-*4274.91-»80.68.V4194M' * •®®- ■■• "■« P"" 16. (a) Since there are less than 30 dam in TPal,™.— -^ • t.rh:,tX"^r.SL^ r £t t"?^«" on M^r«k 9-^ ^ °^ iJeoruary, and, therefore, legally due JeL?t vTdT;, ter •' t °' *? r "■"; ••• *^ J"* "» TUT 01 xTnr> or j^ of the sum lent. ('m SOLUTIONS OP PROBLEMS 18. The interest for 1 vear is « «* +1,-. « , note. ... the interest for^95 days^^^^' of '""T - *'? the amount of the note. ^'^ ^^^' »^ ®^ 171 «^^or^^^"°1 J' "^"^ ,''' ^^ ^"y«- ••• *^« discount is «• of TITO, or 7j»^^, of face value of note. '" 20. The discount is JJ>^ of 1 __ 1 ^^ * , 21. The discountTs ACof"5' <"• 1^',°"«™ ™l"e of note note. .-. seller of note deceives ."S.'.'/flf' f, *"'% ^'''o «* 22. The discount is ,% S\'?" ™ ». ™'°,%°''""»; ""h • Tie ^-.^ "' 'r ™'"-»"™ -"See vallttslsto'' «i4. Ine note is due Spnf 9«f»i . •«. • t vooo.uu. d.ys before it is d„e."%t''Lr„t i; I fVTo"t"'i' 25. The discount was $730x S724 60-«f5 in . *u j- dTcouIrffoLt^SoT","^ «M"(¥/:or»«.8b: *' t from $,00 for 1 ,eaHs m'VZl ^t dSunt'if bV'"'""'" for 1 year^woS t $127 75x' .0 't tr""' '^ /i?" count is 7%. *«'»81.64 X f60.6J, ort3942.26. 173 fa »400 X :Jr,v'''„'J' ^'So '^"' !? ^3 "^y- The inter^t amounts to^404''|o TK-"' ••"?«»<"« at this date of the note on Mar islh is'^Sl" T' ■'i,'^" *»'•»<» JulylOthisll7davs Thl,-5 ?°- ZT' ^"- l""" t» fa »7.40, whioh store tS, *?'"" *^^* ^ f""- '" days 15th to' Sept. 20thT 89d!^^s'"^TT"'f"^'-. F™- Ma'r. Tm X Hi c.r «!1 1 M 4i y°' The mteest is J384.80 x WlMVlr I- -i "rf • The «,no«nt of the note is 1384.85 + ••• the balance of the ^L T'i^l'^Ht -T""' "^ «Je6. Sept. 20t,h f^ nL 0..?°' o- •*'P'- -^''* '» S240.76. Fmm T«i X aC ot «3 v'b f .^^^ays. The interest is 1240 76 n iOLTTTTn^TB or . » 0BLBM8 A ^*' -^^ »'^^'*8t Oil $1000. from Mar. Ist to June let, 92 2Xn'fp* mj^' -^^""n'^ of note on June 1st, 1888, is V:?^Ij ■ ^ilfni'*'^'"^"*^ ^^ ^^°^^- ••• ^h« ^>alance on June let, 1888, 18 $720.16. The interes') on $79,0 iR ' -.m June Ist n^«?9nJf/' "'T *''^" *''® paymrr., i;.o ihe interest on^720.16 from June Ist to Jan. Ist, 214 days, is $33.78 .-. the amount oa Jan. Ist, 1889, is $753.94. The payments $643 94. T»>., interest on $643.94, from Jan. Ist to June 1st. laLoP' if •'^21.31. .-. the atfiount on June Ist, 1889. is TflBQ •• ^5"® payment is $400. .-. the balance on June 1st. 1889 isi>i,oo.2:>. The interest on $2G.-).25 foi 1 year is i- rru''' "a^ance due on June Ist, 1890, is $286 t7 1 * "lotn \°.*^''''';'* ''^ "^^^^^ ^'•o"" ^P'"" 1«^'» 1889, to Sept. 1st, 1889, 153 day«, is $37.73. .-. prir.cipal and interest amount to $lo37 73. The payment is 8.-.;0. .-. the balance on Sept. 1st, 1889, is $1037 73. The interest on $1037.73 from Sept. 1st, 1889, to Jan. 1st, 1890, 122 day.s, is $20.81. .;. Pnncipal and interest amount to $1058..!)4. The payment 18 J^bOO. .-. the balance on Jan. 1st, 1890, is $458.54.' Th^ interest on $458.54 from Jan. 1st, 1890, to June Ist, 1890, $469 9T' '^ '*■ *'^^ ^''^^''''^ "^"^ ^'^ '^"''^ ^^^ ^^^^' ifiio' J^^«^iite'«8t on $950, from Jan. 25th to March 2nd, Iq-aV/ mf' '^ ^^^*- ^"'^^xal and ir erest amount to $906.74. The payment is '25. •. the b mce is 3731.74 • ^VSo^""^'* ""^ $731.74, from Mar. 2nd to May 5th, 64 daysi IS #0.98. .-. the amount on May 5th is $740.72. The pav- menb IS $174.19. .-. th balr .:;e on May 5t> ■s$566.5[; The interest on $566.53, from May 5th to June 29th. 55 days, is •■^10^ "rlX ^T"""* ^"^ '^"^^ 29th is $572.51 The payment 18 i|pi8<.50. .-. the balance on June 29th is ;|385.01 The interest on $385.01, from June 29th, i: S, to an. 1st, 1889 $39874'''^ ^^^ ••• *he amount due a J . 1st, 1889, is onJu' F^®^,*^n*fX®f °'' ^^^°^' ^•'oo* Sept. 13th, 1886, to Ap. foci' ' "^ii ^^^ ^*y^' " ^1^2. .-. the amount on Ap. 20th, l??no '' rl^^?^- The payment is .S800. .• the balance is f o ? . f interest on $2702, from Ap. 20th to July 2nd. is »600. . •. the balance is $2 1 29.02. The interest on $2 129. 02, IN THE HIGH SCHOOL ARITHMETIC. from July 2nd, 1887, to July 2nd, 1888 is «10fi iK . *i. Jan 2nd, 1889. i;$1266 ei' ^ ' ' ^^' ^^'^'^^'^ ^"««" ftsnn ^if V^"^« on Jan. 7th, 1889, after payincr .^Ioqo was S800 The interest on $800, from Jan. 7th to" Ad Tt'h q TrA'\T; The balance is $516. The inteUC^l' isTsotPn/^'u "^"f ^'^^ "°»^h8, is $6.88. The balance IS $3.2.88 The interest on $322.88, from June 7th to Dec 7th,^6^months, . $12.92. ... the amount due on Dec 7th was 174 19. The interest on $600. from Junp -^nfl, isqq . « . EQUATION OF PAYMENTS. 7. The interest on $62.50 for 1 2 days = the interest on JtT'^n for 1 day = the interest on $50 for 1 5 days *^^^ 8. The interest on $200 for 4 months = the inter, on *-?nn for 1 month = the interest on $1G0 for 5 months ^'^fionf J^^^^'-^f on $600 for 5 months == the interesC on ?0 tI T'^'^r'^lt'''''' ^'^ ^1<^00 f'"- 3 months ^loon J /"^"^f ^" ^^^^ ^•^'- 4 "lonths^the intero;t on $1200 for 1 month. The interest on $500 for ' mnnfhf \k interest on $1500 for 1 month. The interest oH Ao 7 I months = the interest on $900 for 1 month l^'^l ' me the interest on $3600 for 1 month fn,' 'f w*^^''^? have $3600 for 1 month ' ' ^ """'* ^^^ ®^^<^^^ «olL J^^ '''*^''^? °^ ^^^^ ^o^ 6 months = the interest on $2400 for 1 month. T e interest o- ^inn Z , i""^®*^^'^ °^ the interest on $1100 tor 1 month." .• lam entiH^r^r interest on $3500 for 1 month • T r^L. ? A 1? ^° ^^^ 7 month' * ' ^ °^"^* ^®^P ^^^ $50"> for ii SOLUTIONS OP PROBLEMS P'ft Llf 12. The interest on BrV ) for 6 months = the interest on S3000 for 1 month. The interest on §800 for U months- the interest on $1200 for 1 month. .-. I owe the interest on $4200 for 1 month, or the interest on $1300 for 3/g months, 175 13. The interest on $1500 for 20 days = the interest on 830000 for 1 day. The interest on $1700 for 40 days = the interest on $68000 for 1 day. .-. Morton & Co. must allow the purchaser the interest on $98000 for 1 day, that is, the in- terest on (or use of) $4900 for Ysofr. or 20, days. 14. The interest on $100 for 30 days = the interest on $3000 for 1 day. The interest on $800 for 40 .lays = the interest on $32000 for 1 day. The interest on .SGOO for 60 days - the in- terest on $36000 for 1 day. Total interest = interest on $71- 000 for 1 day = interest ori $1500 for 47 A days. .-. equated time is 48 days. 15. The interest on $1200 for 10 days = the interest on $12000 for 1 day. The interest on .«800 for 30 days = the in- terest on $24000 for 1 day. Eaton & Co. must allow the in- terest on $36000 for 1 day, which is equal to the interest on $2400 for 15 days. 16. 8ee solution of 14. 17. The interest on $800 for 3 months = the interest on $2400 for 1 month. The interest on $600 for 5 months « the interest on $3000 for 1 month. The man ought to have the interest on $2400 for 6 months, which equals the interest on $14400 for 1 month. .-. He must keep the balance $1000 until the interest on it equals the interest on $14400— $5400 or $9000, for 1 month. .-. the remainder, $1000, becomes due m 9 months. 18. See solution of 15. 19. The interest on $2400 for 30 days = interest on $72000 for 1 day. The interest on $800 for 60 days = interest on $48000 for 1 day. . •. White & Co. must allow me the interest on $120,000 for 1 day. .-. the note, for $1000, is given for 120 days. 20. The interest on $2000 for 15 days = interest on $30000 for^l day. The interest on $1500 for 12 days = i terest on * oOOOfori day. The balance, $1500, must be .pt after 0ebt was due until the interest on it equals the interest on IN THE HIGH SCHOOL AttlTUMBTIC. derwafdJe''^- •'• ""> ""°" "''""W "« P«d 32 days after »12M0^5or'?'Zv' Tl *'"? '"■• •■"' i'y ' "'« '"'-"^"^t «". f , "" " tor I doy. Tlie interest on $600 for 40 dav.-th« Cr rff '"? '"'■ i,''''^'- T'»^ interest on asfc 'so -'2. f'rom March 4tli to June 15th is lOTrln.. . . from March 4th, the $1200 is due in HH, lav? t' ? ?""? on $800 for 30 days . the interest Z $^OoTfor T day"X «2o7ofoJ94'"davf °" «'«''.<'«° f- ' day = the interest on 4th, thl; is! i„™^7th. ^™'°'' '""' " "' ''"^^ f™-" March <5inn 7''",'^'°"," '^V'^"'^ fr<"n June 1st. The interest on 8400 for 30 days -interest on $12000 for 1 dav' T?,^ mterest on |8S0 for 49 days = interest on $41650 f"^' 1 Jit dt°. '"Thf „°:re*s f"! ;T4o5oT = J"r'-='' °" *°««° '^ ' for 52 dlV ! ? «^l280oO for 1 day =. interest on $2450 S/uIy Sd. ■ • ^"'''^ *^"^' ^' ^' ^"^« ^^^«^ J»"« l«t> that 24. I owe the friend the interest on $1 6000 for 1 dav TTa must alow me the interest on .$30000 for 1 day Tou^h? roor th"'' '^.^i^'^' ^^•- 1 ^-y- The whole deU^is ^JOOO. .-. the equated time is 7 days. 176 2 tiL^d^bffori dav^ °Ti!''- 1^' ^^^ ^<^ days = interest on - times debt for 1 day. The interest on yV of the debt for 20 days = interest on 6 times debt for 1 day^ The debt mus? equated time is 82 days ; that is, Sept 10 ^"' ^^' 'if. Ihe equated time of the Debit Kido io «'?•' i counting from M«.v 1«f tu„ ^nl,^ - • ? ^^S .^ays, 82400 f y^^!i counting from May 1st. Jones should pay ^2400 1 63 J days: the payments made are equivalent^ to \m I: SOLUTIONS OP PROBLEMS ei500paid m27| The interest on $2400 for 63f days- interest on $153000 for 1 day. The interest on $1500 for m days = mtere8t on $41500 for 1 day. Jones should have the interest on $111500 for 1 day, or the interest on the balance $900, for 124 days. ... balance should be paid 124 days after May 1st ; that is, Sept 2nd. to jtni'uCrjmw*'"" "* '^ "'' ■"""""' '">" ^f'- ^■"' 29. Counting from Jan. 5th, Smith is entitled to: (1) The interest on $840 for 30 days, or interest on $25200 for 1 day • (2) The interest on $900 for 45 days, or interest on $40500 ^''^AoillK ^K^^^ '''*^''^'* ^" ^^^^ ^«^ 57 days, or interest on $42750 for 1 day; (4) The interest on $800 for 71 days Slfi?9*.Tf T^^^^S^^ for 1 day; in all, the interest on J165250 for 1 day. He has had (1) The interest on $150U nniL'tP' aI T^'^^^ ^" <^'^^^^ ^^" 1 ^^y' <2) The interest on $500 for 46 days, or interest on $23000 for 1 day • in all the interest on $63500 for 1 day. .-. Smith is still Entitled to the interest on $165259 for 1 day. .-. he is entitled to the use of the balance, $1290, for 79 days. .-. balance is due on Jjlarcn 25th. oK^uJh^ ^T"""* ""1 ^^^^^' ^* ^% Pe^ annum, from March 25th to June 1st, is $1304.42. COMPOUND INTEREST. 177 13. The interest for 1 year is .05 of the principal. .-. the amount for 1 year is (1.05) of the principal. . •. the amount for 2 years is (1.05)2 of the principal, .-. the amount for 3 years is (1.05)», or 1 -157625, of the principal. 14. The interest for 1 year is .04 of $525.35. .-. the amount nr.] vr^"" if ^^■^'^^ ""^ $525.35. .-. the amount for 2 years is i-'oVL ^^-^-^^^ •■• **i« amount for 3 years is (1.04)3 of !ft>oJ5.35. .-. the amount for 4 years is (1.04)* of $525 35 I he^ required fraction is (1.04)*, or, 1 -16985856. for 6 years, is $126 532. 15, The amount of $100, at 47 "^he amount of $100, at i% for a years amount of $100, at iX, for 4 yeara is $121,665. The i, is §110.986. The amount m THE HIGH SCHOOL ARITHMrnO. ft ll?°i *'o*^' ''"' * y*""' i>' «n2.486. The amount of SlOO TB !„•!.,?•. ■■•"le total amount is $689.83 178 150 or ^662 45 ^^- ''• ^^« ^"^^""t of $150 is $4.4 163 x ly. Ihe amount at simple interest is 1.24 of the nrincinRl Ihe amount at cK>mpouudinte^^^^ .^ i,262m96oi ?he X c oal T*hf y^ ''"'^ °1 ^^*""^«* i« .02247696 of the pr n- Tmltu^T'^''^'''' •••^h^PrincipaUeiOoi.O?- the princioar'''\h *^-P'' ^J^^l™' '^^ ^"^^^^^ ^' (104)* of tne principal .-.the pnncipal = $1200-(1.04)* =$1025 765 ^1- The interest each half-year is 0^ \f f^i „ ?^^-^'/^o. that half.jw. .-. in f„ur ha^^^L^ the amou^t'^^Mro^ /°? Ite?"'^'- ••• ""^ ""'"""<" »1200 = $lToO.(103)'° halfyeLTo^fo^Z"*-"^ '.'f-n-^- The interest each $1861.49-$Uool?ii,:4^. *'^"-'"'- ••• 'he interests 23. The amount at end of 2 years is $1460 v n nms ti. amouat for a half-vear i« 7 m J,u ?'*<>" M'-fo)-- The .r^«- ^1 ^ ** ^•^'* *** ^he principal for that Iinll ^4 The interest for 73 days, or ^ of a year is 01 nf fl,o 'xW^ xa.miriri?Lii ^-^ \r'? '/'^^^^^^ cents. JtPl.Ud525. .-. the interest is 11.3525 « Vi."4; x(l.Ol) of the principal; that is, 1.092416 of the H hlf I SOLUTIONS OP PROBLEMS principal, .•.the interest is .092416 of the principal, .-.the principal = $400-^.092410 =$4328.25. ^ ^ 26. The population at the end of 4 years is (1.1)* of that at the beginning • that is, 1.4641 of that at the beginning. The ncrease is .4641 of original population. .-. the original tioSs 4T92T '''''^-''''^ - 30000. ... present po^ula- 1 imV^i'^.fT''''* •'' Hy^^'^ at 4%, is (1.04)2 X (1.02), or 1 JS'llMOr""'"'- •'• *h^P"-ipal = ^16989.7728^ /I noM^^^ T^^''^ '"^ ^ half-years, at 2% each half-year, is a02)Sor 1.08243216, of the principal. .• the princinal- $10824.3216-M.08243216 = $10000. pnncipal- 39. The amount is (1.05)'^ x (I.61), or 1.113525, of the the principal •. the interest! is . 1 1 3525 of the principal. • theprincipal = $82.82-^.113525 = $729.53. ^ " 40. In 7 years the amount is (1.1)7, or 1.9487 -f, of the principal. In 8 years the amount is (1.1)«, or 2.1435-f of the principal .-the sum of money will double itself in a little more than 7 years. nn^l'i ^i"? ^^f[^^«^^^« between the interest for the first year Ltieif ^Th ^.ff'°"^ y^' ^' '^" ^^'"^««* °^ ^^^ fi^t year's interest The difference between the interest for the second year and that for the third year is the interest on the second years interest ; that is, is the interest on the first year's in terest and the interest on $1. ... $.05 is the interest on $1 for 1 year. .-. the rate is 5%. 1 n^nJ"t T6 P^'ib^^ear, the amount in 1 year is (1.03)2, or 180 43. The amount at the end of a half-year is obtained by multiplying the principal by a certain fraction. ... the square of this fraction =1.06. .-. the fraction = 51.06 = 1 029 + rate per half-year is 2.9 4- 7. - ^r. .. 44. $129600 amounts to $178506.25 in two years • the amount m 2 years is obtained by multiplying the princinal bv the fraction i;i4^^«. .-. the^mn,,«t fL 7 ;:!?"il._^? by multiplying the principal by ^\iUUU^ or ||2» ; that is, IN THE HIGH SCHOOL ARITHMETIC. per Ittum'' ^^A '''■^'t!^ ^"T^^''^ ^^ T% and the rate is 1711*/ forl'yia ' "^ Henr;f '^^^^ '' '1^^^^'^ *^^ ^^^«r««* on $10.50 1 year, ilence the rate iSTlr^s xlOO or ^SV .%;o/ c amount at end of first vear- ^^^^^ ^ "^ ^™®- • • <^^>e discount from $108^ is ^8.. ••. the discount from $1250.60 is $1250 60 x -^ or f 97.97. (c) $100 amounts to $108 16 in the . '''^' u ihscoxint from $108 16 is «fi lA ^u^®,*'"^®' •'• *^e $1234.56 is $93 14 /%?/^-^^: •"• *^« discount from fho f;«,« ^i''>i*. (c/) ^1 amounts to ^$1 05^s x n n^\ • the time. .-. the present value of ($1 05)« y n nq1 '^^.^ '"* the present value of $17684 95 i, SVr^. n^ ^V?^^ ^^ ^^' ■'• or $14831 07 . ^ ''''V^^-y^^is $17684.95 + (1.05)3 n 03) at 57 r^r annnm fl '^'''^"".' ^^^ $2852.98. L In^ 5 years' ««'«-/„ per annum, the amount is CI 05 » 5 „f .u^^ - . "^ Y^^^ present worth of $1, due in 5 years -L In nt;R"''£'Pf^- •' ••• discount from $l„21.65c^ ^ ^^'^^^ =^'^^35 + 14. The present value of $400 due in I year is ^ (1.06)". " «362.811. Tho present v»l„e of ?400 due in 3 ^"■"•V'' S" " *''''-''5- •■• *e pn^sent value of th«. years' rent is $1089.298. ^.. ^? present gain is $1922.75. * ' ^^^^^0.37. .-. my ^uo uiscount. ... the interest on the ^^ SOLUTIONS OP PROBLIMa /■lo^ol' Jiu''!*'!",* ™'"* "^ *'"' instalment due in 1 vear i. dl£1 tt;sT.T.",^ The present value of the in^^ of the inS ^iffi\n1 tr s ?!-;. J^ir?', ™"'^ The Drpspnf v.il„«» «* *u • ^f , HV*") of the instalment. Ane present value of tjie instalment due in 4 years is ( joo^* of the instalment. .-. |(^oo) ^ (.^oo^, ^ . , ooja ^. ^^ oo)/^^ 4r^n'SQ"i*r-^^l^^^' •••thei„stalment = $3463.09. ^ *i. ^Uy9.15 IS the amount of $1200 in 3 vears • f>,n amount is \HU^ or »8fli _r ..„ „ y^/" o jears. .. the ii.« ^ 1 ?(y or g^j^^j^, of the principal, in 3 vears thfi P w'lF'^' ^'^j* ?'''®'^ ^"'°' '^"^ in 3 years, is equal to of th^ J; ^ sum due in 1 yeitr, whiohlatter sum is theP W of the given sum, due in 2 yedrs. Similarly, the P W of thp gven sum, due in 5 years, is equal to the PW of a sum due in 1 J^r* ftVnA il ^^^ differences, the PW. of $5.10 due ml year .$5.00. Hence the rate is 2% per annmiL ANNUITIES. 184 ^ 7. $250 is the interest on $6250 for 1 year at 47 • Sfio^n jsthe v^ ue of the perpetuity, when theist pay;^;nt is to '^ made at the end of 1 year. The present worth 0^62. oav R «onn^-^5 ^ • ^6250-(1.04)< or $5137.04. * ' ^ ^" 8. $200 13 the interest on $5000 for 1 year at 4°/ • the tL v^ir^t^'^r^^^y "^ ^200, begin'^iing nolfis $5000 made'lutTnd ofT;::rs"TlOot^^^^^^^ frst payment is afc fi°/ m. «9nnn /ears. i|piuu is the interest for year 10. The value is flS«3>ftAA_!-/i nK\a Aimnm /.« IN THE HIGH SCHOOL AHrTHMETIC. 185 of perpetuities ^ "Pressed as the difference of values aniLy':'%7t 'IXTv iil^^"?T^ "°^ » ^0 *-«« the ti.es ^e ^ui^r^t;:^;:^-.: si^f -i^S ' "" (TMy *^"^^ th« ««n"i^y. .-. the value of the limited annuity is /_?L_ 20 ) ^^77.' . ' l-OS a05p / *""®^ *h« annuity, or 0.677 times the annuity 13. Thepresentvalueof thefirstpaymentis$Ao^.ore78.. 4dl. The present value of the second payment is ^ ^^ ^^ «575.d85. The present value of the third payment is i^l or^S73.458^ The present value of the fourth ,^^^1^ (T02)T' ^^ '^69.644. ... the present value of the payments is 3295.92. yeirs ^^%E!;f ^"?!fy ^'" «^«t $4000, at the end of i So.si.' "" '"^''''^ "^^ °^"«t be $4000^(1.05)*. or or ^^teo'Ja ThTpr^tTtl^^^^^ ^' ^^^^ i^ »<> ^ .05. 10 years, is $160^; "(1 05^' « orXillT'"'' 11 '^'^"^^ payments are worth $6177 39 ' ^''°^^'*'l- ••- the annual " OMl' "■"" "■" "'""'"y' <"• I8-7578C3 times the annuity "rSo'll''^'' '"^ »•'» "-i'y^SlSOOO. ... the , . ( , 1 - X -iv raiuu 01 uie ten pay- ments 18 < 1 - _ J__ I y on X- ^„ , , ^ "^ I (lT05l»o / ^^ t*™®^ ^ ^' payment. Hence m PI Ml I 'J I* SOLUTIONS OF PROBLEMS the annual payment will be -1?^^ -i.f/i 1 1 n^l „,/ $800 1 Oo.'>.«li'-(To55.»)>'2'>} ' 20 -{ (105)'0-iT/' ^^*^^ ''^' $63.60. .;l^^5o-.^rfro"^pl'srVJ-^^^^^^^ of the payments is n - _i_ 1 v i fi2 *;«..»« +i, I (T06)«/ ^ ™®® ^'^^ payment. .-. the payment will be $554,214 -r f / 1 - ^ \ v i r2 1 $131.57. ^^ (i:06]»r'^Sj' °^ 19. The present value of the six deposits will be 25 x V'C^M'yf ^'""^^ *^« ^"^0"^it deposited each year. The present value of the partial annuity will bo $1250o/ 1 1 1(1.04/ 1 (T04)^ ^ f • ■• *^® a"»iii'»Hte|;.osit = ^^^-^^^^ ^ (1.04)s "( 1^4) ~(T04)« I 3 (To4y>^i.o4)' - ''^•^^"•^s. 186 «nn ^^® .^*l"« " *he difference of value of a Dernetniftr r.f 600, begmninL' now inrl «f fV,„ Perpetuity of fe luiiij, now, and ot the same perpetuity when de- (1.04)' }" o , — ., ^^ i,ijt; Bciiae perpei ferred 5 years. .-. the value = $i«oo ^ T j $7122.92. '• present value is $=»»xr ?/!>„ ",<'f '«■■«' 1 yar, it. valueisS»»»v 1 ' "^' "''^'<'™'l 13 years, its present aiue IS S.jj^x „.„,„. ., present value of the partial annn- .ty,s$-(_l___^|„,j,,3232. made in 6 months, the p'rS ^.ue'Vo'il/S f/oO rOeT:; or I 3 IN THE IIIOH 8CH00L ARITHMETIC. vKaioS^^^^^^^ months, the present annun.f|aK"^^^^^^^ '^'^^'r'''"^' ^^"^^'^ •'^■^^^ 25% per year.i45600..fLO5062J)-So^ ^5000duefn% H.p f. ^ .oijigjojl..]., present valueof tl'e ten payments v^ interest is $^^^oo, _ *^ 1 ? or $2309.50. .■ the fnf«l r. . T''^ (T.050625)io/, 2G Consider first f h! T^'^f '^'''"^ '^ ^'^360. 85. ^ $1000 due Fn 5 yeat Ind"'^' ''''T "^"^ '^'^ ^^^"« <>£ Tl.« vol . 1 P^^yment of interest is $110. The value of the 11 payments would be l.uo/l. 1 \ or $1046.562. • thptnfni t o '"^ (1.025)ii/, Hence the v*e notril'/sr'rHn'.Tol iirK^lfl-'f ' '*■ -'7. Aperpetuitv eoinl fn ♦>, '-^l-o, or $4146.33. -. would Le I^Sln^SrSS:^ -^^S a perpetuity, deferred 20 years, would be worth -?^ times the instalment. ... the difference 20 / 1 - l''''^T times the instalment is equal to $12000 .u ^7^5)^ / r"(T05]^ 1=^962.91. 28. Thevalueis?12??/i 1 1 ,, •04 r~(r04)T5/> or $11 118.39. 29. The value is ^f 1 1 1 30 Th 1 -^^ ^(^^^* "(T05)B f' o'- 15965.57. <=- pa,.en. .. end o, ,6 ,e„. . ~,. „, ,^7 To" present val.'o of *k^ - , <. . a^^.^" •■^"® or .He payments of interest ial^i l _ I 1 « «™U7. ... the present .«,„e of ^ortgagf, ^7^'^ .05- IMAGE EVALUATION TEST TARGET (MT-3) /IPPLIED^ IIV14GE . Inc .;^ 1653 Ea^> Main Street ■ass -^ Rochester, NY 14609 USA -as-^ Phone: 716/482-0300 ^='.^=: Fax: 716/288-5989 1993, Applied Image, Inc.. All Rights Reserved ^^^ t:\ n^ °;^^^ '^ 5t^ A^* '.^v ""^ I' $ . s '■f SOLUTIONS OF PROBLEMS + (102) •+ +(102,» }.. Working ti five decii^alpli^ the result will be $4178.91 ^^^^ JLT'^ '^v'T '■ ^^^^ ^""^ ^'^'^■y^*'*- " ^^^ rate were reduced to 5%, the interesfc would be $75 each half-year • if ft!^" Pt mV^^ ^ e4uivalent to ten half-yearly payments of »45each. The pi-esent value of tho^e payments is $*^ 34. Each 80IV receives the income every fourth year: that 18, a perpetuity beginning now, in 1 year, in li years, in 3 years respectively. The present values of such perpetuities are proportional to 1, _L _J \__ „- to n 0i\8 (..04)S ,1.04,1. ■••0M1.04,^(1.04,r' •»<••"*)• 36. The value, one year before the payments begin, is ^"y |^"(T:06)»»| «'• ^l^Jt).837. .-. the value at the end. J*^ j^ yc'^»;8 (J6 jears after the time above), is $1456.837 x 36. The value of the 15 payments, one year before the first payment is made, is l-^l ^^Y^ times the annual pay- ment. .-. the amount of the payments at the end of the time is { Izixjj)!.' } X (1.05)' « times the annual payment ; that is, (1.05)»«'-(1.05) times the payment, or 22.65749 times the payment. This amount is $5000. .-. the annual payment is $.)(»004 22.65749, or $220.07. M. We shall first fiad the annual payment, which will amount in 15 years to $4000. From the previous solution it n nc 8 ^^^^ *^^*- *^® amount of the payments will be (1.06)' -(1.06) times the annual payment, or 24.67252 0)1 times the annual payment. . •. the ann- •> 1 payment - $4000 -5- *li uj' ■^"' *"''^® " payingeach year $240 -$162.12, or $77.88 more than enough to amount to $4000. 38. First find the value six months ago. The present value of the mortgage and interest would have been SS??. + (I.OD)' \*» f.XfS IN THE HIGH SCHOOL ARITHMETIC. $30o(-,-„V<,+-T^« + -i- ■ ^ J 11 vJe no. is'^52S"73rx (^025);" o^ ^8^1^ ^^^^ - ^^^ ^ The present value of a |100 Bond and its interest is roij, ""■ $79.03 + $26.21, or $105.24. rJ^ ^^ "^^^' ^''^ following payments at the end of the (1 T28?5^7fi?Sic i'J ^\'' 1'^ ^25, (3) $25, (4) $29.13! loS«« ^/ i^'H^kn^^^^28., (8) $27.63, 9 $27.25 «25sf a;.u^^^^!-^3 <13) $26.13, (13) $25.75T (U $25.38. At the end of the first quarter, the value of the Ist 5th. 9th, and 13th payments is $25 + ^^^-^^ + ^IL^ ' $25.75 1-^5 (1-05)^ + (1.05)3- ^^^s amounts to $99.34. The P. W. of this is fn^^^*"^/,?^,^^^' ''' ^^^-^l- The value of the 2nd 6th m:i $26^'"^^"fe.t8^"' ^^ '''-''' ^"^^^-' ^^ '^2' '' 1-05 + (1.05)=* + (TOSF' '^^'^ ^^ ^^^-^3. Its P. W. is $98.33^(1.025), or $95.93. The value of the 3rd, 7a, and 11th payments, at end of third quarter, is $25 + ^^-• $26.50 105 ■^ TTOSp'^'^^^^'^O. The P. W. of this is $75.70 + a. 0375) or $72.96. The value of the 4,h, 8th. and 12th payments, at end of fourth quarter, is $29 13 +Ml^^ ^ $26.13 $79.14. The P. W. of thi., is $79.14-(\" 05) or^l?'?^? ' ''' the total P. W. of the payments is $342.37^' ^^ '^^- " 188 41. $1 paid each year will at the end of ten years amount to ${(106)10+ n.06)» + . . . 106^ - <.,«^-,. .-. the yearly payment, which will in ten yiars amount to n f'',i. IF '1 \H- SOLUTIONS OP PROnLRMS $1000,ia$1000-fl3.971G,orSn.C7. .-. he paid $105 -$71.57, or 1$33.43 more than enough to amount to |!1000. 42. Tlie heirs receive $1000 insurance, and $82.36 x 1(1.04)5 + (1.04)* + (1.04)« + (1.04)2 + 1.04}, or $1463.93 m all. Had ho taken the endowment policy, the heirs would have receivt'd only $1000. .-. the heirs gain $463.93. 43. If the man live the ten years his bank account will amount to $82.36 v or $1028.38. .-. he is the endowment policy. 44. The amount of {(1.04) 8 «DZO.O( + (1.04)" + + i.38 better oflF than if he had taken , deposited each year, at the end of 15 years, is $|(1.025)3o + (1.025)28 + ^ (1.025)2} or $22.77791. .-. the semiannual payment is $1000 -i- 22.77791 or $43.90. , 45. By calculating the interest, compounded each half year, and subtracting $7500 from the amount at the end oi each //cor. It will be found that the sinking fund will pay the debt, and leave a balance at the end of ten years. 46. The interest on $200000 for 1 year at 5%, is $10000. Hence the tund pays only the interest. 47. The P. W. of 20 instalments of equal amount, p-' ' at the end of each year, is 25 x | 1 - -~r^ | times tut in- stalment. Hence the instalment is equal to $250000 -f 25 ^ t ^ " (1.04)20 I that is, equal to $18395.44. ,04)2 JJ PARTNERSHIP. 2. On $10000 capital the gain was $2800. .-. on $4000 capital the gain was $1120, and on $6000 capital the gain was $1680. * 189 3. The whole gain was $1200. $1200 was the gain on $12000 capital. .-. $400 was the gain on $4000 capital. 4. After paying Sykes$1200 for managing the business, there remained $2000 gain on the whole -apital. Of this remainder Smith received $1500 and Sykes $500. $2000 was at IN THE HIGH flCHOOL ARITHMETIC. ' Zi^'''"''lT^-''^'''\.'-^'>^^ -as thegaiu on $2000 So ^ ®^ invested $2000 ^-Wf $50r)0 was the.ain on $^0000 . ', T"" ""x?^^^^ ^^P^^*^^- ' • 6 A irJrJthi u r-^00'^0 <,apital. .-. D inveKted $20000 or thf uf;oV$Vooot;Tr.l""s' ^''V""' ^- '-^''; for 3 months, or the use of SI "ooo f f""^ *^t "'^^°^ $^^00 of $21000 foJ 1 nVoLh Te Sa '$08oT'^-f ^^l '''' "«^ eOOOO for 1 n.onth A should' reieTv.lfoOO ^" '^' "^" "' mo'ntL ' Vh" us?ir$l?Jo' fo^f "^ ^ ^^ "«« ^^ ^^^««^ ^- ^ for 1 month The use of JSof '""??' = '^^^^ ^^^200 $4500 for 1 month Th!;j^^-fri.^i"'"!^^^«='^^« "«e of tion 48 : 22 . 45 A'J ^hf Ti" h'''^'^^"^ ''^ ^^^ P'^opor- share = J^t of $2400 ^J^qT.^?^ f'^^^ - $1001.74. B's $939.13!^^ ^2400 = $459. 13. C. share^^V^ of $2400- keeper $87°^o"''^''pT^",? ^^^^ * "^^^^h. ««d the book- asTokSpl TlfetTili^'X^r ."^^ manager and ^ToO book-keeper" was ^ofo?. sToJo ^E^ '^' i^^^'^S^^ ^"^ month gave ^,4 «C3So/thel*oflSS)oTf°' ,*'""? '" ' gain i 4 1 •>9 1 ^ . n u 1 j . «>4o000 for 1 month cave I4i22i|,tl5422ij "'^ rece,™ altogether $600 + $70*0 + «2oVforr™„:.r ° i!is?°.^ ^nr^- *"■'«>-' <■»«»' B gave the n« of ».ionn f p "? "' ^^^'">° '<"■ ' "">"«'. gave the us«of <5innrt * io ? JiPoOOUO for 1 month. C 1 month The u e of Sl'lOOoTr''^ "" "'t "'» "' ^^^"O" '« .-. theuseof «48000fofll,?,h°"' ' ■"■".'"> gave gain $6000. 10 On $3loorpfS,\H7tin^7Ji;i?^^ xfi ^r WM TuVbT '" "rr"' ">«V.n would tel "2? 3T» "' «>ow*. Jiut 13 8 capital cainpH ^iSfin . r^_ Ino/x^ for 2 months the gain was $17fi ° «i 7fi t ' "li " * V" *='-^00 capital for 2 mos tf!S«ft» ;i, ^V ^ ^^s the gain on B's F lor^mos. ..$880 was the gam on B's capital for 10 SOLUTIONS OP PROBLEMS P moR. .'. B's capital w»s In trado for 10 months, and .-. A'a capital foi- S months. Since the gain on $2200 for 2 months was ^1 70. .-. the gain on $100 for 1 month was 84. .-. the gain on $2500 for 1 month was $100. .-. the gain on $2500 for 12 months was $1200. .-. C'a capital was in the business 12 months. 11. A gained $600 in 2 months, .-. $300'in 1 month. B gained $500 in 2^ months, .-. $200 in 1 month. C gained $800 m 4 months, .-. $200 in 1 month. $.300 was the month- ly gain from $3000 invested. .-. $200 was the monthly gain from $2000 invested. .-. B and C invested $2000 each. 190 12. The capital for the first four months was $27000. The capital for the fiftli and sixth mouths was $24000. The capi- tal for th« last six month:, was $20000. Hardy received, as manager, $800 for the first, four months, $200 x f ^ x 2,' or $355 g, fur the fifth and sixth months, and $200 x"'^^ x 6, or $SSS^ for the last six months. In all, !ie received $2044-*- for managing the business There was left $3955^ net gain, to be divided in proportion to use of capital. Hardy gave the use of $12000 for 4 months and $9000 for 8 months, which equals the use of $120000 for 1 month. Jones gave the use of $15- 000 for 6 months and $11000 for 6 months, which equals the use of $156000 for 1 month. The use of $276000 for 1 month yields $3955^. .-. the use of $120000 for 1 month yields $17l9iJ'?.. .«. Hardy receives $2044|-f 17l9i«|, or $3764.25 13. Lock gave the use of $2500 for 10 months, or the use of $25000 for 1 month. Smith gave the use of $2300 for 11 months, or the use of $25300 for 1 month. Knight gave that which was equal to the use of $2000 for 12 months, or the use of $24000 for 1 month. The use of $74300 for 1 month yielded $2!)72. .-. the use of $25000 for 1 month yielded $1000, the use of $25300 for 1 month yielded $1012, and the use of $24000 for 1 month yielded $960. 14. B's workmen did 4000 days' work. C's workmen did 3600 days' work .-. they should receive shares in the ratio of 10 to 9. .-. B's share was \^ of $12000, or $6315.79, and C's share was ^% of $12000, or $5684.21. 15. A owned ^ of the value of the vessel. He lost i of his share, or j\ of the value of the vessel. This amounted to IN THE HIGH SCHOOL ARITHMETIC. «1000. •. the vessel was worth $\ 6000. B ]n,t ] „f 1 of tho vrlue of the vessel, or 81 333.334 C la^t X .,( « ^ eJ • of the vessel, or 81 666.66|. *" ^ ^ ' ' ""^ '^'^ ^'^^'''^ ^n^7 ^T"T'^..« Liabilities Ooofis on hand. $40000 Credits .^iirooo Cosh on hand . . 22000 - ** < v^ w I>ebit8 25000 Total Resources, 887000 .-. l/et Capital. . . ^40^00 The original capital was 820000. There has heen a gain .-nual to capital. ... the gainsare : Smith $8000, Jones $.000, cTk Inion f? ;v, L.u^ ^^"'ly^^'' '^^^^-^OO ; for tho fourth year been SI i 100 1' J ^ r,' ^^^ ^^^•. V'' S-» ^"•- < '>« P-iocf ha ^en ^11100. As capital was equivalent to 834r,00 for 1 v ar B a capital was equivalent to $70200 for 1 y.^ar C s cnni al was equivalent to $105500 for 1 year. .-B'h share o^^i i?th rr^7$tofoO^''' l^'J''^'- f' ^^^ -^ '^ mm jear was JJIOJOO. .-. B's share was $13907.04. Jnil^'^«'n^'^;t' ^^^^^' «^'"«^ ^288. ... A's capital, fil500 gamed 8360 in the same time. .-. $1500 gained $600 - $.S 4 month«"' T'^''.. •;■ ^'' ^"P^^^'' ^^1000 S*i"«d $100 in 4 months. .-. B's capital gained $320 in 8 months. 191 mo!?ths^renr$ J20^^ V^ ""'"'^ .^^^ P^'^ *^« ^^'^ ^^ree Zttr; ?hat'is,^^7e:et. ^^^ ^J ^^^.ul'r'J^T' '7^ paid the nextfourmoiths' rent, th. t^l^Ij^altl "^Ter^td Taylor paid the last montli's rent, that is $^>0 ea^h ^y.."''''^ *6dJ, or $133J. Taylor paid $53^ + $20, or $731 20, A had $4000 in businp<5r i Ju- "< *^'^ SOLUTIONS OF PROBLEMS EXCHANGE. 1. He pays $4000 and 1% of $4000, that is, $4000 + $10. or $4010. 2. The draft cost $2500 + 2% of $2500, that is, $2500 + $9.37A, or $2509.37^ .«•■!• 3. the draft costs $800 less i% of $800, that is, $800 less $4, or $796. 4. The draft costs $12000 and |% of $12000, or $12090. 5. A bill of XI costs $4,804. $4.80^x1200, or $5764.50. 6. $100i buys a draft of $100. $7481.29, 7. $4.80 buys a bill of £1. ^^Vu-. or £281 5s. 8. A bill of 5.16 francs costs $1. costs i!?290. 69 + . 9. $4.86| - $4.44* = $.423. On On $100 the increase is 9|. $.42f a bill of £1200 costs -. $7500 buys a draft of $1350 buys a bill of .-. a bill of 1500 francs $4.44^ the increase is the increase is 9|%, $483 10. The cost of £1, in exchange, is 1084% of $4,444, or '"' .-. the cost of £3000, in exchange, is $1°4433J. 192 11. The cost of a bill of £1 is 1094% of $Y. .*. the cost of a bill of £1500 is $(1500 x «-^§ x -*/), or $7275. . 12. i^ of $V-, or $4.80, buys a bill of £1. .-. $2400 buys a bill of £500. 13. £1500 costs $7300. .-. £1 costs $||. $V- is 100% of old par of.exchange. .-. $[» is (100 x ^^^ x ||)% of old par of exchange, that is, 109^% of old par of exchange. .-. exchange is quoted at 9|. 14. The direct exchange will cost 100^% of $4000, or $4010. In Chicago a bill of $4000 on New York will cost lOOf % of $4000, or $4030. In Winnipeg a bill of $4030 on Chicago will cost 99^% of $4030, or $4009.85. 15. A bill for $2700 cost $2673 : .-. a bill for $100 cost $99. .-. exchange is at 1% discount. 16. The amount added because exchange is at a premium o^ i%» is 1% of $2750, or $6,875. The discount, which is usually calculated on the face of the bill, is JA of t5^ of $2750, or $44. .-. I received $2750 + $6,875 - $447 or $2712.875. IN THE HIOU SCHOOL ARITHMETIC. on of ^^o^!:?,?f ^'^"^V^;?^ 9|, £1 yields, in Canada, 109f% Of 9-jf-, or $-VV-. ... 2o.30 franca yields, in Canada, $JgV. • 294000 francs yields, in Canada, $ { '§ x ^^^ x 294000 I \.v ^ f 175 100 ) ' I "36 '^ iiSab "* ^^^^^^ \ ^^^ interest is $2920. $100 the interest is $ { 2920 x -^ x #^ } or 45 16 4. ...rate is 5 16 + %. ^ "' '''''^ ^ or $5.16 + . welL^^l II"' ^^*^J ^^^^^ ^«'^ ^'^'"^d at 934^d. 7869x240 ^ "'^ ** (1869x240)d. weighs 93|i - 02., or 480 oz., or 40 lbs. Troy. JL fnf '*^^' ^^^?) '°''fr"^' ^"' ^ A ^^- P»r« gold. « 22 t:Tl'^ J1^'' ^^ iA PT ^^ J"^ i| of pure gold mt^orui'^r *Hf^- But a4W6V^isnVry "nearly 109*^ of $4.44* .. -By the new par of exchange sterling money is worth 9J% more than by the old par." *^ 193 20. The agent's commission is 77 of $7800 or Jt^idfi ti,^ duty is *(,. . 0, 616 X 4.86). or »3425.*' "^he agenf •. .S r.in i."m6°73.*'''''*"' " ^'"»«"- ••• "»- ' '"^ 21. By the circuitous exchange, $10000=10000 v fi lO rancs=10000 X 5.40 x 'oo „, J3L1OOOO x 5 40 X ^of^ 1 7s 4 + d. By direct exchange £1 costs |( *« x ll^), or «* * fs^£2''76 8r3d ' *■• ^^0^^^ P-d"««« W4l9s"lT • lain ftq?9r.7^'V^^''*'r ^^'^""ssion was 2^% of $12500, or ! /;i^* Jn^ net amount is $12187.50 ^°$99i pays for a ?. U^fn^' •••012187.50 pays for a draft for $1^2279 60 23 11520 marcs banco- 11520 x 2.12^ francs = 11520 x 3'12Jxy»y dollars = $4744. 186. ^^^^-ux ^^4. 1800 francs = i|.^o^pounds-||o,o^v|08^4^q^|342.. 4.0?2/rancf^'^^'^'^^^ P'""'' ^ 38.177 xl-H-- francs- 26- 01 -T.^Wr pounds = ^'.|^o^Vj florins -2.48+ florins. UENSURATION. the wall = V302 - 24=^ ft. - 18 ft. 2. See 1. ri/ht fTH^'T'' '^':^- ^*^' ^ • * ^^'"•^ '^'^^ '•«J'"« form a V8^T74?r'"'"'*' of square. The di. of s<,.« 5 The line joining pt. to centre, the tangent and the radius to the pomt of contact of tangent and circb, f orm a righ W gled A. . •. tangent = VTTTpr f^, ^ 5.74^ ^^ » 6. Apply Euc. 1. 47. 7. Let A = jet AB ± to ground, CD -height man Rn -ground and DE^shadow then ECA is a sX^^ ^£. '^'^^^'«ll = i--«^ = ^^t. = 13Jf....BE-(4.7) ft.=e2ft. * of ^rtd'TfThll '^' '"n"^ '^' ''-'^ ^^"'^^^ *he diameter or tne end of the tree. Diameter = (12-J.aa) ft =*» ft •0000918 in. i * « -so . ^o x 4e4u x y x 144) }. in. - 196 11. Length of string - V2Ta + 182 + 78 ft. „ 30.805 ft. l-i; I! 1 IN THK HIGH SCHOOL ABITHMETIO. 12. Dimensions inside the road are 77 yds. and 48 vds «». spectively. .-. area road = (85 x 66- 77 x 48) sq yds - 10^ sq. yds. .-. co8t = $(1064x •26)-$266. ^ ^" ^^■•-^^64 13. Perimeter of semi-circle -Bemi-circumferminA J. r1,-o,«»* circumference - 2 :■« a. 2 ffc . *"* ';'r^V"'«'^«nce + d'ameter, .-. «tc ^. 7 • <« ic. . . 8emi-circumference = 6^ ft. 14. (Vr + 2r)in.=:80in. .. &o. Ifi" !,^^!f)'=/23- 4840x9) sq.ft. ...Ac. 16. Altitude bisects base. The alt. J base and side form a right-angled ^ 17. See 16. Apply Euo. See 8. 18. 19. 20. 1.47. sq. yds. Length of ground -GO yds. width = (!^^60) yds. See 12. 21. 22, (25000 -J- 360) miles - 69-4 mil»s „, U^ \^^ '^- ?r (27 X 18) sq. in. = Ac. Jd. Ihe diagonal divides the quadrilateral into 2 trianiyl«« ^lose sides are 20, 30, 40 and 25. 32. 40 chains resST.' 24. See 1. 197 25. When the complete figure is drawn, there will hp fw« (i;:i8fft'^°''?..'"r'^^' V^!^^-^*^ ofVe'strtti: ( J4 + 18) ft ; and the lad le,^ %/24> + 18='~ft. = 30 ft ^o. Apply formula. 27. No. of revolutions = 1 mile -f- (-^-,30) in. = &c. (560l-rft "'r^'J?^'-''^^ ft. and ^he outer radius- 22% f7 ' road = (560 -420)-!^ ft., ?n <^V2/)ft. = 12ft. where r = radius, &c. oO. Apply formula. 31. See 9. 32. Radius of circular base = 47 -^V- miles • Inn^fK «* slant side - vr«T^rriT\2 miles. 7r«r^ i /-llJ^H&f' /% :r / A u-iU acres » 20u0'3 ac. " "' 33. Length of side-vIsO yds. ... dimensions of new ' i 4j:!i fir!) •OLUTIONS or PKOBLBMS •pace are ( Vfw) yd,. + 4 ft. 3 in.) and (VI50 yd.. - 3 ft 4 in.) miZf* *!^^/J,'»".®''*he circumference of the wheel equals one "168ft"" *'^*'''*"°'^®''* •• di»-(5-28-i.i<^)ft. 36. See 23. angfej&c *^'**^°''''* °^ * rhombus bisect each other at right l)ond-?«i"q5j( pond-(220^V) yds-SS yd.. Area of ^^U >^^^^ \^\ yds- 3850 .q. yd.. ..area of outer circle, including both path and pond is (3860 + 120) «, yds ^k:^!?'^' ^^^' •••/adius of this circle -(V39^0-.-U) yd8 = 36^641y(|s. .-. widthof road = (35-641-35)yds.«^41 yd! 39 i«Y^?^P^^29ft.8in: = 5ft. lOinfj ft.6 iL/etc': J9. Apply the following rule : of th« fcfi^^ *'^'*i^ °* u^^ ^'^'^^""^ ^^ * «on«. R the radius of the ^ttom and r the radius of the top/the volume is i T A (Ra + Rr + ^=.); or if A and a be the area, of the bottom and top respectively the volume is ^ A (A + Va^ ^ ti; n^I^r" ' 9°":P'«t« *he cone. Let h' denote the height of the part required to complete it. Then A' + A - the height of the completed cone. By similar triangles A+i' ^ ^ . h', f> « + /t = = Now the volume of '• ^' " -=r — and K- r the frustum = the volume of the completed cone - the volume of the cone required to complete it. .-. the vol. of the frT tum = J (A + AV R' - J A' TT r» = J ^ A (^-=^) - i T A(R8 + Rr+r«)-JA(A+ VAi f a). 198 contL?l/r^l^ '^"J*^''' ^ ^*- *" "**»y *™«« '^ *he width contains 2 ft. Area of a rectang e 3 ft. by 2 ft - 6 so ft • 240 sq. ft. contains (240*6) of these Rectangles 2' 40 *.' the length = (n/ 40 X 3) ft. = 18-973 ft. 41. In 88 days it goes 37000000 x 4* miles, Ac. 4 J. oee 1. . i.^ See 7. IH TBI HIGH bCBOOL ARITBMETIO. 44. See 9. 46. Circumference of circular field «(Vx 15) rods « 474 46. Radius of base of circular cistern «( 20-5- y.) ft. = U ft. Volume of water - | 7 x V x (» » )» | cu. ft. = l« \9. cu. ft. The side of the square base = (20 ^ 4) ft. .-. the area of the base = 25 sq. ft. .•. the depth of water = J^tJii cu ft -^•»-. sq. ft. « 8-90 ft. ■ ' "' 47. See 3. Let a;-perp. on chord 12 units in length, then 14- a; :^ length of oth er perp. .: x' +6»=rad.''-(14 -x)" + 8=. • •• a;»8. .*. rad. = v^8i« + 6« = 10. 48. Field is to contain (fxlO) sq. chains. .-. length of the field - { ( J X 10) -=- 2^ I chains = 3 chains. 49. Area of quad. = | |.40.12J + ^.40.9 |« | sq. ft. -etc. . ^\ ?Js'!® . °^ *^"*^® " ^^ ^^' • *• di»- = ^14 . V2 in. = V28 in. -. 5-291 in. 51. Side of sq. = Vyo in. .-. length = (V80 -i. 8) in = 1.118 in. ' 62. Apply Euc. I. 47. 63 . Area fi eld = (40^ 5^ x 3 x 30 x 3) sq. ft. . •. side of sq. « V40 X 5J X 3 X 30 X 3 ft. - 243-721 ft. 64. See 11. 65. See 39. 66. Ground passed over = (3 x 3f x 5|) sq. ft. = e4^\ sq. ft. 67. 3 (side of cube)2 = 1 sq. in. .-. side- ^f in. = -577 in 68. Cost={(21ixl3J)x6}d.=.etc. 59. See 15, 40. 60. Rad. outer circle - ^ x 1 10 yds. = 17i yds., and rad .nner circle = yVx 88 yds.S4 yds. .: area olter circlel V- x^i/j) sq. yds., and area of inner circle = ^ix 14» sq. yds. ••• *^*» path - ? I (17i)« - (Uf I sq. yds= = (? X 3I| X H) sq. yds. = 346 5 sq. yds. ' /o ^\J^ «*^- " ^^ ^^^•' «"^ 62| lbs. = 1 cu. ft. .-. no eals - (8 X 10x9x62^ -10) = 4500. ' .-no. gaJs.- *, SOLUTIONS OP PROBLEMS 62. Rad of circJeld-v5;^^484o^9- £<, ., eircum- rr"j^^SElH?840^ ft. = 1046-529 ft. Side of sq. held = "^2 X 4840 X 9 ft. = 295-1609 ff • «..; ^« * u = e^3>:f^5^-l|f^t. = 1180-643 ft. .dl^' = 134-ir4r' ''^' J4^^Cistern contains (Vx4»x4) cubic feet of ^vater. 200 65. Number of cu. ft. emptied per hour =(^ x 2000 -62^ ^. Area of base = \^ x C^Y \ an n . 4. . " I 7 ^ ^2^ I sq. tt. .-. water must rise per hour [(I X 2000 -62i)- | ?x (|)» | ] ft. = | « ft. 400o' ~J ^^' ..!!• ••• ^ ''^- '"'^ °^ S^^^« represents «? t^^i^l- ?^iles = 444444-4 sq. miles. ^ 67. Rad. of end = (/^ x 22) ft. = 3 J ft. . -. area of end = I T X (3i)' I sq. ft. ... no. of cu. ft. in stick - I ? X (S^y X 40 ]■ cu. ft. .-. no. cords » f x (3i)« x 40--128 = 12,,V. 68. Apply formula. 69. Let r = rad. of pond. . •. ? r^ = 2| x 4840 sq. yds. • r = 5 VI54 yds. Area of walk == t^ | (r x 2)^ - r^ \ sq. yds. -J (4 r + 4) sq. yds. = a^s (5 vm+ 1) sq. yds. = 792-608 sq. 70. Area of dm = 2.3 (32 _ 2M so in - 22 ^ o« • a of outer surface = 2.^.3' sq. J Area ^f^innp^; '"' ."^'"^ 2 2^8 22 sn in _22Q ' • ^ , , ®^ °^ *nner surface =. sq'ia =9Vsq:ra ^^ •'• ^''°'" '"'■'*°»-? (S+18 + 8) 71. Theory. 72. Length of tree = 1 1 2 + v/ 12^:3? } ft. = 48-055 ft. 73. Distance around the pond = ^2 a „ ov .„•, „ . .. quired to driv« = (V^10) h^ anytiL'^^S^' .X"" ^4T-6)tir» &c, ^ a .0i0»4r = SOLUTIONS OF PROBLEMS n^?.i^:r Vl^' ^^"f^ respectively the width and length to the outside of the road .-. xy = 480 sq. rods and (x + 8) (y + 8) - xy = 51 6 sq. rods. Solve for x and y ^ ^ °My + O) 6^353 ods.' •■• '^««^d«=^36' x 3 rods = 36v3-rods, 76 Wt. of sphereof water = (*. ^.S^'-HOS x 62i)lbs. • wt. of iron = 8 X (|. ^^^.38 - 1 728 X 62^)lbs. = 32f | lbs " " 201 77. Vol. of cube = (5 x 2^) ou . .-. 8ide= V572^ in - 2V5in. 78. If side of cube be 1 in., when diminished it is ^ in • ^Xl ^"' ^" =-^ ^"' ^"- ••• '' ^« '"^^^ ^y 23?n/* ""is ^^ ? "-u^ ^ '''■ »-?«P««tively, their vols, are 4;r o^ V '^ ^^'*- inches respectively, or as 2» : 3^ ^ 80. See 47. relented by 2 A^"""'"^"'''' ''^ ' *"• •• '' "' "'" •■» "P' = 13 ft I'in'^"*'' " "** ">• '"' •■• '™8"' = (9 X 144-8)in. 85. See 12. 86. Draw the diagram. 87. Side of sq. = 10in.,and sides of rectangle 5 and 15 inches respectively, &c. "'-itgio u ana lo 46'f 58^ n''. ^^'''''''''' 48"' 60", 72". and inner dimensions Vol. of whole box = (48x60x72) cu. in. = 207360 cu. in Vol. of interior of box=(46 x 58 x 71) cu. in. = 189428 (17932H-1728 xV-7 . 62^/1^." ^^ itU^t n'94™' l'9ff •7)^1/28 x62ilbs.- 11845-5 it. <'°^*'°^ 89. Areao£f.eld = (132-24--I2)«,_1102 ao. .-. ddeof IN THE 'IP HIGH SCHOOL ARITHMETIC. .'. C08t = i field-i Vii 02x4840 vdn •35)=: 8323-326 ^'^'' ''' ««««> = 1?(4xV _^ 90. Page 81. 92. See 12, ^^ e,ual3|hegeaeeat^wh„.e„o.in73^!i,J,|),bi = ;2.^'""''^ 96. See 39, 61. SQ7t*^':l''f ''"'^"*"^^P^P^^^d = 2^18 + 12)ll = 660 98 Se;87 ^ * P"P^^-(660--)h-3«97-7. 99. See 89,' 87. "^ ^T ^^--^ r5 It- ••.circumference 102. I base = (3i^2i) metres, etc. 103. Apply formula. 203 104. Let X. length of comer out off. .. xi/J..ide octagon ocfgoni'dlf t- '"' "™ "' ''■ = '™» ^^ y^ ■■■ a- of 105.3(side)..ie«,.a ,„ide = ^f,...,,p, ,. ,„, - 4i/37 ou. ft. - V ,/3-ou. ft. '^ .06. Inner rad.-^, 1050 yd* .-. „„ter «d.-(^.io50 + SOLUTIONS OF PROBLEMS V \JT' ^ V i P^P®'"- ^*'- °^ «<^"P« required = 2(20 + 12)2- ^T- 04 .-..No. yds. paper= 12^ x24j^ 100. ^^"^ ^"^^ ' Tf - 24. 108. Side of field -ViOx 4840 vda -59n »-,!« t ^.u * wire = ^5 X 4 X 22n\ trrJo a Ann 7 ^ '^^ y°^' length of X w«>; = i|pid2. No. of posts to a 8ide-«4 • »,« v^w required = 4x84 4 qqo /V f ^l^®-*'*- • • no. of posts «(9fi ifi - * X «4 - 4 - 332. Cosfc of posts = $(332 X 08) = «2b-56. .-. total cost = e(l32 + 26-56)-$158-56 ^ 109. Circumference- V X 12 in •knafK/.fo-^ ?« ** X larin. = 15| in. T X *^ m. . . length of arc =^»y x V 110. No. of cu. ft. of ice = 4 x 4840 x 9x1 • «« «* cu. ft. of water in it = 4 x 4840x 9 x"x »o *'. 4^ "?*. ""^ 4840x9x.x|Ox62..2000TtonB!2Y75jJns.--"*-'^*^ 32 X ) ^g:^^:^^:^^ - - -u^o. Ll-i. See 12. nil'. 7.1:&^^ lo'sr r "^j^it,,. ■■ ''-= ^ '"^ " 115. Rad=^x85i n. = y ia. .-. area of circle = ?. (3.). 12; "is^sif r' ''• - ''^•(-'->' '"• = ^'^^i»- = ¥^3TS# 204 are"^ fto^"' %=/^ J^=i« ^^^ Sidesof rectangle (4^'x utirtafr I'e'^^sVS '"• "• - '"'°'» »— " thelnnefrjteSrif («« + H);«.. ft. and o. .22*66^ "' '^' ■ ""** ">• "• •■• "<'• - ^A^neoo-fa. 120. See 49. 121. Apply formola. fo' i'H '1^ i-J. IN THE liiuii SCHOOL AiaxilMETIO. 122. Letr = rad. 4 7rr»=616 8q.in. r'-r^flifi • ■= 49 sq. in. .• ra-7 ;„, v , ^. , * * ' "¥^x6l6 sq. m. == 1437? cu. in. "*• ^**^' ^' ''P*^*''^ = 5- ?. 7» cu^ in. 123. See 64. 124. See 116. (16'4.t)Tq.T:Vs;V"" °' ' quadrants of circles. 126, 127, 128. See lis, i49. Draw figure. 205 129. Space = equilateral A (side 6 f f \ q 1 . ^"^^ ^yP- = 13 in. Space = n32 4.ioa ■ K2 . 1 c ,«x in Peri = 3(13+12 + 5)in ^ + ^2» + 52+^.5.12) sq. 131. Circumference of 0) whose rad. is 40 ft o^l^eo^l":Zs1/C.i^^^^^^^ ^^^^ one = (22xl2--44) = 6**'^ ''**''"• •••°o- of rows 1 ^0. blant side of cone = Vi+/3\f f^ _ t y ,-^ , enceof base-^^^ 2 f f qio * r "•-^^3^*- Circumfer- TJao^ !s , '^^^ **• ^'ant surface = i , ^ 2 x i Vfq „ r^ llfi'^WK ?^- ^'- ••• ^*»ole surface^sL:" • 2 x ^ V13 sq. ^ x9)sa ft ptr^'^r '''•."'.* "^"*^««°^i°»r^^^^ * « iKm"'''™'"^'^^^^^ 139' ■J'^ole area = original A + 3 equilateral A's equi'tert"^:Vsid:8 rr ' ^^^^^ V^^^*' ^^O^nd of 2 206 /738va'* jj„ _ •«x"iijr iv_-^ y(ja. Area outer f^-«v (-TT-) sq. yds., and area inner G)^^xC»o\ „ ""T ^ ~ ^ ^ of road = I' ^ (IM)^ - /« a \» 1 „y 7 VITA ^l' y"^' • '• area 111 «• ' -1 ^^^ S v-TT-; r sq. yds » 123-^'" m vds - - . . aia. „* »^atara is -Tiuyy ti^jeg ^^^ ^^ ^^^^^ U, I IN THE HIGU SCHOOL ARlTUMETia 142. Dia. of sphere = edge of cube, etc. ^^143. Solid contents = j | -.23 -|.e^.,3 ^ ^u. in. = 29^ 144. Vol. by first pipe : vol. by second pipe : : 3«: (uy . time=2x^--l-,hrs.=|hrs. 145. Side of-courtWl96 yds. = 14 yds. = 42ft. Side of walk = (42 + 4|) ft. ., area of walk « | (46^)' - (42)^ I = m X 88^) ft. ... cost = $ I 4J X 88J X -20 I =$79-66 146. See 139. ^ ^ cuniference (rad. 3 in ) + etc ^^' ^^^^■>- e<>c- Pen. = i cir- in.K'et^c'"' = ^^-^^ °^^^^« ^-^- n -)^i circle (rad. 4^ 150. Sector =i|| circle, and arc of sector- 'i« p,v.„^* e«ce 0, but ^=275 sq. in. ... r = 5^T jT^^ ''''''f''' 151. 1 gal.= 10 lbs. ... 6 gals.- 60 lbs., and 62| lbs = 1 cu. ft. = 1728 cu. in. .-. 6 gals - -^ ^ 1 79« • . ^ • ~ 6^1 X 1 *^8 cu. m. Area sq. m. Diff. in area= j U. r^si^-qfi /q i • \^ll } r-' [jf) -db^3 > sq.ia= 40-737 207 153.^ I^.^^er rad.-r. outer rad. = R Now 2^=796 ^d- ■-r = j.^.,96yds.=i^03yds. ... R«(i398 + i3)- ^f- tli:/,^'- .•'• '*^«-"^°-t' V{ (^!^)»-(lff^)« jsq. yds. - lUo79f sq. in. ) n SOLUTIONS OP PROBLEMS \li' ^^P«® = (Vx 7913-360) miles = 69-n8 miles. 1D5. Ihe length of the degree depends on the length of the parallel of latitude at 60^ north latitude. The dia. of th.8 smal circle is ^^1 miles. . •. length of degree of longi- tude - (-V- x 7-«i,U3 h- 360 miles) « 34-5^ Vs. 156. See 155. 157. See 151. 158. External dimensions are 36 in., 24 in., 18 in. Inter- nal dimensions are 34 in., 22 iu., 17 in. No. of cu. in. of material = |36 x 24 x 18 - 34 x 22 x 17 jcu. in. = 2836 cu. in. 159. Area of end= | ^t^ x 20 Lq. ft. » 4oo gq. y^g. Contents wall = (loox 1500 x 1760) cu. yds. etc. 160. No. cu. ft.-|v •(D'xSoi =212f 161. Space= area ^ + ^ circle (rad. 5 in.), etc. See 148. 1d2. See 149, 161. 163. Sector=i arcxl5 Rq.ffc. = 90 sq.ft. .-. arc=12 ft. Ib4. Whole area = area of 2 ends -f area of 3 sides 165. rft. = rad. front wheel .-. (r-'r^)ft. = rad. hind wheel. Circumferences are 2. V r ft. and 2. V . (r + i) ft. respectively. ^\o^ revolutions made by wheels in going a mile are ^-^^ and ^"^^ r..r. f 1 5280 5280 ana -^ 57—; — r. respectively. 2 .^ .r « 50 eta ^-'^(r + i) •2.fr 2.^(r + ^) 208 166. Let A = height of part cut off. r = rad. of its base, and R = rad. of base of whole cone. . •. — = -^(sim. A 's) ; and .7 i/r i.^.A = ^.E;.14...-|, = 4-=^ ,.,. :V (|)». h cu. ft. R'' ^ ^ 14 167. h ft. = height cylinder, .-. vol 151. 168. Area of semi-circular section = (i. « W) so i/rf °,r- y^^'- = ^^ • T • 10^- 9 X 80) = 1396825. ^ 169. Vol. of cube = 33 cu. in. = 27 cu. in., and vol. of one money = {j. \^j .^i- "u. m. .-. no. of coins See ft. 27-{V(f)».i}-=488A. Ans 488. IN THE HIGH SCHOOL ARITHMETIC. T ( (U) - (I)' } sq. in. = « sq. in. No. cu. ft. =( « ^ H4x20);...weight = (-x,- xU^«-)lbs. ... cost-s(' ^x^-JfV^x -08} = 025099. ^^x »/3 .-. 8ide = -?lft • '• area 171. Side of ^.altitude = 2 1 28 -*x-7=r-xl4sq. ft,=etc. 172. See 25. 173. Dist. trav. in 1 rftv — 9 n r rx nn . - 33 ft. Train goes inTsec 33 f7 *: ' ^\ ^b ^"^ ^" ^^ ••«^- X 60 X 60) miles £ 22rmiles ' ' ^" ^ ^'•- ^' «°«- ^i^ttir ^174. Dia.of = 4in. ••• no. in. of wire = (2 . V . 2 + 4 . 4) 175, In 1 hr. it trav. 2 ^^a 11 f^ . : i j 176. See 151. 177. Vol. 8ov.= I yi.(r)i 1 1 I T vtfV • TF f CU. in. • vnl fionnn 80V8. = 80000 X V ^ (7 ^2 1 ,„ -^ . •; '^°^- ^"^^^ «____ ^ • ^tf) • T^ cu. zn. ... edge cube - 209 178. Height cone = i/ 12*^2 ft _,/«?r*f tt i , l/80-cu.ft. = &c. «tt._,/80ft. Vol.=^.V8' 179. Apply formula. 1 80. Length of cylinder = (42 - 2 x 21^ in Q7 • tt , - cylinder = V ^4) ^ 37 p„ ,•„ ^ 1r , /z; »»• = 37 m. Vo . of ..•; &a "^ ^^' • ^ '"• *''• ^°^- «f ends = 4. V(|)3 cu. in. 181. Vol. of sphere = 4 28 .0X8 ,'« 1. * = lU Vl^'t^^ ^lbs.2r27li?bV •••-^-^g-'^Powder l»j. Vol. of water replaced bv stonft- aa lAtA ISI r<,*- 2 ^ * •*~^'.^o X 1000) 02. =tt^p lod. Circumferencfi of wheel - 22 - 1 /i?. , fv" traversed in 1 hr. = (44 v 50 1 fin -V.o Jf ^^ -7 *^ ^^- I>istance 184. Distance travell^ i„ 1 ? " ^^80 miles = 2.^ miles. nee travelled in 1 second = (35 x 5280-3600) ft I t:m ii SOLUTIONS OF PROBLEMS .-.diameter of wheel = { (35 x 5280-3600)-4 x^ ). ft. = 49 185. See. 39. ^186^ Wt. of v.ater = (480-31) lb8. = 449 lbs. ...no. gals. wi^?" ^^^* ^^ °"® ^^ faces=i/(34«)2+,l7Ai» ft -laa ff Who^ area = (2.35. -- + 35=) sq.'f t' . 3943^ s|.^f t ^~ '' 188. Let. r = rad. of sphere. .-. 4.^. r^ = surface of cylinder = | V • 4 . 12 + 2 .V.4«) sq. in. = V • 64sq. in. ., r- VJin. ... vol. sphere- f . V (4^- T)" cu. in. = 758-556 210. 189, See 188. rad^'oo^jrT .'^9' ^f'"-/ Thearea = 2 sectors ( ^ 150° and ( 90O andrad 2rftw^''?r^^'^^.- '^ ^^'^ + ^ sectors y- yu ana rad. 2U ft.) + equilateral triangle (side 40 ft.) - {V(100^^ + 60=.^ + 20=>.i) + 400 V3l sq.ft. = 29397*58 sq. ft. -' 191. Vol. of sun = * . y. 44T500' cu. miles ; and vol. of 192. See 186. ' 193. See 191. ^^^•^/^*^' ^^ °"- ^°- of lead=(208xl60x J,^-9nfin . cost = .$(2080 X 6-5 - 16 X -07) = ^59 1 5 '^" "" ^^ " ^^^^' " 195. External vol. = (40 x 30 x 20) cu. in, and tho internal vol. = (37 x 27 x 17) cu. in. V. plank contain' I (40x30x20) -(37x27x17)} cu. in. = 7017cu. in. But asq.ft. of plank contains (12 X 12 xU) cu.in. = 216 cu in .-. no. sq. ft. = (7017-5-216) = 324« ' 196. Apply formula. ^ 197. See 72. 198. See 194. 211 cu. ft. .'. no. of gals. =etc. Ans. = 1738229ygals. 22 IN lliK HIGH SCHOOL AltlTHMETlO. 200. Area of face of stone which can be used |(W-(f)^J8q. in. = V. 35.30 sq. ft. Each man uses t' T--35.30 8q. ft. = V 175 «„ f* t^*. i n when fi.t Jn h» tei^M? tr,^\:hirt,lt''?; ffc.=36-496ft. ••AC = ,/36a^g8 202. 6 (edge)^= 2 eq. ft. .-. edge= i/48 in =etc 203. See 186. •^^om.-etc. 204. 1 cu. ft. of iron weighs (62 J x 7-7) lbs. .-. 64 lb shot is equal to { 64^(62ix 7-7)| cu. ft. Let.=the rad" of ball. .-.4 28.r« ^-■ Tf' T '={64-(62|x7-7)|cu.ft. 1 - 'I . . r = Sue. 205. Page 81. ^ ^t cu. ft. lost in drying, &c. * ' ' GENERAL PROBLEMS. J. 9 of A's days = 10 of B's. ... 10 of A's days- l"" of R'« ..tleu.oneyn.ust be divided in the rt io oTl"- H ? or^ -«. Amount of policy =^ of S4000-«oinn a . b premium = ^-^ of ^•)40() - <«/« ^ ^^' A*""""* of $2400) + $48'or8l6!S ^ • •*• ^^^^^s lose ($4000- 3. Page 167 (40). 4. Page 188 (2). 5. 3 lbs of tea at 40 cts. per lb. = $1.20 ) 5 " " 48 tl =|2 4o} =^3.60. «3 60 ^'' °^ *!? '°/* ^^- ^^- • •• S lbs. of tea will sell for ^ o « ^. come from $--^--1 invested or |196 ' * ^'^ '" *^^ "'■ 7. Page 149 (3). 8. 4000 yds. of carpet at 4^8. pw yd. - ^f^22^^ = K ^^)x 4.87 = $4383. 9. Page 199 (58). viied bet!ee'n A tTB^T ^f'^' f'. ^^'^^'^ ^^^ ^e di- $(201- 12)r$ll!f75 *^''''^' "^ ^=2- ••• Ageta J of goeV-4i^?rmes''' "thYfirst". '^"1 T""^' "^"^ *^« ««-^d irst glins 1 round h^fof 5 LT' ^ A^ .' ''"^^ ^?^ ^' ''' '^' before they are toapfV.r ft ^-i, As he must gain one round way -u/droVXlkti:!^^^^ t o^ the "!2' l^'i^^T ^" *^^ oPPosi^eXecdon^ °' ''^ "'^^' "^^^"'- = 69^ct?'^~*- ••• tof cost-78ct8. .-. C03t = a of 78cts. 1 spaoe^fn ^2^ ?n ^'th!^ '"^'^ T ^^^- ^he hour hand goes Pa^„ ^ ' /,^ "!'''• ^ The min. hand goes 12 spaops m lo m-n hand travels at one lm?f the rnf^ ^f'l'"^u*° \'''' *' ^^ ^^^^ «'«' one naJt the rate of the hour hand it will al- ,l>' h 1 1 $1 SOLUT10.\.-J OK Pl(0ill,EM8 Thr^''*''* ''"^ "^'*'''' ^^''^''" ^'''^ ^fe'"'^"* '^"'' ^ho hour hand. 1 e rate of t..e extra hand is J .sj.aco i.i J 2 n.in. The min- ute h,,.a u.ust ga.n 20 .paces r,ef,.re it coincides with the ex- tra hand, but ,^t gams Hi spaces in 12 min. .-. it will gai'. 20 spaces in ^ ^.^ of 20 min. or 20,' « „in. When these two hands coincide the minute is doing what is required of it in ^^K The work requires 36 days' labor. One man works — 2~" days or 15^ days. Now 36 days' labor cost $126. .-. 1 5^ days' labor cost $ •// x 1 5 1 or $54.25. 213 eacMb"" ''"r,?u°""V?^ ^"^o^:^^ '''''' ^ ^""'^ 23 cents on 54 cen; ,:'f r"> '\ ^^"'^ ^-' ^ ^^^ «^"t«- I" selling 54 cent tea for 5ri(- cents I gain 3| cents on each lb. • to make up the whole loss on the 'eO cent tea I must sell {(2^x34)-3|j lbs. of 54c. tea or 26 lbs. 16. Page 188 (2). 17. $85 invested gives $4 A income. 41 V r.inA ^ gives H 19 20 21 22 23. Diff. 85 18. Page 181 (7), 5100 — income or $270. $5100 invested 147 (35). 212(13). 199(58). 212(10). 1 ^^ .t ^ ^ 'i''''' ^"^ i °^ *he work in 1 day. B & C can do onedav ■ V A? """^ ^^ (|+tV + A) of the work in A 7r ^ n ^ 1'^ 'r ,^° ^Vo of the work in one day. 26. Pa^e i6^;:N). 7i-2S5 1428t>7x5 5 27. .714285 ^~ 999999 -142857^7" 7 IN THK IIKill H(;iI(jOL \RITIIMKTie. 28 For every dollar A has, B has $2 and C $3 • A J. H toG;oth,.r get $J700. .-. A g.ts /, of $2700 or $«J00. " "^ "^ " IX Out of every $100 worth of wheat sold, the ag^nt keeps 4 and remits $96 .-. when $9G is renntted the ag.nt 30. I'age IGl (17). 1. Page 212 (5). '^* 2. With carpet | yd. wide and a room 11 ft. wide, it will 8?tT ^ •irV'f ' ''''^''^'' '- '-'■' ' "^"P^- Wi^^'^ - pattern! Jy 8 ft. in a 20 ft. roon. it wiJl require 24 ft. of carpet in a strip . no of sq. yds. carpet require.l - V x J x 5 = 30 ; and „o '2(^0-2417-5?/''^ ''^'^"''^- •■• "^- ^^- ''^'' ^^^ 3. Page 212(14). min %1^3r^^?\ Rate of extra hand is G spaces in 12 and the hg 3. When the hour hand and this extra hand uZfV" '" '^ f' ''""^^ ^^'•'^'fe'^^ J'^^ ^'>« J'our hand wm bisect the space between the min. hand and the fig. 3. The extra hand IS 7| spaces behind the hour hand and gains 5 spaces on it in 12 mm, It gains 5 spaces in 12 min "it will gain 7i spaces in Jllll min. or 18 min. 15 Of ^370 = $148. But he pays $164. .-. B must pay A $(164 -148) = $16. 6. Let $100= cost, . . selling price = $135. .-. gain = $35 or 35% of l?^'"lVor. ■ ■ ^ S^'j^"g^P"Ce = ^9^ of $100 = $90. cost. 7. Page 165 (21). 8. Page 161(7). 9. Page 182 (8). 10. Page 181 (12). 11. Page 154(15). J/r^%K'^ ^ \^'J^^- ■'■ ^ ^"1 have * of $12 or,.„ ?6'^35T;!i'.r/ ^ '' ^ \'^^'''^r- a^ -i" hL $(r2: ■ '13. Page¥03 (108]? """''^ ''' *^ "* *'^-'' ^^ ^^• 14. Page U7(ll). ft] r'f I I* ' 'f, ' 1 SOLUTIONS OP PROBLEMS 215 fel^' ^^^ ^'1 ^'?*''r «!*' (1 - 1 - J) of the farm - ^-^ of the f^kA r U ""K^^ ^^'"^ ^^^*^ ^ISS*- ••• f ^"1 cost $(1884 ^ j\ X 2) or $2826. } 6- ^AV X TTiW X I « X 10 J, 1000 ^ 32. 17. A can do | of the work in 1 day. B can do I- of the work ml day. C can do | of the work in 1 day. .«. the money should be divided in the proportion of 4, I, » or 42 35 30. .-. A gets tVt of 121.40 or $8.40 18. Page 167 (40). 19. Page 161 (6). 20. Minute hand gains 1 1 spaces in 12 min. .-. min. hand will gam 25 spaces in ( 1 2 x 25 -1- 11 ) min. = 27 A min. Aoni" ^i, house be worth $500, it is insured for | of $500:3= fo ;on?'"''°'m"°'"T^^ of $300 = $9. Loss = $(500 -300) + *y = $209. The loss is $209 on a house worth $500. .•. the loss is $522.50 on a house worth $55^|^M? ^r $1250. lloi^Sl^ ^io^fPr^^'^^ ^ 1^ ^ 17 = $327.25. .-. total cost = $327.25 + $253 = $580.25. Selling price = $48 x ll-$528. .-.loss = $(580.25 -528) = $52.25. 23. Page 199 (58). 24. Page 212 (5). 25. Page 116 (178). 26. tVIt of 1st sum = ^^ of 2nd sum = ^J^ of 3rd sum. If 3rd sum - $3000. . •. 1st sum = $(^«, of 3000 + J/ » ) = $4000 : S^ni!"^^ ^"^ " ^^^^^- • *• total sum = $(4000 + 3200 + 3000) = $612 = $240"*' *^^ ^""^^^ ^"""^ ^ ^^^^* •*• ^'* ^^"^"t^Vtt of 27. Page 188 (2). 28. Loss on one bushel = (87| - 77) cents- 10 J cents. .-. on 87| cents he loses 10| cents. .-. on $1.00 he loses ^-^ x 100 cento •■12 cents, 216 ^ 1 . A and B can do | of work in 1 day, A can do A of work nl s^-, •••?.*^*^^o(M)orJ'^ofworkinlday A and B do 3 of work m 3 days. B can do | of work in ^ x *« days or 5J days, a s j 2. Page 142 (1). 3. Page 146 (17). i». ■' IN THE HIGH SCHOOL ARITBMBTIC. If 83370 is invested, 4. Page 189 (6). 5. If $72 is invested, income fs $ income is ^^ of $3370 = $140.41| 8 Page 196^(12) ^^^ •*• ^"^°"^* = ^348. 9. 1 day A's work = f of day B's workl . •. 6 of A's == 4 of B's 1 day C's work = |- of day B's work L 8 of ^s = 10 of jI'h Divide $42 in the proportion of 4, 7 and%.^ fg ^ets^' ^f $^2 or $8. B gets ^ of $41 or $14. C gets $20^ '' ^^^ share ^369 nt[o/^ "i «*>^ ^^ ^'^^^-^- ^'« shares B's = mfm4 SO i^ia^ ^' share + $4.20 =$369. B'sslare fTyOt^d64 80 = $320. .-.A's share $49. • iPiV f "? 4fl^°'^- 95%«fcost + $1.05 = 1077ofcost !l"/^n ^^^""^ " l^^ ^'=^^'- B's share = f of farm C's share !x"nf^f« ^^'Sr^^'°^- 293j^ ac.+i« of farm = farl^ if of farm = 293 J ac. .-. farm = 7 20 acres. " 217 ^s 81 yd,. B g«3 100 yd. while A^rDTJ ^t tti^ Bank disc, off $508 due in 44 vr, at fi°/"«r!l » ' i^°^' $137.16. Difference = $137.11 -'Sos-Vao'fe ' "^ * " "'' = 15. Page 203 (107). <3P^' ?i^i?rli/'° °^ ^^^^- ••• cost = 86°/ of $600-^516 S.P. = 114% of $516 = $588.24 ^° oi ;»wu-i^5lQ, 17. Page 179 (38). 18. $100 is present vahjo nf ^iriQs j,,- .•_ n , ^a- i_ _i ,. J ....?_ auc lu b mos. at 0/ ,• present value of $7470 due in 9 moe. _ i^ <,f $7470 = $7200. Amounb of stock sold at 72 to produce $7200 is $10000. II ■ll't' i 'ff Ui- 1 h 4;i rfl SOLUTIONS OF PROBLEMS »3. .-. he makes isT^J? °" "'"''' '''''■ ^^ ^el'^ f«f for J, »«""'l'«s 9(610-640) more on each 75 bbl,. he sells wonh fof Jlol'oflsif :" •itrato"^""'"! °'?,^-, ^"' «•'•'! -$m ... 44 45 is ga°Ll"|35,75 oSr"*i if?o '""' on $643.50 cost. ^.«^cosc . . ^M4.10 la gam 21. Page 144 '(29). realfzes''8%'" "'"'"^"* °' ^^^^ ^« -^^-s $32. ... he m'- Bj^tJ^'tl.tn^f^^ \\^^ -Id have fives and 42 four dollar bills ' ' *'" ^^^ ^1 teas. 28 24. |- of distance B travels in 1 hr - 7 mlo . j- ^ avels in 1 hr = 4 ^f ^ _,„ *", \T' " ' ^Is. .-. distance B liRf - .^"^- Tirb- or amount of stock .^ .'l!i9nn 1. 2. Page 142 (48). Page 205 (133). 3. Pcige 108 (40). 4. Page 212 (1). Page 212 (1). 2l8 7. Selling price = 1 307 of coqf nf 3 ^i! , price = n of 1 30)7 or Q7^vJT\ \ ^ °^, ^°°'^'- ' '• celling 8. Pa|e 215(16) '^ '°'*^ ^^ ^'^^^^^ ^°«« ^^ H%. lloVtf"'^^'''^^ ^""^ materials times amt. paid for labor 1 lU^ ot 3 times amount paid for labor 4. Qav «# / • , « labor = $3637 92 • 4.91°/ .f t ■ t.^° °^ ^™*- P^^d for • AOnoTyr' '■\% ^° ^^ ^^^- paid for labor = S3637 QP • . 400// of amt. paid for labor = $34 32 9P^t)d7.y2. 10. Page 196 (15). ISnnvL^edTn tlfe'5 n^' ' ^7 '■''''' «™ ^'^ ^^'^^^ of $4. or ii 9 ?i^ 1 «. P^'^ ''^''^^ g'^^« an income of AV of Is or m^j. a/y ,s difference when $87 is invested ilf ^.^if ference when 1I>^1* of fl!87 nr (Ksfi-^ . ^/"vebcea. ;|pj7 is dif- ^ o TT_._ , » or !|}»87 or $88 1 4 is invested. -pais t?^-;^T «xs^«^>r.f ;r L^^^^^^^^^^ IN THE HIGH SCHOOL ARITHMETIC. be opposite min^tZF f3J TT ""'■ ^'^ 'P^^'"^ '' ^i" (360i - 12) spaces in 1 o' .^*''! hjnd gams on min. hand in spaces i/S|.-/- pastTeve?^^' ^^^"^ ^^-- ^^^ 13. Page 189 (6). ^^^ coslt$m5T^.^%1f^li;^?^^ price-90% of voice price -$260^ '^ ''''^''^'' price-$2925. .-. In- 67 L whofe'^^^'sTf l?n^'^°"^^*- ^e wants to gain $300 worth for $424 - $94 or %30 T ^^^^ .'' ^'' ^^"« for $110. What formerlv .oMf ^.o'. ^^ '^'" ^^^*^ "^'"'^^ What formerly soinor^iS, ^^1 "^ '"''^ ^^^ ^"0. wonh ^Oct:l dollar^"''? ^ ?^ '''''' ^^ ^^^^ to be ofassets = |of iSabilitie'rSTflbiil^^^^ •'• ^ onb^pa^ A of liabilities or 41| c^ c!)! S^^^^^^' -•• ^« -- « ga s. of wine @ $1.12^ = $9.00 14 . ;; " 1.25- :- 15.00. }4 '« 1,50 =21.00. 16 " " water 50 gals. 50 '« of mixture cost $45.00. " '* sold for 50.00. Gain = $5. SOLUTIONS OP PROBLEMS o .(P^c»-rbuic. . . o mos. work 13 worth «7 n . i work 18 worfh <^'^ . Q _ J , . worm ii^io. .•, j mo. .-. suit is "ol 120: • "• ""'^ " ""^'^ a45 = $25 + suit. • ?on'y^'^°^ '"""T^ ^^''^^ ^' $^865.50-400) or S465 50 ; ■ 100% of income taxed is J-Oo „f ^a(.~ -.^ y. "'^ ij^oo.ou. income is $475 + $400 or $875^ ^^^'^•^^ = ^475. .-. total 24. Page 215 (20). 220 iJ'u^lot''T ^^^ """^ T^'^' ^^^^^ ^^ «tock. For invest- ing iig4d00 broker receives $25. For invpqtinr, 79jO invested income is l^so ^f ^^« ^^^en A 140 yds. A has gone 290 vis th^ "Tj'^ "^ ^'^ ^^^^ o^ B wins by 10 yds ^^'•' "^^'^^ ^ ^^^^ gone 300 yds. at 7% per inLjluOoTl^^^^^^ tJ'n^V'"' ' ^- ence is $6.01^1 ^ '' **00 = $90.01^|. Differ- 12. Page 217 (U). 13. Page 158 (19). 221 Juo°n.!? ""'.'"'"''"'■ "°T'"'°"'^* J- O"«"76.50 . sent to t, ^„„.,,,„^ ^^ ^ ^^ 'of 15176,50. oreaSO. 16. Page 219 (21) °* 16. In4daysJAworks4x6hrs.,or24hra. 1 B "orts 4 X 7 hrs., or 28 hrs. In 2 days B and STeaT 11^20^' "-.'i""- ,. worts 48 hrs. C works 52 ts DWde«4"°*'^* ■!"• » uiTide?62in proportions of SOLUTIONS OP PROBLEMS m 12. B gets T*^ of $62, .-. cost =70% of selling e 1' 24, 48, 52. A gets ^Vf of $62, or or $24. gets ^%\ of $62, or $26. 17. Page 146 (13). 19. Page 155 (21). V9. Gain = 30% of selling price. .. «u«c price. ' / of selling price= 100% of cost. .-. fg ^f selling ^In^fr f '■''' "'•. ^'^"/^ «^ '^''- ••• gain =-42"^% of cost^ • £'i^^°'"^^ receives 35 in., but pays for 36 in! • On every $36 worth the customer buys he loses .'i^l. .-. on every * 0? '^^'■^^ ,*n^^ customer buys he loses -V/gO of $1 or {53.33i iSl. Page 196 (12). -^3 22. Page 144 (29). 23. A can do work in 7 days, B in 10 days, and C in 14 days. .-A can do | of work in 1 day, B can do Vtv of work in 1 day, and C can do ^V of work in 1 day. In^'two days A and B do(| + ^) of work or f J of work. .-. C has U of work to do. C does ^V of work in 1 day. .-. does A| of work m i*gy^ days or 74- days. ^ 24. Page 148 (42). 25. Page 220(11). 26. Page 144 (24). 222 •^' M^.o^'^r^^'^^^f- ^ ^^- Troy = 5760 grs. Marked price = 1 50% of cost. Sells 7000 grs. for M.P. of 5760 crs .-. sells 7000 grs. for 150% of cost of 57G0 grs .-. sells 5760 f^%!o; V^ °/ \'fA:^^ ^"^'^ «f 5760 grs., or sells 5760 grs. for 123^% of cost of 5760 grs. .-. gain is 23^%. 2. Selling price-= 130% of $40. Adiscountof 25% has been •'''^ i ono> t'^i.T'^*^^ P"*'^ = ^«"'"g P^i««- • •• I of "larked $69.m ^ •■• ""^'^^"^ P"''^^^ °^ ^ °* ^^^' ^' 3. Page 199 (59). 4. Page 149(11). 5. Page 161 (7). Q«°; ^"""^^ value of property + 2% of policy (premium). .-. l^'f^''n,r^'7''^^^^^- ••• 2% of policy = A of $7140 or $14o.7lf Premium paid is $145.71#. 7. Page 188 (2). 8. Page 146 (24). 9. Page 212 (13). IN THB HIGH SCHOOL ARITHMETIC. 1 8 «fii^;rli1n^*!f expected --= $80. . •. | of gain expected « f of i?oA %^?-o«®®"'''S Pnce = $5300. .-. value of farm $5300 - $120 or $51 80. 11. Page 179 (31). 12. Page 198 (40). 13. Page 161 (4). 14. 140% of price of one lot = $1400. .•. 1007 of price of = $1400 -$1000 or $400. Sold cheaper at a gain of 50°/. .-.». -f.of cheaper=150% of cost=150% of $400-1600. Sold 70°J%tn.r2t^n^^- ••• ^- ^- «f dearer = 70% of cost=: 70% of $1000 = $700 .-.Totals. P. = $1300. Total cost = $1400. .-. loss = $100. 223 15. 80% of total profits is divided between A and B in pro- portion of 27 to 37 •. A's share = f| of 80% of total pro- ^J8. .-. II of 80% of profits = $675. .-. profits =«* of '«« of $675 = $2000. On $6400 capital profits are $2000^"^. •. $7oO capital profits are^ of $2000 or $3U. .-, profit8-3U°/ 16. Page 220 (4). ^^° 17. Page 148 (50). 18. Average price paid was 92 cts. a day. Average price P^k fe^ ^l^i'- ^ ^ ^" ^^•^^- ^"^ 8 Ws received 50 cts. X 8 or $4.00. Difference = $3.36. .'. $3.36 is amount paid to men over the average price paid. Diff. between price paid and average price is $(1.40 - .92) or 48 cts. for each man. 48 cts. 18 diff. for 1 man. $3.36 is diff. for 7 men. 19. Page 141 (43). *,?oq-tIa^'^*''® ''^^^ ^^^i- ^^^ scares cost 150 of $79i- $lloo7.50. * 21. Page 219 (19). • ?'\^l ^^ P^^'i^-.^ *^® ^'^'^ ^^^ ^ 'bs- *he quantity. He if ^^'J^^^°g^^'^T>f I4or$1.20, butgainedonly I of $1.20or $1.06f . He gained /^ of $3 on the 3 lbs. or 90c. •. he gained 16|c. on the last lb. or threw off 13Jc. .-. he lowers the price 13^0. when tea is $1 lb. .-. he lowers the price 10c. when tea IS 75c. a lb. 23. Page 182 (18). he has $300 invested at 6%, income = $18. Total income = SOLUTIONS OF PROBLEMS 22. $22 is income when he has $400 invested. /. $76.42| 76 42* is income when he has ^^ ^ of $400 invested, or $1389.60. 22 224 1. Gain = $715. A's share =$275 + | ($715 - $-75), or .'. A's contract B used $1300- $495. .•.B'sshare = $715-$495, or $220 used III of stock -III of $1300 = $900. $900 or $400. 2. Page 198 (40). 3. Page 143 (15). 4. Page 221 (14). 5. Page 217 (19). 6. 252 lbs. @6| cts. a lb. cost $16.06^. But he paid 17^ cts. more than this, or $16:24. If he had bought 252 lbs. @ 5| cts., it would have cost $13.86. Difference -$16.24 - $13.86, or $2.38. But diff. is (7J-5|) cts., or If cts., when there is 1 lb. at 7i cts. .*. diff. is $2.38 when there are 2.38 -QjTo lbs. at 7^ cts., or 136 lbs. in hind quarter. 7. 97% of debt = $1008.80. .-. 96% of debt = || of $1008.80 or $998.40. . •. after a reduction of 4% debt becomes $998.40. 8. Page 215 (26). 9. Invest $187| in each. From $134 invested in 6%'s in- come is $6. . '. from $1 87 J invested in 6% income is l^of $6 134 or! 'tA- From $187| invested in 8|%'s income is $8^. Dif- ference 'in income - $8| - $8y\3_ or $^5\. $ J/^ is diff. when amt. invested is $187^. .". $10.60 is diff. when amt. invested is $10.60 X Yt- X 187| or $19023.213. 10. Let $2 = price paid per bush. Sold 435 bush, for 113% of cost or 113% of $870 or $983.10. Sold 325 bush, for 111% of cost or 111% of $650 or $721.50. Total S. P. is $1704.60. If sold 760 bush, for 112% of cost, S. P. = $1702.40. Diff. is $1704.60 -$1702.40 = $2.20. $2.20 is diff. when $2 is cost per bush .'. $1.10 is diff. when $1 is cost per bush. 11. Page 222 (14). 225 12. Marked price = 75% of cost. Selling price = 125% of marked price = M4 of 75% of cost=93f%of cost. 4%. loss is IN THE HIGH SCHOOL AKITHMKTIC. of 13. Page 225 (8). 14. When he gave 5 cts. to each he had 14 cts. left. When he gave 8 cts. to each he had 22 cts. too little. .-. when he gave 3 cts. more to each the difference in the amount required was 36 cts. .-. There were 36 -r 3 or 12 berr-rars l^iiAr"''*' '^ ^ inos. = $334.40. Amount in 16 months c=!tfd4D.bO. .-. interest on sum for 7 mos. = Sll.20 • in- $334 40-$14.40 = $320. interest on $320 for 9 mos. is or $6 . ''raf ^-'^^o °" ^^^^ ^°'' ^^ "'*'''• '*' ^^^ ""^ ^' ""^ ^^^'^^ 16. Page 198 (40).' 17. Page 148 (46). 18. Rates are 5 7 and 9 mis. per hour. 2nd man gains 2 miles over 1st while l.st ,oes 5 miles. 2nd man gains 80 rods over 1st while 1st goes | mis. or 20U rods. .-. 1st and 2nd will hrst be together when 1st has gone 200 rods. 3rd man gains 80 rods over 1st in going 1 mile or 100 rods. .-. 1st and I , r« . ^^ together when Lst has gone 100 rods. L C M of 100 and 200 is 200. .-. the thr.e men will first meerwiiTn 1st man has gone 200 rods or 2| rounds. .-. they meet A way round. ^ j 19. Page 218 (9). 20 Wholesale price=115% of cost. Retail price- 1107 ?9fiTo;^ P"^^- •.•• ''^tail price = f JO of 115% of cost = ^'^H/o of cost. .-. gain on cost is 26i7. 21. Page 217 (19;. ^^° , nr. 226. 1. Page 161 (9). . ^' ^i'^r^^^ ^2 ^"'^ ^'« ^3 ; B invests $3 for C's $i» • C invests $1^ for D's $^^. ... the moiiey should be diviJed in :f:v'oT3iror $480^ ^ ^^^ - - ^^' ^^^ ^^^ ^^- ••• ^ ^^^ 3. Page 215 (26). 4. Page 121 (219). 5. Page 146 (17). 6. Page 162 (23). 7. A difference of 20c. a lb. in the selling price makes a difference of $10 on the total. .-. the number of lbs. is $10- 20c or 50. On 50 lbs. the gain is $7.00. .-. on 1 lb. the gain 18 Uo. .-. cost = (70-14)c. = 56c. ** ;v :,i I SOLUTIONS OF PROBLEMS t a i: r.' >i 8. 45% - 39% - 6%. 6% of A's money + 2 times. A's money = 8164.80 or 2 A of A's money = $164.80. .-. A's money = VA of$164.8C = $80. ^^ 9. I of last army -6000 men. .-. last annyi=7500 men. .'. number before reinforcement =- (7500 - 2500) -. 5000. .-. f of Ist crmy - 4000 men = 5000 men. .-. 1st army- 12- OOO men. 10. 25 acres @ $120 an ac. = $3000. .-. had he sold the whole he would have gained $(3000 + 200) or $3200. On 1 ac. he gains $80. .-. no. acres = 40. 11. Page 178 (21). 227 12. Paje 193 (23\ 13. Page 171 (24). • 14. Page 159 (12). 15. 95% of total taxes = $9690. $17 are taxes on $1000 assessment. $600000 assessment. 16. yf^ of f value flour -$36. .-. value of 1 barrel = $6. .-. selling price per bbl. =|A4 of + 7j^of $36=:$(6.90 + .12) = $7.02. 17. Page 126 (277). (18) Salary left after paying board = -/^ of $1200=, Salary left after paying rent = | of $840 = $672. Salary left after paying clothes = ,86^ of $672 = $57 1 .20. Salary left after paying books = ($571.20 - 71.20) r. $500. Salary left after paying loan -- ^"^ of $500 = $300. . •. per cent left = -^^ = 25%. 19. The numb er of bush, squared = 1849. .-. the number of bush. = Vi 849 = 43. 20. Time past noon + time till midnight = 12 hours, and § of time past noon = f of time till midnight. .-. time till mid- night =^ of time past noon. .•. | of time past noon » 12 hrs. time past noon ^ 4| hrs., or 4.30 p.m. total tax^.s--^ $10200. $10200 ETo taxes on value of flour = $1800. 21. {^ of cost = vessel cost $9(i00. .• 22. Page 184 (41). 1. Page 146 (13). 2. Page 162 (25). 1700. .-. cost = $9000. vessel cost $30000. 228 3 of I of a 3. B has to pay at end of 9 mos. $5000 -f- int. on $5000 for IN THB HIGH SCHOOL ARITHMETIC. fe * V^ "*" ^^^^2- ^^' ^^h B pays now = M of ^000 «4750. B receives at end of mnn ^o^^n ^ • * *^ f; """ = 6. Int. on 1st part for 4 vrq af ««/ a ,.* i ^ on 2nd part for 6 vrs at 5°/- 3 ^4 o ^7 ^"^ ^'* P'^''^" I«<^- part = ,3^^f 2ndparr;$?2 ^-TsJpL" Co .•"• ^^1^«* ••• I of 2nd part + 2nd Dart-«ffiO ^l^"^^'''^ + •^^^• pa^t=$300. "^-''''* part-??450. 2nd part = $200. lat woricf At Cs' ^63 h ^l?"" ?' ^ ^^^ ^-- I »>- B's work = hxl) hrs C's ?0 ^ ' 7^^ = " V ^''- ^'«- ^ t^''- A's . v^r ^ F^ '•rs. L/ s. 30 hrs. A's work = i*o hfa r<'- required proportion is 1*2. . 147 . no =280 • 4Y1 fifin "' aV wages = $28. B's = $44.10. ^. = 66 " * '*• ^ ' $600 - 'sT'^if "" = ^-^^ ^^ ^^^^ = ^*S- -^<'t«al gain = J^ of Q To^:": ,^^"^'' gain greater by $2. ^ ^^ °* A 7 g . ^? total rate on investment = 5 • °/ . *«* „i ^„ i.^ , . o, Sil IS income on $100 money, $3 is iScome in^Sfi Z^ " ^^'^' market price = .^6 - A = 551 ''^' * income on i^5b money. . •. 10. Page 216 (11). ^* 229 11. Page 182 (18). 12. Total money owned by A and B-«7fi game 4 times B's monev-^78 • St ^ " *'^^' ^58^. .-. A has won $2.50. ^ 13. Page 21.'> (17). 14. Page 216 (11). 15. Page 199 (60). 16. Page 161 (15). 17. Page 226 (10). At end of A's- 18. Dishonest gain = ;i^ pnc 13 ifa Ct Jn.v:t!iai D. X", g*^^ = 351 of marked price ||co8t = $124.80. Cost=|451 |451.20xf = $12. = (tX|?) CORt=«0cost, ;jy of marked entire gain ■ 20. .'. dishonest gain = JL of [I SOLUTIONS OF PROULEMS m- m i i 19. ^ distance B travels in 1 hr. = 5 miles, .•. distance B travels in 1 hr. = 3J miles. 20. 3 hound leaps = 6 hare leaps. Hound gains distance of 1 hare leap when he takes 3 leaps. Hound gains distance of 75 hare leaps when he takes 225 leapa 21. Page 218 (12). 230 1. Suppose ho sold Isb lot for $200 and 2nd lot for $300, coat of 1st lot = -V'«*i of $200 or $^J!J}^ and cost of 2nd lot = \^^ of $300 or $^V-. .-. gain - $(500 - ^^^ - ^V") = $.«^o^o^o_ Qain is $V,^V- w^'e" proceeds are $500. .-. gain is $16 when proceeds are $1656. 2. Average gain on whole = 5% «> t^, .*. -j^j cost = $68.50. .*. cost = $1270. 3. Page 217 (25). 4. Page 144 (29). 5. Page 178 (22) (23). 6. Page 199 (60). V. Page 215 (26). 8. Page 215 (26). 9. Let $100 -the cost. . price. yVtr o^ $1 1 2 - $13.44. 10. Apply formula. 11. Page 161 (17). 12. Page 220 (5). 13. Commerce dividend - $8 x 250 = $2000. When money is worth 7%, 8% stock is worth $^^^ = $114f S.P. of 250 shares at $(114f-i) = $(J-f ^^ x 250). No. shares Toronto stock bought = $(J-f f ^ X 250 ) -f $205^. . •. Toronto dividend - $(15 »3 X 250 X jf X X 12) = $1661.105. Difference in incomes = $338,895. 231 14. B's 8.p. of 1st lot = 120% B's cost. B's s.p. of 2nd lot -75% B's cost. .-. 45% B's cost = $153. .-. B's cost = $340, ^l of A's cost - $340. .'. A's cost - $400. 15. S.P. of remainder =^ of -^ cost = fl cost. .*. loss 4%. 16. At end of 1 hr. 3rd is midway between 4 and 6 miles .•, he goe.°. 5 miles per hour. 17. Page 226 (10). 18. Taking bankrupt stock worth $1 at wholesale price. $12 ea gain and Percent.=i|3f*xl00 12 = selling 112. i M '■r^ pnce. IN THE HIGH SCHOOL ARITHMETIC. 10 i! 1 ymn-f.2H4. .-.gain % = V.* x 100 =Y7|i 19^ ^ total votes polled = 240. .-. total v7tes polled = uU No. who did not vote - 1 800 - 1 440 - 360 yd8.\*38| !ef" ^'''''' ^^ """"" '"^ ^ ^''' ^'^^^^ train gains 187 r^H-J^ • '^"t^ received »V/- = 5^ Actual value of butter = (^s-Xt(tX li5/c. = 75i|c. Loss = 242% 22. Let $a = A'8 stock. .^$(l^)0 + a)-B'8 stock, and $(a- 30G) = A s gain Gam on $a for 3 mo. = gain on $;5afor 1 mo. Gum on 8(loU + a) for 4 mo. =gain on $(600 + 4a) for 1 Ja a -306 ,,^ ' • 450. mo. 600 + 7a '400' 23. Page 214 (12). a' 232 1. Cost of 2 apples, one of each kind - a + A^c ff 0. To lose Ic. he S.P. must of 2 apples = *c. Loss = {% - *)c. = /^c. have bought 30 apples at each price. 2. Page 146(13). 3. Page 165(11). 4. Page 162 (21). 5. Page 229 (20). 6 Incomes below £100 + incomes above .£100 = £500,000 ..gV incomes belo«v £100 + ^V incomes above £100 = i25,: --%R^rl ^^^ 'T"".^' ^^^^"^ £100 + ^V incomes above £100 -i.io,/oU .-.8 mcomes below £100 = £6,250. and O, incomes below £100 = | of £6,250 =£8.750 ^^ 7. Page 218 (12). 8. Page 144 (24). 9. P. W. of $1654 in 9mog. at 4r/ = Sl654y 8oo_*ifinn 11. P.W. o£$200m8mos. at 8%-$200x|f = $189.87341 ^400 X If = $37o. Loss = 5.86 cents. ^° 12. Page 189 ("^ (7). 13. Page 156 (33) 233 U 6% cost = $200. . •. cost = $3333 J. I E I H l» ■ 'ii Is.-, ,i SOLUTIONS OP PROBLEMS 15. Page 217 (19). 16. Page 215 (26). 17. B's contribution = $30 more than A's. C's. contribution = $70 more than A's. .-. A's, B's and C's contribution = $100 more than 3 times A's. $100 more than 3 times A's = $3100. .-. A's contribution = $70, B's = $100, C's = $140. 18. Page 217 (19). 19 Gain is $6 on $75 = 8%. Area = (^ x \'-) sq. 20. Length of side = -\^ yds. = ^ yds yds. = 240|^ sq. yds. 21. Page 221 (19). 22. Page 215 (20). 23. Page 204 (24). 24. Oiiginal cost of mater!al = 2 times cost of labor. Total first estimate = 3 times cost of labor. Actual cost of 1st half of material = cost of labor. Actual cost of 1st third of labor = J cost of labor. Actual cost of 2nd half of material = f J cost of labor. Actual cost of 2 thirds of labor = || cost of labor. Total actual cost = 1"^ cost of labor. .-. amt. saved = tj^^ cost of labor == $10. . •. cost of labor = 83000. Total first estimate = $9000. 234 (1) $4850 St. at 87^ yields -J of $4850 cash = $4243| Amount of money invested in 2nd stock = $(4243f - 46) =; $4197f . Amount of 2nd stock = $41 97f x 1^^ = $4350. Total 96| amount of st. handled by broker = $(4850 + 4350) = $9200 or 92 shares. For handling 92 shares the broker receives ,*. for handling 1 share the broker receives $.^. (2) Take a risk of $400. Premium = Amount reinsured = | of $400 = $300 $300 = $9. DiflFerence in premium = $(16-9) = difference on a $400 risk. .-. $27 is the difference on a of $400 risk = $1542.85^ (3) Page 189 (7). risk. jt-^ oi $400 -$16. Premium =- y^^ of $7 is the 87 (4) (5) (6) (7) 40 daj 153 (42). 175 (22). 217 (19). .u .. TT *^i. cv a t 40c. -• i.u, man received only $7.60. "00 -7.60) = $8.40. For .-. through every 00. jjut the work" being idle he loses day he loses (40 +16) $16. of SJ ^a IN THE HIGH SCHOOL ARITHMETia cents. = 56 cents. . •. no. of days idle = 4*^ = 1 5 days he worked = 25. ^ff" **'• (8) Page 178 (22). (9) f cost of 1st hor8e-$200. .-. cost = $160. na of .'. gainaa $240 On u 1 . * .T^ ^^ second horse = $( 200 + 40) . $240. he lost $40. ... on $100 he lost $16*. (10) Page 161 (8). * 235 11. Page 229 (21). H- ^^' TT]%«JPP? = $2646 = amount spent in paying liabili- r$264'6jWoo1*'^'^'^^^^^^^ = ^^^^^ •••^^^^^^^^^^^^ ^'^sfcents!' ''' '''' '' ^'' ••• ^^ ' ^^- *'eXy\1|o-efte ^14. Price of st. = $(72 x 46f ) = $3357. Cash val. of draft = — tof $2500 -$2478. 12 J. Cash balance = $(3357 - 2478.12J) = $878.87^. 15. Page 176 (27). 16. T^^ of $4200 = $294. 17. Page 157(13). 18. Tli3 int. on the sum for the eiven timft nf /fil r\o/ .»(618..35 - 558.60) = $57.75. The SI at^% t&.^^4 the int. at 6% =^| o( $57.75 . I|»38.60. .-. 8ttm-$(658.60 ttoe!i?l=f*X_5^f,°" «*20 for 1 yr. at 6% = $25.20. .-. 19 The 2 extra pay $(4 x .75) = $3.00. .-. one pays $1.50 .-. total rent-$l.50x 6 = $9.00. ^ y»<^^.oyj. 20. Let $100 = cost. ... $70 = new cost. Let r-rate of selling price = 100 ( 1 + r) = 70 ( 1 + 2|r). .-. 100 + gain 100r-70+175r. .-. 75r = 30. r = — 2 ■ 40%. 21 3 lengths require 2 cuts and 4 llngth8°3 cuts. For 2 cuts the cost per cord is $2. . •. for 3 cuts the cost per cord is 22. Pvate of man + rate of stream is 3 miles in 30 min. or 6 miles per hr. Rate of man - rate of stream is 3 miles in 45 mm. or 4 miles per hr. Twice rate of man is (6 + 4) miles per , 11- ' Jl- SOLUTIONS OP PROBLEMS ,«.''! 51 ;. hr. Rate of man is 5 miles per hr. .•. rate of stream is (6 - 5) miles = 1 mile per hr, 23. 4 times the number = 36. . •. the number = 9. 236 1. Page 217 (19). 2. Page 178 (22). 3. Page 164 (47). 4. Page 221 (19). 5. Page 229 (21). 6. A gains 2 yds on B in 1 sec. .-. A gains 10 yds. on B m 5 sec. B gains 3 yds. on C in 1 sec. .-. B gains 15 yds. on C in 5 sec. .-. all will be together in 6 sec. 7. Page 234 (7). 8. Tlie protit on 83200 for 6 mos. = the profit on $1 9200 for 1 mo. The profit on $4000 for 5 mos. = the profit on $20000 for 1 mo. The pi-ofit on $2500 for 7 mos. = the profit on $17500 for 1 luo. .-. the total pro its = the profit on $56700->for 1 mo. .-. Cs share of profits on account of the money invested = i-ll of net total profits = 1^5 ^f _9y total profits = y\'^ of total profits. .-. Cs whole profit = (^^ + ^\y^) total profits = $428.40. .-. total profits = $1134. A receives 4^ of A of $1134 = $345.60. ^^ ^^ 9. Let the amount of st. in each be $(96 x 101). Income from 4% St. =T*^ of $(96 x 101 ) = $387.84. Income from 37 st-=T§(7 of ^(5>6 X 101) = $290.88. .-. total income = $(387.84 + 290.88) = $678.72. $(96x101) st. transferred from 37's at 96 to 4%'sat 101 =$(96 x lOl) x T-Vr = $9216 st., and°a similar transfer from the 4% 's to the 3%'s = $10201 st In- come from 4% st.=-^A^ of $9216 -$368 64. Income from 3% St. = j^ of $1 0201 = $306.03. Total income = $(368.64 + 306.03)--: $674.67. Diff. in incomes = $(678.72 -674.67) = $4.05. .-. $4.05 is the diff. in incomes when $(96 x 101) st is held in each. .-. $12.15 is the diff. in incomes when $(3 x 96 x 101) St. is held in each. $100 st. costs $96. .-. $(3x96x 101) St. costs $(,»'^ X 3 X 96 X 101) = $27924.48. 10. ($600 -$500) is the int. for 3 yrs. on $500. .-.$600 18 the int. for 3 yrs. on $3000. Again, the int. on $500 for 3 yrs. is $100. .-. the int. on $100 for 1 yr. is $6|. 11. Page 217 (19). 12. Page 216 (11). 237 _ 175 total of IN THE HIGH SCHOOL ARITHMETIC. 13. 2000 lbs. at 6 cent,q a Ifi — il6i on i. /-w n of 2000 lbs. = 1 920 it. 15 ~ot self Zc ^"T''^ '"^^ = lbs. sell for « xir,vlQ9ft I ^"^ 6 cuts. .-. 1920 $(122.88-120/112 88 ^^^^ '''''' «^ ^122.88. .-.gain = 14. Page 201 (86). gion = -2 of ^1 000 ion^ A ^ XT ^^^^^- ^®^^"'g comniis- ";"". 1(717 or j!MUJO = !^20.40. Net proceeds = -"« of «s in on Buying commission - -8 of 08 „f .ftinon «>in^^-? ot .>1020. commissions =.(.$20. 40 -ll 9 eW 1 80 2 Qn^ ^u^"..^^^- ^" S1020 goods are consigned $12 i. H,«!i ff '' ^^^ f ^^ ^^^" 1020) loods are consJlned^eissOO "'''" ^^""^^"^ " .-. ill eLn?3 T ^'h -^^ = f 50- 1 -Oman earns = a.75. au earn !jpd. J5. ... the no. of men = Ap-Ao x 2 - 28 21. I^comefrom$4470st.in4|%'s=-I of$4470 = $201 15 .•.Income from 3%'s = $(201.15 - 16.87.J)= $184,271 The money received from sale of 4J%st. =$(4470 x ^^^)^$iU70 X 1^^). $184.27i is the income from 1(4470 x » ' «^ in^^of ^ .-. $3 is the income from $74A invested ^'?. $(74|-|) = $74|. ^^^^ested. .. market price = 238 2. 50 acres would occupy 1 man fnv 11/^ 1 , , 168 days. .• 7] acres wo.fu « , ^ ''^^^ ^"^ ^ W for 1 boy for 24.36 days But nT"^^ ^ "^T. ^^' ^^"^^ ^^^^^^^^ 94 da-o ^p^ I W^/:.. 1?"? ^i ^^'«« -ould occuDv 1 min fn. V9.36'd;';' wti for a boT'- .'lo-l' '/f ^^^ ^^ ^ -- boys, &c, ^' • • """"^^ °^ 1 ^^aa - work of 3 m i\' ■V i, I «»^f f'P' P ! ft SOLUTIONS OP PROBLEMS 3. He sells 388^ yds. for the cost of 77 7 V 38 he must sell the rem. for the cost of { 777 xY- Si J f « ' « ;/ yds. But he sells th« r.m„,-„^.. L ttt^ .1 ^, ^^stXt^I .'. each he must yds. But he sells the remainder as -^ix «« yds U#4^ yds. 'fiUT yard of the rem. is sod for the cost of laark it at \l\ of 100% above cost. s4 50^' ^'/r w*^' ^? ^^'- ^^ *^^ ^^^«^i«r kind would be h ooo/- • *^«^i„"/enor is worth 35 cts. per lb. 7. 32% on 80% of the goods= 17.6% on all • 17 67 nf the invoice = $633.60. /o "" »"• ..i/.t);^of 8. Page 217 (19). 9. " 150(16). 10. time. 11. 12. 13. 14. 239 The interest on $120 is $5. and .-. $10 for twice the '' 215(26). •* 146(14). 14 lbs. of each cost $1015 • 14. iKo ^^ $4.90, &c. **- «>iv.io. . . 14 lbs. of green cost 15. Page 204 (124); 196 a 2). oranges^''''* *^' '°'* ^^^ '""^^^ P"^" ^* ^^ «q"*I number of 17. He sells the barrel as ^X+ a ^ or 2S6 „£ „ r „ , , .-. gains ^^ or |o%. Vtt + tt; or -^is of a barrel, and 18. The premium is $33.75 on $2700 19. Page 189(9). 20. " 219(21). 240 1. Page 163(40); 157(5). botle'^rbroke. '^ "" ' ^*^- P^'' ^°*««' *^- -re 1.464 Jt;elT«-^3^crf7'''^fnnm°^^ «^«- ^''^y- ^ <'«• ™™ 7 'ttuit) cu. ft. .•. 1000 kiloffranimfi = T noo - * s 7 1.- .^ (fff§)' ozs. troy. " "^^ ^ -*«5~ ^ 4. Page 217 (9). ¥,. IN THE HIGH SCHOOL ARITHMETIC. 5. Page 213 (29). 6. " 147 (38). mif^li^in^jT'' ^ ^Tr ^''' *^^ ^^'•"^^g^ ""om be f-^^ 10. {pj.75 IS the wage of 2 women and 1 man • ^917^ la the wage of 18 women and 9 men ' " '* 8. Page 148 (42). ?i).%T?^it) n'st or "■' -"'' "' '* """ ^" """'• 241 11. The value of ^ of |* of the patent is $756, &c. capital ^^'"'''"^' ^'^ !i^290054.57, which is 4.^% of the «n^-SRfi5?'°'*£V^'''^'5^*'*'- •••total cost = $77^. Total s.p^=$85|. .•. $61««a.p. of 200 lbs. ^ gains ^f oflst "''' ^'' ""'' "' «^^'^ '^^ ^^^'•g- ^64. .•. 15. Form^r rate is 2|%. ... he must pay 3% on $2500. .u d.^3nn?^^''^*'^P'*^^^ ^""^ 1 month is $78000. • A's must be $58000, .-. A's withdrawal for the 7 mos. must be equivalent to $14000 for 1 month. of wate^.*'*" ^^^' '''''*^'''' ^ ^"^ ^ ^^^' °^ '^^''^' ^'^^ •■• >^ « g«l- }?■ Jit?f *^^ "'l^'?^T = ^^^^- • •• ^"*y i« 24% of this. 19. If both rates had been the same as first the interest would have been $8.24 less. • mterp<;<- on ^Txt ol ViT f of the first = $39.35. ' ' ^ ^^^^ ** *^® ''^^^ 20. Page 234 (7). , ^ 242 1. Page 182 (27). 2. His assets are 65% of $3000 + .1^50 3. Page 185 (3). 4. « 204 (122). 5. The net taxes on $1000 are $18.62. 6. Page 191 (2). 7. " 178(18). 8. If they can pay 12% on 10% of their capital thev can pay on^y |o of 12% on 14% of their capital. ^ ' ^ 10. If he buys $65 worth of goods he pays only $64. He & i' W 111 If H.fi. r SOLUTIONS OF PK0BLEM8 TooL for ^cT^ ""^ ^T^' ^°' !^*- ^^ ^^^"« ^65 worth of goods for $66/^ ., he gains $2^^ when he pays $64. He gains $15 when he pays $372.50. ium^on%40o! *^^ P'^'"^""' °" ^^^OO. . •. |25.50 is the prem- 12. Page 236 (8). ^^^ 13. " 192 (13). 14. 68 days' wages = sum and int. for 4 days. 72 davs' wa^es = sum and int. for 6 days. .-. int. for 2 da^s = 4 0^1' wagfs days -^^ ^ ^^'' '^^^^'' '''' *^^ ""^S^^ "^ 2 men for 4 15. Suppose 12 lbs. cost $1.00. .-. s.p. = $l 12 At the '76.T4et75'(2of ^' ^^" ^'' ^^'^^^' '' ^ ^^" ^"^ ^l'^^. 18. Page 158 (19). 19. The 6 boys earn $ 63 x 6, or $3.78 below the average . . the men must earn $3.78 above the average. Buteach man earns 12 cts. above. .-. there are 31^ men, or 31 meuXu time, and 1 man half time. 20. Page 232(1). !^- ?o^oH'ltPj'^^^^^^-^S«*«^'^^t)eforethe dividend. .-. cost IS j^^ of $1317.58 X [« 2 „ $.1267.86. 244 1. Page 185 (20). 2. On $100 worth of wheat sold he got $5^ com. .• the are ^ZTt ""'" ^^ '^^ 'l^) °' ^^^^- ^^'^^ *^« ""'^ Proceeds ^70?!!' *'^^ '^"\J.',.^^i- ••• ^1^^^ *be net proceeds a.. $<790 ihe com. is $210. 40 ft. frontage is b^t. for $210. .-. 1 ft. frontage is bot. for $5.25. 3 Selling price with a just measure =« of $1500 = $1875 Real sel]mgprice = $(1500 + 750)or$2250. By dishonesty he disposes of $1 875 worth of goods for $2250. . •. by dishonesty he disposes of 1875 gals, (true) as 2250 gals, (false). .-. by dis- honesty he disposes of ^f|« gals, (true) as 1 gal. (false); or 4 gal. as 1 gal. o \ /» ^ 4. Suppose $400 is the value of the house. 1st r.rom;„yv, ^ TsU of f of $400 = $4.50. 2nd premium = | of i^of $400 - IN THE HIGH SCHOOL ARITHMETIC. 6. Page215"(2?/ *^' ^''"^'""^ ^^ ^^t- " 193(21). " 145(35). " 243(14). 6. 7. 8. !L-^7^\;^:4^^'-^--^"^^i^^ Page 182 (27). 245 11. alloy required * '« ^f 1U9 i?n + ^^ °"- ^^ = ^^G oz 15. « 195 (8). dtol^A ^''®^®''^''''® ^^""^ receives (SV-Giy) „f *i^n/>^n $2850 more thau the averH

rl| 80LT7TIOI7S OF PROBLEMS and the diflference in income $16.40J. sain The income from invested in 34% st. at 77 - S-^^-' ^ 3^ - ^^\Vt- For every share of 4% St. held .%Y/i is added to the 3|% st. income. .-. if the 4% St. yielded only $(4-^VA)» or ^Sjl^f, the $^\Vt would not be required to increase the 3^% income, and the diffcience The income from would still be $16.40^ or $V/- vested in the 3^%'8 at 77 = income from the stock paying P-Wi~- ••• no. of shares = $lAyV-{.l -r ^.^ttfft per share = $15510 ^ 165 = $94.' "(Another solution.) •12 1 3 00/ IS 43,2 '"» $(Hv 165. $15510 in- .•. the -V--4x $16.40« 165. price I "3 .•. price , , Letx«- no. sh. 4% St. .-. inc. = 4x. Stock sold for $(15510 + 3^x). be- cause each share brings $3| more than was paid for it. No. of 8h.of3i%st. = li^l^. Inc. = l^£l^-x^. Diff ^ 15510 + 3|x 77 $15510 + 165= , 5. Net sales, when no guarantee = ^«„% of 83400 = $3298. .-. net sale with guarantee = $(3298 - 96 + 19.50) = .S3221.50. .•. com. with guarantee = $(3400— 3 J21.50) = $178.50 .•. rate =Hiifo^-x 100=54. 6. Page 191 (5). 7. A's gain = $(4600 - 1000) -$3600. $36f0xS the gain in 12 mos. on $12000 capital. .-. $1000 is the pain in 8 mos. on $5000 capital. . •. value per acre = $5000 + 37| = $1 33 J. 8. Whole gain % = ( 1 2 J + 7 1 = 1 9^. | » « of A'W of sales = Vinr o^ cost. 29.66%. sales = 1.2956 of the cost. 253 advance = 9. Page 246 (8). 10. " 216 (1). 11. Dist. by coach -s^ of /g. dist. by sea=^^^ dist. by sea. Dist. by rail =/^ dist. by sea. .-. dist. by coach, rail aud sea are as 1, 3, 20. . •. dist. by coach = ^^\ of 480 miles = 20 miles. Dist. by rail = 60 miles, and dist. by sea = 400 miles. Fare by rail = f of 8c. = 1 2c., and fare by sea = J^" of 1 2c. - 40c. Total cost = (20 X 8 + 60 X 12 + 400 X 40) cents = $168.80. 12. 5 hrs. .2 min. diflf. in the times. For 1 hr. diff. the nlacRa aro distant, ^A° fmrn *>npli nthp** • •f'-"» R It*" ) distant Pl 10 2 HI. [aces other, and as the time at Callao is be! id Greenwich, Callao is 78° west. DiflF IN THE HIGH SCHOOL ARITHMETIC. 13. Suppose the apples are a pennv each wo .r^f o^n i for^a sov and at tkZ,uc.a p.L7.fTl0; Zl fo^/pphft areTd ekdf ^ ''''f:;i^'''^ ^'^ obtained for a sov. wh^fipp apples a" ]d. each '""'" ^°" ''' ^^^'^^^^^ ^^ '^-V' -^ea 14. Page 192(15). 16. " 179(38). 16. " 188(2). 17. Crew pumps out 6 x 24 tons in 1 hr i«i * from s,,or: at .6, L'uroriroivi'" '" "^^ ••'«"'-- 18. Page 242 (10). 19. " 159(5). • (90 ™f.nnr *^r ^''^, /^ °^ ^^ ^PP^e« «old, or 44 apples • • S ^ V Tsr) '^PP^es when sold bring equal prices A \^tt? i : price 5. apples sell tV.r in a<. *u o,^'i"'^',P"ces. At the 1st sold = 42 appl, s. ^ Ts-F <^t (90 - « |) apples 2. Page 199 (61). m 120 must be , guard Th^ P''*"' ^-are, i.«., evory prime vois.- ^^ (^) +62 {..Icuin. .•.V(rad).-V = 2j3 /o\2 + 6^ 15V2 in. 6. L.C.M==JJj.V-lor858. 4V&3W!^-Itl'i'^ = ^f '^•'^- «*°«^ purchased = ■'^3- % .50= $135, Rate = 2i or 6°/. «97nrf ^ .ifoTi^n'''^,.*^® ^^"^^ offer ''he should « n 34°/ $2700 or $94.50. But he loses ^21. .^50 . .ull l7u,.^*/^ of are $(94.50 + 21.50) or $116 loses $21.50. ... the wSr^thlesa sdes SOLUTIONS OF PROBLEMS ill l4iiki 9. Page 251 (13). 10. Page 215 (17). 255 11. Page 244 (3). 12. Expenses = $(11 00 + 400 - 1187.50) or $312.50. 2°/ of $1000 = $20, and ^Urr of $1100 = $16.50. .-. ^^ of | of val. farm + j^oi^ of val. of farm = $(312.50—36.50). .-. tUj. of val. of farm = $276. . •. val. of farm = $6000. 13. Page 208 (170). 14. The true discount is the interest on the present v^orth, and bank discount is calculated similar to the interest on a sum of money equal to the amount. 15. Page 253 (12). 16. 2ndnumber = il4pJ = 132. 17. Page 191 (5). 18. « 213(29). 19. Cash cost-.^_ of 20 of 25 of $4.37^ = $2093.30. Cash sale = |oo of 20 of 25 of $4.62i = $2267.15. .-. cash gain = $173.85. ^ 20. Int. on $100 is -^^% of | of $100 = $17. 10. .-. $38.90^ ik int. on ^^ of $100 = $227.50. 21. As assignee, A receives 3|% of $7290 or $255.15, and as creditor, 4 of $(7290 - 255.15) or $3126.60. 256 1. II lb. troy gold and ^ lb. troy alloy are worth 45 guineas, but T^ lb. alloy is worth ^V of ^ lb. gold. .-. (|| + ^VV of tV) lbs. troy gold are worth 45 guineas, or f f ^ lbs. troy gold are worth 45 guineas, or |f^ lbs. troy alloy are worth ^y^ of 45 guineas, or 1 lb. troy alloy is worth 2^ guineas. .-. 1 lb, avoir, alloy is worth |?§^ of 2^ guineas or 2^ guineas. 2. The sum of their rates is 39 miles in 4 hrs., or 9| miles per hour. The difF. of their rates is 5J miles in 7 hrs., or | miles per hour. Twice rate of faster man»= 10^ miles per hr., or rate of faster man = 5^ miles per hr. 3. Letx = perp. .-. hypo. = 50— x. .\{50—x)^<«x^ + 32^, ..; X = 14.76 in. ,-. .area = | x 14.76 x 32 sq. in. - 236.16 sq. in. 4. A the end of the 1st year he is worth ^ of original capi- tal - $1000. At the end of the 2nd year he is worth (|)» of ^ IN THE HIGH SCHOOL ARITHMETIC. ^e^ he U^::l-^,,i3Tr »100q. At the end of the 3rd ilOOO - llOoT At the STl.';?"^' -p. "* ^"^ - i of otsr^5rrhSS?S7ro.lfoo-t ■■Uj) - 5. ^ original capital = (^.zi of $1000. .-. original capital = 128938.24+. '^~ 5- Page 112 (130). 6. One cent is the unit, Int on . 3'> of 60. X 7200) 4. Bk. disc. = i^of bill True disc. = (c^-^m^)^'^-' lUO 101^ 5. No. barrels = Oc. bilL lOlf ■ ^ll^ZZOV-g, $36 + $108 = of bUl. 100 IO2J of $6150 + $6 = 1000. Total cost . $6150 + $250 = $6400. Selling price = ff of $6400 » $7360. Price per barrel -» $7.36. IN THE HIGH SCHOOL ARITHMETIC. 6- Amount received - j^^ of gl of $8000 = $7880. 29. 7. Whole capital = $13000. C gets i of $13000 = $4333i • •. hepayaA$8000-4333J) = $3666r ^^t^^^*. A t-n?.^^ ^'' «^T= ^^ ^'^' ^*' ^'^'== lU of B's steps. ... A ^vms the mce In running 11 J yards A gains A yd. .-. in running 100 yards A gains 1 yd. ^^ 9. Page 165 (7). 10. Cost of farm at end of the year=.|«« of $4000 = $4240. Cost of taxes at the end of year - ^( ^-^ vie ^ 3 ^ V 100 TITCTT) '^ r ^ ^^^r^ol^f%l^''^P^:L''^ '''^^''' ^' *^'« ^"d of the year rJli°^ $500 = $5J5.00. Total cost- $4800. 14#. Gain- $(5.';00-4800.14|)or$699.85f ^ TO of 11. Premium received = ^^^ of risk. Premium paid ^. risSjnIo. "'^" ■'■ ^°^P''^"^i"^ = TO of risk = |r30: 265 1 2. Duty == 2|c. X 3« 8 X 3825 = $84. 1 5. Total cost = 4«. v 3825 + $36.25 + $84.1 5 J$273.40. co8t = 4a x 13. Page 260 (9). 14. '« 260 (10). 15. No. who receive honors = 8i% of 120 = 10. No who ^.-'W<' = »»- ••■No.wSi°fail = 22. .[parcentag: -T2"o- or 18i/^. 16. Pa-r^ 253 (12). ^^ "^* ToOO^ ^^^ tender = $11000 +1.02". P. W. of 2nd tender = I -V -1. Diff.^ $340 nearly. 1.02'^ I .02 18. Pago 261 (14). . J?- ^?^°^°2^ ^^*% ^^ ^^^^ sales = $2736.25 + $179.10. .-. total sales = $3000. . •. no. barrels sold = 4000. 20. Page 224 (6). 266 1. A at the rate of 6 miles an hour goes 12 miles in 2 hrs , the distance B travelled before lie met A. . •. the time it took B to walk 12 miles -.-(12-41)hr.s. or 2|hrs. Awalks(5x 2^) miles before he meets 6=13^ miles. B takes ( 1 3i -^4) hrs to go this distance or 3J hrs. A's time = (2| + 2) hrs =4| SOLUTIONS OP PROBLEMS Distancea(12 + 13J) hrs. B's time = (2| + 3 J) hrs. = 6 hrs. miles =25 J miles. 2. Page 148 (47). 3. " 236(10). *• T^ of cost = $(210 - 199.60). .• 6. Page 253 (12). ' 6. The square of the greater number = 35643 x 3 greater number = 327. 7. 6 mos. credit price = | of cost. 12 mos. credit price - ^^ of -l of cost -11 nf /'nof no.,U — :„ 7 -i. «« » .'\ cost = $175. the t of cost - I « of cost. Cash price = 1 of | « of c jst - * » of nrkei - nf 4fi fi« ^'r^''-°^ goods =|^ of $6.66. 12 mo. credit price = 1^ of %^M. Gam =\l of $6.66 - $2.58 + 8. No. lb,. ™g«r = ;-|of ?i|o(»i|of «4000^6o. =64870 lbs., 2 oz. 9. There are (48 x 30) days' work = 1440 days' work 48 men in 3 days do 144 days' work. .-. at the end of 3 "days there are (1440 - 144) days' work to do= 1296. .-. Tt the end aM^^^^J ^hl\ ^'^ ^1296 - 138) days' work to do &c At the end of 39 days there are 36 days' work to do and 22 men to do It. .Mttakestheml^Vdays. .'. total time = 40^VdTy8 267 10. Page 216 (11). 11. Int. on $1 at 1% for 25 yrs. = for 25 yrs.=$l. ^ f2llir9fi.n ^ITnT^f'^^'P^^' Premium received = $13800 ^^^ ""^ "'^ " ^^^^■^^' ••• "«J^ = •• int. on $1 at 4% Premium on $6000 13. P.W.= -^(i__^_^^^$2482.16 + . the 100 92^ IN THH HIGH SCHOOL ARITHMETIC. eq. ia '°° '■'"'"'(■' + 8) -twice are*, .-.area = 384 ao.ou„^^j523ej, .-. time ot credU itmonuf " "' '^ 19 Timo of T»- A . */°' ^'^z ™"^8 on the dollar 20. Take 4 lbs. of better quality of tea Rf «T o ik n . of mixture -= $4 + 84 - $4 «n Y.ii • «•* ^1 a lb. Cost ..gam -^.70. ••.gampercent.-^ori4/^o7 1 r, 268 1. Quantity of coal oil = | of 210 pais - T571 „ i of remaining liquid- 157i Lu . S^^^T^^^ S^^^- .'. | 3. " 262 (7). Every eiOO^hat he handll at 4«/ 11°? "'"' ''".•" *% «»"• 66^0. Gain per cent. = -1-5290/ 64 ~ 57^• (16 + 20) hrs. or 36 hi ' ' * ^'^^' '*• ^« ^^^6-. 8. Page 143 (16). 9. " 267(19). SOLUTIONS OF PROBLEMS li " 10. C08t = or 269 Cash selling price = $4. Credit selling prico Gain = $]|. gain % = ^ior50%. 11. j^j^ of seMing price of house = — ^ of $6000. . •. selling price of house = $6168.36|4. 12. Page 148 (48). " 262 (3). " 267(11). 13. 14. 15. house 16. 4|% of policy = $87.50. .-. policy =$2000. .•. val. $2000 - $87.r>0 - $125 = $1787 50.' I o^ iVff '^f assets + yfy of assets = $1575. assets = 17 Rate of train for 8 sec. - rate of man for 8 sec. = 88 yds. .•. rate of tram in miles per hr. = i'6A 18. Page 267 (20). 19. Val. of tea = $' 1956-1 29|)« $1826 «. .-. buying com. ==^V of $1826« =$9m. .-. com. for selling butter = $(129i - 9Hf) - $37f . •. val. of butter sold = -Lg^ of $37| = $756. 270 rad. of circle = l//^ x 9 in. circ. of circle = V- Vsr x 9 in. = 10.63 in. 1. Area of square™ 9 sq. in - ^, 7V_, 2. A s rate = 5.23 x 2.4 -4- 3.7 miles per hr. == f-^y. x f f) miles per hr. Time A takes to go 10 niiles= 10 x /«<> ' 34 tj^.g_ Time B takes to go 10 miles = 10 x /JV hrs .-.' no. of sec. start = 10 X /Vt x A| x 3600 or 3752. 3. Page 236 (9)? 4. Cost = $1.15x 3000 = $3450. Sale = U of $1.04x3000 = $3432. .-.loss = $18. 5. A's gain at the end of the year = ^ of -£ of $4050 = $1443.75. lQQ-1-am't realized from 1 share. .-.$406 = am't realized from 50 shares. 7. The Bk. disc, off $100 for 63 days at 8% - $40*. .-. P \v . = «/ 1 00 - |0i) = $i|9 « c. _ ., ^3 1 o|6 has a face" Value of $719.92 has a face value of $730. $100. j I IN THE HIGH SCHOOL ARITHMETIC. 8. Page 179 (37). 9. 40lba. at37|c. alb.-$15. 64 lbs. at 45c. alb. =$28 80 -S505%Vlf ^^kn-*- \^'^^r^^^-S price =^ of $43.86 tor ^4J.<5, or 1 lb. for 53.4375c. 10 When the customer pays for K, oz. he is cheated i oz. ;:,ri'-n ^ ^''^,' ''"^ fV^ ,^^ ^« ^^^^^^^d ^h ••• wl^en he pays out $.jO ho 13 cheatefl 78Jc. *^ •'^ « V*. S't; -^ ^/ir "'^•' ^f'^2.75 = $893 75. Selling prices 271 }^' frfnaRnn'^''?'^^^^^/^ "^ ^^^^^ " ^^OO of 25c. =$9107. - 4 375 + :n '''''*' ^ '"'' "" '^^ "" ^^^^ "" ^^^^^ ^"^ '^ ^ '^^^ 14. Page 264 (11). 15. Income after transfer = $(^ of ^%\ of 4850) = $200. .-.income from 1st stock = $194. .• rateL '.«* =4°/ 16. Invoice^price = $100. .-. cash price i $90, and selling price»^^-jof ^of $100-$125|[. ... gain = $35t|, and 31526 gain per cent = -^^-I or 39^3%. 17. Page 188 (2). .^iL'^dTb^sttS1f''^^*°^ ^ 19. $93 invested at 6% yields $5.58 inc. Again $9 is the mcome from $100 invested. .-. $5.58 is the inc'ome from $62 on -d" ■ "■ ^^ ^ discount of ${ 1 00 - 62) = $38 20. Rate of man + rate of stream = twice the distance in I'hr Rate of man - rate of stream = f the distance in 1 hr. . •. rate = 2mfl™s" ''*^''''^''' ^ *'''•" ^'""''P®''^''' •••distance 21. Page 223 (18). 272 1. ^ val. house = ^ val. lot. .•. val I val. lot = $1 400. . •. val. lot = $400 2. j%\ of cost = $24. 60. .-. cost = $30. houses^ val. lot. vs or 20%. gain per cent. 'li s fi h ' Is- inIC' ' ■i^Hw' MUf \ Wlu ' i wL i SOLUTIONS OF PROBLEMS 3. 5 COWS cost $ir)0, and 1 horse costs $120. .-. 6 cows and 1 h<,rse cost $270. .-. no. of such groups -$10800 ^$270 = 40. .•. cows = 2ti0, horses = 40. 4. The extra man works 4 days, and does 8 days' work. The whole work t.ake3 320 days fur 1 man. .•. the fraction of the work performed in 4 days by extra man = ^, or, in 1 day -j-iSjy of the work. 5. Page 212(10). 6. The int. on $100 for 16 years at 5% = $80. The int. is $20 less when sum is $100. .•. the int. is $90 less when sum is $450. 7. Dividend = -| of $2304 or $36. Money received from Loss - 8(2304 - 22G5 - 36)or$3 , vol. cone . 1 A 22 %. -J-. sale = —J of $2304 or $2265. 8. Rad. of base .^ /^ o^ ^ ft- = ir ^ t- (|f;>cu. ft. = 8ii^cu. ft. 9. Cost = $10"0, gain = $20, selling price = $120. .•. gain % = T%'Vorl6§%. 10. No. gals, coal oil in 15 gals, mixture = | of 15 gals, or 12 gals. .'. he cheats the customer 3 gals., or 45c. 11. Page 225(18). 12. " 218(12). 13. Net assets = T»j^^ of $540 = $523.80. .*. no. centsonthe dollar=&-a3 80^17 40 ~sxr-- 273 14. Suppose each contributed 66§c,, thp amount would be $16. .-. the $8 must be raised by some of the people contribut- ing $1.20 each. When 1 person contributes $1.20 instead of 66§c., 53 Jc. of the $8 is raised. .•. no. of people required to contribute $1.20 each = ^or 15. 53^ 15. They approach each other at the rate of (4 + 2J) miles 39 per hr. or 6 J miles. .-. time = ^ hrs. or 6 hr«. The traveller 2 from Toronto will have gone (2^ x 6) miles = 15 miles. 16. Page 272 (7). *^- DiflF. = $46|(1.045) -(1.09) |= 92o. IN THE HIGH SCHOOL AKITHMBTIO. 18. 19. 20. 21. 22. the number a 110. *. on^lOOinvest- T of I of the number = 166. Page 223 (22). " 218(12). " 225(18). ^A u ^'^ i^j^O invested he ru*-.". $8 a year. edhegetseiOavfiar. uyy ' 23. Page 271 (20). ^°' 24. '« 231 (19). 1. Page 116 (171). ^^ ieSO T i' """L'y '" ^ ™°- ^" ^^a'-e would have been ITof $4o^oV& = '^'' ^ ^^^^ '''-"' '• ••• «'« --7 comefitso^ assessable income = $345.80. ... assessable in- T?a*ge ?*64 (48) ^'^^^"^^ = ^^^O + $350 = $750. stream = nt"?if'''r"^^^:!-^i).,"^^^« P«^ ^'' R»*« "P 8 ream in ^ 1;^^^^""''"' ^^^ ^- ., ^^ ^^'^ »<> 1 "^"e dowS scream in ^j- hr. He can go 1 mile up stream in A hr • he goes up and down 1 mile in (A + A) hrs - •« hT« • \^ 7 hrs. 24min. he can travel up aS do^^f^^it ' ' "" $1 50 ni"y^ T ^^°-' '^^ 17. Pag?217 (19). 18. The A train takes 8 hrs. and the B train 6 hrs. to go the whole distance .-. the rate of A train : rate of B train *^ ? • /:,. ^1^. ^^ ^'^- *^® ^ *^^^^ ^^3 Sone I of the distance. .-. f of the distance left after B train starts. B train covers J of this distance before they meet. .-. B train covers i of » of whole dist. or ^\ of the dist. B train goes A distance m 1 hr. 17| min. Time = 17| min past 12. 19. Liabilities = $1200, and assets = $400. But $100 of assets realize only $40. .-. on $1200 of a liability he pays only $340, on $1 of a liability he pays only 28ic 20. Page 158 (19). J' S • 2 1 . jV^ of greatest sum = j^^ of smallest sum. . •. greatest sum = | of smallest sum. .'. | of smallest sum = $47.50 smallest sum = $142.50. .'. greatest sum = $(142. 50 + 47 50) = *i^'or'^d.#o''* ''*^^'' Bum = T\nrof $190 or other 8um = i 22. Page 218 (19). 23. If both A and B worked as long as /'— +Z*\ VlO 12/ more of the work would have been done, i.e., if all worked Cs time (1+^8^ + ^3^) of the work would have been done or V(^ of the work. But A, Band C do (^V + ^ + ^v) of the work ml day -. A, B and C do -V/ of the work in 6i^ days. O worked 5^ days, or the work occupied 5^« days. 280 1. W of cost of 1st horse = T»^ of cost of 2nd horse. And fV of cost of 2nd horse - ^ cost of 1st horse = $4. . •. cost of 1st horse = 3. 4. OQA '■')' 273 (14) 260 (9). SOLUTIONS OF PROBLEMS '-1 !itl 5. Duty = £72. . •. total cost = £792. Net amount of salas -iV^of $4200or$3990. .-. gain = $3990 -£792 = $3990- $792x4.86f or $135.60. 6. Net income from 1 share = ^»^«^ of $4 or $3.92. .•. no. shares = .j«^f. Amount realized from sale = $^«§£ x 98 = $15000. Net income from new stock = T^»/,r of tIq of $15000 = $656.25. .'.difF. = $56.25. 7. i'age 261 (14). 8- -2^17 of /^«j. of sales =-$6.41^. .-. sales = $1350. Agent received j^ of $1350 + $6.41 J = $73.91^. 9. 1^ of 9 gal. 1 qt. 1 pt. is alcohol or 72 pts. 72 pints is 84% of 100 ^i 72 pts. = 85? pts. = 10 gal 2 qt wa er added = 1. gal. 1 qt. -f pt. 10. Selling price of oats e « of $1500 = $1800. .-. selling price of wheat and barley = $7596. He sf lis $100 worth of wheat for $94. He sells $300 worth of barley for $327. .-. he receives $421 for wheat and barley when wheat cost $100. He receives $7596 for wheat and barley when wheat cost -^'""'.27. If pts. 281 11. Page 155 (21). 12. «' 217 (4). 13. A and C do -^ of the work in 3 days. .•. there is f yet to do. A, B and C do ^ (^ + ^^ + J^) in 1 day. . •. A, B and do I in 4|'^ days. .-. whole time = 7|f days. 14. Page" 112 (130> 15. P. W. of $5 = ,;?^, of $5 = $4.83. ter way. 16. Page 212 (10). ^7- (w)* of the original value -$4197.6U. value = $5859.37^. 18. T-»^ of * of selling price per lb. -if|. of 5c. price per lb. - 8Jc. 19. Amount distributed = $74537.50 + $94567.50 -$107- 963.00 - $7397.00 = $53745.00. Rate = $53745 + $895750 or credit is the bet- original selling 20. Page 218 (9). 252 1. Income from latter stock = $(4 x ^^x 156)-i$480. IN THE HIGH SCHOOL ARITHMETIC. income from former stock = $(480 - 12) hU^uOr3%. See page 261 (14). 2. Cash value paid for goods = :r7^rr: of = $468. ratoi 125 -104 ^100 P^^^-^Too^^ioo^^-oIJ 3. Av. of first 101^ of $304.50 = 1390. $304.50. Selling two = 76|c. Av. of last two«=96Ac. By selling a bush, of 1st mixture for 80c., there is a gain of 3Ac., and a bush, of the 2nd mixture a loss of 16^e. .-. 16* bush, of Ist mixture will balance 3i bush, of 2.nd mixture, or 33 bush, to 7 bush., or 33 bush, of each of 1st two kinds, and V bush, of each of remaining kinds. 4: Cash price = j%% of |i of cost = i«^« of cost. j^§^ of cost, or 7|%. 5. Page 214 (5). 6. P.W. of $618 for 4 mos. at , Cash price = ^9^"^ of $618 or $593.28. 7. Page 116(171). 8. « 158 (19). 9. T-^, of $150- $6 = $49. 10. A receives tVtt o* the divided profits = $2100. .-.the divided profits = $6000. .-. total profits = $(6000 + 800) or $6800. Per cent. = $6800 -r $10000 or 68/. gam = 9% = i§^of$618or$600. 09 = T^of B's diff.=$6.72. money. .•. B's money 283 11. Let $100 = invoice price. Purchase val. (cash) = 100 Gain = $(100 - If «oo) = $|oo. Q^in % 104i of$100 = $-'.°oo». d^ooo . ©20000 .->^/ii ^2^77 - ^-^u»^ or 4^. 12. Page 275 (22). 13. Suppose he borrows $120. Int. for a year -$7.20. Gain on stock = $(7 + 5) or $12. .-. total gain = $,12- 7.20) = $4.80. .-. net gain = y«5««y of $4.80. .-. ^%% of $4.80 is the net gain from $120 borrowed *" — ' ' borrowed. 14. Page 282 (1). 15. 2 lbs. at 30c. and 1 lb. at 60c. make a mixture worth 4Uc, Tx: lbs. at 45c. aixu 1 lb. at GOc. also make a mixture worth 40c. .-. 2 lbs at 30c., 4 lbs. at 45c. and 2 lbs. at 60c., or any mixture in the proportion of 1, 2 and 1 make a mix- •. $51.74| is the net gain from $1320 .!>. I, ill' m I ' SOLU'ITONS OP PROBLEMS ture worth 40c. a lb. .-. for 144 lbs. of mixfcure ^ of 144 lbs. or 36 lbs at 30c. &c. (Different sets of answers may le ob- tainec! for this problem.) IC 98|% of the policy = the value of the house. | of ♦ihe policy = value of the house - $750. . • |^^ of f.lis pi: licy -- ^ of the p 'Hey + $750. .-. policy - $4000. .-. premium = $ >0 &c. 17. Paga 212 (IV 18. A goes (2 X 9) rajles ~ 18 miles. .•. B goes 18 miles in 1^ hrs. .-. rate- 12 mi'cs per hour. 19. Selling ptics .= ^•(GI7A x L37i) + $260.62J +$7ll.93f = $2130.37^. ,-. se-'lio,; prior per yd. =$3.45. 2S4 1. Page 116 (171). 2. yV?At ^^ taxable in(:om^ = $470.36. .'. taxable iniCCTOe = f}^^. .: total income = $87 7|^f^. ■^^ of investment = $877i''3 T^y the iu vestment =if^ of $877|^f = £j^-.- of 10 o£ 877i^ = £3005.4s.3.6 + d. 3. Page 168 (3). 4. 5. 6. 7. 8. 9. 10. (( 212 (10). 270 (7). 154 (15). 238 (3). 214 (5). 282 (3). 163 (36). 285 11. Page 155 (17). 12. " 212 (1). 13. « 155 (9). 14. For 2 lbs. coflfee he receives 48c. balanced by the higher price of tea. for tea, when he sells 1 lb. of each for tea when he sells 6 lbs. of each. 15. Page 216 (11). 16. « 116 (171). 17. " 282 (2). 18. « 282 (9). 19. ^ oi cost of one lot-^ of cost of the other — ^ the di£fer|ence in the cost of the two lots = $ 10. .'. this sum must be He takes in 8c. r"»ore '. he takes in 48c. Oa-; f'^ 6 lbs. tea. 8 lbs. cw>irati f of tb? « OM* IN THE HIGH SCHOOL ARITHMETIC. of the first lot + ^ of the coat of the second lot = $208, or | of the cost of the first lot + f of (the cost of the first lot — ( $208. .-. the cost of the first lot = $120. 20. Page 118 (196). 21. 150 (22). 1. Page 158 (16). 286 2. 3. 4. 6. 6. 7. 145 (10). 282 (3). 282 (2). 163 (36). 171 (19). Cash value - J-^ of $2100 + 15^ (t of $2100 = $4048.40. 8. A's wages for 14^ days + B's wages for 14 J days = A*s wages for 25 days. .•. A's wages for 10| days = B's wages for 14^ days. .•. A's wages for 25 days = B's wages for 34^^^ days. 9. Page 281 (18). 10. Av. cost per bush. = |- of |^f of 76 Jc. or 62|c. See also page 282 (3). 287 .0 8^ nf cost. gam I i_JLi_ 10000 of 11. Page 212 (10). 12. Selling price = i^f of cost = $22. . •. cost = $5000. 13. Page 146 (12). 14. « 216 (11). 15. 8 times A's capital : 6 times B's : 5 times C'sso$72 : $90 : $1 1 2.50. A's capital = | O's capital and B's capital = § C's capital. .•. whole capital = (f +| + 1) C's capital. .'• fy C's capital = $1550. C's capital = $750. 16. Y^^Vff ^^ taxable salary -- $491. . •. taxable salary = $500. .'. whole salary-: $900. 17. $3| is the income from $100 invested. .-. $83.12| is the income from $2557.69^ invested. The stock and the in- come tax do not afiect the solution. 18. Page 189 (9). 19. •* 151 (32). 9n « 216 (9), 288 1. f of I of cost + xVir °^ T ^^ ^^^ " rit ^^ ^^^ ^^^ zbv ^^ SOLUTIONS OF PROBLEMS 2. Page 112(130), 8 ^' f ^f ^pives j\^ of original flock + 20 sheen • +»,. AV of original flock - 20 sheep left B recdL •I'a*^;'^''! original flock - 20 sheep) + 70 shepn . ^t ' Tinr (/A of of o™i flock - 20 4^- roX' fe 'Xtc^,' W sheep) -70 sheen !► 79^Jk2 ^^?*;^?.^^ original flock -20 6. " 212(10). $1596^''*^VI!^k'^*>°pV «^ $20000000 + $3500 + $1760 - t snn ^^^^^•^'•« salaries - $15260 - $1150 -VlOOO l]il?o ^*^^«««"«°ted from salaries = X of Lf mno^ t^'^ r«nnTn;rpTl7Ta3n + 1150) = $88S8^ ^^^^^^^-^°^-32-3500-1760 9. Bot. 50 oz. for 49c. Sold 50 oz for so «p rn 49c, he gains ^oop lOn 99 ror ^^ of 60c. .-. on 4i^U " ^ ^^'' •■• ^^^'^ %=!!<'•- 49c. =. 1 woLtzi:i:s:if ofi bo%Tain7i2r ^Th^?- ^^ isr^;^hi^i$t^t-Je^~- ^^' ^tt^^'V^ '^^ st. = $4690 + $87|. ... selling price = $(^~- X 90; « $4824. Income from SIV st - ./4690 V ^'^ .P^-g^xyuy«jp4»a4. Income from 3^ /4690 „ A V 87^ " */ ~ "^^ ' ^•-^- •'• lacome from 3% st. = 8(174.20 IN THE HIGH SCHOOL ARITHMETIC. -1.40) = $172.80. .-. no. of shares of 3% 8t.» Si 72.80 -*.«3 \^IK •■• ?"'" P"" share = $4824-571 or $831 ' * 12. Page 154 (15). t * y ^«^t" ^ ^°,' o^^^Pf ^ ''''*"' ^^ ''^^'^«- ••• 8 ac. keep32oxen 10 Hk'd^ "'• ^''P«' ""''^ ^^ "««^«' ^' 320oxen are kept 1 Avk. by the grass on 8 ac. + 10 wks'. growth of grass and 464 "Ts "^^ '?S4 ' "'• '^ '^ ^T^ °" « -• + 16 XrgVowth of grass. .-. 144 oxen are kept 1 wk. by 6 wks'. growth on 8 ac .-. 24 oxen are kept 1 wk. by 1 wks'. growth on 8 ac. Tnd 24 oxen are kept a wks. by a wks'. growth on 8 ac , and 45 c^en are kept a wks. by a wks'. growth on 15 ac. .-. 32 - 24) or 8 oxen are kept by 8 ac. of grass for 10 wks. .-. (70 - 45) or 25 oxen are kept by 15 ac. of grass for 6 wks. ^ ^ B^^y ot 8um = ^170. .-. sum = $1600. thlL?Jl7 - 2 X 18 yrs = 39 yrs. .-. 39 yrs. = the sum of their ages 18 yrg. ago. .-. ages 18 years ago are 13 and 26 y^' ••• present ages are 31 and 44 yrs 5 1;,^ ""k" "^^ ^'^ ""^ "^f J?" 2^ ^^y^' °^ *h« ^hole work in 5 days. Now 4 men and 3 boys do ^ the work in 1 day and fC" ^^*^' ''"'•^/^ 1 ^^y- •• 12 nien anS 9 b^s 3o the work m 1 day, and 12 men do | the work in 1 day 9^boys do i the work in 1 day. .-. s'boys do /, the wo^rk in 18. 1 cwt. cost 19 X 21 shilling.^ or 399 shillintrg Sellin<112x4i shillings =l78| shillings. °.': ra^e of «"^"=-i99— «^20%. 19. Page 112 (131). 20. 5 cents buys 1 qt. of mixture, .-. 28 cents buys 5| qts of mixture. • to 4 qts. of milk are added If qts. water • water is to m{ik as 8 to 20, or as 2 to 5. ^ ' ' 21. Page -37 (15). 22. « 198(40). 290 ul'nf J^^"^ ^'J fortune =:^ of * of |«« of original fortune - m of ongi val fortune. . •. rate of gain « ^. Sr 16^%. f'i SOLUTIOKS OF PBOBLEMS Income '. altera- 2" n( 2. Income from | of $3750 st. at 5% = 014O.62i. from 3%8t. = VV of i^ of $3750x |f.^-«i •" 62A tion in income =-. |9. 3. Net earnings -^jy^o^ A of sale -Or of sale. $(450xl25)-$1462.50 ^ 4. 1 <)i sold for $28, 2 cows sold for $34, 6 sheep sold for .•I lot sold for $107. .-. 7 lots sold for $949. .-. the drove -- T oxen, 14 cows, 42 aheep. 5. Total assets = $(l|x 365) + $100 = $647, 50. Expenses -$fiJ + $25 + $33 + $llxl2 + $l7.50 + $2xl2 = $291.50 • net assc b8 = $356. 6. Wt.of water = (|o x 30 x 5 x {^ x 1000)oz. = 113636,Voz. 7. Page 289 (14). la / n *. 8. «• 145(10). 9. " 145 (35). 10. Cost of sugar - 6c. x 1 50 + 25c. -. $9. 25 ■ (iw X ISO) 01 6c. = $11.16. Rate of gain ■■ '20||%. Selling price ' 1.16-9.25) $9.25 291 the rep^'iining 1 1 must •ed the count would be ftS. 11. 9 marksmen score 12 counts, score 28 counts. If 11 ci itres are so 33. .'. there were 5 ters a-^d 6 cei 12. Page 214 (12). 13. 2 men and 5 boys do ^V^^^^r^ in 1 day. 2 men and 16 boya do ^^ work in 1 dn v. .-. 1 1 boys do yVo work in 1 day. .*. 1 boy does the whole work in 180 tuys. Again 5 boys do ^ work in 1 day. .-. 2 men can do y|jy work in 1 day. .-. 1 man can do the whole work in 90 days 14. Page 214 (12). 15. Clear gain = 82|% of T^j... :tl7<^'' = $27.50. 16. From March lO&h to Oct 25- ^229 days. no8t = ^ of |i2 + |of 40c)x229 = $b<.32f 17. Prime cost = $(25 x .75) - $18.75. Freight = $(25 x 1.75) = $43.75. Specific duty = $(i x 25 x .60)=.$3.75. Ad val. duty = ^ of $18.75 - $3.75. .-. total cost = $70. Selling price = $(25x 3.50) = $87.50. .-. gain = $17.50. 18. 14 bush, wheat at $1.50 = $21. 19 bush, barley at$.48 = $9.12. .'. 33 bush, mixture cost $30.12. and bv selling 33 bush, at 65^0 he gets $21.61^, .-. he loses $8.50J. " By selling / 2* m THE HIOH SCHOOL ARITHMETIC. no. of bush, oatsi 19. Page 2 18 (9). 292 takes $12 ess to buy 1 share <.f 4% st. thnu to buy I share of *i- St. . . It takes $1044 less to buy 87 sliares of 4°/ sf fhnn s": t^srfiT 1 '*^f ; /•• ^^' ""^ w «7l'i of 4^% St. or (87 + 12) shares of 4% st. with his money. Income £r ^^n'*To?^^^4) = ^396, and income from^A7 sr^T ^^o^ ^*^)-^39l.50, .•. diflference = e4.oO. ^^° 2. Page 281 (13). 3. " 145(35). 4. " 290 (6). 5. 3% of cost = $10, .•. cost = $333.33*. 6. Page 282 (9). ' vni/'^^K-^'"''''^''^"^ candidate had been promised as many votes his opponent, and had received 757 of the extra numbex. ae would have added to his majority 757 of 200 or 11 'to V '°^ ^' .f' ^''T P^°"^-^ *^« deft aJed cand" ^.aZ? ""o^nA •■• *^® ^""^^^ number promised the defeated candidates 6000 votes .-. total .nmbe'r of votes promised- nl^L oVvotfr^sfieru;""''^ ^^ voters= 11800 .. total 8. Page 212 (1). 9. It is assumed in the problem that the society is allowed to reduce the amount of its obligation by $11200. the price ^oooOO^^ '': '. ?r^"' ^""^ *^^^^ -" ^ collecteron 5 2' i^® *'*''* "^^ *^^® remannng 8 houses. Int. due at the PnH n? . y"*^=™ Of $18800 = $940; taxes due at the end of the y , tr = ^ »^ of $20000 - $300, . .. total outl. y in -Iseo!'" ' ^^^^ = ^(12 X 6 X 25) = $1800. .-. gaiS 10. Page 163 (42). 293 11 _a« «< voice price 8 A -« xl. '-'•j.oo'^'- i'iiO iuvoice priud = $111.80. tue in- pi •OLunoNs or problems 12. Duty-$(1452.38- 14.63 -1278)- «1B9.76. ^'''■^'orl2J%. rate of ^" $1278 13. Pago 289 (20). 1 4. 88% of proceeds « $550. .% the proceeds - 8625. 15. Page 217 (19). ^ 16. " 275 (20). 17. « 146 (13). *^^;.^®/^°^^^^^=*^^®°08<^» ••• 112%of theco8t-|Uof ^«A of 8700 or 8689.92. ' /<> " t** oi ^^^ 19. Page 177 (15). 20. No. of square yds. of broad cloth after sponginff = ( • » of 50 >J^ of 1^), and one yd. of lining after sponging is ( lI yds. - 2f in ) or 1*^ yds.'wide and contains {» x UX sq. yds 21. Page 143 (9). 22. « 287(15). 1. Page 156 (33). 2. •* 290(1). 3. " 217(13). 4. Suppose he invests 8(78 x 112|). of 3^% St. = 112A, and the number of .'. inc. from 3,^% st, 5%8tock = $(78x5) = 8390. real diff. in inc. = 87.50. xll2J)x 2 = 817550. 6. Page 214 (5). 8(112Jx3A) = .-. diff. The number of shares shares of 5% st. = 78. 8393.75, and inc. from in inc. =83.75, but the sum he had to invest the 6. f of the cost -83. 15. .-. the cost -82. 52. .-.the gain at the latter price -$.81. 7. Page 260 (9). 8. « 275(20). 9. « 146(12). 10. « 218 (12); 212 (13). the part gained - ^^^ = ts- 11. Page 143 (11). 12. « 177 (15V 295 IS. " 179 (38> .*. rate of 325. \noij%% ingis (1^ ^sq. yds. sq. yds. -~ of shares ;st.-78. inc. from , but the 3St" le gain at IN Tdl HIGH SCHOOL ABITHMETIO. + \o. t""- !"! n' ^^^^T^"'' ''^ ^^« «°d of the year.(lSl amount collected = «/fi i r. ^ i o72?v ^.5„^ ^ . r ^^^^\ •• ^^^^ amount collected assessable property 15 + 124.81) = |639.81 of $630.81 -$142180. '. the value of no. of lbs. = ^ by seUing 1 lb. of first mixture at 49^c. -14, or 7 lbs. of each. $862.50. ^ *' •' "® gains ^108.75 on stock which cost 17. Page 257 (12). + 2«S : U8*8 "?• M^ '1'';"« ''"*«■• he obtain, (1240 $(3720 + 396 80)-«4nfi «n n- V ' * *°**^ outlay = |4n6.80;7936 J642V/ •■' ««li^"«P"ceper bottle =1 of 19. Page 239 (14).^^ ' 1. Page 274 (3). ^ 2. " 178(22). ;!PJ8.J0. .-. sellinff Dric« nnr i,^ _ ca-ssn ®oo, *^ * y"' ■ »2 3.S0 _ = 83Ja $28.20. . •. selling price per yd. 4. Page 296 (3). "^ ^ ^ 5. P. W. annuity. =$_21__/ (1-005)30 _ j | 6Th«« ; V005)«o| ^05 1=$694.72, men cInYo i?in 3 dIV '"\t .""' *° '^ ^^ ^ ^^^^^ '' 7. Income from rent - S36o F '"''' ""^^ ''"^^^^^ ^ ^^y"' insurance $25. ta"e"o^ho?eV7XT^t^$13r5T ^''' come from house = $167.50 V 8? „r /„ ki ^^^ "^^■ $(1156.10 inn_i«'r Km . ' * TY^^ ?^ taxable salary = salary -.$iu007 '""'* '• '^^^^ ''*^»^y = «600 on total 8. Page 189 (8, 9). SOLUTIONS OP PROBLEMS 9. «* 283(13). iH ^\ 297 10 16 gal. at $1.80 a gal- $28.80. 46 gals, at $.90 a gal. =.$41 4a .-. 62 gals. co!t $70.20, and 62 gals, at $1.14La gaL ^ $70.73i. .' by selling these kinds he gams 53K ^^ foses L. by selling 1 gal. of $1.15 wine. .-. he loses 53^c. by selling J)8 gals of $1.15 wine. 12. A fb. If mSre consisting of equal parts i^/ortli 40c. The first mixture is worth (12 x 45)c. = $5.40, and 3 lbs. of lach ktdTmixed) are wortll $2.40. .-. 6 lbs. of one kind are worth $3.00. .'. 1 lb. is worth 50c. 13. Page 237 (19). ^ 14. " 178 (22). IQ. 88% ofthfirockery sells for 125% of the cost. .-. rate = 11 or 4°2tjV%- 17. Page 296 (5). 216(11). 212 (13). 212 (1). 298 1. Incomefrom 1st chance = $2000 + ^^ of $10000^^^^ Duty = AV of ,^8800 = $211 ^^les = ij^ of $88^^^^ tl2OO = $14760. Net receipts = -iV5*^* TVffO^'**^'"" ^ .. -«11769 78 . gain-«1769.78. .'. loss by not acceptmg fl"rftoffer = $(26b0-1769.78) = «830.22. I Kn4\u.t be^^^vid^i j^the .tio o. 4 ^ ^^^^ ^^ 18. 19. 20. « ^'rXTanny - ,U ot A "t army = 504 men. .-. the a^y " sTn r"of the miKture is «ne and | water. In B i of 6. In A, -i ot me m ^^^ quantity irom r+Tori';;": »°y ?roVB.-the wine drawn ott-1 gaj. tii of the quanvit/from A+ I of the luanUty from B = the wateT drawn off = 1 gal. .'. the quantity from A - 1^ gals. lO a gal. Ic. He 53^c. by rth 40c. 1 lbs. of kind are •. rate = $2600. )-$2112 accepting 9 sells 80 St. at 120 + 40} = will give the army InB, fof ttity from ff=l gal., )m B = the If gals. IN THE HIGH SCHOOL ARITHMETIC. 8' P.W. of amiuity = i ._8000 . 1 ^i . 12x.05 (^"i;05i the annuity is better for the purchaser 9. Page 178 (22). 0= $5908.84 299 25 35-11|ir9^^n";i-^V.^^*2 francs - £3.39 x 98424- Qfl.o T ^^ 2s 11.7d. Afterwards 9842 roub]es-3?7v 9842 francs - £3.37 x 9842 - 25.625 =£1294 6s 10 3d • fZeX ''^'''' '' n.7d-^1294 6s lo'sdi'i'l 16s j^^ ft ^v^iTt^rr?if t i.r ^^ 12. Page 281 (19). 13. « 223 (22). yields ^6^0 ''^. S?^ ^'J^-i?* ^i*"* ^i'^' ^^5 invested 15 Com -Vil n? '^-^"^^J^^" yi«ld J^ of 06.30 or $7^. .10/ ^°™' ■ ^i/ of sales. Net com. = (5A - U)°/ of sales^ oMs^L-m ^TJo* --itted = 94i%^ol saleT Buf 11% 1 c^r *?^» • • 94|% of sales = $1 134. *^° 16. Fixed rent of 450 ac. at $1.50 an ac =S675 00 lOn alth'-SmJ'V. tn- =^7^-^«- 40 b:;h.S?'at 28'c' a t>ush.-Jfl9.20. 75 bash, oats at 35o. a bush -.tSfi m ■ «orS757.50 G^n = tof *'''\"L*'''of«8^To'or*ir7'S CostperTliA^ r625'''of i8^33l.*'-r'^i^ ^r he^=«(ll.?6|-8.33i) or **3.l2% of the total oost = t600. .. the cost = «5000. 1257 of the invoice price -«5000. .-. the mvoice pnce-^^ of «BOO0, or *4000. 4. Page 164 (49) I C^ltof'th^eLdattheendof the/-=«(iV/>f36''8 ];p?gt^^ltpn. = $1.0alK^.^^^^^^ $10.00. Selling price = \h% of npiu.uu, or ^i i.uv. W THE HIGH SCHOOL ARITHMBTIO. .'. 5 doz. )- 11.60) Assignee's Bates ndled, .'. b of water tnd 60 qts. a the first he second, each. y sq. ft. or Gain = ^ of jlling price lead = y\ of -8.33^) or lOOO. 125% ice = ^ of lb. is sold for 8 lbs are sold for $11.50. ... afterwards 1 $1.43|. .•.gam% = 43f. 9. Page 217 (25). * 303 tate was worth 6 x ■"» J HOTnlJi loV ? 1. °°^ ""' «'" T-TF^"' •«"»-, ••■ outof the estate worth l»12fi7l)IJ„l, 11. Total cost = $160x2 + «I3i.J333}. Let r^r.t^ ., total cost = fi^^ + |^^ = S333j. ,.,..„,2„^ .-. it goes 3i German Sr/s' mS" "^ ""'°' " ' ■"• 13. Page 275 (2li). •""ST- 1*. " 193 (2n. =30. ... $800^k?e fhe" p J4 L"|,*fefi^ »» «>920 'or 6 «l920gain in 6 mos. 1480 ° «1 920 „ n • '°°'- , ■^S""" «960. .-. $1 1920 + 9601 or kssnru" '?'" "' '^ mos. 17. Page 242 (5). 18. « 145 (10). 19. «« 246(9). 10 4 X 368 X 63 + ^x368 or 118.40. , of 10 lbs. « Afterwards n SOME PROPERTIES OP NUMBERS. 304 1 Anv no ending in is a multiple of 10 and is .'. divisi- ble by 2 and by 5 ; I. when any no. is divided by 2 or 5 the remlder is determined by the digit in the -nits place 1^ the units digit is 2, 4, 6 or 8 the no is divisible by 2, and if the units digit is 5 the no. "divisible by/; . , . ^q ^ LtibtSned in dividing 7 by 3 ; ^nd similarly for any other multiple of 10. Again, 100 - 3 = 33 + J, .;. ^.^O - 3 = 7 x 33 + ';.-. the same holds true for 100, and Bimilarly for 1000, and so on for any power of 10. .'. if any no. as 5724 is ex- pressed thus : 50^+ 700 + 20 + 4 it will readily be seen ha Whatever rem. may be left after ^i^ding 5724 by 3 wilUe Eult from dividing 5 and 7 and 2 and 4 by 3, but j+^ + ^ + ^ '"L^S^ and .-. the rem. is that obteined by dividing the sum of the digits 5, 7, 2, 4 by 3. 3. When the last two digits of a no. ^f <>« f « ^^l ," * multiple of 100 and is .-. divisible by 4 and by 25 ; .-. when Ty no. is divided by 4 or by 25 the rem, is determined by the dltTin the tens and units places .;. if these digx^« «<>^^ » no divisible by 4, the whole no. is divisible by 4, and if these digits are 25, 50 or 75 the whole no. is divisible by 25. 6'. An/no. is ^divisible by 8 which is divisible by both 2 •"fpoweTs mt when divided by 7 ^give remainders as fol- lows:- 10- gives U^f^^^^^ l^^"^ fpi^at^dt' the and we have 100000= 14 Jb& + T. ■;°^V^ j. .,. „ o ^:^pg 1-, that is. the remainder obtained by dm^^^^^^ 100000 by 7 is the rem. got oy ^^^^^-S " ^^^^ " -^^^ ^'^^ smdiarlv for any other raultipls of 100000. In the same way IN TE ilGH SCHOOL ARITHMETIC. + jO + b, the rems. will be -^il. 1><3 6x5 3x8 1x2 viT.). • the final rem. is 6. -. 435826 wheVdivSed bVr wHl givTr'e rn'' in the last ip\Zs ilthJl^V '?,<'^,»«™'"^'' by the digits 8 the whole n'o Tdiv] by 8 a'dl ty llfs^ ,°,°; "^ce lib is a multiole of 11. gives the same rem. as a-b»sum of didts £ th« LJ places - sum of those in the even places ^ ^ "^^ and'4.^8^i:i'r'^' '^ '' "*^^°^ ^« ^^^^^^^ ^ both 3 io»f o^X^r^:^ Si^; -^ 3^1^^.^^ rt ~»u)tS?^f^^^^ • f 4h iV ^ '. v^ -tuV ^'""^ ^ "^"^l^^- of 13 leaves no rem . . If 4h + 3t - u IS divisible bv 1 3 sr. alsn io *• k^ ^^ * "" ,' this is a test ot the divisibility of a no'^f'i diah^hrr tit'^A^'' not give the rem, when the no' is n::Zl^S!^Tli':LZ pr TnmTTgui' i I ?U! i \i t SOLUTIONS OF PROBLEMS ! liii 846 when so treated gives 12, but as this is really- 12 the rem. is 13 -12 = 1. 13. Let a, b, c, d, &c., represent the Ist, 2nd, 3rd, 4th, &c. periods (i.e., if the no. is 736428579361 then a = 361, b = 679, c-428, d = 736); then since 10», 10», 10", &c., divided by 13 each leaves 1 for rem.; and 10^ 10*, &c. each leaves 12 for rem. .-. rem. on dividing any no. by 13 is that obtained bv dividinga+12b + c+12d + e+12f + &c. =a + c + e + &c. + 12('b + d + f + &c.) = 13(b + d + f + &c.) + a + c + e + &c. -(b fd + f + h j. in + ^ u . 10 t + u is the no. formed by L last 2 ^^ ."^c *^"' ^"* 9 ^u M ! iTk 'I "^^o ^^"'^ ^y ^^^- 9 intS h + 1 + u which - y (u-h) + 10h + t-8u = amult.of 9 + 10 hit «T, •* 10 ; + 1 - 8 u = 0, there will be no rem. ''' ^ + * " » « ; • ".if o2. The rem. is obt. bvdiv 13 infa Q t^-lia «. . i .« , . ::-%ti°'*'f;a»t"3f^'^''^°«"^^^^^ ,3^11^.^^tn:;L\»nto1l\^^:^-l-^ Sfi' ^'ir"'^''- ^' ^^^^"^^« ^y 14 («ee 16), and • by 7 fhe>"diL:^^',%Te:/jfi':t:^t^^^^^^^^^ ly that of no 10 • bnf if f},l «!: * j-' ^ • ® ^*^® ^^ precise- ^ 37. That IS, the first n- 1 dibits follntrofi u„ u^^e .u. _. . , „ mgifc represent a number divisible by 1 1'-" we'lia^ "^ "-'"^'^ that the remaining half of the mi/djj dig" foCed V^h': SOLUTIONS OF PROBLEMS I: 'ai! \'4~ n - 1 dig. to the right represent a no. div. by 1 1 j but this second no. is simply the first with its digits in reverse order, and is .-. div. by 11. See 36. Thus suppose 3874783 to be the no., on dividing 3874 by 11 the rem. is 2, .-. 3872 is div. by 11 J /. (by 36) 2783 is div. by 11 ; but 3874783 - 3872000 + 2783; and since ea. of these is div. by 11, the whole no. must be. 38. If t be the tens, and u the units digit, the two nos. will be 10 t + u and 10 u + 1, and the dif. between the squares of these is 99 (t» - u') or 99 (u' - 1») and .-. divisible by 99. 39. Affixing two O's multiplies any no. by 100, .-. we hare theno. xJ-J^ = no.x25. 40. »no. xi^=no. xl25. 41. =no. x(100 + iof 100) = no.xl25. 42. The first partial product is 8 times the multiplicand, and 40 times this is 40 x 8 {^ 320) times the mult'd., and .•. the sum of these is 320 + 8 ( ■= 328) times the mult'd. 307 43. First multiply by 12, placing the prod. 7 places to the left (=120000000 times the mult'd.); next mult, this prod., omitting the O's, by 12, placing units under units, &c. ( = 144 times the mult'd.) ; lastly, mult, this second prod, by 12, plac- ing the result 3 places to the left ( = 1728000 times the mult'd.); and the sum of these will be the complete product. 44. If n is any whole no. then n + 1, n + 2 are the two nos. next greater, and it will be found that n (n + 1 ) (n + 2) + n + 1 = (n + 1 )'. Or, if n is the middle no. the other two are n - 1 andn + 1, and(n-l) n (n+l) + n = n^ 45. This will appear from a consideration of two such frac- tions, say, 4, |. Here the |, the smaller or proper fraction, is less than 1 by f, i.e., by a fraction whose num. is the dif. of the two terms, 5 and 7, and whose den. is the greater of these terms, whereas J, the greater or improper fraction, is greater than 1 by f , i.e., by a fraction whose num. is also the dif. of the terms, but whose den is the smaller of these terms ; .'. the sum of the fractions exceeds 1 + 1 by the excess of f over ^, i.e., by a fraction whose num. is twice the excess of the greater term over the less, and whose den. is the prod, of the terms. 46. If n and n + 1 are the two nos., the dif. of their stjrB. is 2n4-,l = n-{-(n+l)BBum of the nos. W THB HIGH SCHOOL ARITHMKnO. but this irse order, 733 to be 72 is div. = 3872000 nrhole no. 3 nos. will quares of -. we hare Itiplicand, 'd., and .'. d. bcesto the this prod., 2c. (»144 y 12, plac- lemult'd.); le two nos. + 2)+n + o are n - 1 such frac- p fraction, Ls the dif. greater of 'raction, is is also the lese terms ; excess of ^ cess of the rod. of the leir sqrH. is ^*1:,,^IJ°^."^"V in 6i«of the form 10 placed to the right, Or, m + 6, and its [m + l) with 25 tiJproT:tx5' 67'7o\o''';Ao' ^n"^".."" "' the four pa. the right ^ ^ "" ^ "^^^'^ ® ^ 7 with 25 placed to jy to 53. See Arith. p. 54 et seq. ' 65- TirV|-AWAV;&c. 57. Multiply both terms by 2997 : Ac 58. Muhiply both terms by 37683 ; Ac. 59. Let ^ be the fraction : then ^ Arith n RR\ f],« «« t '9 ' »■'>' «■"■ m ; and .-. divides 10- with i^m I'Zl " dayides the no. consisting of n - 1 9's without rT^^ a^d • In b^mg pnme and not a mult, of 9) divides the no! Zis^ k.f *i. , , suppose n to be a factor of r I's where r b P r 1 s ana o' q (n - 1 ) 1 's (where p, q, are anv whole nm > . k,,* p, q can be taken such that the dif.LtC/p r andT(n - 1 . SptU"3nX^:rshrtti;- 1\^^^^^ 308 periL^ thu^qq" q"? '^'^*' ^j '^'e' *^" ^^- ^'^ ^ ^'^a^ged in penods thus 99, 99, , and each period is divisible by 11 StheleTtofthf''^ ^i!^.*h- 11 -ill divide withouLeLii rem t-1 ^-f 9 "^5/?'* ; the 9 in the units place is .-. the rem. t.e., if 2 were added to the no, it would be a mult of 11 61. ^ _ L lO'^ 10°- 1 1 1 io^" +1 10^ ' 10- -n - lo^ ' io5i . j^rri. .-. the repetend consists of n O's foUowed by n 9's. 'tSf^V SOLUTIOy-^ OF PROBLEMS »• 3 ; ii H 'i ;'N' I • h 62. See Arith. p. 66. 999999 -4- 7 - U2867. f " UUvi = TTi TTT •'••' 1 1 1 1 1 1 is divisible by 7. 64. Multiplying 111111 by 4 introduces no factor common bo 7 .*. there must be six 4's. 65. Every six 4'8 will divide by 7 .•. the no. of 4'8 must be 6, 12, 18, &c. 66. We know that ^ gives a quotient of 16 digits before the rem. 1 occurs. .•. the least no. composed of 9's which will contain l7is 10"-1. 67. In ^V *h6 rem. 1 occurs after 28 digits in the quot. have been obtained ; te., lO'^-l ( = 28 9'8) is the least no. of 9'8 that will contain 29 : .•. since 9 contains no factor com. to 29, a no. consisting of 28 I's will be a mult, of 29. 68. S7 is a far v : of 111, ,-. of 999, .-. of 10^ - 1, .'. in re- occurs after 3 digits have been found in ducing ^ the iv the quot. 69. Since I<. - 71. See 6 and - 1, this is a particular case of 76. The successive rems. are 1, 10, 26, 1, 10, 26, er of 2 as . .ctor, and is . •. an even no. If 2 occt rs as a factor of a coiuplete power it must have been a factor of the root, since no new (simple) factor can be mtroduced by raising a no. to any power, .-.if the power lb even the root must be. - * » t^yrvr 83. Tl.jdif. must be a mult, of 9 (see 27), .-. the other digit must be 4, making the dif. 54 or 46. 93 and 39 willgivl one ; 94 and 49 the other. "*8ive 3X0 84 This may be inferred from 82. If 2 does not occur as a factor in tho root, It cannot be introduced by multiplying the root by itself any no. of times, .U. uiupiymg 85 The diagonal form results from the omission of the O's which properly belong to the right of these partial products, but are omitted because the value of the digits is indicated b^ of reTnite Tj!'' ''''''' P^"'"^^ ^^^^^ «^-« ^*^« P-^-' rr^fo!' K^^^i?''*^ P"^"^* i' *^^ '"^ °^ *^^ Pa^^ial products and ran ed ^^""^ '"^ whatever order these products are ar- 87. 46987 4967 187948000 42288300 2819220 828909 233384429 Remember that the 4 in the multiplier is 4000 ; the 9, 900 : &c 9qfiOO^^„^l^'^\ r^nnn^^frS 3 digits is 999 whose sq. h! 1 QQQ A' ^u''"" ^2?^^^^ (*^^ ^^^^* ««• containing 7 digits) by 1999. Or thus : The greatest no. containing 3 dig. is 10^ -1, whose sq. 10»-2.10«+1 is less than 10« by 2 10»-1 Ihe second part follows necessarilv from t^^ „bo"- ' 90. The total no. subtracted from the original no. is the sq of the no. represented by the digits thus far obtained in the IMAGE EVALUATION TEST TARGET (MT-3) A V «p /, ^> '/ •/. 1.0 1.1 150 tii Hi US lU |2^ 14,0 2.5 2.2 12.0 i !'■« 1-25 i 1.4 1.6 150mm /APPLIED J IIVMGE . Inc 1653 East Main street Rochester, NY 14609 USA r.^ Ptione: 716/482-0300 = Fax: 716/288-5989 ® 1993. Applied Image, Inc., All Rights Reserved jk '%\.. '^O q-" ^ ■,' f'f ■ i f it I'. ( 1' ht BOLcnoNs or problkms sq. root, (see Arith. p. 73.) But the subtrahend immediately producing a remainder = ab + b», where b h the no. represent- ed by the digit last obtained (thus far) in the root and a, those previously obtained. 91. Between the sq. of the part of the root already found and the no. whose sq. root is to be obtained. 92. Between the cube of the part of the root already found and the no. whose cube root is to l>e obtained. See Arith., p. 77. 93. Disregarding the dec. pt., the first 6 digits in the root are 331662, i.e. the part of the root already found is 33166200000 ( = a suppose) ; and if we denote the no. whose rt. is required (11 and 20 O's) by N, the next complete rem. is N-a»(=a 317756 and 10 O's), and the next trial divisor is 2 a ( = 66332400000). Now if x denote the rest of the root (« 47903), .-. N-(a4-x)», .% N-a» = 2 a x + x«, and we are required to show that the rest of the root (x) may be obtained by dividing 2 a x + x» by 2 a instead of continuing the ordinary process. The quot. so obtained is x + x- which gives the re- ^a x' maining part of the root x provided h- is a proper fraction. Now since x contains 5 digits and a, 11, .-. x' must always be less than a. •*. k" " ^^^^ ^^^^ i- ^'^ *^is division by 2 a the contracted method may of course be used and the whole operation, retaining remainders only, is as follows: — (The 30974 under the divisor is the quotient in reversed order. See Arith., p. 70.) 11 (3.3166247903 63 200 661 HOG 6626 43900 66326 414400 663322 1644400 663324 3177fi60 30974 5242«4 59940 243 46 nmediately , represent- root and a, eady found Bady found See Aritb., ;he root are 166200000 is required N-a«( = is 2 a ( « e root (« id we are •e obtained le ordinary ^es the re- r fraction. always be by 2 a the the whole wB :--(The rsed order. IN THE HIGH SCHOOL ARITHMETIC. 9 wfthol;tiv^in'gar':iv,i^^ r^r ^^ *^^ -- * -^ •nade up of two flctora one of wh; K *^' '"*'°" *^^<^ « » any twofatrsXe^rSr^^^^^^^^^ ^ o'^-o^ setof fIcS>rT;a'"be1:^^^^^^^^ ^-^-. -y other the factors already obt^bed h J J-^^'^^ "P «" combining broken up, nor can anv ^ZIk- VP"""'! ^*^*^" <'an»ot be prime factor. ^ combination of factors produce a 96. SeeA.-ith., p. 135, no. 379, 380. 97. See Anth. p. 67, 68. 98. See Arith. p. 69, 70. to the 3 in the units'^|j™e g?ye ] 9 '^°''* '* """ "''• «'d«d 4 X ?'„ Jtif r^r«tr'' ■ ' ''""™'*' * X 2 u^t, ; „d 1, 4 X 312 103. 5 + 4x6 + 3x6»+2x6«+6*-1865. 104 i8» = ^ 10 successively an^'i:.^'t1i:il,'^.J5=^'''/• 126 4« 1 + 1 + ill! ■! ni SOLUTIONS OF PROBLEMS as 10 will not divide into 9 the next dividend is 9x7 + 1 = 64, giving the first rem. 4 ; Ac. 109. The sum is 27376 in the scale of 9. 110. The dif. is 2767 in the scale of 9. HI. The prod, is 11578813 in the scale of 9. 112. 41625 254 230436 302364 113553 14643006 113. The partial prrxls. are 3421, 12540, 255200. 114. The first rem. is 288. 115. 110)1000002(3030 330 1000 330 116. 1 2 3 4 5 6 102 2 4 6 11 13 15 3 6 12 15 21 24 4 11 15 22 26 33 13 21 26 34 42 6 15 24 33 42 51 117. 8 = 2'- 1000; 10 = 2'+2^ =1010 ; &c. 118. This is a particular case of the general theorem estab- lished in the algebras that the sum of the digits of any whole no. (radix r) divided by r - 1 will have the same rem. as the whole no. divided by r - 1. Or, the reasoning in no. 2 may with the necessary changes be applied. The same will be true of 2 and 3 since they are factors of 6. 119. Place the Mb. wt. in tho scale pan with the sugar and the i-lb. and the 4-lb. wt. in the other scale pan. 121. Place with the quantity to be weighed the wts. 3', 3 , 3^, and in the other pan 1, 3, 3', 3*. 122. Since 6x6 ends with 6 and the successive powers of L-lJysr- 5»« A •«A^oooo»'ilTr involve the TTiultinlication of 6 by 6. .-. every power of a no. ending with 6 will end with b. 9x7+1= 0. eorem estab- )f any whole ime rem. as in no. 2 may factors of 6. be sugar and 16 wts. 3', 3*, ive powers of tiplication of 11 end with 6. nt THK BIOH SCHOOL ARITHMETIC. ^ 123. Let n and n + 2 be the two nos. ; their prod, n +2i) 18 less by I than n'+ 2n -f- 1, which is the sq. of n + 1 124. See Arith. p. 131, no. 323. 125. If-y'S^-I^-— _____ ^hich is less than VV since |/3 is greater than ij (since the sq. of 1 i - 21 only) 126. Tiie sq. of 12345 exceeds that of 12344 by*'2 x(12344i + 1 ; subtracting this leaves 152374336. " 127. If the even no, is in tho right hand the mult, gives even by even wh. is even, and odd by odd which is odd : and the sum of odd and even is odd. If the odd no. is in tho right hand the mult, gives even by odd which is even, and odd by even which is even, and the sum of even and even is evc'n. 128. The first of the two nos. must be of one of the forms 3m, 3m + 1, 3m + 2 ; and the second, of one of the forms 3n, u "*" ' j"^ ^' "°^ ^^^ ^^ *^® ^^^^ ^^^ ^® ^'^'^^^ with one of the second in 9 diflferent ways, viz.: (denoting the nos. re- spectively by a, b, c, X, y, z) ax, ay, az, bx, by, bz, ex, cy, cz. m!1 u o* ^^^^ ^'^' ^y* ^^ ^^> *^^ °"® ®^ *l'e nos. is divis- ible by 3 ; in 2 of the cases bz, cy, the sum, and in the re- mammg two, by, cz, the difference, is divisible by 3 129. See Arith. p. 131, no. 335. Let 2n-f- 1 represent any odd no.; its sq. is 4na -f- 4n -<- 1 ; the two nos. nearest the half of this sq. are 2a^ + 2n, 2n2 -f- 2n -}- 1 ; and thee are the two sides and the hyp. of a rt. angled tri. The ratio of the greater 9n 2 J. On of these sides to the less is "^"^ = n + n which by • • .u . , 2^+1 ""2n-hl' giving n the successive values 1, 2, 3, &c., produces the series H> 2f, &c. 130. This is equivalent to multiplying the no. by ^ (1 -j- 1 + 220'''' ' * "^''Tirs^iT^ I 220« 1 -,ri- "yii 131. This is equivalent to multiplying the no. by (1 +i + irv + T^) X TTnnTOTy = TnnMrsxfi but nrwir = Winmnr x + , — .^ -Ac.) .'. the ro» 1+T 1 0000' 'soooo\joo ^(1 Tiy ooo 10000'^ ,1 i^ ^ h r #1 KW '■■(■ TO BbLUnONB OF PR0BLKM8 •nit obtained by the method given exceeds the true result bj less than Yuhra ^' either. If the result obtained be diminish- ed by YvhiV oi itself a closer approximation will be obtained. 132. yV - 1 ^0 01 "= t^SJtf ^ TVVTT " ThUn ^ i _. i ■■ ~ 10000 (1 - m^oo ) very nearly ; hence the rule. 133. ^ = x^Vrr - «fcc. as above. 2N+a* 134. (1) ^ — r-r a is greater than a, if 2N-l-a* is gr. than ^ a + 2a' ° 8 N-|-2a>, «.e. if N is gr. than a*. (2) Let VN-a+x ; then o V . * xu 2N + a» .. a + X . ^ ..^ 2(a + x)» + a» 3VN IS greater than j^-j;^, if -^ is gr. than ^^^^^.^g^. Uifi gr. than ^*'^'^+^^,^' ±^, if (a+x)» + 2a»i«gr. than Sa* + 3a'x + £*, if 2a + x is greater than 0. 135. If we take the whole no. in the rt. as the first value of a we get -= — ^ ^'~v'* *'• the next approximation is 1 36. In the algebras it is shown that the sum of the natural nos. from ltonis|(n + l); and that the sum of their cubes i9^(n+l)«j .-.Ac result bj diminiBh- obtained. 1 TTmnr gr. than ■x; then pX)»+S» x)» + 2a» igr. than rat value nation ia e natural eir cubes