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AND R. A. THOMPSON, B.A., Mathematical Master, Collegiate Institute, Hamilton. AUTHORIZED BY THE DEPARTMENT OF EDUCATION. ROSE PLliLISHING C0M1»ANY (Ltd.), 189L ^^ ^t\A03 fe\b mit Entered according to Act of the Parliament of Canada, in the year one thousand ei^-ht hundred and ninety-one, by tha Rose Publishing Company (Limited), at the Department of Agricultur*?. trintbd and bound bt Hunter, Rosb & Company, xCIiOKTO. PREFACE. This Arithmetic i,as been prei«i.-ed for tl,e u-se of nu„fls o t't, r f ";' '"'"''""' "'^«'"'-' -" ■•''- to be not f ■ T' "'■ ''"""« ««"-l^- It is imended to be not n.crely a eollection of problems, I,ut also a serie. ;s r-eSr""" '" ^^•""■" "-' "-'""^ ''-■•^' °^"'«— The worlc is largely a transcript of notes on which we me based our practical teaching for son,e years. Jfe v of the ongnial problems and methods of solution will te recognized by our former students whtb '«?'■'; "' ''T' '""' "^""'^^ ""« "«''" iutro '0 i-lude .11 Oapters on Compounrlllr, Z^lZ^'l "iT.:,':: s:n;-:;:pr:r-----^ uuces tiie piijiji to a, series of proDosifion« ^f ^i ".atical reasoning, with the relu "":; l' , T.h: d" more or less familiar already 3lt>c>9lot. vi PREFACE. of t ,11 ?■' 'T'"'"'"*'' ""^ <'"^''"- ^™*- A nun,l,cr paperB ' "" ^''°'" ""* '="'™"-=« Examination we?!,!r"""/'''"'"^ '■™"' ""^""^^^ Transactions, as well as tliose on Koots of Nambe,-s and Mensuration, lave logical development of the principles involved. Under General P™Wems we have included one thousand examples arranged in two-page sets of approximate Oder drmculty. These will furnish material for review of all previous sections. We give a few pages of problems illustrating some supplement to those given on pages l:!0-13'l In the Selected University Problems we feci that we 1 ave preserved in a more convenient form a number of the best prob ems that have appeared on Toronto Universitv examination papers for the past thirtv years "^^""^ ..rJsentin!f''""f' "i'''"™'"''"''" ''^^''' «" addition to presenting a number of carefully prepared problems will aerve to acquaint the student with examinatL standards W. H. BALLARD. • A. C. McKAY. K. A. THOMPSON. CONTKNXS. Glossary : Terms, * • •• Tables of Leiife^h, Weig-I.t. Volume, etc. Metric System of Measurement. . Powers of Certain Numbers. Vulgar Fractions. - • • #. Decimals and Decimal J^^ractions. - Powers of Numbers. Roots of Numbers: Square Koot. ... Cube Koot. - Mensuration : Rectang-le. - . . Triangle. - . . _ Circle. Cylinder (Surface). ... Cone (Surface). - Sphere (Surface). ... Rectangular Solid. Prism. - . . Wedge. ... Cylinder (Volume). ... Pyramid. ' - . . _ Cone (Volume). . . . _ Sphere (Volume). Problems—Metric System. , PAGE. • 1 6 10 12 14 42 71 72 75 78 79 83 85 86 87 89 90 91 91 92 94 94 95 vm CONTENTS. Miscellaneous Kxehcises : Fnictioiis and KSimpio Kuk's Units. - . . . • *• • Tlieory. •/ - '■ -■ m PiioiiLEM.s Arising From I^usiness Transactions: Percentn.;:i:e. --..... Trade Discount. - - . . Profit and Loss. - . . , Commission. Insurance. - . . „ ^ Taxes. .... Duties and Customs. - Stocks and Investments. Simple Intej-est. .... Bank Discount. - Parti'il Payments. - ^ > . , Equation of Payments Compound Interest. - Present Wortli and True ])lse(mnt. . Amniities, - . „ ^ Partnersliip. Exclian^'-e. - Pkohlkms in Mknsukation. General Prohlems. --.„.. Some Properties of NuMiiERs. Projjlems Selected from Toronto University Ma- TRICULATION PaI'ERS. Examination Pa reus. Answers, - paoe, - 102 - 121 - 129 - 139 " 142 - 145 • 149 •• 153 - 157 - 158 - KiO 164 ■• l(i8 " 172 ■ 174 • 177 180 184 188 191 194 212 304 315 343 373 ARITHMETIC. OtOSSABY-Tems and Tables. '^"mio";^;or.''"''°" """""•''=<«• «» t~n«»<=t business for Annuity.— A pcriixlical payment made annually or at more frequent intervals, for a flxed term of years or during the life-time of some person ^ numberoftl,equantities, will givean^gyegatet mtl to the aggregate of these several quantities ' Hank DiscoDNT.-The amount deduetcd from the face value of a note when sold to a bank. It Ts reekoned as simple interest on the face-value for the ttae before the note falls due. Bill op Exchanoe.-A written order directing some i>er sTm^o^SioX.^' '" '^' " "'"'' "'''' Pe-nTLTed BoNDs.-Written or printed contracts, made under seal penir^cit- :,'-^:t.^^ rd-^ot^rTot^^^^^^ BROKER.-A man who buys and sells stocks for others Brokerage.— Payment made to a broker for bnvin^ .. ^''thfCott' ^ ttrsS' — -" p=a °: "'"Z'VJr:^;^^'^'^ "- ^-"'"'"^ -" points ARITHMETIC. OoMMisaioN._A,i allow,,,,,..,, ,„■ |,„,v„K.,t „„„le t„ „ com "shu's's "'/;']'''•'''''■ ;,"■ •■'*■?■'" '■"'• ""-' "■^•»»°'".nf M,si„.!».s ; ,t IS Hsiii.liy c,,!,.,,!,.,,,,,! US a uctwiitnirc o„ the „,o„ey ,.ct,.,Uly hivclvc.,! i„ the t. JJcdo,,* Banks usually aiUl Uiu iutcvst l;„ thu Di'lnchnl every SIX inoutlis. l>nncij),ii CONR,- A solid ,vl,„se base is u eiivle a„.l whose ci've.! «u,.fa,=e tape,-s «„ifon„ly „. ,, ,,„i„t ,„„„;,";;;;; ,\'.';;;<' CoNsioNK,o._TI,e ,,erso„ (,„• li,.„,) t„ who,,, «.c»„lsarese„V '''^t=;;;^;::3;r-''' --'«---. eoi,,,,,issio,; the sjiiue us those of ,i siujfle jjerson. "y ""Ptt-ts Coppojj An interest cei'tifieate a,tta..|i,.,I to a l,o„,| . f„ ,,,, fslui! """ ^^'■"''"""' *'"'■ '^''y'"^"' -1"'" f.l>" i.;.:ere»t CouRSK OF E.v,T,,«oE._TI,oi-at« con„<.cti„j. the units of value of two eonut,-ios, aecc.liuf. t<, w| "h wl nl exchange ft,., the time bein^ are n.^.^^ * Cremto„._A person (or h>n,) to whon, money is due. CtJBE Root. -One of the three e,,ua,I factoids of a nu.nber ce,ft"..f ,!f'f f " '!"'^ ';? <;"I'^"I.-itol as a eo,-tain per- A .S7)ec//H- duty is assessed upon the ,uin,l,ei- wei-ht or measure of the ^oods, without regai-d to «Uue OLOHSAllY. 8 to a com- i.'icr.it)n (»f Mita^c oil Ion. l>i'inci[>al wlicii it |)rlncl]j.'il * curved i vertex. Hi'e sent. milssloii -'«-'--o'--ney """ bei^gT«w"Ii^ 'l?f# f '''^'' ^-h '■-« other obligation. P^y™""' "f a note, bond or zSe f ^"--'^ -- tifoJt:„irbi- PARTXERSHiP.-Tlie association of two or mnr. with joint capital, for the carrvW L ? ^'"'''''"^ ticular business. ^'^rrymg on of some par- Perpetual Annuity, or Perpftuttv Ar, continues forever. ^'^^^^tuity.— An annaity that ^'^''^-f^''^^^^^' Pamllelopiped havin.- two op, PObite iuues Square. '" P GLOSSARY. 5 ^"""^'Ss^ce.'' ''"'''' '^^reement or contract guaranteeing Poll Tax.— A tax levied equally on all taxable persons. Polygon.— A plane figure bounded by straight lines. Polyhedron— A solid bounded by planes. PowER.^When a product consists of the same factor re Preference STOCK.-That part of the stock of a companv on which a certain percentage must be paid before any dividend can be declared on the ordinary sto^k ^'"''ontTkel' ™r' ^f^ '■"'■ '"^°™n<=e. Also the excess 01 market value above par value '^^r afsreVu!;: ^^ ^^^-^ -' ^ -- ^^ --y ''TJ^^lt^^Sd.^"" '' "^"^^ ''' '''' -^ '' -^-h - Prism.~A polyhedron of which two opposite faces are parallel polygons, connected by plane faces at riSh? angles to their parallel faces. ^^^ Pyramid.-A polyhedron of which one face is a polvffon vertex? '""' '"'" "" '"'^^'^ ^^-'"^ ' — n Recta.^gle.-A right-angled parallelogram. Right-angled Triangle._A triangle, one of the ang-les of wlucl) IS a right angle. The side opposite the rf ht angle IS called the hypothemise. ^'^^ '''''A^nfber ''' ''^'^'"" " '"' ""''''' equal factors of a ^^'^indh'n^^ Circle. A plane figure bounded by two adu o. a circle and the part of the circumference in tercepted by these radii. ^umierence in- i'ts into which j a company is divided. capital of 6 1 I ARITHMETIC. Stocks T'Ho i^ called the centre. '' ''^''^''''' P^^^t witLJn, Square Root. One of th.. +« is cailec. t>>o:Z^ZZTetZZ' "'''"'""''- T« St, t: r '^^ ""-'^ ^ -<=^e. .„t When "per cent, om" "'"''"^ expressed as such a """'Ser^ '""'-"'"''' P'"- «^"- "aving two sides equivalenrto the imerc'st on til '"'' "'"" '"^'^<""« ^^ debt. iiitcicst on the present worth of the Wedge._A triangular prism. ^"'^^o'^XK'eho.t.""' "''' ''"•^•'^ ^-'-"-"l between 1 1 I TABLES OP LENGTH, WEIGHT, VOLUME, Etc. lp:ngti[. tlJ^lT''"'"' '''''' '"' "^« "--sumnent of length is lycml =3feet = 3G inches. 4yards^l rod (pole or perch). 1760 yards = 1 mile. ^ 1^2 yards = 1 chain ==100 links. -'20 yards = 1 furlong-. 2 yards:-! fathom. Of h!:;ies.""'''^' '''''"-'^ ^--^ i" measuring the height TABLES. AREA. A square foot =144 square inches. A square yard = 9 '' feet. A square rod = 30 J " yards. An acre = 10 square chains. = 160 <^ rods. = 4840 " yards. A square mile = 640 acres. ' Also, 100 square feet = l square, used in raeasurinff roofing, flooring", etc. VOLUME. A cubic yard = 27 cubic feet. A cubic foot =1728 " inches. A cord (of firewood, etc.) = 128 cubic feet. CAPACITY. The standard unit for the measure of capacity is the GALLON^ which is the space occupied by ten pounds of distilled water at a temperature of 62° F. 1 gallon =4 quarts = 8 pints = 32 gills. 2 gallons = 1 peck. 8 gallons = 1 bushel. 31 1 gallons = 1 barrel. 63 gallons = 1 hogshead. Also, 8 bushels = 1 quarter, used in reporting the Brit- ish grain market. For Apothecaries' Fluid Measure the pint is sub-divided as follows : — 1 pint = 20 fluid ounces. 1 fluid ounce = 8 fluid drachms. 1 fluid drachm = 60 minims. WEIGHT. The standard unit for the measurement of weight is the POUND AVOIRDUPOIS. 1 pound =16 ounces = 7000 grains. 100 pminds - 1 cental. 2000 pounds = 1 ton. ARITHMETIC. m poands of flour -~ 1 baiToI. |00 pounds of ))ecf or of pork = 1 b-irrol . ^80 pounds of salt = l li'el An ounce Troy. 480 grains ApoundTroy=i2TV;^nces. m, „ „ . ^•''x 60 grains, ces named ; «i^shel of each of the substan- 14 lbs. of Blue Grass Seed. 34 '' '' Oats. 36 « << Malt 40 '^ - Castor Beans. " Hemp Seed. ;; Barley or Buckwheat. J* lax Seed. " Indian Corn, or Rye <( 44 48 ^' 50 '•' 50 60 70 u a VALUE. value Of about 23J gr^l^-nrorpurc golT""' "'"'" '^ "- 1 dollar. 100 cents = 1000 mills an?aSL^ t^f dt'eT= lot^;^^^^ t^t ^ ^ ^ '' ^^"-)^ are not reg-arded in business on^r^r ""'^ denominations of U. S. money is practicaut Tj''^''''''' «« ^hat the table 2. British mone^^ ^"^ ^^'^ '^"'^ ^^ «^i' own. 4 farthings = 1 penny. 12 pence =. i shilling. 20 shillings =1 pound (the unit). 21 shillings ^ 1 guinea.' ^ asTotr :!^'^ ^'''''' '^"^ ^^--^-n money are connected $73 = £15. or, |4.8G2_,i;i. TABLES. 9 January, 31 days. February, 28 March, 31 April, 30 May, 31 June, 30 << << (< TIME. The unit for the measurement of time is the jjoan solar DAY. 1 day = 24 hours. 1 hour= 60 minutes. 1 min. = 60 seconds. 7 days = 1 week. 365 days = 1 common year. 366 days = 1 leap year. The calendar year is divided into 12 months of different lengths, as follows : — July, 31 days. August, 31 '' September, 30 " October, 31 " November, 30 " December, 31 " In leap year February has 29 davs, making 366 days in the whole year. To ascertain which is leap year, divide the number denoting the year by 4 ; if there is no remainder the num- ber denotes leap year, while remainders 1, 2, 3, indicate respectively the 1st, 2nd and 3rd year after leap year. Thus, 1891 is the 3rd year after leap year, and 1892 and 1896 will be leap years. If the number denoting the year ends in two ciphers it must be divisible by 400 in order to indicate leap year • thus the years 2000 and 2400 will be leap years, biit not the year 1900. ANGLES. The circumference of a circle is divided into 360 equal parts, and each part subtends at the centre an angle called a degree, each degree is divided into 60 equal parts called mmutes, and each m.- te into 60 seconds. 1 degree = 60 minutes = 3600 seconds, y V viCgrces =,1 quadrant or right Single, 10 ARITHMETIC. MISCELLANEOUS. 12 articles. 1 dozen. . 1^ dozen =1 rrro^^ The • I^KNOTJL micro denotinp- i • » • u (< (( u centi de^ji deka iiccto kilo myria mega 1 metre ^ 1 metre = 1 metre = 1 metre = 10 metres = 100 metres = 1000 metres = 10000 metres = 1000000 metres = 1 TOdo" T(T?r I My 10. • • • ,.. -^■''^• J re. rREMENT. ^7 inches iiearlv) e imit, the folJo'w- cviated u. 111. c. d. D. H. K. M. %. ' microns, /*ni. mm. cm. dm. Dm. Hm. Km. Mm. Mgiu. ^0 one sqiiai'e METRIC SYSTEM OF MEASUREMENT. U VOLUME. Istere = l00ce4itisteres, cs 1 store = 10 decisteres, ds. im'!^''^'= Idekastere, Ds. 100 stores = 1 hectostere, Hs. lUOO stores = l kilostere Ks cuMrtTmTtil^^^^^^^^ to one of capacity. ' '^ "'^^'^^^' ^'^ "^^asure 1 litre = 100 centilitres, cl 1 litre = 10 decilitres, dl 10 litres = 1 dekalitre, DI. 1^ f®'' 1 hectolitre, HI. 1000 litres= 1 kilolitre, KI. = 1 store. WEIGHT. to ^he' wethToT ofe e^^^^ ^^^ -HAMME-oquivalent 1 gramme = 1000 mil%rammes, niff 1 gramme = 100 centigrammes J' 1 grammes 10 decigrammes, ^' 10 grammes = 1 dekagramme, 1 hectogramme, 1 kilogramme, xvg. 1 myriagrammfe, Mg. V"-:: = 100 Kg. 1 i^iHier or tonneau . 1000 Kg. melt Ss";"^ ''' approximate values of some of the 1 w.^i. = 39-37 inches nearly. I inch, 100 grammes = 1000 grammes = 10000 grammes = 1 quintal dg. %• Hg. Kg. 1 metre 1 centimetre 1 kilometre 1 hectare 1 litre 1 store 1 kilogramme 1 inillier or tonne, au = 2200 ff mile, 2| acres, 1 quart, i cord, pounds, pounds, 12 ARITHMETIO. POWERS OP CERTAIN NUMBERS. 52: 1 :i 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1-02 1-025 1-03 1-02000 1-02500 1-04040, l-050Gb 1-06121! 1-07G89 1 -082431 1-10381 1-104081 1-13141 1-03000 l-060!>0 1-09273 1-1255] 1-15927 •035 1-04 M2616 1-15969 l-1940r, 1-14869 1-18869 1-17166 1-21840 1-19509' 1-24886 1-22987 1-2(;G77 1-30177 1-31392 1 1-21899 1-28008 jl-24337 1-31209 ll -26824 1-34489 il -29361 1-37851 1-31948 1-41297 1-34587 1-44830 1-37279 1-48451 1-40024 1-52162 1-42825 1-55966 j 1-45681 1-59865, _ 20 1-48595 1-63862 1-80611 21 |1 -515671 1-67958 1-86029 1-38423 1-42576 1-46853 1-51259 1 -55797 1-60471 1-65285 1-70243 1-75351 1-03500 1-07123 1-10872 I •11752 1-18769 L -22926 L -27228 I -31681 -36290 -41060 -459971 -51107 •56396 -61869 •6753r, -73399 1-91610 1-97359 1-54598' 1-72157 1-57690 1-76461 1-60844 1-80873 2-03279! 1-64060 1-85394 2-09378; 1-67342 1-90029 2-15659; 1-70689 1-94780 2-22129 1-74102 1-99650 1-77584 2-04641 2-35657 1-81136 2-09757 2-42726 1-84759 2-15001 2-50008' 1-88454 2-20376 2-57508' 1-92223 2-25885' 2-()5234 1-96068 2-31532' 2-73191' 2-28793 ■79468 1-85749 1-92250 1-98979! 2-05943 2-13151 2-20611 2-28333 2-36324 I 2-44596 2 -531571 2-62017! 2-71188: 2-80679! 2-90503' 3-00671 3 11194 3-22086 04000 •08160 12486 16986 21665 ■26532 ■31593! •36857 42331 -48024 ■53945 60103 ■66507 73168 •80094 ■87298 94790 2-02582 2-10685 2-19112 2-27877 2-369921 2-46472 2-56330 2-66584 2-77247 2-88337 2-99870 3-118651 3-24340 3-37313 3-50806 3-64838 1 1- 1' 1- 1 1' 1 1 1- 1 1 1- 1- 1 1- 1 1 1-045 No. 1 1 1 1 1 1 1 1 1 1 1 1 01500 -09203 •14117 •19252 -24618 •30226 •36086 •42210 ■48610 55297 6228.'^ 3-794.321 69588 1-77220 1-85194 1-93.528 2-02237 2-11.^.88 2-20848 2-30786 2-41171 2-52024 2-63365; 2-7.5217 2-87601 3-00543 3-14068 3-28201 3-42970 3 -.58404 3-74532 3-91386 4-08998 4-27403! 4-46636! 1 ') 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 19 30 31 32 33 31 POWERS OF CERTAIN NUMJJERS. 13 POWERS OP CERTAIN NUMBERS. No. I 1-05 106 I 1-07 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 20 30 31 32 33 34 1 '05000, 1-06000 1-07000 1-102501-123601-144901 M5763 1-19102 1-22504 1-21551 1-26248 1-31080 1-27628 1-33823, 1-40255 1-340101-418521-50073 1-407101-50363 1-60578 l-47746|l-59385 1-71819 1-55133 1-62889 1-71034 1-08000 l-16640i 1-25971 1-36049 1 -46933 1-58687 1-71382 1-85093 1-99900 1 1- 1 1 1- 1 1- 2-51817 2-71962 2-93719 317217 3-42594 •68948 1-83846, . . , ,^^ 1-79085 1-96715 2-15893 1-89830 2-10485 2-33164 11-79586 2-012202-25'>19' 1-88565 2-13293 2-40985'' 1-97993 2-260902-57853! 2-07893 2-39656 2-75903 2-18287 2-54035 2-95216 2-29202|2-69277'3-1588'^ 2-40662 2-85434 3-379931 o-.;^ou-. '2-52695 3-02560 3-61653 4-3l575| 5-141 rr l2.9?5t%-mlt'ifnf ^-^'''^'^ 6-10881 f ;;r r!.'^ - ^4 4-43010 5-43654 1 -09000, 1-18810 1-29503 1-41158 1-53862 1-67710 1 -828041 1-99256 217189 2-36736 2-58043 2-81266 3-06580 3-34173 3-64248 3-97031, ....^,, 3-70002| 4-327631 5-05447 3-99602| 4-717121 5-55992 10000 -21000 '33100 •46410 61051 •77156 94872 2-14359 2-35795 2-59374 2-85312 3-138431 3-45227{ 3-7!)750 4-17725 4-59497 '3-07l52,'3-8l97.' 3-22510 4-04893 |3-38635:4-29l87 4-74053 6-87146 5-072371 6-34118 5-42743 6-84848 7-39635 6-65860, 7-25787 7-91108 8-(;2308 9-39916 6-11591 6-72750 7-40025| 8-140271 8-954301 9-84973! 10-83471! 11-91818 ioiwao y-o!J916 11-91818 2{\ 7-98806|l0-24508 13-10999 27 8-6271l!li.i(>7i4 3-55567 4-54938 5-80735 3-73346,4-82235 6-21387 '3-92013 5-11109 6-64884 o-«^m i n -itti^ 5-00319olS'9 3 'i r7\';??J5T«'''33i21.n378 5..)noorJ7 ..^y^;! '^-'^•^^il2-()7()0.'),17-18->03 '^3-'»f\i r. :5^-3,7^^^103^1i_^^^ 14-42099 15 •863091 17.44940! 28 29 30 31 32 34 14 ARITHMETIC. VULGAR FRACTIONS. wiTv!''; ''^'f^''^ ^hich have 80 far been considered be^nn T^Jr w1 r ""l increasing by 1, eacli number being bers 11 /,n f''\ '^/'^ "^"'^'^' J^«^ ^^*'«^'^ i^- Such mini- beis are called whole numbers or integers. Besides these there are other numbers, some of them such as one-half, three-quarters, being less than 1, and greater than one whole number and Jess than the nexl higher whole number. These numbers are called Fractions, those less than 1 iSpE^tr^^^^^^^^^^^^ ^"^ ^^°^^ greater than 1 Tni!.''i!^f«rf * "^'^ ^^f'''''' hereafter that whole numbers S'^mnron^r T^f "^"^ '^' *''^'^'^^"'' ''^^^ '^« ^"^^ ^^e classed as improper fractions. PROPER FRACTIONS. mrL''ir'S^^'^''''l^^J''*'' *'^^ ^^l^^-'^l P^^^ts, each of these ITuL I ^?"!? '\^'^'-^'' '^ "^'^ ^^^^'^^ ^q^^I P^i-ts each is h^; T.^.^ '"^ ^'^""^^ "^'''^ ^^^^"^^ i^« n^^nie from the num- ber of these parts required to make up the whole unit. wnt^'lr '^^f ""^^'^^ "^"'^^^' ^^*'^^«« P'-^rts in the same way that we do of a number of feet, pounds or dollars; but the notation employed when we come to write them s very different. Thus for three dollars we write $3 sLb'^nri^T'^'' r 7''^ ^ '^'- '""^'^'^ *h« character oi' s>^mbol denoting the kind of thing under consideration is placed to the right or left (and sometimes directly above) that denoting the number of things ; but for three-fourths wc write I ; the -1 in this expression, which tells the kind ot thing under consideration, mimG\y, fourths, being placed directly below the 3, which tells the number of fourths This symbol f is called a Fractioti, and indicates that a unit of some kind has been divided into 4 equal parts and that 3 of these parts are under consideration. VUUiAn m ACTIONS. IK unit has been divi,l,.,|) i, ";,,,"' i;"''' V'*" """ich the ^tev the,n) i. called the «, 'X Th^" ^'^ "" '"«•"«''- (lenomiiiator are called the r«'»V^f 1 nmneratof and If the numerator of , ♦?. i T "'^^ fraction. the dcnom~Mtl"ev S"th:r T'""' ""•""- "'«" taken is less than the n.unhe, of ./ •" '"""'''"' <"■ P^Ms «-as divided, and tlu " Se tlr t' 1' T" "''"''" '"« "nit ■lumber less than th,/ „," t n , S '.''"f"" ='''"'"«« » eording-ly called P,;,per FmrfiL. '■'"""'"' '":« "«- IMI'HOI.KK FRACTIONS. If we take four fourths or five fifths nf tliat we take the whole of ttl?, ,, . 'V'^' """ '* '» "'ear -5l;,!"-'«'0" - ^ J or' « °J tlrto'l '"' """■«'•"- --T a ^ b7o'.feX".i eo™ e^r ul r ^"^ ^'-'^ ~ than the whole \snph nv-^^^ "^^^^^^^ something- less clicating a^:,,%f'a"u'i\™^^^^^ ^^-^-^ - I, noVS! If now we suppose two or n?' ^!^P''^Per Fractiom. into five equal S! ^L consJdf '' '^ ^^"^^^^^ ^'-^^^ parts as we please, and thus woTn !' ^^ "^^"-^ ^^ *li««e f f, ¥, etc., Where the numer oi^ • '""'^^ '^"^^ ^^'^^««"« denominator. These fm- J "^"^^^'^^^^ ^^ greater than the called impropeSic^^ ^^^.^V''' '''''^'' -'-«^'o are .ailed Simple FracLi,^^ '^ ""Pi'oper fractions MIXED NUMBERS. Ave HfJ'l^sr etiSrr'' 'r^"™ "' - '^"O- that fore equal to 2, .Zl c2eo, ;;.? '1''" *'" «^"'^ "''c there eciuivalent to 2™,<1 •, ""1"^ "'"''* ">« fraction W ^ faction is equi^'l'eLft:: XTl^^r^"-''"'''^'^^- whole number and a pro«er f, n!l '5' <"" "^'^e to a number can be obtained bvfind^!,^' ""'* """ *''e whole inator of the fractionTs c^Sed^in^^ <>"«" '^e denom- also that the rcn.ainder, Xr d?vM 1 't ""r'''*""-' ""d numerator of the urope ■ fine ion* ' ""■''^ "'"' '* "'e 10 AlUTITMETIC. Such mimhfu-K jiro rxi>ross('(1 l)y wriliu^- tho wliolc, niim- hev lii-sL with tlu^ fniction cJoscafbT it, ; thus, tn'oamlfhrcc. fifths would lu' written 2;,'. Such nunihcrs hh! cjtIN'fl Mixed NumhiTs. 'V\\m W(5 8oc that cvci-y imiu'oix^' fniction mny ho cxprcsHcd cither as a whole or us a mixed number. EXERCISE. Express he following improj.er fractions as whole or mixed numliers : (1) 1 ;( t • ;» a 7 1 a • /'0\ 4:(H7ft (-0 (13) «iM^-. (14) -^-;- 27 :1 • 00 I 1" 47 7 • 10 1 101 I ni Conversely every mixed number mav be expressed as an improper fra=ction. For if we consider the number 3^ then, since 1 is ecjual to live fifths, therefore, 8 is equal to fifteen fiftlis, and tlierefore 8' is eciuivalent to fifteen fifths and four fifths, that is, to nineteen-fifths, which we express thus, -'/-• We have, then, the following RULE. To reduce a mixed numbcir to an improper fraction : lAIultiply the whole number by the denominator and* to this product add the numerator ; this will give the num- erator of the improper fraction, while its denominator is the same as that of the fraction in the mixed number. EXERCISE. Reduce the following mixed numliers to impvonnr frac- tions : (1) (2) (3) (^) ('>) («') (7) 31 1 7 1. To l\X. (8) (y) (10) (11) 9Q18 3476' 969|.». (12) 576^111. (13) 1010{«i. (14) nm- ' 1 O'' 3 4 7 6 (15) (16) (17) (18) 10001^J>^. (19) (20) 9009 "007 1000 AVtf- 8736J-?,. VTTI.nAH FnACTIf)Ns. 17 s "vvliole or (•OMI'OUNr) FRArTlONM. U iH cvidci.t tliat if -Wouiths Ik, iv prated 7 ti,T,r.« fl,n Hence W(! have the tblNming HULK. To mulliply n rniction l.y n wliolc mmibcr: Multh.lv tho nuniermor of the fV.eti,,., by the whole nui.l.erCte --.erato. of the ,..Kluct and retain the .anie X:nomll^" KXKliOISK. (1) Multijily « !>>• 4, (!, 8, 10 and K;. (2) Mulfii.ly ],: by!), i;), i>;i,,i,.xr k ^i ¥, a,,d if - be dlvicled l^y f ,0 i^Iu' t' wilT '» "i"/'? IS evident that this onemfimi r^.^.c/ • ! ^ ^<' ' ^^^^* ^^ started Wit,,. It foii;,wr "rX Th^t"^''',?"/':''' -r r eqiiai to one another. Henoe ifi oth™, ^ "n.st bo BE MULTIPLIEO I,y TUF VvmI' " '""'^EBMSOK A FRACTION FRACTION REMa/^ t"p samL "™™''' ^-^ VALUE OE THE Tims, 4 = j'.. = ^»,^|,^ ^^^ given denominator '^ ' denominator eontains tl,c times 7 is contained in ^H ImI^^ l^^^"^"^ "f EXERCISE. >«'eqS'uot t^i'-^'ir-"*" '""' ''--'"".-to.- .vbie,. sball thei'r d^nZiir * *° ■""'^"™' ^^■'-" ^"■''" >'-- :-'« for samectnSir'jf ■ *" "-^''''""'^ -'-" ^■'-■^" >.ave tbe A^'dSSil """ * '" ''-"""""^ ^^■'>-" «""!> i>ave CO 5: fedt'f F '? fr?rt" "•■"';""•-' '•"'• "-on,inator. ator. "' " "^ *° ''-actions liaving 18 for denoinin- ^. Reduce '^ I ^ ^ i i - wl>iel, s„all Im;/i„i^aS,e den^li.i;:.^:"™'''"' "^''''""^ '!. Ii 20 AniTH]\rETIO. oi)t.Mu „ Aviuch J.a,« been shown to hv. e(|ua.l to ' •" • Immk ^ IF BOTH TKRMSOF A FHACTI(,^^ HK DIVIDE ,V ^^„V 'amp NUMHE., THE VA.UE OE THE EHACTIO. KEMAINS THE ;AMr Tims, »^' = ^s = B, When both terms of a fraction have been thus divided by a numbers which will divide them botS hL'Son IS said to lie reduced to its lowest terms Hence we have the folloAvinir To reduce a fraction to its lowest ternw • nurM. i ^i numerator and denonunator of the Ivlcl^i:^ ;, , ^^^^ u rVn.^Monis::i?-^t' f"^ ^^^^"-^ tlus^pSn factor ''"''^'^ '''^^'''^ '^^'"^^ 1^^^^« ^« Common '(dividing by 5) = ^]^ (dividing- by 3) . |^ = |. EXERCISE. Thus, Kcducc the following- fractions to tlieir lowest terms : 1. 5. 9. 13. i'> •20' 4 5 T5' 2^5 (i 7 (i 8 • 2. 6. 10. 14. 1 dirided 2 into 5 equal parts, and we find that each part consists of 2 fifths ; that is ^ iof2 = ^. Also since the fifth part of any number is obt.'iined by dividing that number l^y 5, it follows that ^ f = lof 2==2-f5. V\i 22 ARITHMETIC. It will also be seen from fbi^ ^■^.^,- , ^ may be expressed -is • fV?.. f^'^^^^^^^V ^'^'ole number ator ; thus '' ^'''^^^^" '^'^^"^8- 1 for its denoinin- 5 = 4, for « = 5-1. 5. ^nuj:^:;,:^';::^^^ ^- expressed as a P-rasa.aetio^.EtS;;— ,^^^^ 7 = 7 _ 3 S I — 5 • EXHRCKE. lo'lC^!"""'"' ' "' " ''•''>'•"»" "•'>o^« clenomtaator is 6, 7, ^^2.^ Kxpres, 39 as .a fraction whose denominator is 14, BEKOmSb' ™*'"°'''^ ™ °'"'^''' H'"-'-0 A COMMON Suppose n-e have the fractions I, J and 5, tl.at this denomi^uoT'™: ^T^r' '' l« -*'«"' must be a multiple of •/ ^ n„rt 7 "', ■''"'' ^' "'•'i' '«. etc., and therefor^ thel^uSetr; IvoiK"' '''■ ~J-^^ _'!_3 ."3 ior.> io5> lojt or A-f-. A2« fio and so on eve^!, "tS ttlfenrt^sXr' f "<""'"-<"■' "- tl.at will contain all trlh-e" de,,!^"' ''"' ''"'"'""'ator take the least eonnnon mu i e « ZXT' '"■^' "^' «° Hence we have the fcllowin." denomniators. 3 QUOTIENT OF VULGAR FRACTIONS. RULE. 23 ^' MnlHn?v l' H 'f- "^'''" ""^ denominators. of^info^iwr ''• ""''"' "*■ •"*<''> "''Of ion ''v the number of times ,te denominator is contained in tlus'l. c. m i'^x. To reduce 6 1 _7_ fi to a common denominator^' ' " ^' The J. c. m. of 8, 6, 12, and 9 is 72. ine first denominator 8 is contained in 79 thn i ^ times, therefore the first fraction fbecomes ' " "'^ ' 5 X 9 _ 45 8 X 9 ~79 Similarly, the second, ^ beconres i2 the tliird -jV,- « V| and the fourtfi << lo Tlie resulting fractions are, therefore tI^Is, 41, and |o ' EXERCISE. thf ftre^nrrofnitttr " ^'''^'™"'"' "-- -'* tlie^.''Jlm;;7fi,'^ 'f""?"^ are not in their lowest terms W tlte rut''' '"'"''"' '" "^«"' '»-«^' '--ms before appi;i 1.-1-8 1 2 3 4 R o' 4, 6' 0* 10, loTT* iooo» TDo^ij-' 5 _;}o^ 1 li 1 V 111), 51^ 3' 1; T' 2, 3. 14, -L i23 1 9 , 15, 18, 2Ti tiT 4. 5. J- Ifi 29 • ^i>' 2 1, TTi- G. ^' 13 Of I, 11^1^. 1 of y of3-«. 10. ^ of I of ^, 1 of I of ,^,,^ of A of A. lo COMPARE FRACTIONS IN MAGNITUDE: Suppose we wish to know which is the xrro'.fer of fb. tAvo tractions, $. and ^ gi cater ot the enulenomhnth'f ''■'"" """" '" '°"^' *^ ^"".^ '-^« ^iff- squai part3 are greater or ] (!ss than r> out of 7 equal r ! 24 ARITHMETIC. ively : "t^iiuiiuiicitoi, tliey become respect- 11 and p. and we can nt once sav thnf •)« ^^f .^o- fl-re'ifrM- H,.,i, .>- Vwi ^ -^ ^^^^ "^^^ equal parts are EXERCISE. Which is the greater ^ or " '-^ Find the least of the fractions ;;, i and ' ^ Arrange 1^, i in order of magnitude, beginning with tlie greatest. *' *' I, tV' M- in order. ^ ^^ l> tV and -^ of 2. ' '' * and "" 1. 2. o O. 4. 5. 6. 7. 8. 9. ti 11 n It 1^, ^f, A and 1^. "J"' ^«' !•*■-' and 2 ,\. By ^t::;^ " ^^^ ^^- the oWgll^} ^^ o/ two fraction^ 1^ t' eate^ 'Tl!;;: ^ ""^''^'^r^^^^^ IS-, we see that a« ma v^be formed W^lr'^^^ ^f and the tenns of - and tllJSj^^t ^^^?g.i ^^^^^ ^^ the..efore the s^^t^^^,;^ ^^^^ "^ ^-^^ -ci Agani, m comparing the fractions ^t^W^i^ir^ ''^"^^^" -ay be formed from inator. ll ^1^^^^ ^ ^^r o"' ^;^;^^ ^^^^^^- wiU be obtained. ""^ "^ ^^'^ fraction ,«^ , „ is greater than 4 bnt 8 1 <( 10" \) "10 u (( (( ii From these examples we mj.,y infor. s 6 ^ginning- with VULGAR FRACTIONS. 25 ^J^^ni:'/z,X'z:^ i7Jrf "^^ --^^^"^ the ^|-i- - '---'V'^ *^"~^^ then the then the ft-aetlLtmotle^rtlnr/tCn/r'" *"^''"°"^ •>. ihiit It a tract on can ho fhrm,„i f , adding a ffi-cat,u- nuuiho, tn ^,c ' f™'" anotlici- by denominato,-, then the frtction .n """'«f »»'• 'h™ to ite the other. "-"°" '" '^»'™ed is greater than EXERCISE. J'Sng^serr'"' ^^" greatest f.-action in each of 1. 3. 5. 7. 8. 10. 12. and vv readily reduce fe ''" .t;"'!>;"'\'«'l.">an= V, ,^ tliat J-.J or •' is the srrc'itest-nf fr ' f *»J. '*l'0''<'"'g at once This method i prefo , , f ""'"<' rvaction,. hyge denon, i- '. fo, i ' ' '" ""^ '^''"^ "'' '■'•a<=ti«>>s ha,vi, I. 2 1 « 18 ll 1 2 *irf« 1 r» T 9 • l:j 7 4. 6 17 z 7 ^5> 4 5' 7 1 (i 8> tI (change T toH ., "• 1T» US' then compare J-l am .1 .2 3 4 t^ m IT* a7' 8 3» ?> -BT" io> ii> r^j 1 1 .*{ If 11. ■i 5 r, r » 5' 0' T> SJ !>-• ivmg 26 ARITHMETIC. EXERCISE. Arrange tlic following fractions in order of magnitude beginning with the least : iiui^niiuae, 1 ^ ^ 6 ./ 8i 7 8 * 19' 31' 37' 3 5 U 7 "ill' 92' 105' 3. -, , 97' 161' 353' L>L>r>' In this hist example the fraction may he Avritten • 1 1 jL_ r 321' 3o|' 32 ^y 39 f' in which form the comparison can be readily made. ADDITION OF FRACTIONS. If 1 dollar 2 dollars and 3 dollars he added together their sum will be G dollars, and so if 1 seventh, 2 sevenths and 3 sevenths be added together their sum will be b sevenths ; or as it is written^ Thus it will be seen that if fractions have the same c enommator their sum is a fraction whose numerator is the sum of their numerators, and whose denominator is the same as their denominator. Hence we have the following RULE. To add fractions which have the same denominator • x\dd their numerators together for a new numerator and under this i)lace the common denominator. EXERCISE. Add together the following fractions : ^' 5' 5» 5* Q 10 10 ^' ^yy' yya T\ iii; yo' 3 100 10 1 "• ^yy yyy Uylf* "• 13^' Ty> T7i« 6. .%, I of T 1 of 7 ' of iiuignitude, VULGAR FRACTIONS. 27 . ^jnho^fVnctions to 1. added h.ve not the s.n.o deno.u- S 2^::^:^:^^;^^^^ ^-^^-s havin, their ^HKl their sum can then ']tfo;ih ., g-iven, tlias : '"^ ^""^'^^^^ ^^y <=ho method already Hence we have the following- RULTC. To add fractions which h'.vo ,w.f ^i Keduce tlie g-ivcn f c ions ?n ^ f'^« ^denominator : in,r their U^ist'connno ] ^^ ^ ^.^^^^^f ^^ Lav- - and under their .un. Z:^Z':Z:^Zo^ Vinrl f1,^ EXERCISE. J'nicl the sum of, 1- I and I. 4. ^andV". 7. I, f and t. 10. « " " -" 2 4 and" ^- #and{. 1-3. I -^ 1 " 1 1" 1 in ■2i7> TJiT' 6. ^# and |,«. 12. i, 1 -V 1 7» ir» 27> ST' f- Y^and'^«. i:i/rf^i Sum = 3| + 4^ + 5}. =3+1+4+^+5+1 =3+4+5+^+^+1 = 12 + 1 + 1+1 * -12 + ]:> " ^ = 12 + 1^ = 13 '- 15. 6J and 71 \ ' V,"" '"™"^ '"ff^'her. 17. ii^ Hnd !1--. Yr fr '-'il" -ii. !«• 17#, H, ^i Iiiirt i. ''■ *- «- ^J ^"1 ?• lP.Srs befo^.^'-^P-- *■'•-*'-- to nuxed au-nbc-s, then O 28 , ARITHMETIC. EXERCISE — (Continued). 90 ir.i 7^ 4" 1" 10 2 7 21. Find the value of ^4-f'Tof^oflOf + |of{|of|«. Here we must first reduce the compound fractions t( simple ones, thus, •■'- of 7 of 1 0- — "^ of 7 of 7 2 _ 2 4 _ 9 2 4 of ^of 3 -i. • • • ^ + TT Of ^ Of lOf + 1 = 2 + UA + _ q 89 Of 8 Of 2 8 22. Find; the value of ^ofi4-^of4H^of(| + f) + yof(| + |). Here we must first add the fractions within the brackets, thus, Thus the expression becomes reduced to fofi + iofa + Aofi| + i;Lof^|. Next reduce compound fractions to simple ones, thus, 3 nf 7 _ 21 . 1 of 41 — ^ • -5. of 1 i - ^ 6 . 10 of •■** — 3 2. J Thus the expression becomes reduced to 2 1 , 3 I 2 5 , .-5 2 3 •3-2 +4 ■T*T}(y "T^SoTT* which -Sg^j^LV Find the value ot 23. f + 53?T + liof2| + 6^. 24. -^ofx^^ + iof^V + |of(J + H) + V^of(f + |.) I fractions t( the brackets, VULOAJl FHACTIONS. SUBTRACTION OF FRACTIONS. 29 tl'oro will „e .•, nintl. loft, o.- ll^L "rwHtt':;'""^ '^'^ "''""' 7 IT 4 11 .•J is the (liffcT(,„ee of th.. ,•.,„ ',:./''' '■'"'','^''°'*'' ""mcrator tlic following """' "«''">»"'ator. Hence, we have RULK. tl.o'^Lt^.tomSrtST !,";? '™^'''»"^ -"'"h Lave numei'ators for a new nume,- n,- '''"J''""'* ''«'«'««n their common denominator '"""'''■'''"• '""> ""'1«'- tM» place the EXERCISE. Find the difference between 1' i^aHA ?' i-^d I. J^md the value of ^^ 7. i.l__6_ 3^ 11 _ 6 10. 7#-| __S7 11 ' lilf 3. f^andji. 6. xVV and ^i^",. 9. 3|-i 12 7 s Suppose Ve require tho'dilerenee between' TTT i^ and ^ weseeatoncethattheir'difffe^Li^ceis Hence we have the follo^^ng- RULE noUhe"lm" + 11. IG. iJ+,\of3nandl^4-|-of ;r;of I|. Find the value of 17. I - 2. 18. .|« -.17 10 28 2 7 20. :^ of II - 1. 21. 'l-\. /'^ - ^1^. 22. hi - 31. In examples like (22), wliere the difference 'between two mixed numbers is reciuired, it is not necessary to re- duce the mixed numbers to improper fractions, for, since 3 and I are both to l>e taken from ^, we may first take away 3 and then take i- from what is left.' When 3 Js subtracted the remninder is 2i ; we have next, there- fore, to lind the value of Now, it does not matter whether I is taken from 2 or from I so long- as the whole quantity 2\ is diminished l)y v we may, therefore, take I froni 2 (anil this part of the opera- tion can always be done mentallv), loavino- li, and also the 1 which must now be added to If. This" is done in the way already explained, and therefbre the result re- quired is li+l which = 2 2 Find the value of '" 23. 17-V-15;;. 24. S\\^-2\^. 25. 13()4,-120-^Ai^. 2G. y^-lX '' (Change the improper fractions V and J>- to mixed num- bers, then subtract). 27. Find the value of 1_2,1G_1 1 mi • 1 8 „ 5 ^ 1 6" t ~ *• J nis value may be found in several ways : 81 VULGAR FKACTrONS. KXKucisK-^ Confhued) 1- >Ve may pi-ooccd thus to this .',(1(1 ; ^vhich irivcM J..H ' ^'•oni this t/ik('V\vl,iMi I '^^' * N X = J- i ; I -'Irwj; i;;;;' /!,£;'!!;'■ £'^";v" <■ m- «..» or "7«t first 1,0 found ^ Hi Xn; '" 'T'^''^^^^^ ^''^'n" value subtracted aceordi,,^ !,s . ''^' ?'^""^ >'"^-t 'h. added r '^^foreit. "^ •'" ^''^ ^>''tcket J.as the si^ni + or- and 20,' - 7'"''L t')?:U'' ^1- i^^ind tl,; v;d.4';i^- -^ «^ 1 - + « - /V Of I of 4^ ^^ ^^-it,!!:.;teui^^^^ ^•iMcieuee between K I 32 ARITHMETIC. EXERCISE — fCotitinuedJ. 33. Find the sum of the greatest and least of 27 28 20 3 0. :ilso the sum of the other two, and then find tlie differ ence between these sump. MULTIPLICATION OP FRACTIONS. If 8 be multipled by 12 the result will be 96 ; if 8 be multiplied by 6, the half of 12, the result is 48, which is half of 1)6 ; if 8 be multiplied by one-third of 12 the pro- duct is one-third of 96 ; if by one-fourth of 12 the pro- duct is one-fourth of 96. Thus we see that in multiplying- by whole numbers,, whatever part one multiplier is of another,' its product is the same part of the product of the other. In the example cited, all the parts taken give whole numbers, that is, one-half, one-third, and one-fourth of 12 are all whole numbers. We assume the same law to hold where the parts taken do not give whole numbers ; so that if any number is multiplied by one-fourth of 3 the product Avill be one-fourth of that obtained by multii)ly- ing the number by 3. But one-fourth of 3 is |, therefore, to multiply any number by | we must first multiply the when multiplied by f gives ?,^ as the product ; or, as it is written, x4 = 15 ^. .. 5x3 . ^ 5 3 5x3 ^^"^" 28 = 7T4'*^"'"^"^'" 7^4=7^ And thus we see that the method of finding the pro- duct wlien one frnctioii is to be multiplied by another is precisely the same as that for reducing a compound frac- tion to a ijiii!i>lc one. VULGAR FRACTIONS. 33 If we multiply I by 4 we set X^ fn. .k We may first take the nrndiml ^f 2 ^ then take the product of.^Zt whfehls t "1"'' '^ ''-' -'x^.^^ie 2x4x2 "'""■'*'" 357 105 ~ 3 f; — 7 - hT4%;: 5Sa'/"y --"be; Of- fractious together, Af 1 • RULE. denominator. ^lenommators together for its ITir,^ ^i, EXERCISE. i^md the product of 2- T and |. ^- l» ^ and f °- iV»^and|. "• ¥> TT> I and i« ;:?;'°''^°''*--iil.f.«'if;-,«,. !*• H J and i + 1 , f f J' ° J. 3f and 5i. ' 16. J of C4+ •■>=?„ ,„ l^-i + sandl-i ' Rn/'the^Iu"r"'-"^i'--<'f<'fxV 20. 1x1- r^^- 19. I|x4^x 3 Proceed thus : O o - 1. land ^. 3. « and |. ^- tV, f and |. ^- i, I # and «. 9. 4^ and J^ IP 34 ARITHMETIC. ,; 1-Ji P( SI I'Jt EXERCISE—^ Continued J. In such cases as this the order olTeduction is as follows : 1. J^md the value of the fractions in brackets 2. Perform the nmltiplications indicated Thus'!!'''"' '^'" ^''^'^^'^'"•' ^'"^^ subtractions indicated. 1 I 4 1 () ^ a + 7 - 4 7 X ^« ^ 14* 25. 4 i X 3tV - 2^ X 1 s j. h 13 "• li><(i-e;)+A.ixG..+ ,»,) 2S- A+.VxSi-ax-H--^'). " 29. Multiply tlio excess of U over the sum nf i i i i, the difference between -4 of g and •* of s" "' » '"'" ^' "^ Cnotirr """""• ''''^"'^' "^y n -'« «-e 13i for :)• tlie difference between 4 of (li 80 What number divided' bv 7| will o.,Vp I'^i <■ quotient? ^ * 8^'™ l-^s for I I DIVISION OP FRACTIONS. If CO be divided by 12 the quotient is .5. one-half of 12 the quotient is twice 5; Z" 'n '' " 3 times 5 'I- «»mi,er bj- ;, and then nmltipl'y the resuU ^1. ""*' as follows is. Heated. VULGAR FRACTIONS. 35 Thus, to divide ^ by |, We first divide';; by 3, which gives -^ • We next multiply ./,- by 4, ^^ Vo ' 1 herefore, -^ wlieu divided by " J o-Wes ^ o ,« JI* .. or, as it is wj-itteu, * ^ -^ " '^^ *^® quotient ; Since -^^^x^S r^^"' :^1 ' 7 X .-i ~ 7 "^ 3 ' tiierefoi-e :'*-^^-''^^i 7 •4-7'' 3- RULE and i, by i 13 i for 'ill g-ive I is twice 5; 3 times 5; 4 times 5. U'th of 3 ained by f 3 is f; St divide 4, aivisoL and then proceed as in multiplication. EXERCLSE, 5. 7. 9. 11. 13. 15. 16. Divide ^' ff by ^^. 1« bv 1 •" <^' % 7«. 2o|by|o^. U by i of 6^. 4. 6. 8. 10. 12. f by ^ li by ^^. #^ by ,--•,. i;^ofl7iby- jlof.-Vby|^ofl^. To X 7 Of 14 by 99. 14 /V , 4 'Y\ x," V 7, n^- iof (0^ + 31) X (| of 5^) 1^ 4^1^^ ^^ ^ <>f (A-^). M ^»d i^l' by the product of Ui and Tlie product of ■•jfi Find tlie value of, 17. 19. 21. r. ^ 8 8 • Iff 9 9 !) _j_ 9 t) 10 • TOO' vof ^J-Jof^V- lS. 20. 1 1 _^ 2 2 '9 • .'is- k Of U 1 0* f.:-^^^ i ii i I ■ i 1 11 J i ; 1,1 •1 IJ! \ \w i 1 i-S i 36 ARITHMETIC. EXERCISE —f Continued). Proceed as follows ; 4^ v 3 2; • *T(r- 1. Find the value of the fractions in brackets This reduces the expression to 2. Simplify compound fractions this reduces the expression to ^^ ' 3. Invert every fraction before which the si^n- occurs • this reduces the expression to "^^uis , X -r X y-g X 18 V, 113 70 4. 3Iultiply the fractions together, and the expression be- comes =1. I' vu uc It may be noticed here that there is an important dif- ference between "Of " and " x " when placed between two tractions. The fractions between which ''of " is placed form parts of the same compound fraction, whereas, those between which " x " is placed, are two distinct fractions! means that « is to be divided by 'the fraction | of a that is by y% and the quotient is, therefore, ||. ; whereas means that |. is to be divided by f, and that the quotient so obtained is to be multiplied by | ; that is, we are to in- vert ^ and then multiply together the fractions « ^ and 1 the result in this case being ^. '* ^ ^' In the iirst case the sign -'occurs before the fraction J of ^- ; m the second case before the fraction |, so that in both cases Ave have followed the rule and inverted the fraction before which the sign of division is placed 23. Simplify | -=- 1 + •« x |- - ^ .^ f of 6. Here we first reduce the compound fraction ^of 6=^. Next invert the fractions before which -=- occurs. This reduces the expression to ^ V 1 4. i> V » 7 y a &' ^ if 1 -7- occurs ; ression be- ortant dif- ;ween two is placed, 'eas, those fractions. Pf, that is eas ! quotient are to in- I and i fraction 50 that in rted the 3d. s. This VULGAR FRACTIONS. EXERCISE— f Continued). m pre'I^ronSTcots"^"^'^^^^"^^^^^ '^^^- ^he ex- 15" + _ 7 24. 1x1/^^21x3^. 25. (34 + 51.--/ )x(4i 34)^ riJ"' 4-5)1 o <. i ,s 28. ^offX-^ ]|of9JV-f4^Sf =^^^^'9^ nfl.^ 29. ^ 2| /| of (f . 3i : if^l/f^T^^.^^ ^^> COMPLEX FRACTIONS. Since every fraction expresses the quotient of t^o of a fraction. ThusTwe^Th To SItVL' tf™ may and.eate the operation in the fo„rof : ?JaotiL 'hu7 ■'5 7 (f i !» 5' The same method is emrilmr^ri ^^ • j. of one fraction by rnothe?- ?r 4 '".'""I'o^'^ the division form of a fraction tZs: ''^^ "^ expressed in the !i 4 ^l«o(| + |).|of Gj^maybewritten I of 6-^ catllt this' w1f".hf rn,[r"™ V -other is indi- Complex FkaSv .n,i f, .^ expression is called a called respectlvTthr.u.ne'i-atr/'r,'""'* ^" "^^ •'^'•« complex fraction: 'Ais n tl e Tl ^'«".»'"'"^""- <"'"=« called th(Miiii.i..,-.,t-,.. . '^''"^e fraction J j- a ig - -n.. nuniciatui, and 5' „f oj ti,e denominator ° I n I i m 38 ARITHMETIC. Complex fractions may l)e reduced to simple ones bv performing the division indicated thus, ^ ~ -J (T ~ "5" _ 12 -1 V 5 = -'A M 8 or by first reducing the numerator and denominator to simple fractions, and then performiuL^ the division nl ' 1 I of «! ining the division thus, 217 2 0"~ rr -■-* X -''^ 2;{ Every complex fraction wliose numerator ;ind donomin 1/ i = ii X :? ^\ vy '> o X .> 7x2 prod uct of highest aiKUowest numbers, prod act of the other tvyoiuim be rs. Simplify EXERCISE. 1. 4. 7' 8 ir i> 7 7 6 3 ,^ 1 4 u . 2 U • 2/ 7| ■3 '■ + 6" of (; of 1 3i •> ^ f "• i + J 8 XXr^7 10. !• T^ a -^ 7 ()■ 1 2 2 U. -±JL, •■• ^ - ^ ■*" i> S ff 7- +,^ G. y. 1 of i X 2 ^of3| f ^ 4- — 4 + 7 12. 14. 15. 16. 17. 18. VULGAR FRACTIONS. EXERCISE—^ Confhved). 4 !> 4 13. 7J 94 '+111 7|[_on2f 2i- of 4X '"2^oFl5f-3x^~^, H±8K 2^4-^ 45|. "^ 1 (3i + l|) + i of ii-_i^-ii^ «•'' of 4 33^ 3 + -^ 1 « 20. C + ^ -j7 19. 21. + 2-2, 4- -^z 23. Tg- 39 24. 4^}fx^-^lii::4]£x ^^ ^ m GREATEST COMMON MEASURE AND LEAST COMMON MLLTIPLE OP FRACTIONS. The a C. M. of 9, 12 and 1.5 is 3, and the G C M nf q quarts 12 quarts and 15 quarts is 3 quarts In the same way the (>. C. M. of 9 se'venteenths V^ L^r r!"r'^r.^^^^^^^^^^^^-^^-^^ ^ seventeS.'''Ti^:t TT> r and I » is v*^. TT* \ 40 AIUTHMETrc. So that if any number of fnictions have tiie Hame de- nominator tlieir (}. a M. is found by taking*- the L. 0. M. of their numerators and placing under it the common denominator. If the fractions have not thesamedenominator they can be reduced to eciuivalent fractions whicli have a coiiimon denominator, and tlien their G. C, M. can be found in the manner already stated. Jf any 7iumher of fr act torn in their lowest terms, hav- t?ig different deiwminators be reduced to equivalent frac- tions having the same detiominator, the G. 0. M. of the numerators of the neio fractions is the same as that of the old. First, take two fractions, \i ana -%. The equivalent fractions are " That is 4.1 a „«^ 3' TiU '*"'^l TsTjr 14x8 21x3 i-F~- Q and .- — - n 15 X 8 40 X 3 Where we see that the first numerator is multiplied by a factor of the second denominator, and this factor cannot occur m the second numerator, since the fractions are in their lowest terms, and therefore the factor introduced into the first numerator is not one which already occurs in the second numerator, and therefore the new numerator of the first (112) and the old numerator of the second fraction (20)^ have no new common factor ; neither, by the same reasoning, will the new numerator of the second and the old numerator of the first. Lastly, the factors 8 and 3 introduced into the numer- ators can have no common factor ; for if they had 120 would not be the L. C. M, of 15 and 40, but 120 divided by the common factor of 8 and 3. Hence, we conclude that the process of converting two fractions to others having a common denominator leaves the G. C. M. of their numerators the same, and this will be true of every pair of fractions if there are more than two. *rom this it appears that in finding the G. CM. of frac- Ik! H'cimo. dc- tlio L. C. M. Iu3 c'oniinon tor they can 3 a common mnd in the terms, hav- V)alent frac- \ M. of the f that of the Itiplied by Litor cannot ions are in introduced y occurs in numerator the second neither, by the second he numer- y had, 120 JO divided irting- two tor leaves his will be than two. M. of frac- VULGAR FRACTIONS. ., find tlie G. C. M of S n 7 ''"""""na'or and then we can Henee we Z^lt^f^J^^^'^'^'^ '"^tead of the net RULE. For findinL'- the (I r m <• in their lowest terms'. ^"^ """^^«i' ^f fractions, Find the G. C M ni' ^y.^; place the L. C. M. oflhlirden^Sr. "'"'' """'"■ ""^ TO .rND z.. c. M. OK ..^ ™,„IONS. d^xroJht^Ts :'reM'::e''r'r' "'"^'^ ^-«- Thus : — T ^ ^ s= 3 w 6 "^ ^ y\ -fr Tl 2 ' "5* dividrg fraetionslmrthSr l'c TT'^T "'"'^'"^ '•™™ arising from the division of the ;5' "^ *?* '""« as those tions by the reciprocal of their I ^o'lJ' ' "^ ""'^'^ ^'^<^ fraction less than the L C M wni ' "•' ^"^^ ^ince no follows that no fraction grl^^''^**^ '^"^ ''™««°"^. " the reciprocals of these fractions are TheG.C.M.ofthesela!;tLons,is ^C.M. of 3, 6 and 9 ^' ^- ^^' of 2^51^51 ¥■ 1 1 i 42 ARITHMETIC. ,, (I aTidl) .•• tlie L. C. M. of . „ , Iff and ^^ IS forincrt bv tHi<:iiiL'- the ?. P at ^^ n • Placin, „„cie,. it t„t <1 Ci' M^^f t, L't^^^Sr:" """ Hence we liavc tlie folloAvino. '^"^HiinatoKs. RULE. ^^IECIMALS AND npPTur it t.... ^»«JJ i>hCIMAL FRACTIONS. 43 I'V 3 fa one-tenth of o o- el tl I , '"f'Tf ' ""'' '»«"»'«d nnits are therefore call,.,l t^drX./' "^"■"'i^' '"ese one:!,u„a,.;'Sh ^"•^' ''"" "'"•' "«"»'"' by 5 is one-tenth of wonhl denote? to, m'«Sa 'a "'S "^"« "^ "■« 5, thev ~> s., What n.nher thetoup'X^-rrXntr if 4 be taken as the uniK-wflit .i 3 tens, 4 units. 2 tonSs ^,?I k'^ ^,''""P ^"' ^present t|tken as the units', S To L^''"^"^'^' ' ""'if 2 be placing a dot afte it tllus Ti1'f« 'l""' t"" "'"ers by wMe in 342-6 the units' dig!" is 2 ' '' ""* "°"^' "^^"^ ?; [^^ t'S '"""*^ """>hermu: J'lacc, tliough not necessarily in l/'/il 4 i 1 fid 46 ARITHMETIC. te .u U^oi, r r '' "''■;'" "^•'^'■>' "''"""^l '""^t have a not m.t .i ",' ""' i'l«° '' 'I'ffit in the tenths' place, though Wc"m X -I i" I""' °^ "undrclths, thousandths etc"^ vvc might in fact go a step furtlier and assume that ■!« t'St^t h'^^'i d' """V"™ =' "'^'^ '■" 'hrrts'pl let s 1 rthr,^'..niH ■,'"'•'' '?"" "-^P'-c^s^l orundemood o tit nni£^« " '''*,'""»' I'"'"' '» '''"'-^V's to the righ .5£^s .:^2:^t,::; dSsr^h'rnT- th: •3487 • l()-31)t; 3-043:>4 74G-28 ' 70(5-06794 t)oinrThl7J'-'l ^'''''' '"^ ^^'' ^^^^^ P^^^^^' f^'^^ the decimal bSmv tt lirro '' :;"f "^'f'' t' ^^" *'^^^^^'^^'^ ««t it down Pi ee we find 7 ""' ^"^ ^^^^ ^'f ^-thousandths. In the fourth place we hnd 7 and 2 and their sum 9 is also set down -is L78''Tndf 'h '" ^'^ ?'^^^^'^ ^^' thousandth? ptle w f mndit; H? 'i r\ "''''' '' ^^' *^^^^t is 17 thousandths or 1 imndiedth and 7 tliousandtlis. We therefore Dlace thp ImnClths r T h'' ''^^'f' '^'''' ^^^^^^^« ^q^^'-^l to 26 mmaicdthbor 2 tenths and 6 hundredths: we set down are^^^^^^^^^ '''''T'' '" ^*' '^''''''^' ^^^^t .s 10 tenths ciicc(|ual to unity we i)h,ce a under the column of tentb^ and i^serve the 1 to be added to the column o? units wh^h Add toij^ether EXKHCI8E. ?• f;i ru>'.r'^^' ■^^'"^^' i-^-^ and 1-4142. 3. .-141.59205, 2.500, '4771213 and -000048 ..>nin-> in r.,. -- '-00004848. 3010o, UX -5230, 140-11 and 2'71828i8. •1, -901, -0001, '000001. -0000091. nust have a ace, thoug-h ths, etc. me that, as iiiits' phice, -inderstood, io the right me method under one nber in the ;ive points. '8, arrange le decimal et it down the foui'th t down as ' place we iandths or place the idd to the [ual to 26 set down column of 10 tenths I of tenths lits which Edition is MO.MAT.,S ,N„ mCMM. FUArnoN.S. SUBTKACWON OF DECIMALS. 47 r)7-2!)577 armnging- the munho.vs ~wHr^7Z~\ as m addition. '' '^^' '^^""^'^ to tlie decimal point to t^^'^l^^ll!j.!l;^~^^^^--t if ciphers be added 57-L>!)577UO() 3-141592(;5 .,, ., . ^'1-15417715 M,TI.T„.,KV.TIO.X OK OKCT^fALH. Take tho mimlx* n-mZ '" '""'"I'Mod hy lo. ono pl,u.c to the riij-ht and wc '^If!',';;""^" the decimal point n . 670 -40^ Comparing- this with tJie orio-f nT. i ' 8 ten-thousandths V. '""''''' -« observe that 2 thousandth" ^\f ^'^'^?^"^ t thousandths. 4 hundrcdtlis u ,, ^ hiindredths. r> tenths u ,, ^ tenths. 7 units ,, \\ ^ units. 6 tejis << ^" <" tens, so that each dig-it roDresonf > T .^' ^^""f^i'cds, new number as hi the o d ^^ I? 'T'' '' "'^^^^^ ^^ tho ^« ten timcss the old one ' "^ '^''"^^^ore the uow number cwt^^r^Hf ^^"T^^ ^- P'-« to tho .i„4,, . . 3 Places. u;^)t:i^r:-^;^^^^^ !.. i 48 ARITHMETIC. r.io T u ^^ evident, too, that if we move the point one pace to the left Ave .hall divide a, number by 10 if two places, by 100; if three places, by 1000, etc to the left by placing ciphers between the point and the nearest hgure, thus: -67 when divided by 10 becomes Obi J and this when divided by 10 becoiiics -0067, and Biu on. ' Suppose we wish to multiply 37-643 by 3-86. We have 37-043 = 37-^^^ '"^''^''^ and 1000 1000 ' 3.8fi ^^^' '^^^' '^ ^^^='^100= 100 • Therefore 37-643 x 3-86 _ 37643 386 I ~ 1000 "^ foo 37643 X 386 100000 145301 !)8 100000 = 145 -.301 1)8 Now in this result we observe that if 1;he decimal point be omitted we shall have the product of 37643 and 386 which are the multiplicand and multiplier with their deci' mal points left out, and that the number of digits after the decimal point in the product is equal to the sum of those in the multiplicand and multiplier, there beine- 3 in the multiplicand, 2 in the multiplier, and 5 in the product Hence, we have the following- RULE. To multiply together two numbers containing'decimals Disregard the decimal points and multiply as though the numbers v/ere whole numbers. Then point off as many decimal places in the product as there arc in lioth the multii)licand and multiplier. It will sometimes be necessary to place ciphers to the I MCIMAM AND DECIMAL FRACTIONS. 49 10 10-100= 06- burl'hc";!^? rclire'sT' '/""^'^'^ "^ °- y243andbvl77U7 ' " "'' •>>■ ■'59-049, b,vl9-(i83, «• iC-SOl by 3430. ' ^ ^'*°^' ^-y 2401, by .0034.3. 3430)10-807(4 _13720 "087^ •004. ined by ing the lumbers 3tient is uber in in the i point tl to the in the vidend be no • -064. 9-G83, r -625. )0343. DECIMALS AND DECIMAL fRACTlOKs. §1 shouldTctr" "' " ^^'^^'^^ '"'^'^ ^- ^ho dividend we 3430)16-8070(49 JL3720 30870 "^^'-^ 30870 It follows, therefore that if in \ ^ ^• does not terminate c pheA mnV? ^''L ^i'''^ ^^'^ ^'""^^'^ou the dividend unt 1 tl^divi^ioTdot'; f ^'^ '' '''' ''^^'' «f sufficient number of dilTts in th ''""""^*« «i* ^"^til a obtained. ^'^' '" ^'^« quotient has been Divide i. s;{5;.r»5 s*' '»■»■ .f. 1771 -SI)! by 1-4641 in 'u^S.^ ^^^d^ rhe^^nr"^-'^^ ^^^-"-' P^-- may add ciphers To tL^r^^^^^^ ^^T^^-^d, we number of plaees equals the nmiUs in ^I^^^ until the then proceed as before ; thus- divisor, and l-4641)1771-5611(l-)io 14641 ^"30746 29282 T464T 14641 f>nd the quotient is therefore 1210 Divide 10. 16796-16 by -006, -216 •V>9a .m-7r • ?^9-f V7.776; .01296,'-:^:;65^ 12. 1 by -1, -01, -001, -0001. 13. 1 by 2, 5, 8, 25 .% 64, 125, 256, 62i 52 ARITHMETIC. liowevei- t . IL r^- ' ''"'''' ""' g-eiiorally liappei, ^i'r ^;i^^f ^eSf ^V'^^ ^? V ^^--• quotient is to com^dn 7t^^^^^^^^^^ P^^^^^ ^^^ the as it contains Tu Ctwt dphe^^^^^^^^^^^ bH^r^ ^" J' ^^^^ process of diviclin,,' will S fo^^^^^^^^ .^' '^^^^^' «^ ^^at the 2-718)3-lil5!)00(1.1558 2710 1285 2718 1517T) 1351)0 1581)0 , lf^590^ 23000 21744 Divide to 3 places of decimals : 14. 3-1416 by 57-20. 15. 57-2957 by 1730 1<). 2-4495 by 1-4142." And to 5 places ; 17. 1-09801 by 2-302585. 18. 1 bv 2-31258. 19. -30103 bv -43429. 20. 113 by 355. TO CONVERT A DECIMAL INTO A VULGAR FRACTION. Since -9375 may be expressed as 9375 ,,, 10000 V^ e may reduce this fraction to its lowest term^ in fi.. manner already indicated. Thus, if .'uHWcl^ll^lLe^i:!:: ly by 5 we obtain ^:l . IG Similarlv -5104 -.^_ ^ ^^9 10000 625 ' DECIMALS AND DECIMAL FRACTIONS. 58 KXERCISE. etc., have .^o ^.,^C[ S ^^^l^^^'^^^ i^OO, TO OONVKHT A VO..AH .KAClON TO . ..eiMAL. We may divide 11 by 16 thus: H5)li 0000(6875 9 6 1 40 1 28 120 J_12 80 80 "* ^■neethe quotient obtained by dividing ii hv ,fi • expressed by the fraction ii it mZ. tj ' " 11 j-g = -6875. This excnplifles t„e ™et„ DEUIMAI, rRACTIOXS. S6 fraTuonrwl.iZrl^i-r't',"'; ""'' "" """"^ "''"'^ vulvar Since ■> i, a faZ of 10 '' "'''""'' '"""'' of decimal.. Therefore 2 x 2 is a factor of 10 x 10 There ore 2 .2 .2" ., lo. 10x10 ^^Ihat >., any power of 2 i« a factor of the same power remainder 'I'^iMble by any power of 2 without Ce^r intVr : ~Tf t"^^ «- Stion whose decimal. ' "^ ^ <='" Induce a circulating And the same is tnie if power- non-t:rmtLing dec,- 7)3-0OOO0OO0G0OO ^ , •428571428571 # vem:&iSrar '^ '" '■* ''"™"'"' ^ «"'! «-' *e -. «. -i, r>, 1, 3, ■i hf V 56 ARITHMETIC. and the .livi,lcn.ls must therefore be .after the first) 20, 60, 40, 50, 10, 30. wiU^TS Ve'':tired'"'"^ '' ""^ ^"""^ - *-« dividend s.moT^:^!!:, qi'S'r *""""'=" ''•"''• ^ -"< "« the samerr fltt rr^- '"^''''"^" '■'-' '« -"■ >>« the Therefore the next dividend Avili i... fi next to the first, and so on '"" '^^"'^ ^« the one Therefore the second set of fi i^in-u • ., be the same as the first set of 'S" andZt:'""^^ reason the third set will be the sin^f f« Vi? *.^^ '^"^® Thus we spp tv.of fT. \v- r'^.^'^^e as the second. in the same order " "'^"^ '" '''« 1"°««"' ™«^t recur c.i.4^e^';:^iLtti?i,']U«I- - be o„,y 6 diffits in the repeati, g par oTthe 15 ''1 "^S ''" °"'y « the denominator of thf Ction 1 7?h'"''"- ^'^'i'^'-'y. if digits in the repeating. paM-if It In' "'"" ^ ""'>■ '« only 18 digits, and so on ^Vti . *u o i-opeati„, pfr Co'X:tnrH;!,f ^<'^<'- -- 9*23 3. 24-3 ~8 = •e - -375 = •291666 . = •2916 131 2 15-5 -"3 = •2 4 -6 = •86 -•375 5a ARITHMETIC. m ••■ '^' 274 7 • 275"25"^li N()TK..-lf oitlKT of the Im-torH 2 or 5 is common to both terms ot any fraction it in <,l>vious that tins i^^Z" be strucic out ])efore applying- tlie test. Tiie reduction of a vul^^ar to a decim.-U fraction may generally be mucli sliortened, as in the following c^xLmpE 1. Reduce ./^ to a decimal fraction. 19)70(-3C,84210520.31578 67 IM) 114 152 "80. It is evident that the result obtained by dividinL" 10 into HO Will be half that obtained by dividing 19 SlfiO 'T may therefore bcffin at the tig-ure obtained from 160 as dividend and divide by 2, and continue this operatbn Thus we shall divide 2 successively into 8 4 2 1 10 5 12, 6, etc., and obtain the quotients 4, 2, 1,' 0,' 5, '2, 6 etc. ' 2. Reduce || to a decimal. 13)120(-li230769 117 3. Here since 3 is ^ of 12, we may commence at once and divide 4 into 9, 12, 3, 30, 27, 0. Thus, whenever in the operation of reducing- a vulvar fraction to a decimal, we reach a dividend which is -in exact divisor of any previous dividend, we may commence dividing at the digit obtained as quotient to that previous dividend. ^ ° i'i DECIMALS AND DECIMAL FRACTION'S. 69 KXERCISE. Rc(luc') to chicinuil fnictions : j^ 1^8 12 10 (5 8 1) 8 • * Vy Vf 17' 17' f3' 23' 21- 2 y 1'^ '^ •'^*' 1 !•> ' 23' 2U' 31' rr U' 53* 3. 80 17 7(; 100 08 1000 JU' 63' 77' 251)' 181)' 28li)' 1001* If we reduce 1, ;•, i?, i, y, y to (ludmal«, we luive 7 = •112857, ^ = -428571, 2 r 7= -285714, ~= -857142, 7= -571428, -^- = -714285. Ill these cases the siiiiu! di^n'ts occur in all the decimals and in he .tnie or-ives the result '00571428. From the value of -^v S'iven on page 58, we can find the value of any other proper fraction having- li) for dcnomi- na,tor. Take for exauiple /v, : On dividing, the first digit in the quotient is 1, but as 1 occurs twice in tlie repetend, we must obtain by division another digit in the quotient ; this is 5. Then we are at once enabled to write down the remaining digits of the repetend, namely, 781), etc., and therefore the required result is • 15789 173G8421052()3. TO REDUCE CIKCULATING DECIMALS TO VULGAR FRACTIONS. We know that 2 . - = '2 9 3 . Therefore, a pure decimal in which only one digii re- peats, is reduced to a vulgar fraction by placing 9 under the repeating digit. Again, 99 2 99 = •01 = •02 i' r tbn (Ic- (Icciiiuil ibrc I is 1 at once result is ; there- lal i)oint ,}\m six ves the find the icnomi- , but as iivision c arc at 3 of tlie eq 111 red .CTIONS. igij re- ' under ( y; It DECIMALS AND DECIMAL FRACTIONS. 61 Now suppose we wish to reduce a decimal in wliich two digits repeat, to a vulgar fnictioii, for example, -37. •37 = -37373737 = -30303030 - - - - + -07070707 -. = 10 X -03030303 . + -07070707 - - -. - = 10 X -03 + -07 == 10 X + --^ 1)1) !)9 30 7 ~ 1)1) "^ iTi) _37 Therefore, when the repeater consists of two di-its the corresponding vulgar fraction will consist of th "se two digits for numerator and two D's for denominator. Again J^.-OOi, A = -602, ^c. Therefore •()73 .: •G73G73G73 = -GOOGOOGOO + -070070070 + -003003003 = 100 X -OOGOOGOOf) + 10 X -007007007 + -003003003 = 100x -bOG + lOx-OOt + 'OOfi •- 100 X -1'— 4- 10 y " a. ••! !'!M. ^-^'-'^ irT.l) + U!.-lr _ .[! , 7 , :; !• !• it ~ U n !7 "I* H'ifW __ 35, •35i4L>857, 7, .0;](; .u- vHum^r, tlsT^ '' ' ''''''~' ^-"^ ^^-^ ^^- - •235;]oa •mill F7~7TG58 The difference between this result and the true result is ess than one ten-thousandth. That is, it differs f mm tho tnic result by less than 1 in the fourth decimal place! EXERCISE. Add 1. -7, -73; .810, .|i342, •(;3205 to 4 places. 2. -Qi, 'Oili, -0067, •1)37(5, .0400 to 4 places. 3. -7342, .1)4705, .3704i(}, .fOOioOO, 38.74 to 5 places. ■1. 100-36, 1)37.4, Sy-OOl, 387-ir)<)i, 101. l6i, -' go; )-i 000 J too places. 2 When absolute accuracy is re(|uired \/e iuay either reduce the decimals to their e-juivalent i> . DECIMALS AND DECIMAL FRACTIONS. 65 viilgcar fractions and then add— which is apt to bo a tedi- ous operation— or we may proceed as follows :~ Kequired to add -30, -539, -032437. First make the repeaters all hegin^t the same distance from the decimal point. Now, since -36 = -36363036 any of the following- ways : •363 -, it may be written in •3636 •3636 •3636363 •363636363, etc. For all of these, when the points are removed, ffive the same result, namely, ^ •3636363636 Thus we see that the place where the repeater beg-ins may be moved to tlie rig-ht, but not to the left. We must therefore select that decimal which has its first repeating others ''' ^"^ ^^'"^ '*'^'^'*' ""^ '''''' ^""'"^^ '"^ arranging- the This is -032437, therefore the others become -36363 '^n<^ -53939. Secondly, we must make the repeaters all end at tlie same distance fi-om the decimal point. We see that the llrst of these must take some such form as, •032437 •03 M37437 •032437437437 That is, the number of digits in the repeater must be a multiple of 3. Similarly in the other two the number of digits in the repeater must be a multiple of 2, and since 6 is a multiple '■'■.'I. 1, 1 :ii •- 66 ARITHiyrETIC. •032437437 •363030363 •530393039 •935467746 to Zlr^:: t"?" ttf/ oo^"" T'"-"-' -mcmbcrins carried as tl e ivsult 'Ti'" """ ''>',."'« "^bcr to be be to the right of tb if^w "*^- "'? "'^'"^ ^^•'*'> ^™uld ifeiii. oi tins It the decimals were extended. Add ^'^^'"='^'=- 1. -3, -265, -037. 2. -36, -ael, -9873. 3. -OOOO;}?, -4038, -ti, -27. 4. -398-157, -00375, 'ooois. M^LTIPLICVTIO^ AND DIVISION OP CIROtTLATIXn nKC.™u.S. 1. Wlien absolute accuracy is reqtiired ^ J^dace the decin,als to tLir eq'uiv^t.t v,.„„r frae- Perform tlie required operation : 'mo,!"", ™f ""'■"«' '■'■"^"■°" to » decimal. 2. When .absolute aeeur.aey is not required me LTo'f'rtr'S^i'mni'iliT'''^-^ ""^'''^ ^^-'-' ">■ «'« mals oxemS below. ''''''"''''°" """ ""'^''''"^ "^ 1««i- CONTRACTED MDLTIPr.rCATION OF DECIMALS d-.^.? Thoiilldi^::; t;;;L;^,.r ^'r^^^.^^- ( .- iuA\ piotiuct cuiiicjuu \vit,li the old? DECIMALS AND DECIMAL FRACTIONS. 67 2. Obtain the sixteen partial products whose sum forms the complete product of -GTSI) bv -5432 ; set them down in order of magnitude, beginning with -30 and add them to- gether. What digits multiplied together aflFect only the first four places in the product? Only the last four? Find the sum of the remaining partial products. What decimal / places does this sum occupy ? What products could be omitted without affecting the first four decimal places of the result ? 3. Multiply .5G78 by -9243, obtaining only those partial products which give digits in the first four decimal places of the product. What digits have you multiplied by 9 '^ By 2? By 4? By 3? What pari of the multiplicand have you multiplied by 9? Set it down and multiply it by 9 m the usual way. What part have you multiplied by 2? Multiply it by 2 in the usual way. What part bv 4? By 3 ? Multiply these also in the usual way. What IS the denomination of each of these four products ? Add them together. How far does this sum coincide with the complete product of -5078 by -9243 ? Explain why this result is not correct to four decimal places when none of the omitted products give digits in the first four places. 4. Explain the following operation :— r>(;78 _ 3429 51102 1134 224 15 •i)247r> Why is 2 placed under 7 ? 4 under 6 ? To how many decimal places is this result correct? 5. iAIultiply when the numbers are ari-angcd thus • r.G78 3129 Note.— Tn this case there is no digit in the multiplier l)y which 8 can be multii)lio(l. nor any di<>-it in the mnlti phcand which can be multii)lied by 3. The products 8 by 68 ARITHMETIC. 9, aiul 5 by 3 which occui-rcd in the previous result will tiierefore not appctir in this. AVhat other products have also been omitted ? To how many decimal i)iaces is this result correct. 0. If the multiplier had l)een 1-9243 where would the 1 have been placed in tlie process of multiplying? If tlie multiplication of 38-3I2G5 by 7-8 III* be arran^^ed as ibllows, how many decimal places will there be in the product ? 38-3 12 OS 6487 2683982 30()73() 15336 2298 * Under which decimal place in the multiplicand is the units' di^rit of the multiplier placed ? Where should it be placed to ^ct three decimal i)i{ic(;s in the result? To get live ? Obtain the results to three, four and live decimal places. Find, by comparing it with the complete product, how far each of these results is correct. Generally, unless the multiplier and multiplicand con- tain many digits, the product will be correct in all but the last two digits, so that if the result is re(iuired to be correct to, say, three decimal places, it will be necessary to multii)ly so as to obtain five decimal places in the pro- duct. That is, to place the units' digit in ilie multiplier under the fifth decimal place in the multiplicand. EXERCISE. 1. Find the product of 47-3846 by 3-14159 correct to three decimal places. 2. Find (1-06)5 correct to two ])laces. 3. Find ^186-75 x (1-07 )-i correct to cents. 4. Find the product of 1-07, 1-035, 105, 1-025 correct to four places. 5. Find ii^597-67 x (l-OS)"' x 1-025 correct to cents. 6. Find 37-25 x 37*25 x 3-1416 correct to three places, 7. Find \ x 3-11159 x (2-37)- to four places. Ji DECIMALS AND DECIMAL FllACTIONS.* 69 8. Find ;» x 3-1416 x (238-r))'' to four places. 9. Find 57-2!»r)8 X 3-11 k; to three places. 10. Multiply l.'}-tlL>8r)7 by !)•!>; 17-87 l)y 385-04397 ; and l)-28i')7 by -0139, each correct to four places. CONTRACTED DIVISION OF DECIMALS. Divide 2343987G0 l)y 23450789 to six digits in the (piotient. Omit the 9 in the divisor and divide to six digits. How Oir docs this quotient coincide with the first ? Omit the 8 also and divide to six digits. How far is the ([uotient correct in this case? If you divide by 2345r) instead of the complete divisor how far will the quotient be correct? ' What rehition do you notice betAveen the number of digits in the divisor and tlu; number of digits to which the (luotient is correct in each case ? Note.— If in any case we divide by, say, the first seven digits of the divisor, the (lUotient will generally be correct to at least five digits, and may be correct to six or seven, 'rhus in every case we liave oidy to take as much of the divisor as will contain tAvo digits more than the number recjuired in the (luoticnt. Divide 2-718281828 ])y 3-141592(;5 correct to five digits in the (luotient. Take the first scA'cn digits for divisor thus : — ' 3141592)2718281828(8 25132730 "2050082 As we have now to obtain four digits in the quotient, we divide by six digits of the divisor, thus .-— 314159)2050082(6 1884954 T65128 Similarly, we next divide })v five digits, thus :— 31415)105128(5 157075 8053 70 ARTTIIMETTC. in Next dividing similarly by four and tliivc dibits of tlio divisor wo ^vt tlio other tAvo dibits of the quotient, Avhieli is therefore •Hf.oL'r), the position of the decimal point being determined h^ ti,e same way as in ordinary division of decimals. The complete 'Operation may be arranged thus : 3-14ir>J)2n5)2-718281828( 525 08 25132730 2050082 1884954 105128 157075 "8053 0282 1771 t 1570 201 A convenient arrangement is to place each digit in the quotient under the right-h;md digit of the divisor used in obtaining it ; thus, 8 was olitained by dividing by 3141592, and is accordingly placed under 2 ; is placed under 9 of its divisor 314159 ; and so on. This enables the multipli- cations by these digi-ts to be rcidilv effected, and also serves to indicate the next divisor to be used. The quo- tient, when ol)tained, may bo arranged in proper order in its usual place, and the position of the decimal point deter- mined. f:XERCISE. Find each of the following to three decimal places : s.. 30__ 7000 .ox ^•■^•'^<''> ^ '' 1-00017' ^"^8-i0834' ^^^ 58^995 t 29-995__ _ 13031-301 ^ 1-001294 ' ^'^^ 1002-700 ' ,^,- 110350-81 „^ 1-04315 ^') 9-815 x-7iM^'^')l.oot2x252-?8G; >) 19-9508 X ^^^^-x '^^ X ' POWEIIS OF NUMliEllS. 71 its of tho t, Avhic?h nt being isioii of t in the used in 141592, [er 9 of Liiltipli- id iilso he quo- rder in t dvter- !S : ... •r,9;u.i7i ^ . , (^> 2-30258509 '^ '''' J''^""' ' ..^,1-0986122 ^ . , (^^) 2^30258509 '" ^^^ 1^^^^^"^' (11) 1-^2-30258509 to eight places. POWERS OF NUMBERS. I. Write the following products in the exponential form, and find their values :— 1. 2x2. 12. i-i- 2. 3x3. 13. 1 X 1 T T* 3. 10 X 10. 14. a ^ 3 4. 17x17 15. 5x5x5. 5. 20 X 20. IG. 10 X 10 X 10. 6. 100 X 100. 17. 6 X G X 6 X G. 7. 300 X 300. 18. •1 X •I X -1. 8. •1 X -1. 19. •02 X -02 X -02 X ^02 9. •4 X -4. 20. ii V i^ V •' V •' V ■' ■^ y\ ■„ ■'^ -^^ '^ "o A ,y. 10. •06 X -OG. 21. Ifx^lfxlfxif. 11. •235 X •235. II. Find the value of: — 1. 4-, 5-, 23-. 2. -3-, ^04^ -0162, .5182^ 4. (-21689)-, (-0l:G35)-, (4-623)-, correct to four places of decimals. 5. (10)-, (100)-, (1000)-. 6. 7^ 14% 163-'. 7. -1% -02-, -00013. 9. (%932J ', (•02475)-S (lG-4CJ5y , correct to five places of decimals. 10. 42 x33, 17- x53, 122x10'. 11. 7*, 3°, 2^ 12. (1-046)9, (2-0482)', (1-OG)^ ', correct to four decimal places. 72 ARITHMETIC. i m i It' ROOTS OF NUMBERS. 1. Obtain the s(iit/irc roots of the followinff • 1 4 16, 25, ,30, 4«), (51, 81, KM), lU, VM\, 2r.(;, .324, ivl ^SL 3bO0, 8100, 16 X 2') X 4J), 04 x .3(> x 100, 040000, 81000000. I 2. Obtain tho cnI)o roots of the i'oll()\vinir:_i g 07 C4 ' 12.5, 210, 34.3, 512, 72!), 1000, 8x.3l3, 27 x 512,'l2.5''x VW 8xl0a), 12,5000, 512000, 27 x 210 x .31,3, 27 x 720 x lOOo' ,8000000, 721)000000. "3. By resolvin}? into prime i'nctors, ol)tain the square roots of the following- :— ,30, 04, 100, 570, 72i), J024, 12!)0 1701, 0100, 108!), 57(500, 040000, ni;;0OO, 78400, 11025. ' 4. Obtain the cube roots of tiu! following- in a simihu* manner :^8, 27, 512, 1000, 210000, 27000, 1728. 1)201 1331, 1331000, 27000000, 42875. 5. Obtain the lifth root of tlie following- in a similar manner:— 32, 213, 3125, 1024, 7770, 3200000, 4084101. 0. Obtain the square roots of the following- in a similar mauiier:— k i *' ./j' «j__ l 1 () (! 4 ;0' TT(r2 5» 4 48 •01, -04, •09, -64, ;)-70, 10-24, 17-64, 110-2.5, 1-102,5, -,^^^. 7. Obtain the cube roots of the following in a similar ni'?"? V;r'; fi\ ^' ^^^' '^^'' ^^^t, ^.^v^, -ooi, -008, •004, 1-728, 1-331, 9-261, 42-875, -000001, •001728, •012875! SQUARE ROOT. I. 1. Multiply 35 by 35. 2. In this example, how many tmifs are there in the product of 5 by 5 ? 3. How many units in the product of 3 by 5, and how often is this product found in the whole operation ? 4. How many units in the product of 3 l)y 3 ? 5. What do we obtain if we add the results of examples 2, 3, and 4 together ? 6. What is the sum ? 7. If the product of 3 by 3 is sulitracted from the proper part of this sum, what is the remninder? 8. Could any square integ-er g-reater than 3- or 9 be subtracted in a similar way from 1225 ? HOOTS OF NUMUERS. 73 -1, 4, 9, (')! X 81, 0U()(JO(). \ 27, 64, 25 X 721), D X 1000, ) S(|U}irc M, 12I)G, 1025. siiiiilai' ^, !)2()1. similar 4101. similar •01, -04, similar )1, -DOS, 0^2875. in the id how amples •m the r y be TTnw tlioii cMii the tens dl^nt of the square root of 122i» he lonnd ? , U). llow are the (li^rits .'} ami 5 eombincd to form the rem.'iinder m exaniph; 7? 11. If twice 3 tens is dividcul into this remainder (since twice 3 tens x 5 forms the li^reater part of it) ; what is the 12. What should Ix; julded to the divisor twice 3 tensor GO, to make the division exact ? 13. Divide G5 into the remainder of example 7, and find the nnnainder. li. As tills remainder is 0, what has been subtracted alto^-cther from 1225? 15. What is the t(Mis difvit of tlu^ square root? The units di^-it ? The Aviiole S(|uar(! root ? 16. How was the Wm di«-lt of the square root ol)ta.ined ? Ir. How was the units dj^-it of the square root first ob- tained .-^ '' 18. AVhat was added to the trial divisor of example 11 to give a divisor which would dividi; exactly ^ 19. If the number had been 1230, after going through the sjime operation to find its square root, what would the remainder be? 20. AVheii we obtain 5 for a remainder, what has been subtracted from 1230? 21. What is the number subtracted from l'>30 the square of? 22. Extract the square roots of 625, 631, 961, 970, and give the rema.indor in each case. 23. In an example in lo)?g division, what is used as a tria dmsor to obtain the ditfercnt digits of the quotient? ^4. IJocs the trial divisor always indicate accurately he number of times the whole divisor will be contained in the dividend? Why? 25. In a case where it does not give the rig-lit dioit what is your next step? ^ "" ' 26. Should the trial divisor in example 11 always ffive a^quotient which will hold when the complete divisor is 27. When it does not, how should you proceed ? 74 ARITHMETIC. II. ' 1. What is the square of 3 ? . 2. What must a number be multiplied by to increase it ten fold ? , 3. If 3 is increased ten fold, how many fold must the square of 3 be increased to obtain the square of 30 ? 4. If 30 is increased ten fold, how many fold must the square of 30 be increased to obtain the square of 300 ? . 5. If any number is increased ten fold^ how many fold must the square of tlie number be increased to o])tain the square of the new number ? G. How many di^nts in the product of a units digit by a units digit? of a tens by a tens? of a hundreds by a hun- dreds? 7. In obtaining any product, wliat is the digit of lowed denomiiiatlon which is affected l)y the product of a units digit by a units digit ? of a tens hy a tens ? of a hundreds by a hundreds ? 8. What part of 508516 can be omitted, if we wish to obtain only the hundreds digit in the scjuare root? 1). What part can be omitted if we wish to obtain both the hundreds and tens digits ? 10. Find the hundreds and tens digits in the square root of this number. (Use only the digits of the number wliich are reom Avhat place should we begin to mark off the decimal in periods containing two digits each ? 6. Extract the square roots of -09, -0049, -000081, -000004 \?h }'^^' '^^"^^^ ^^^-^1' 6772-41, 4-5369, 297-6423, 354-5,' ::b49-5^ -4, to 4 places when possible. CUBE ROOT. I. 1. How do tlie digits 7 and 4 enter into combination to obtain the product of 74 by 74 ? 2. If this product be multiplied by 74, how often will 7 tens cubf'd l)c found in the prod act ? 3. How often will 7 tens squared multiplied by 4 be found 111 the product ? How often Avill 7 tens multiplied by 4 squared be found ? How often 4 cubed ? To obtain tlie answers to the foregoing examples the work will be simplified by proceeding as follows :— 70 + 4 70 + 4 70x4 + 4- 70- + 70x4 70-+2(70~x4) + 4' 70 + 4 74 74 296 5180 5476:^742 74 70- x4 + 2("70x4-)+"4'^ 703 + 2(70- X 4) 4- (70x4-) 21904 383320 70^ + 3(70- x"4) + 3(70 x 4"-) + 4'^ ^ 405224 == 74^ 6 76 ARITHMETIC. I U I !i i 4. Under what digits in the number 405224 must the cube of 7 be phiced to obtain the same remainder, as when 70 cubed is subtracted from it ? 5. Coald any cube greater than that of 7 be subtracted from 405224 in a simihni- manner ? 6. How then can the tens digit in the cube root of 405224 be found ? 7. How are the digits 7 and 4 combined to form the remainder in example 4 ? 8. If 3 X 70- is divided into this remainder [since 3(702 X 4) forms the greater part of it], Avhat is the quotient ? 9. Wiiat must be added to o x 70- to make the division exact ? 10. Divide 15556 into the remainder in example 4, and find the new remainder. 11. As this remainder is 0, what has been subtracted altogether from 405224 ? 12. What is the tens digit of the cube root? the units?' the Avhole cube root ? 13. ,How Avas the tens digit of the cube root first obtained? The units digit ? 14. What was added to the trial divisor in example 8 to make the division exact ? 15. If the number had been 405250, after going through the same operation to find its cube root, what woald the last remainder be ? 16. When the remainder is 26, what has been subtracted from 405250? 17. What is the number subtracted the cube of? 18. Extract the cube roots of 13261, 13269, 830584, 830507, and give the remainder in eacli case. 19. Should the trial divisor in example 8 always give a quotient which will iiold when the complete divisor is ased ? 20. When it does not, how should you proceed ? II. 1. Every time a number is increased ten fold, how many fold must the cube of that number be increased to give the cube of tlie new numl)er? HOOTS OF NUMBERS. 77 dilit?"'' '"•'"'^ '"^''^ "'■•^ ""''■« '■" '■'" "'"« Of <^ ""its ^.3.^ How „uu.y digits are tl.ere in the cube of a tens digit?"'' many digits are there in the cube of a hundreds units digit? of the tens dfeit'? 'Xlu'^, e,^'';? .'^i^r"" "v't'iirr "^ >•"-ireds:d^grirthrr•rlr -^•^ 8 Find the hundreds and tens dig-its in the cube mat ?n ^wT '''^^/,^'^''' ^" ^ hundreds + 4 tens ? sand;, ^s\^::^r''^^^^ ^- ^^^^^^ ^^o. 12 When we have obtained the remainder and the tof-.l number of tens in the cube root wliat is thP tl i for finding- the units dig-it ? ^'^''^ '^''^''^^' 13. AVhat is the complete divisor? ]t' wl'""! ?^ ^^''' i-eiiiainder after this division ^ 10. vvuat IS the cube root of r)048'\S3'U ? 16. Every time we found a dig-jtln the cnbp m... i ;;eJ:so^':™;.£::i-;:----ir-e\--;'s the operation of finding the different digits of i?s cube root III. ^i" ^Y}^^^ ^^ ^^^® ^'^^^^<^ of 1, -1, -01 ? I What is the cube root of 1, -001, -Onoooi v a. How many di^nts must tliere ],e in a deciuial to give 78 ARITHMETIC. 1 docima I place in the cube root ? to give 2 places ? to give 3 places ? 4. For every digit in the cube root, how many must there be in the decimal ? 5. From Avhat place should we begin to mark off a decimal in periods of 3 digits each ? G. Extract the cube roots of -008, -027, -000008 •OOOi")!'^ 1-331, <)-2(;i, 380-017, 14-(513, 2l)-4, -1, -08, •00027, to 3 places of decimals when i)ossible. MENSURATION. RECTANGLE. If 1. Place 3 units (square feet) in the form of a rectangle. 2. What is the length of this rectangle ? 3. What is the width of this" rectangle ? 4. Place a second rectangle, equal in all respects to the former, with a side contiguous throughout to a side of the former, and in the same plane. 5. What figure do the two rectangles form ? 6. AVhat is tlie length of this rectangle ? 7. What is the width of this rectangle ? 8. What is its area in terms of the area of the first rectangle ? __ 9. What is its area, in square feet? 10. Place a tliird recta,ngl(\ equal to either of tlie others, alongside the second, in a manner similar to that in ex- ample 4. 11. What is the lengtli of this figure? 12. What is the width of tliis figure ? 13. What is its area in terms of tlie rectangle ? 14. What is its area in r^quare feet? 15. What is its length in yards ? 16. What is its width in yards? 17. What name do vou give the fiinire ? 18. How many s(iua,re feet in a sc^uare yard? area of the first Il MENSURATION. 79 )f the first II. 1. Place 4 square anits in the form of a s(iujire. 2. \A .lat is the length of a side of this square ? * 3 If this square unit is one-half foot long and one-half foot wide, what is the lengtli of a side of the whole square V 4. llow many of tliese units in a scjunre foot '^ 5 Make a rectangle whose lengtli is 8 and width 5 of the linejir units corresponding to the square unit in ex- ample 3. (I What is the area of this rectangle in those square 7. What is the area of this rectangle in square feet ^ 8 WliMt is the area of a rectangle 4 feet in lengtli and .'f feet m width ? Find the areas of the following rectangles : 9. 7 rods 10. 3| yds. 11.7 yds. 2 ft. 4 in. " <' 2 ft. 7 in " 12. 6-947 ft. " " 3-4()ft. - 13. 2 mi. 7 per. " " 46| ft. " 14 Prove that 80J- square yards are equal to I square rod, being graiitetl tliat 5i linear yards are eciual to 1 linear rod. ~ " TRIAXGLE. Let A13C he a right an- gled triangle, with the an- gle ABC a right angle. Let ABDE l)e a "square described on AB, and VF A OH a square equal to one in length and 51 rods in width. 2| ^'ds. descril)ed on BO, and let l^D and FD be cotermin- ous at L. and in the same straiglit line. Join GC. Ixemove the triangle GFC and place, it in t]\o. position of (;i IK. Also remove the triangle ABO and place it as AEK. t::i it M ARITHMETIC. -i lie figure AOOK, Avliich ifi tln^ scjiuirt^ on CA, is com- posed of the same parts as tlie tAvo squares ABDE and DFGU. .-. The squares described on AB and BC are equal to the square described on CA. Note. — If a piece of papcu' sliaped like ABFGHE be cut along" tlie lines AC and (JG, tlien the parts can be put tog-ether in tlie sliape of tlie fig-ure ACGK. ANOTHER PROOF. CG, DTI, AE, FB are all equal. Tlie triangles AEF, FBG,GlIC,andDEII are all e(|ual. The tria ng-le AEF = ?/ the rectangle AK. . •. The 4 triangles mentioned are ci^ual to the 2 rectangles AKj and KG, HEFG is the square on FG. BK " '' FB. DK is e(iual to the square on BG. IIEFG + 4 triangles, AEF, FBG, IIGC and EDH, are equal to the square AC. BK + DK + AK + KG are ccjual to the square AC. But the 4 triangles Avere shewn equal to AK + KG. .-. HEFG is equal to BK + DK. Sw i.e., The square on FG is equal to the sum of the squares ^on BF and BG. Find the length of the hypotenuse in each of the fol- lowing right angled triangles, given : — 1. The sides 5 1, and 12 ft. respectively. 2. '' ft. '* 40 ft. 3. " 13 ft. " 84 ft. 4. " 8 yds. " 15 yds. (( (( MENSURATION. 81 na-le APJF 5. Tlic sides 14 jn. 2 ft. 2 in. 3 yds. 2 it. 7-61 ft. 7 A yds. 17-56 in. and 2 ft. '' 4 ft. iif. '' 5 yds. l in. " 2-i) ft. " 4^yds. ^ 9-4 in. respectively. (( (( (< \ Find the tliird side in each of the following right angled triangles, given : 11. The hypotenuse 10 ft. 12. 13. 14. 15. 16. H7. 18. 19. <( a 41 ft. 113 ft. 14 y(1h. IJIJn. 4 yds. 2ft. 7 in. 9-72 ft. 11-345 ft. 4-68 yds. one side 8 ft. (( 9 ft. 15 ft. 6 yds. 14 in. 3 yds. 1 ft.9 in. 8-6 ft.> 107-69 ih. 3-7 yds. Example.— Tlie sides of a triangle are 3, 4 and 5 units m length respectively ; iind the le-ngth of the perpen- dicular dropped on the longest side from the opposite angle. In the diagram let AD represent the per- pendicular, and let it contain p units. Also, letDCcontain x units. .-. DB contains 5 - X units. In the triangle ADC, And in the triangle ADB, p-+x- =3-(l) p-+(5-x)- =4- (2) Subtract (1) from (2) and we get (5 - x)- - x= = 4- - 3'-^ Therefore 25 - lOx = 7. lOx , =.18. X Substitute this value for x in (1) and p- +(11)- -3-. Therelbre p- = -\,V. _ 1 S _ 1 4 ~ 10 ~ ^6* 82 ARITHMETIC. Ns I! Find the segments of the base of a triangle into Avhich the perpendicuhir dropped from the opposite angle divides it ; also the length of the perpendicular, the base being given layt in each case. 1. Sides are 5, 12, 13 units 9, 40, 41 ft. 26, 168, 170 yds. 4, 9, 8 ft. 6ft., 7 ft., 19 in. 3| yds., 9.^ ft., 231 jn. 5-2, 7-9, 4-63 yds. 6-4, 5-93, 8-45 in. 'I. 3. 4. 5. 6. 7. 8. (( (( in length respectively. << <( (( <( u (( it a n a IN I i 1. Draw an acute angled triangle having an altitude 10 units, and a base 7 units in length. 2. On the l)ase dcscrilie a rectangle having the same altitude and towards the same i)arts as the triangle. 3. Into how many parts does the altitude divide the triangle ? 4. Iiuo how many parts does the altitade divide the • rectangle ? 5. Yfhat is the area of either part of the rectangle in terins of the corresponding part of the triangle ? ('». What is the area of the wliole rectangle in terms of the whole triangle? 7. \\'!iat is the area of the whole triangle in terms of the whole rectangle ? '^^ Wliat is the area of the rectangle in square units? An:l wliatthe area of the triangle? I'ind the areas of the following triangles : 9. i]ase 4 ft. and altitude 3 ft. K). " 2-79 yds. '' 8 " 11. " 2-59 'in. " 4-68 ft. 12. The sides being 3, 4, 5 ft. respectively. 13. " " " 7, 9, 10 yds. " ' 1 J. " " " 11, 15, 17 in. l^^ '* " " 2vds.5ft. ir)in. " which the divides it ; ciiig given spectively. u il (( (( (( (( iltitiide 10 the same i^-ie. lividc the livide the itang-le in L terms of rms of the re units ? 77-. ,) -_ 27TY MEXSURATIOX. CIRCLE. The circumference of a circle bears a constant ratio to Its diameter ; this ratio is 31 nearly, but closer approxi- mations are S-UIG and f^i^, and is always denoted by the EXERCISE. Given 7r = 3i, hnd the circumference : 1. If the diameter is 4 ft. in length 2- " " - 5f yds. ' 3. " '« ■ 4. " <' Find the diameter : 5. If the circumference is 9 ft. in length. '* 4 yds. 2 ft. 3 in. in length. *' 7-4G8ft 7: 8. 9. 10. (( (( " 7 ft. 3 in. '' 0-46 yds. " 17-43 yds. " •0469 per. ^" '< 27^'* in <' <( U << a Find the circumference : 11. If the radius is 12 ft. in leugth 12. '' " 41yds. 13. '' " 2-794 in 14- " " 4 yds. 2 ft. 7 in. in length. 15- " " -04(37 rods POLYGON AND CIRCLE. * Take a number of equal isosceles triangles ^ 1 Express the sum of the areas of two of the triana-les m terms of the area of one of them. ^ctngies 2. Express the sum of two bases in terms of one base. 3. ^^ hat relation is there between the measure of tlie sum of the areas m example 1, and the measure of the sum of the bases in example 2 ? 4. Express the sum of the areas of 5 of the triangles in terms of tlie area of one of them. 5. Express the sum of 5 bases in terms of one base. ^ b. J^md the relation wlricli exists between the measures 111 examples 4 and 5. "iccibmeb 84 AUITirMETIC. ^ t I 7. If the tria,i)^'-les arc of such a slwipo that when plnocd 111 n, ].l.-iii«", with ,'! coiuinon vertex, their hases will form the i.eriiiieter of a complete iigurc ; what is this ili>ure called? ^ 8. If tiiere arc 20 trians-les in tJiis polyg-oii, express the area of the poly^^on in terms of one triang-le. 9. Express the perimeter of the polygon in terms of one bfise. 10. Express the measure of the area of one of these triangles in terms of the measures of its altitude and base, 11. Express the measure of the area of the polygon in tei-ms of the measures of the altitude and the l)ase of one triangle. Note.— The base and the altitude of one triangle are respectively equal to the l^ase and the altitude of anv other. "^ 12. Express the measui'c of the area of the polygon in terms of the measures of its pei'inicter and the altitude of one triangle. 13. What is to be noticed regarding the number of tri/mgles required to make a com])lete j.olygon, if the bases of the triangles become very short, while the sides remain the same length ? 11. What other figure does this one resemble ? 15. If we round o(f the corners of this figure, so as just to destroy the angles, how much of the figure is removed? IG. What is the ditference between the areas of the figujes in examples 14 and 15. 17. What is the name of the figure in example 15? 18. What is the difference in length between the side of one of the triangles and its altitude in example 13 ? 19. In exami)le 15, what is a side of a triangle called? 20. In example 15, what is the perimeter called? 21. Find the measure of the area of the figure in ex- ample 15 in terms of the measures of its radius and cir- cumference. 22. Find the area of a circle Avhose radius is 3J, feet and circiimferenee 22 feet, 23. Find the area of a circle whose radius is 4 feet. MENSURATION. '86 21. Kind tho urvn of n, cirvlo whoso mdlus is 5 yds 2 ft -'^' " " " " ivha. p6- " " " di.'inictcr is 2^ in. 27. " '* cii-cumfcrcncc is U ft. 28. a a ft. 4 in. SECT«1R OF A CIRCLE. 1. In a cirolo avIioso circamfcroiicu is 144 in., wluit part is an arc of 12 in. of tiic Aviiolc circunifercncc ? 2. In tlie same circle, wliat i-.irt is a sector of a circle, which stands on tliis arc, of tlic whole area ? 3. If tlu' circumference of a circle is 1 tO ft., iind tlie len^-th of thearc on wliich a sector, whicli is one-tlfMi of tlic whole aren., stands. 4. In a circle wliich is 72 ft. in circiinifei-once, jiiul the area of the sector which stands on an arc *) ft. in leiio-th ? 5. Find the area of a sector of a circle whic , stands on an arc 10 ft. in len^'-th, if the I'adius of the circle is 7 ft. 6. Find the ai-ea of the sector of a circle vhich stands on an arc er(unl in leni^-th to the radius ol the circle- given the radius 3 ft. ' CYLINDER. 1. Take a sheet of foolscap paper, bring- the two ends into contact, and have each side in the form of a circum- ference of a circle. 2. If the space inclosed were a solid mass, what would it be called ? o. What does an end of the paper represent on this mass ? 4. What does a side represent ? 5. Find the area of the surface of the paper required to enclose this solid, if the end of the paper is 8 inches and the side 14 inches in leno-th. 0. Find the area of the curved surftxce of a cylinder the heijrht of which is 7 i't. and the circamference'of the base 12 ft. 7. The radius of the circular base of a cvlinderij.: -U ft and the height 3 it. ; lind the area, of the curved surl^ice. %, «>. v^ IMAGE EVALUATION TEST TARGET (MT-S) /. i// M. '^ < .^ ^*' C/j ^^ 86 ARITHMETIC. 8. The curved surfa OP nf « i- , and tlm height is 4 ft find fL''^' "''^'' ^^^"^^"'"^ 24 sq. ft m CONE. eir.o, ^l^,^: K'i4 \!'4f n «^ -^ -tor of. it he called ?'^""' '"^^"'"'^ ^'■«''« ^ «oIid ,nass, what would face. "•' ^"ct the area of the curved sar Of Lt;r \t"^ ^f ^f/ -e „ in. „„, „„ „,,,.„^ '^ce. ' ^""""^ ■■»•«» of the cun-edsur- 10. Given the irp'i r,p ^i 2 in. and the «Ia'„t tide fin""S ,^r^"''« "^ '"^ eone 49 '"0 base. "'■' nnd the circunifereuce of ba".- '" "■" 1'™-'^'"^ 0-a.np.e, fl„d the radius of the the Ll^f t "£f r ZZ T'^'' " ^^- -- -d 13. Given the slant ^fdJ*:.P ^^'^ ^^^^"^ ^ido. '•■e "- U in.. «nZ\r;e? -i-t!7 ~.i-e^^^^ of c3 ^^ ^* c MENSURATION. 87 18. Given the mrtir ncu\ '">5' "'^ sl«.nt side. 1!>. Cfiven the perpend c„^^ , :' f* ''"''^'^''' «»rf«cc. ■■"dins of the base i'^L '"fi'ti , f ^'" ^'l'"- '''"•l "•« surface. " """ ">« area of the curved ^^re^^^'^Z^f^J^^!^^ « ft., and the eir- surface. '^ ^^ "'' ^"^^ the area of the curved cn^SiTo?^ JS"^-^--l>ei^ht 12 in., and the cir- surface. "" ^^ '"'' ^"^ ^he area of the curved SPHERE. ic Let AB and CD represent Hv« „stra,^^htlines. Pthei'S^S ^^^^l^^wAM,PS,BI.perpendicu. and AR perpendicular to BN JN axfs! "^ '"■"""'* <-^» "«' an " K P'^,™f*™''«°^*'ieareaofthe su.-es of AB and the ch-c, llnce T.T'"'' "f ""^ "«'- describes. "-.luiieicnee of tlie circle which P Who; JIb and KsVt'fnd' f ""T "^ ""^ "^^ '^ 2- >< AB . PS these lines. "' '^"'' "'" ""^^™'-«« «*' the lengtL of Tl|^trian|les ABR and PSQ are similar. ■"■PQ = AB- •;. PS-AB..PQ.AR,andAR = MN. 1«1N IS called the projection of AB on CD ) "" ^ I I i^ 88 ARITHMETIC. 1. Take half of the perimeter of a reg-ular polvffon of nn 3. Coiupai-o the lengths „ftliese porpendkiUars. dial;on:r'™ "" "''"■P<"reon in exam,,lc 1 arouud the 4 feljualt"'"'"' "' "'" ""^•■'^•■^^ '^f *« l-^lt^ in example . C. What factors are unequal ? 7. What is the sum of tlie unequal ftietors ? 8. A\ hat IS the measure of the sum of the areas of tho belts in example 4, in terms of the measu x^s of thf pendicular from the middle point of one of the sides and the diag-onal of the polygon ? "" 9. If the sides of the'poiygon in example 1 become verv shor , and increase in number, wldle the dialoSrenn'ns caHed? '^''' ^"^^"^ '''^ '"'"^^^'"^ ^^' ^^'^^^^^ i« the solid 11. What doe^s the perpendicular from the middle point of any one of the sides in example 1 become irexam^^e 12. What is the measure of the area of the surftice of n splK^ m terms of the measures of its radiJ^^ aM d' VS. What is the measure of the area of a sphere in teuns ot the measure of Its radius? ^ 14. Find the area of the surface diameter is 7 ft. 15 l^^ind the areas of the surftices of the spheres the radn of which are respectively 7 ft., 4 in.?9 265 in 9-t)4 yds. ; 4 yds., 2-50 ft. *' .u.e';4™:i:^;!;lf °f ??« tTr^^^r'^r i ^^■t'^ 478-DO s,. in. ; 4I.C.03 sc/ in. ' ' ' "'^ '''"' ' '"• "^ ' of a sphere whose MENSURATION. 89 RECTANGULAR SOLID. 1. Place 5 units (cubic feet) in the form of a lectane-ular 2. What is the length of this solid ? What is the width ? 3. Place an equal rectangular solid alongside of this one, so as to form with it a single rectangular solid ^ 4. What IS the volume of the new solid, which is formed m terms of the first one ? ' 5. What is its volume in cubic feet? 6. Take 4 equal rectangular solids like the one in example 1, and place them on a plane so as to form a smglc rectangular solid. 7. What is the volume of this solid in terms of one of them ? 8. What is the volume in cubic feet ? p. Place squarely on top of this solid another ecuuil to It in all respects. 10. What is the ncAv solid called ? li. What is its length, width, height? 12. What is its volume in terms of the solid in example (P 16. , A'V hat ]s its volume in terms ^f the solid in example 1 '^ 14. What is its volume in cubic feet? 15 Place 8 equal cubes, edge ^ ft., in the form of a snigle cube. 10. What is the name of the cube so formed ? 17. Find the volume of a cube whose edges are 3 ft each. ^ 18. What is the name of such a cube ? 19. How many cubic feet in a cubic yard ^ 20. Find the volumes of the rectangular solids of the lollowing dmiensions :~ 3, 4, .5 ft. respectively • 2-i 31 8 yds. respectively; 2} yds., 4ft., 13 in.; 3-46, 5-9, 8 ft. respectively; 2-G2 yds., 4-33 ft.', 12-i) in. 21. If the volume of a rectangular solid is 1728 cubic mches, and the height is 12 inches ; find the area of the of it^* ^^^^^^ ^^^^^ ^" example 21 be a square; find a side i« fL ^^ *'if ''''!'""'' '' ^^' ' ^''^*- ^■^•' '''''^ ^''« '^'-^^'^ of the base lb 108 sq. tt. ; find the lieiglit. i IS ff kJ ■.I 90 ARITHMETIC. S 1 24. Given the volume 725 cub. yds., the heig-ht 7i yds., the length of the base 20^ yds.,' iind the width of the base. 25. Given the volume 847 cub. yds., the height 25| ft., one side of the base 56 inches, Iind the other side. PRISM. 1. What is the volume of a solid 1 foot in height, whose base is a polygon containing 9 square feet? 2. Place another solid equal in all respects to the former exactly on top of it, so that the base of the second solid will coincide tliroughout with the top of the former. 3. What is the name of the new solid formed ? 4. What is its volume in terms of the volume of the first ? 5. What is its height ? i\ What is the volume in cubic feet ? 7. Place 8 solids, each identically equal to the first, on top of one another in a manner similar to that in ex- ample 2. 8. What is the new solid called ? 1). What is its volume in terms of the volume of one of them ? 10. What is its height ? 11. What is the volume in cubic feet ? 12. Find the volume of a prism 10 feet in height whose base contains 14 square feet. 13. Find the volume of a prism 3 inches in height, whos ) base is a square, with an edge 2 inches in length. 14. The sides of the rectangular base of i\ prism are 4 and 5 feet respectively, and the height 'is 7 feet ; find its volume. « 15. The sides of the base of a triangular prism are 3, 4 and 5 feet respectively, and the height is feet ; find its volume. 16. The sides of the base of a triangular prism arc 7, 8, 9 feet respectively, and the height is 11 feet ; find its volume. 17. The sides of the base of a triangular prism are 7 yds., 19 ft. and 30 inches, and the height is 7-463 ft. ; find the volume. olume of one of in height whose MENSURATION. gj 18. If the lusc of a prism is a polyc-on of a Ln'O'it uv^u arcL.ooJh^f'off rrriitt"^vf' t T"!,* ^I""' P™"- now solid fonued » ' ' '"'' " "'" "^""= "^ *e sm^t.^!;';„n;;i'eiet-/'-'^'" '"^^ •^'=- — <» "^ Jd^iri'::!;:;;:'!^-^-- '"« vow. or the WEDGE. faces.''''' ' *'"''"^"''"" '"■'^™ "" °"« "fite rectagularsur- 3. Pmd the volume of a trmniraUvr^r\. Given the volume 108 cubic inches and the area of the base IG square inches, find the height. 17. Given the volume !)8 cubic inches, and the radius of the base 4 inches, find the height. 18. (riven the volume 39G cubic feet and the circumfer- ence of the base 35 feet, find the height. PYRAMID. 1. If the bases of two triangular pyramids are equal in area, and the pyramids have equal altitudes, how are their volumes related to each other? 2. Divide a triangular prism into 3 triangular pyramids. 3. Compare their volumes. 4. What is the volume of the whole prism in terms of the volume of one of these pyramids ? 5. What is the volume of one of the pyramids in terms of the volume of the whole prism ? C. AVhat is the measure of the volume of the prism in terms of the measures of its base and altitude ? ■" r MENSTTKATION. 93 iidur in terms of square feet, is 7 le radius of the and tlie area of Lnd the radius of d the circumfer raniids in terms 7. What is the measure of the voliimo of ouc of il.o i,v,-.i mids in terms of tlie measures of th(. base and altitude of tlu! ]»risni r* 8. AVhat is the altitude of the first pyra.mid whieh >vas taken from the prism? !). AVIiat is its base in terms of the h.-isc. of Mio prism '^ 10. Wliat IS the mensiire of tlie v<,Iu,m(. of tliis pyramid 111 terms of tlie measures of its b.-ise and altitndcV 11. WhatistlunnensureofthevolunKM.fanv trinn-nlMr pyramid in terms of the measures of its b.se and altit^.dc' v IL. Jf w(! have a number of (.,,iml trian-nh.r ],yr..,mids of suchashape that Avhcn resting- on thrir bnscs they have a (M)iiinion apex, and one ed^c (common to all of theni Avhat eomplete solid do they form v ' "'"'^ in. What is the volume of this whole j^yramid in terms of the volume of one, of the trian«-ular pyramids, if it tnkes b triang-ular pyramids to foi-m it? _ 14. What is the area of the b.-ise of the whoh^ pyramid in terms of the base of one of the triangular pyramids? 15. What is its altitude ? ^ u lb. What is the measure of the volume of the whole solid ni terms of the measures of its bns(, and altitude? q ftl; ? . fi contains 20 s(,uare feet, and its altitude is y feet, what is the volume? la Find the volume of a pyramid b feet in height which stands on a square base, whose vd^v. is 4 feet hJXf?^of7'' ^^■^' Py^/^^'ii^l i^ 72 eubie inehes, audits bn^Pi. cl^ '^"J ""'' f '' pyramid which stands on a square iJi^iA^^'T'''^. ^'^^''''^^ '^'''''^^' ""'' "" rectangular l^ase r, volume "^ ''' ''^^^'^^' '' ^ ^'""^ ^^^8-l> ; liiid its .idf; Z^'^^ ^'T.^'^ ''^'' ^^•i'^n.^^^^l^i- Pi'ism is 9 feet, and the H w!L n '^^ triang-ular base are 3, 4 and 5 feet respec- tively ; hnd the volume. ^ 28 The sides of the trinngular base are 7, 9, 10 feet re- i 94 ! :!': * III ' 111 I ARITHMETIC. CONE. 1. Take n pyramid wJiose base is a polyg-oii of a very great number of sides, and cacli side very sliort. 2. Wliat does the perimeter of tlie ])asc resemble ? 3. Smooth off tlie shmt edges of this pyramid. 4. Wliat is it now called ? 5. What is the difference between the volumes in ex- amples 2 and 3 ? 6. Find tlie volume of a cone, given its altitude 9 ft and the area of the base 27 sq. ft. 7. Find tlie volume of a right circular cone 8 ft. high, the radius of whose base is 7 ft. 8. Giv^en the volume of a riglit circular cone 77 cub. in. and the lieight (> in., what is tlie area of the base? 9. In the preceding example, what is the radius of the base ? 10. In a right circular cone the volume is 47 cub. yds., and height 4 yds. ; find the radius of the base. SPHERE. Take a number of equal pyramids. 1. Place two of them side by side with their apexes at the same point. 2. AVhat is the volume of this new solid in terms of the volume of one pyramid ? 3. What is the area of the two bases in terms of the area of one base ? 4. If a number of them be placed so as to form a com- plete solid figure, with the common apex in the centre of the solid ; what is the name of this solid ? 5. If it takes 20 pyramids to form this polyhedron ; what is its volume in terms of the volume of one pyramid ? 6. What is the area of its surface in terms of the area of the base of one pyramid ? 7. What relation exists between the measures in exam- ples 5 and C ? 8. What is the measure of the volume of this solid in terms of the measures of the altitude and the base of one pyramid ? PROULEMS— METRIC SYSTEM. 95 olumcs in ex- altitude 9 ft me 8 ft. high, [ns of the area 0. What is the measure of tlie volume of this solid in terms o the measure, of the altitude of one pyramid, and the surtaee of the whole solid ? 10 If the bases of the pyramids beeome very small what 18 to be notieed regarding the number reci'uired to lorm tiie solid in examph; 4? ^ 11. What other solid does it now resemble? 12 What is the dilferenee between the altitude and slant edge ot a pyramid like those in exanipU' 10 v 13 If the edges of the surfaee in example 10 are smoothed off, wluit js the solid then called ? 14. What is the difference in volume between the solids in examples 10 and 1.'} ? 15. What are the altitudes of the pyramids in example 10 termed in example 18? ^ IC. Find the measure of the volume in example 13 in terms of the measures of its surfaee and radius 17. Find the volume of a sphere, having 'given the radius 7 m. and the surfaee 01(5 scp in. 18. Find the volume of a sphere whose radius is 4 in IJ. Jnnd the radius of a sphere whose volume is 38808 cubic inches. 20. Find the volume of a sphere whose surface is 154 square feet. * 21. Find the area of the surface of a sr)here whose volume is 480 cubic leet. ^ ^ ires m exam- PROBLEMS -METRIC SYSTEM. 1. Express 1000 centimetres in metres ; 1000 centi metres in decimetres ; 1000 centimetres in dekameti-es 2. Express 1000 metres in kilometres ; 1000 metres' in dekainetres ; 1000 metres in decimetres. 3. Express 12 kilometres in dekametres: 15 kilometres m metres ; 21 kilometres in millimetres ""'^^»^s 4. Express 123,450,781) millimetres in decimetres- in metres ; in kilometres. ' 5. Express 8-50 Km. in centimetre^ ; 5-(;32 m inmilli metres; 124G8 mm. in microns. -^nmnn 96 AHrrilMKTIC. 1 i •il: • I r! i 1 HP 6. Add 14-G 111., L>:>7cin., 1i;l>-;j Dm., l(;;u Km ; express tlio result HI metres. 7. Express in centimetres the difference between i)-(^tS Ivm. and 13i;4-8i) Dm. H. Multiply 12 Km. 5 m. 8 cm. by IM). '■>. (Jiven tliMt (It miles is very neiirly eciuul to lOa Km.. cxpivss 1 yard in ceiitinietres. 10. A train is runiiin^- at tlu^ rate of GO Km. per hour llcny many metres docs it «•«) at this rjite in one second. 11. A tram runuino- jtt the rate of (10 Km. i.er hoar passes over 20 spaces betnven tc^leomph poles in one uiiiiute. !< iiid the distance betw(>en two consecutive poles in metres. ^ 12. Usin^'- the approximnte vmIuo, 1 metre - 39-37 in express the hei-lit of n, nnin, f. ft. 10 1 in., in centimetres! 16. Usin^^ the same ;ipj)roxinuition, exi)ress in milli- metres the hei«-ht of a barometer which stands at 2*>-5 in 14. Fmd the leii<.-tli of the two parts into which n strin<.' 10 m Ion- IS divided, ^-iven tli.'it one-hnlf of one part and two-thirds of the other make a h"ii«^tli of GOO cm 15. ilow ioii^- will it take a, man to walk from Toronto to Hamilton, a, distance of G5 Km., at the rate of HO m per minute ? ' ^ IG. The radius of a wheel is 1-4 m. ; how many times will it revolve in ^-oin^- ao Km. ? (tt^ -.-.) 17 Takino-u centimetre as ^r of an inch, find the number 01 millimetres in 1 yd. 18. How many ien^-ths each 1 m. 5 cm., can be cut froin a len«-th of 1 Km. ? Express the rcmainchu- in mm. l.». 1^ iiid the number of sipian; metres of carpet ro(|uired to cover a lloor, the dimensions of which are G m. 1-75 dm by 4 m. 12 cm. 20. Find in metres the side of a square of which the area is ir)227-5G sq. metres. 21. Express 1200 ares in hectares ; 1200 ares in cen- tiares ; 1200 arcs in dekares. 22. Express 12315 milliares in ares; 5G78 centiares in dekares ; 1 hectare in centiares. 23. Find the sum of 12-G4 ares, -0408 hectares, one mil- lion milliares. Express the result in ares. PR< )nf,KMS— MKTRIO SYSTEM. 97 '21. KxyrtvsH in dckaivs tlicdiflerenco Ixvlwecn 1 Ila., and 25. Multiply 7 JIa. 5 Da. <> a. 4 da. 5 ca. ]»y ''7 in; |'',„d i,i hectares the area of a rcctaiig-ular Held, of lcn«-rli 4 JIiu. and of breadth 2 Ilm. 7 Dm. 27. The area of a square field is 18--i0 Ha. ; lind the periiuetrr in metres. ' 28. Ex])iess 1 s(iuare deeimetre in milliares. 2.). iMiid in centiares the whole surface of a cube the voliim(M,f which is 27 cubic metres. ' HO, K\]nvi^H in S(|ii;ire centimetres 1 a^e 5 da. .'51. A jwiralielepipcd whose ed^'-es are r)roportional to •> .5 and 4 contains ;j cubic metres ; iind its whole surface in centiares. 32. A block of Avood contains 4-5 cubic metres • find in ares the aiva it will cover if cut into sections 1 cin. thick- .>o. JMnd in cubic metres the volume of a sheet of ice covering- a ].ond of area 20 Ha. and thickness 3 cm. .5 !. i^ iiid the cost of painting.- the walls and ceiling of a '^T milli'n-( '"^'' ^ "'* '^''^'' '""^ ^''' "'• ^''^^'' '^^ 75 cents 35. How many arcs arc there in a square field whose di.'i^-onal is 10^/2 metres? 3(; The sides of a rcetaiifralar field are proportional to - .md 3, Its area is 24 Ha. ; find the length of its diagonal -111 1111^1)1 Oo, 37. UsiiifT the a,pproximation, 1 inetre= 39-37 in find the numlxu- of acres in 1 Pla. ' 38. A rectangular garden 15m. long and 12m. broad is surrounded by a path l-5m. wide ; find the area of the path m centiares. 30. A circular area of 7 metres radius is divided into two portions m the ratio of 3 to 4 ; express the area of the smaller portion in ares. 40 Using the approximation 1 Ha. = 2-5 acres, express 1 metre in inches. ' ^ 41. Express in steres the volume of a cube whose edg-e is o metres. ^ 42 The volume of n r^^he. h 1 K^. ; find the length of Its edge in centimetres. ^ 98 ARITHMETIC. 43. How many cubic centimetres in 1 stere ? 44. How many steres in one million cubic centimetres^ 45 A pjle of wood is 45 metres long-, 1-5 metres wid; ^Om Wdlnd'^'n' ""^ ?''''^'''''^^S ^ cellar 12 m. long, 47. Express 1 ds. in cubic decimetres. 49. Also express the stere as the fraction of a cord 50. liow many steres of wood in a rectangular iloor 40 Th^e^ .ZT^^'f'' ^^i" '^^"^^'' ^^'^ ^l^"^«n«ions of which h. JdLlerions' '""^ '' ^"^ ^''' ^^^ '' ^^"^« P- «^-« 5 52. If a wall 15 m. lon^, 1 m. thick, and 3-4 metre^ n.ffh contams 51000 bricks, how many bdcks of the same is emv!l'lo\ho ^^fiT' '^'^ '"'^^^^ ^*''^ ^^^« ^h««^ cdg« ientiare. "" ''^"'''^' *^^ ^'^'^ ^^^ ^^i^h is one 54. Given that 1 cubic centimetre of water weio-hs 1 Sri^'^lnies'l^h- ^'^ ^'''^' ''' ^"^ stereT^earth wmcn IS « i> times as heavy as water 1 s"f te.'^^'"^' ^^^^ '" '''^^"^^'' *^"^^ ^^^« ^^^^-^t of 5G. Each linear dimension of a block of mctnl i^; in creased -002 of itself by heating- find the vXie of ," block which before heating was 1 stere sheet 10 m'bv hT 'r 'I'f ^ f- i'"""^ "'^^ a rectangular sneet 10 m. by 8 m. ; fl„d its thickness in microns. •> if lo?li^'nd't 'r ""''''^^'^ P"^^^"^^^ "^^o an empty tank 59. An empty t^mk is 4 m. long and 2-5 m. wide • w.ter 60. Prom a cylindrical tunnel 30800 steres of earth were tlie number of he weiiJfht of PROBLEMS— METRIC SYSTEM. 99 rcniove(i ,• tne diameter of the cro^^ ^oo^^r^r. • t • find the length of the tunnel (L!! . 'f'''^^ ^« ^^ ^^^^'^^J 61. If 27 steres of wood Insf- ^ TnT^dfi^o t, ^ the length ofa pile 1-5 iTwide ■> m hth' .^^^T""* ^^ C5. Compare tl,e cul.ic centimetre and tl,e MJitre OoS g.S°:,es v"' ""■" "' "'«'^'- '^' ''"C- -"• weigh 1000- n.^^p:!Z;;tt^jStt^::t:i^^^ weig-hed in air ^ ^•' *^'^n when e|e^!;:f,^:KrSel.rs;t=^^--'-- litrii.S?L.t '"*"'™"'' "<^'^^'- tl.e quart H„d the 72. Compare the kilolitre an,! the stere. 2 m 1 5 m r"f I't-'"" " '■'"''"nffulai- tank are 3 m in 3 sec., and the other 5 Dl. in what )f earth were the Olio pourinc- in 20 sec. ? ^ ic.^th of 1 uu. be cut oft* to contnin Cl -(j ste,- " P ' is ^ife^wtT t l^ofT^Jl^ t^^^^^^^ ' ^"^- ^^^^'•^^^-- in the box *'^''' ' ^'"^ ^^^« '^»^^'""t of material inillion litres. ' ^^"'^" -^^'^ contain four — r (I ]■' lis I, I:! I k h f 1 J -I i U ! fl P ' hi I 100 ARITHMETIC. 77. Using- the approximation, 1 metre =39-37 in find the number of litres in 1 cubic foot. ' 78. Find approximately the weig-ht of a kilolitre of air under certain standard conditions, given that water is 770 times as heavy as air. 79. Tlie area of the end of a rectangular box is 1 ca • And its length if the volume is 1 Kl. ' ' 80. Find the number of cubic centimetres in one litre. 81. Express the kilogramme in milligrammes. 82. Find the number of grammes in 1 tonneau. 83. Find in kilos, the Aveight of 10 cubic metref-- of water at 4**C. 81. What is the weight of 1 litre of mercury in kilos given that mercury is 13-5 times as heavv as water? 85. A litre of sulphuric acid weighs 1810 grammes- compare the weights of equal volumes of sulphuric acid and water. 86. Given tliat an ounce Av. =28-35 grammes, iind the number of kilos, in a ton (2000 lbs.) 87. Using the above approximation, find the number of grammes in 1 lb. Troy. 88. Find the weight of a block of iron (7 times as lieaw as water) of which the dimensions are 5 dm. long 4 dni broad, 3-5 dm. thick. ' 89. A watering cart is 3 m. long, 1-5 m wide and 1 m. high ; iind the number of tonneaus of water it carries. 90. Read 123156789 grammes as kilogrammes : as ton- neaus. 91. Given that the kilogramme is equal to 2-679 lbs. Troy, find the num})er of lbs. Av. in tlie kilo. 92. Find in centiares the area of a triangle, the sides of which are 1-3, 1-4, 1-5 metres respectively. 93. The parallel sides of a trapezium are 2-6 m., and 1-4 m. ; find the perpendicular distance between these sides, given that the area is -75 centiares. 94. P'ind in litn^s the volume of a conical vessel, the diameter of the base being 21 cm., and the height 1 dm. 95. Find in kilogrammes the weight of a pyramid ol' lead, 6 dm. high, having a square base the side of which is 40 cin. ; 1 cubic cm. of lead weighs 11-4 grauimes. 5 =39-37 in., find liar box is 1 ca. PROBLEMS— METRIC SYSTEM. IQl 06. Find the weight of a sphere of tin of radius 3.^Hr« 1 cu. cm. of tin weighs 7-3 grammes ™ius3 5 dm.; e»rpor sTZr"^' ' ^- P'^'- "»"■•' -P-- '"is rate in tair250?er '" '"'''"" "" ''''^ "' ^ '-- -^ich con- 103. Express 5280 feet in metres. (2irO]bl7'''' ''' ''"''""''""' '' '''' '"'^'^^"^ "*' ^^ ^^"^ t«^ 105. Tlie pressure of the atmosphere is ohmn- i v a square centimetre- express fhilTr! i ^ ^^- ''^^ incli. ' ^""P^^^^ ^l>is m pounds per square 106. If 3 kilometres are as much under -^ miles n« ^ worth 10 francs the litrp lom f* ^'"'/'^ '"1"'^' ^■"^h is ^,^^^ ancs the litre, IJOO francs being equivalent to '.nra|:;y-^f-i--:~-pe..t^ Letre = 30 -STOT? inches. '^''' """^ "^'^ ^"'"^ 1 111. Consiclerinff tljc earth sDheric'il win. ., • ' i '^ 102 ARITHMETIC. J i ■f J MISCELLANEOUS EXERCISES IN FRACTIONS AND THE SIMPLE RULES. 1. Divide 3-003 by 148-28; -003003 by -014828; and 300-3 by 1-4828. 2. Find the value of •7325G - -Gib ~ 74534g~'~^of 5 miles 27 yds. 3. Multiply 1-728 by 3-146, expressing the result as h circulating decimal. , ^ i-lof-K^ 4. Express '^-~:j:r, — of i;i5 17f«s. as the decimal of a dollar, assuming the value of £30 to be $146. 5. Find, by the method of abridged multiplication, the product of 26-42783 and 523-23856 true to within -001. 6. Divide -37848 by -456 ; 3-7848 by -0456 ; and 3784-8 by -00456. 7. Reduce to decimals ^{i|^, ^\, and express as vulgar fractions in their lowest terms, 3-0561, 15-6013789. 8. Keduce the decimals -21316, -31219, and -8934 to their equivalent vulgar fractions. 9. Find the value of -414 + -0352 + 6- ioi, and divide 2-980i by 7-456. 10. Divide 91-863 by 87-56. 11. Find the difference between 3-14159 and 3 + „- -, • ' + iV 12. Find the difference between V6 and the product of 1-732 and 1-4142. 13. What fraction of £174 16s. 6d. is ^'l of £34 4s. 6d. ? 14. Add togethei" r\-, 'I-.., ,■*,_, _^_._. 7(7 15. Simplify 1^ of 2 + s^ + i ^^7 ^ U 3 J 4- 1. 4. 1 2 + ;] + 4- •101, and divide : of £34 4s. 6d. ? MISCELLANEOUS EXERCLSES IN FRACTIONS. 103 Ifi. Find the product of 325-62534 and '>7.4Sfi7 ..« f.... three places of dcciniil« y.^r <->.„^ :. 1 7 ^ ' *'^ ^^^ ''^^ tiplication. ''^^""'^^«' ^>^ ^he method of contracted mul- 17. Find the value of . -t^L?:?^ + Ma1:5^6, IS T^^wi fv 6-2 X -857142 3-75 + 1-7" clelaV tL'ter^dlS^s o; ^r^. P-^^i^ vilue of a •8397. ^'^'^' ""^ ^^^«h are known to be 10. Find the value in decimals of ^ . 7 20. Obtain the quotionf wimt. fi,^ "^tV, •^'^23 . . . ., is Slrl h .1 "^ recurring- decimal •28752875 . .'. ^''^'''^ ^^ ^^« recurring- decimal ^l^n^t:;de^Ji.;^S-^^--^^^P^y the dedmai by ^^22^ DMde £120 7s. lO^d. b^SI, and multiply the re- riLfor:Ut. oT^^^^^^ '' ^-- '^ -in- to the 24. Simplify * '^ ^ + ii-|ofl!4of4--iiA ^^i' 26. Simplify T?±^Tr_9^i4 ^ ' 97 p/ T^-i^ 2 + 2i + 1761^-1650^p. of ; mn^^'' "^"' ^""^ "^"^-^^^ths inches to thJdecimal 28. Add 1 -«, _4 ., « m^'l by 2f|. ' '"'"'' -^^' ^^' ^"^ air; and multiply 29. Taking- the length of the vear p^ qfi^.9r; ^ press 73-05 days as the decinial of a year ^^'' '^" — - 41- # of •'5 " . ' 1| + 2S- i + i + i-i f! 104 ARITHMETIC. 31. Reduce Jfi^Lil.^f of^-VV 32. Dividea.i,«^^)^^^l_,^^^^^^^^ 33. Simplify 8i - 4| + ^ - 1 1 of ^ ; and 1 + I 2+ — -- 34. Find a decimal whici, shall be within .^hJoAi 10 00(7 "'4 2' 35. KeducejI+^li^nd^iJ^^ 36. Multipl/.^2i"b; 48.921; ^M d'^ivido 4-O.S bv -1407 38 RXftn'"' ^^'-7»- ft'^^tions -037185 and .637185. rf». Kediice to a snigle decimal •04275 4-21 6 'yj •^•05 .342 1-5318 39. Simplify^of|xfofK(r+|of20). " 298 33 '"''^'^ ^'^"^' ^'^'^'''' ^' 7899.114 miles, and is 299~^ ^^ ^h« e(,uatorial diameter ; find the equatorial diameter. 41. Find the value of 7| 111-2^- 6f "'l4T2|^^^^^-'*• 42. Simplify ^•-l--J-^^of7-25ofl.2 3-lx.lQi 4-1+ 5.8 3.25^ "^ 2.i,5 " 43. The speed of a railway train bein^ 05-84 fathoms in mfleToffh; d'f ''''\' '' '''' ''''' ^^^^^' «-^ the Ihnit: miles of the distance traversed in one hour 44. Reduce to their simplest form : 1-^V r+i+l 3 5 l-lof( A_ + ^) s--i: 6- •> _ 1 ffwI'tir^eVreri^'ir''"",""'^ ^° *"' J'^>«»>">ato.- will give me neaiest approximation to "">' ? V V o "nominator will MISCELLANEOUS EXERCISES IN FRACTIONS. 105 46. Find the difference between :- •2(5 + -2 of 3-7 4-3 + r,-6 /•-^b + .2of3-7 4-3 + r,.6 ^ V-48-.014of 20 "^~.^~STi) ^^^'1 1^«- 6d. 1 /^of 4 of 74 * -91 j.i"8 H + 4^ of .V"^?5Tl50j^-74^) of ^1 ^'>s. Od. pi«h:VlIufof '^^^^ '' ""'''^'^^ --* --'^y ex- ^ = lia^lSk^_37^'- 38 48. Find the value of; •23-(|>?.n '~^- TT— - Of fii ^ -571428 - --L •45-. 1136 -,V "^2jrrr-:Tt <^fl guinea. 49. Subtract I of -^^t . i nf I -- x i from 101 times the sum of ^ and * of Z'of ^. ^ ^ 50. Find the least fraction which added Tn'.h i, 1^ and - shall make the res^I a^ tnttj" '''' '''''' '' ^'^'^^.^^^^^ ^^^^^'- tl.e same ''^-^of^^ofL.^lOl 52. Simplify ' ^H + n '^ 61 -loi- 53. Reduce 5^+ -21605 - -005 ' ., ^. , -00240623 *o a vulvar fraction. 54. Find the value of Kr. wJV ^i^'~^''^ (^+wh), to two decimal places 55. Which of the three quantities ^'>« ^r-, •*' f f^^*' nost nearly expresses the\llue or#.fi^|fe^^ 56. Fromg --^JJ + r , ^ . . ,, ^^ ^^^^ , ^^.^ ^.^^ ;_ j_ _ ^ 57. Simplify 6 + ^ «-i . IT ii_ofj A+i%of3^-(|-offfr^) |_of ^ + •• of 5 4 -If-' »^ 106 ARITHMETIC. } i i- 1 I P r 1 l! 1 1 58. Ascertain whether -52 or "5 111 more nearly repre. sents the product of -83 1 and -ll^o. 59. What quantity must be added to 3^ of 2f ^^ If of 32f ^ 3f of 4| to make it equal to ,^ J .;, of 3f of 3 lofU + ^? fiO. Simplify i 1-4 '^3.ii/'\(;i'iu + 2> 61. What nuniher multii)licd hy 57-20577 will give a product differing from 180 by less than -0001 ? 62. Simplify ^±A^3zS:^^^ijll). 64. Calculate to seven places of decimals the difference between V^ and yo +i7S5' 65. Find the sum of (l+l), (^-f). (fx|), (1-^). (t-4). 66 What fraction having 17 for numerator equals -I? 1 of fii -^ of 24.1 1 _ 4.1 n V 3'5 •" _i. '-?•"' 7 67. Simplify '-"« __ i^-rfiZ-L^ lil_^^l:. /^ JL? v 4_8_ 68. Explain what is meant by the expression 3| of i of f* How is it reduced to a simple fraction ? Explain fully the reason of the process. 69. Simplify 1 + J + i O-ir i-s (1.; l^ot 2|.-_ 3^41^3 41' 4i 3 (^•) ^-^^n 4- 6 - 1 ~j-?xlO|. 4-i 70. Simplify ^J^g^±^|^ x ^ of 365. '0 iKrjirly ^opr<^ MISCELLANEOUS EXERCISES IN FRACTIONS. 107 72. Simplify the fractions : (1.) '■'?---iA ; (S.) (2.) (f of 1Y\,)^?|; 4- ^ — 1 --^ 73. Find co.Tect to four decimal places, a number which as a multiplier may be substituted for 3 -1 41 sTas divisor 74. By what decimal, correct to five nlacecs mo.. .« ' number be multiplied so as to ffive thrsame r;<,,^f ^ ^^ viding that number by 2-302585? ^""^^ ^' ^'■ 75. Reduce to its simplest form- er ^ + 5^^ - 4V)(41 - 31), divided by 1 s + 24. - r2 « - 1 ^^ 76. Reduce the fractions ^^ and is f^ „ '^ ^-'iiiA erator, and determine wMcri^ the |?e\ter """"^ °"'"- 77. J^nd, correct to six decimal places, the value of •3472 + -03172 78. What error do we make in takintr 208 ^^ +i,^ i of the square root of 17 ? ^ "^^" ^^^ ^^^ ^^^^^^ 79. Find, correct to seven decimal places, the value of fV5 + |^/2 + ;^ Che?. IV^r^^rt ^^ "" '''^^^' pendulum is 30.37070 in- of an inch wTll hlT ^T^ '^ ^^ y^^^^«' ^^ ^^at decimal ^omone mitre? " '"'''''' '' "" ^^^^"^^ P^^^^^- ^^^er 81. Reduce to a decimal correct to six places + 1 + &c. deci 1x3 1x3x5"^ lT3ir5T7 !t^^ '^' '^''^'''''' ^^ *^^^^^1^ ^y -00159082 to seven o.mal places ; and reduce .7002457 to a vulgar fraction. 108 I f l\ iri ili AlUTHMETIC. 8.*]. Fiiifl the value of 5 3 X 5' 5 X ') • 7 X T) ' \) x 5 ^ Troeccd thus : — 1 2' *>7 7x;")" 7x5' X 2' 7 x 10^ 1 of -0000128 = etc. 7x10" and similarly with the other fractions. 84. Find the value of 1 1 !••''> l'3-o 1-3-5-7 4 ^ 4-8 "^ 4-8-12 "^ 4-8-12~lH* So. How many terms of the series in prob. 84 will ^nve the square root of 2 to two decimal i)laces ? ^ 86. Calculate to 12 places of decimals the vmIuc of the series : ^"^^■^1^"^1.2"^3 + 1^ + ^^^^- 87. Which is the more correct value of a kilometre •' of a mile or | of c'l mile ? 88. AVhat numlier multiplied by 6.-]!)4-427 will o-ive -i product coinciding with G384-257u"to four decimal pfacesV 81). Keduce to its simplest foi'ni (•075)'''-|-(-025)'» (•075)^ -"(•075) (-025) + ^•025)^ * 90. Expi-ess ^f. and ^^j;. as decimal i'ractions to Jive places ; and ])rove that their dirt'erence is about i— or either of them. """ 91. Find the value of (1) r(p^, (2) rr"^^; \o,j-7;>/' ^ ^ ^ \-0.'i375>/' •2 5 (3) f/^i2'+ 1^/-003375 fS-- f/-00i MISCELLANEOUS EXERCISES. r, 109 ^^^;.;:i Th.„um,H.,-of Canadian.,,,,,,,,. ,•„, ,,„„„„ ,,„,„^^ .^ ,f^--ot 4-4 ; and tlic nnnihc,- of llniKvl St,,,,,os ,,„„,„.« ,„ ,^ round st.^i'linif is 11 of <" v -.tivi .i value of ±i(xx)i,.iv,,,,c,'i-v',';,;i';;;'iy;i mc^:'' '""' "- _^ .^^..^ H^,w Coso an approxi.nation is -V'J- to tl.e sc,„aro ^»n' l^tst di,.it, show tlmtt^,e metre to 1"^'^^^^ ?«''^ ''' rlaees, most neaiMy S-^SOSllil^^^ ?m ' ^^""'^ '^^"''"^''^ 97. Find to four places of cloeinials tlie valiK. of eorlttT""" "'■ "" *°"»-"'*^^ltltoment. i. n,o,v noa.-ly 10 , „) !Foo9 = ^'''' "'• jT^-n-OOS. »»• Express in <,ecin.a,s accurately to 5 places tl,e s<.,.ies 100. The volume of water '^t thf. k • • " 1-04;J15 . ^ '^^ ^"^ boiling- point is J:qq\1)~ of Its volume at 02°. If a Donnri ^^ A ^^1^ *J 't pouna of water at n-'" tTtSiSllt '■" '"""'^' """ ''» ™-- ^^'-on >'eate:, to!^n,inrnJa;r^^,l:^h%l"rZi,'''"T'°"" ^"•" -'"«•« limits to the reve.tin^tl^:^'^^^::^''- """ ""' K'^.>'in..t,,e1^5^f;,SLia,als,of o \'^ -- X 3 ^5% ^3 no AIlITirMETIC. I It H I lOa. (;iv(Mi the. \vuii;t\\ of a ih'^vvx) (Uli milcH and the IciifJTth ot'ii metro ;;i)-:{7 in., what is tlio orror in t.-ikin^^thc metro to bo •UOMKM)! of the tli.stant-o from tiio ocimitor to the polo? 10 1; Find tlio value of ^-f - + '"•} +^^ • 7 7"^l> 7'^2 7' Find the diffcrcnc(! between your result and the square root of ,'.. 105. if '21-2:} fr.tncs-i'l, tind the v;ilue of a franc in cents. lOG. A starts from Kingston to walk to lielleviilo, a dis- tance of 45 miles, at .'U miles an hour, and B starts from Belleville 3 hours earlier at 2}j miles an hour. Where do they meet, and how i'liv will 13 1)0 from King-stou when A arrives at Belleville? 107. If 100 cubic inches of air weig-h 31 ^rnins, find the weight of air in a room 32 ft. long, 25 ft. wide and 11. V ft. high. 108. Wnter in freezing expands one-tenth. If a cubic foot of water weighs 1000 oz., tind the Aveight of a cubic foot of ice. 101>. If the imperial gallon contains 277-2 cubic inches, and holds 10 lbs. of water, what is the error in saying that a cubic foot of water weighs 1000 oz. ? 110. How many pounds of tea at 4s. 2d. per pound can be bought for i'12 10s.? 111. If rain fa lis to the depth of ^ in., how many gallons will have fallen on an acre of ground? 112. My farm contains exactly 18 1 ae. 70 sq. rd. 24! fH{. yd. There are 3-85 ac in garden and orchard ; iM47 ac. of green croi) ; 7(')-!l ac. of grain ; 23-(*>()8 ac. of me-^- ow • 31 ac. of pasture ; and the remainder is uncleared Di ' : what fraction of my farm is uncleared? 1 13. A farnuir sells to a merchant 3015 lbs. of hay at $1(5 per ton, and takes in payment lbs. of tea at 80 cents per lb. ; 22^- lbs, of cofteo at 20 cents per 11). ; 33 lbs. of sugar at 12 11). for a dollar; 32! lbs. of raJsins at 18^- cents per lb. ; 11 lbs. 13'iZ vti' bacon at 10 cents per lb. ; and the bal- ance in cash ; hQ^\ mtiAili cash does the fa.rmer receive? tit. If 4 men in fiduvsof <> hix o.^.i. '>'-,^^'-ass, |h;w many iu< ,rwi I, 1 ''" T''' ^^'''''''^ 11 lirs. each ? "^''^^ ^^ ''^'f^- i'> -' day.s of ^^^'^4!^[stn'.l:;''';;:lr'"^^^^^^ .It first. ^ "' ' "'"' ''•*' "'"'»lH3r of acres |,e J.ad >j''. A ])in( contiiins 'looo .r,..,;,,. „.• , , .'•.•.ins in 17 husa. 8 U \ " 1 of i^'?' ^''"' ^^'^^"'•' ^'- one after nn<.tlier? ^ ^ ^''^- ^'^''^^^' J^ I'laced '""'"1 ti'=^t i>y stn.tchi,,.- ,.,,11 *° ';? ^•■':n"-K''i. it is I""'- i>i«'o..s, ,v,-,uh rc,s^,. .;,:.' V,i'^.vV-^'; ;' ">; ""' <"' li yil., will cxml.lv tit h,. I,., I , ■*,? ' ^''■' '1 >■''•• 'I'i'l the li.M ? ^ ■'•""■ '^«'^''"«m. wliat will itco.sttoe;u-i,c.t 111'. I boiis'lit a, Imsli f-im, ,un i , «1 faiMu cost n.c por acrj^^ ' ''"^^'""«h l,as tl,e iiup,.„v. 120. A fannci- scills a ii)ci-,.|i.ii,* ja i , , mcvliaut sells the f "m, ' 5^ v,, '7 " "' one-fifth; the pel' .v,l., IC, ras. of cal , , s „' I ' "* '^''"■"'"'"'li »t 8;i.(;o cotton olothat 1 /,' , 1 ,,: f i " P"'',^''''' ""'' 4* y""- «f fourth, ^\•hieh As, . '"'■'''*" " P™"* «'■ ""<■- Low much? " ' "'" '""'■'= ''J-tl'« ti-ansaetion, a„,l l-l. A train cfoino- '>^i iniinc. «„ i !'■ "1. on a tri|, of 2^0 ", lie .t", I"'"' ''"'*'' '" 1 o'^'oek IH'."' starts f,.on, the sa pla,e"" "". ^^S 37 miles an p. ui. ; at what time will tl e i.^st ti-,if, , P"'',^ °^''"'='^ 12i. If £1 = 1,1 fl„ri„, " '^'»" •'"'I'oove.-tak.my 112 ARITHMETIC. i :l I! r *■ 1 1 123. Find tlio least sum of inoiioy tluit can be paid both by a wliole number of faith in^^s -uid bv a whole number 01 liiiis. 12 t. A sidereal day is 2n hi's. 5(; min., and the mean solar (biy is 24 hrs.; reduce the difference between the two to the decimal of a sidereal day. 12.'.. A building-.lot 30 feet in front by 120 feet deep is sold at .^100 per foot fronta«-e ; how much is that per acre ? •.r.nn '■ n '''''^''J ^"''''' ^^ ^'^- ^^'"^ '''''^ " ^'^- ^^^^0, contains .^0( gallons ot Avater ; find the depth of the water in the tank. 127. If 3 men can reap 8 acres in 5 days, working- 8 hours a. day, in how many days can 8 men, working V> hours a day, reap 1!I2 acres ? 128. If marble is 2-71<; times as heavy as water, hiid the weio-ht ot a block of marble 18 inches square and 10 feet lu<>-h. 12!). If j^old is lyi times as heavy as water, find the weiffht of a cubic inch of gold. 130. A clock that is 3 minutes slow at noon is ffainine- at the i-ate of 11 seconds in 5 hrs. ; when will it indicate true time ? 131. If 2 men or 3 boys can do a piece of work in 5 days how long will it take 6 men and 10 boys to do the same amount of work ? 132 If 27 men mow a field of 00 acres in 7 davs, work- ing 8 hours a day, how many men will be required to mow 200 acres m U) days, if they will work 10 hours a day ? 133 If a cow gives 12 qt. 1 pt. of milk every day,^ and 1 lb. 8 oz. ot butter can be made from 25 qt. of milk how many ll)s. of butter can be made in one week from the lujlk of IG cows? 134. How many sqiVAva iWt of surface are there, if the atmospheric pressure (at 14-8 lbs. to the square inch) on a mail is 29000 Ibw. ? ^ ^ 135. If 200 men can make an embankment 2 miles lono- m 20 days, how much over-time must 120 men work in order to finish an embankment 3 nules long in 24 davs 12 houi's iH'iiig a day's work ? " ' MISCELLANEOUS EXERCLSEa 113 111 be pn.irt both whole number , and the mean >etV7een the two 20 feet deep is i that per acre ? wide, contains le water in the ys, M'-orking 8 on, worlving 12 water, iind the i*e and 10 feet ivater, find the loon is g-aining- ^vill it indicate A^ork in 5 days, to do the same I 7 days, work- iquired to mow ours a day ? very day, and :. of milk, how week from the re there, if the lare inch) on a It 2 miles long- • men work in ig' in 24 days, I i:-56 Atrainisjust 27 minutes in passing tlnouo-h the Alont (cms 1 unnel, the length of which is 1122U metres : liiKl the speed oi the train in miles per liour. i;]7. A roller 28 inches in diameter and 10 feet louLr makes .)0 revolutions in going- from one side of a Held to tlie other ; hnd the number of acres in the Held if the fuller has to cross it 70 times. 138 If 12 men, working 8 hours a day, do -i of a piece of work m 20 daxs, how many days will 15 men, woi?di^ 10 hours a day, take to do | of it ? 13!). If a harrel of salt weighs 10 ()uarters, and 4 such (luarters make the iMiglish humlred-weight, and 20 such hundred-weights make the English ton, find the number of pounds m the English, or -long" ton, as it is called. x40. liow many such quarters make a barrel of flour ^ Find the numbcu- of barrels of flour that will make the least whole number of long tons. ^ih' '^ ^^;^^'^'^'^^it *"'iil« owing $10,500, and pays his creditors 35 cents on the dollar ; how much will l.e lost l)y a creditor to whom one-fifth of the debt was owing '-^ 111. Mnd the greatest common measure and least com- mon multiple of 8021) and 237'«' - 1^^'«" il iiii^'M 11) sec. 14o. If 18 men do f of a piece of work in 30 days of 10 irs., in what time would 15 men do the work, workinrr 9 hrs a day ? ' & -^ 144. AVha^ is the relation between a pound Avoirdupois and a pound Troy ? Express one-eighth of a ton in Tro^' ounces, and 57(;o pounds Troy in ounces Avoirdupois. ' 140. 4roni a cask of wine, worth i)0 cents a gallon a sixth part IS drawn and replaced by wine worth only 50 cents a gallon; what is now the value per gallon of the wine in the cask ? r & m tuc iV^': '^ f?l^ '^ ^"^'^"^ ^^^ ^0 I^ ^^"^ .«'^^»^«d one-eighth, B sold It to C for ^30(5 and lost three-twentieths ; how much did the lot cos**; A? 147_. A plot of land containing 20 perches less than 5 acres IS valued at £1081 17s. (kl., which is three-twentieths iH.>re than it was worlh a yc^-ir ago ; l,y how much lias its value increased per acre during the year? i 1; \ ^ 1 114 ARITHMETIC. what fraction /hi^r; ruM\:^r;s ^r '^' '-"■ much can A beat C^in'?. "S^l^:^^^'^ ^^ ' "^ "- 150. If 1 pound of thread makes 3 yds of linen 1 1 ,rH. w,de,__how many pounds would make 45' yds°'of ren\';fd; and^il' T^'fl^'"'^'' ^^'i^"^ ""^"^i- dollar weighs 412-1 svnim 85 cents what i/a doiiarraS:;^-,^; 'w:;;if.r'-"' into\'-sq1,:rc bartoler?'"' ""*-"" '^^^ "« ' " '^'-'"^'l J^agt£ii-ns^-^-r^^^^^ 154. An express train leaves Marseilles at fi qn in fi . evening- and arrives at P'n-i^ nf o r A ^'"^'^ ^" *^^^ to 3281 feet iccKoning the kilometre as equal 158. A railway train has a ionrnp.r ^/p? '\^'tiion .-> Sx^eed to arrive on time? '^ increase its 15!). A house and lot are tog-ctlier worth ^^mn fourth of the value of the hnn<.o7c l .1 *-100; one- valine of thP '^r - 1 fi 1 ^"^ ^'I"''^^ '^^ one-thii-d of the >.»,iue or tne lut ; und the value of each. s ; it is rolled ours ; if the increase its MISCELLANEOUS EXERCLSES. jjg IfiO. A man bought a ou^nfitAr ^^ * done up in packages of 1 Ih t !i^/^ *f^ supposed to be §04; on weighin'them ^ow? ' ^^I'^'^^^^h he was to pay ,oT l1 ^ri'vlh^'^„<;^,^!;^,^f ^-=- and .9 Of it is pure is the value' in dX°l of .n Z""''"^ ''° ^isreg-arded what 1C3. A water taS cVn L « w',"r"'- '"'P"'''' ^"'^l ? and by another i 2 hours If '^ ^ """ *"P '" « h«n'-« tap in 1 hr. 24 n.in The ;„ni i • " ''"'P'"''' "^^ » "'"•'• ai-e opened at one"' in wh.f h '% "'"P'^ "II three taps pend upon the xwioo nf Ami,, i. i ^^^ ^^ bread de- if a 10 cent loa be increased w^he^fl '""'=^ '^"''"^ "^« P^ce in value ? '"creased when flour rises two-flfteenths son 'ta pro,Sn?o1Sr' Xs*' tl'^-'"^ -'■"""^ '"^'^ P^- how far will a railrnnrl Z„- . ° '^'^ Pcr second ; ■nilesan hour have' o,febefirt.''"""'''5 '* "'« ''''^ °f •'^0 be heard 5 mile; awiy ' ''«'^»' « «>« «onnd of its whistle can of air weio:, sTl ain „ ' T"*"' ' "■ l** «»bie i ,ehes salt water w<'slf?' "'""'' '^'" ^"^ ''"«« "'ches of ■'owL"i;d^:s r'lSnnt^^"'""' °'' -""' '»" -"»". sold? '""""''''ant gram on each 1000 gallons I'O. The sum wh'Vh "in -^ *• Will pav B's was-es for 81 1 .^■■' f" T"^'''^ ^"^ «i days, it pay the wa^^es'^f A'a'.d' ;l'l-oI4tf"e';.'""' '"""■^" "^^^^^ -"' 116 ATU'irrMETrc. • fi if. 1 1 r 'I II if mud, doc^s B\wn V "' '' ^^^^ ^^'^'- ^'^^^^5 by how 17l\ a pain dealer buys 5225 hmhcM of whc.f- -.f *i n»^ per l)u.s hel, and nnvs <^i->>-; fv... ,• ''^"'-"' "^ wiicat dt *1.U5 m.'o'l8,;lf tov'.K.il'lS' 1?,:;;:""' ^'f (li"-)-- comed f?ol-l in a .ovoreiffn ' ""'°''''' "' 8''"'"^ "^ P^^ 174. A and B tOi>-e(;]ier e.-m p-im -py ip„ • o i an.I C f,o,.ctl,or c,ur.,u. £7 sj i f? d fv^ -1^^ '. P h„f 'In I;r" *'"°°''' '" " P™"' of «n«-tcntl, the prime cost ■L onrof'tl.TLTr"* '■"i"'-^ '"'>"""" I throw oa one': co.t ,;;, I'y'i,'"; '""""'' ^"^^ - ^^'""' ''action of the pri>ne mil"in'K.n.';h"8'^f ^'^ll build a sidcwallc . quarter of a V ..•» ^' n:!:;:;:: s::: x .z ^vii;;!^.— ::; what , nil the ,,„H„er cost at .¥17 per thous M ?oet ' ' ft Mn ''^ 'h'"" -" "■ ""'• '""«•• "'ft- 8 i"- wide and l- ft. .. ni no;!,, there are three windows each5i ft hith „;,■; S ft. wide, and two doors each 7 ft hi-h ',"-'» *^.'V the l,asc^,oa,rd is 1) i„, wide; how u„ ll';: ',,1 ^.^^^ WKlc, wdl be required to cover the walls and'ceim,t^""'' 17!). An orchard is 2« rods loiiff and I,"-,! r« s wide- at 1 1 cents per cuhic foot what willit cost to di^ ., ^^ i Hround it 3 ft, in. wide and four ft. H^l '" ^'^ '' '*'"^" LSO. T he s,|uare of a nuniher diHers IVom the sauare of the uex „„n,l,er by CiH ; ,•„„ the two 1™,,,^'^^ °^ l>il. It the earths e,,uat(.rial diameter is 7'l-)0 ,„i|p„ find the len-th of a .legroe on the equator. ' MISCELLANEOUS EXERCISES. 117 1«2. A runs a mile race with B n.ud loses ; had his sDeert been a tlnrci gutter ho would have .von by 22 y , rds whiit iraction is A's speed of B's ? "^ ' 188 A boy's a«-e now is one-fifth of his father's In siv l^Z 1!;;"^^ '^ ""-^^'"^ ^^ ^^^ *^^^^-^'« Pi-csennje. 'll^ 184, The diameter of a cent is one inch. How mnnv cents can be placed ed«-e to ed«-e on a surface of six i • e^, square so as just to prevent any falling- oft" '^ ov ,.;i\ '"'''' ^1'''^'"^' ^'''^ one-fifth of his capital is worth exactly as much as another man who has kiined t^nve twentieths on his capiUil ; the second ni.n's c nTt.l v^^^^ or^nally $.000. What was the first nnu" ^^ -i;;;'^ ^"" Mf 1 ;>V. 1 ''V'"? "''"'^'^ ^''''''^' «^'^ «l«^'k af-o tog-ether 187. A hiborer while working- is able to save 75 cen s , day During, ten d. ys of a certain month he I's un'bk to work in consequence of which he is worse off 1?. 050 than he expected ; hnd his daily expenses. " 188 A plot of land is sold at £1200 per acre Wh-.t i^ the price in francs per s<,uare metre, if £ I 25 franc 189. Three clocks which tick 70 80 and 90 Hp v. y)ectively, a minute, heat coinciden^y'^fem n thn^: tTc^M ^;' "^ "'"' ^'"^ ^^^'^ ^^-" beat 1::^ 2r: 1 JO Gold sixteen carats line is broug-ht to the minf what fraction of its original weig-ht wilf it we o-V 1 en' nujde mto coins eleven-twelfths hne ? "^ 101. By selling- cloth at $1.26 a yard I o-nir. 11 rumf. more than I would lose by selling-^ntl^l 05, v , wha. ..^uld I g.,in by selling 800 ^irds at tl a wd 192. A barn 80 ft. Jong and 60 ft wide is built on • 1 n', ground 308 ft, long- and 204 ft. wide. Tl e rest of ' • Ploc IS covered with cordwood to a depth ot' 8^^ h - many cords of wood are there ? 1 ^^ © n. , iio;n 193. If 15 men be necessary to excavate 9(56 cubic vnnU m 8 days, working 10| hrs. dailv, how many men wo ill be required to excavate 575 cubic ynrds in 12 dy w Sl^la^ ''"^'' ' ^"^^^^ ^"^^^ ^-"^^ taken cm du^ii^: tit: 118 ARITHMETIC. It 1 1 U! 194. The sum of ?648 is to he divided among 24 men l) on the avIkjIc transaction ? 1911. Assume that G men can do as much work in '.n hoar as 7 women, and 8 women as much as U bovs u d hat 5 men can do a ceriain piece of work in Wlmurs h,>wlong will it take 1 man. aUmenand SJbovs tog-eth:; to do the .same piece of work ? ' "fe''^'"- ' 197. In what time wouM a field 80 rods bv fiO rods o'lv for underdraining lengthwise at 2 cents a foot f t^,! S S^e d'ldniiJ' « VT '• ""^'"'^ '""•'»"•« ■"'"•«"'-' Dtioic duunmg? The drains are 4 rods apart, and the first dram runs down the middle of the Held h„ ,•//""', ^""^^ *" ^ ""'-'>"'>■ 199. A farmer employs a number of men and 8 bovs • he pays the boys $-G5 and men $1.10 each pel day. The c~i .fj" r" 'r ''" ""^ •■'^ "'"'^" - if each'had .'- eeived$,)2perday; how many men were employed ? tiof A ,rf1, "f '«Sf" 'er and walk in the same direc hour. At the end of 7 hours A turns and goes back • how many rni es will B have gone when he me'ets A? ' wi?^^' ^"^.'i P^^"^^ '^^' ^^^^'^^ '"^ ^^i<^'^ ^^^^tl "^ l^ovs are emploved who do half of it in six days. After this 'one nwe m'ln and one more hoy are put on, and one-third mo^4 is done in d days ; how many more men must be put on that the work may be completed in one day more ? I MISCELLANKOTTS EXERCISES. 119 11 ore man 203. A clock which was 1-4 minutes too fast at a nuarter to J 1 p.m. on Dec. 2nd, was 8 min. slow at 9 am Dec 7. h when was it exactly ri^ht '^ ^^ ' haying: been paid rtnrin- that time? ' ' L0:>. The areas of the continents in square miles are n, folows: Europe, 3,780,000; S.America 6,TO0(^N Amenca, 8 750,000; Africa, 11,500,000 ; Asia 16 500000 i by boys, and the worlc goes on for 11 da^ more .t fh» end of wljich the whole length dug is 1889 a How'lny feet per day do the bovs do ? ^ 207. Water flows into a tank from two taps which run Hul uf;!'"^'- T"""*, "' 'i ""'• -^i hours^-especViyeh fill It up to a certain level; at this level a wastc-niDi w.',';^;/ H*^P' '■"•" '•"""'"^' '" ^^hat time wllX se™«;e y^Lf """'"' "*'*^^" "' "'^''^ *•»"■• '-"dred con- ^^09. Multiply 365 days, 5hrs., 48 min., 47-5 sec, by ■ 210. What is the dilTerenee between the results in the Us;o preceding exam pies ? m how many years would this difference amount to oiu; day ? J- .y <" s wouia tnis 211. A person buys four houses ; for the second he srives halfas much again as for the (irst; for the thii^ l,a|f af ",t;.T'" :!J f«Vf « r«-l \ -'" fb'' the fourth is m,ch What i^'^iie'cosrof laihV"'''""'' '" '""''^ '" -"' ^~'^'' K I m 120 ^ AHlTirMETTC. 2V2. HoAv iii.-iny ounces of ifwcllcr^ (mWi iq rtn.-,.,ta bo ,ua.l F^"" \,r!^^' /'''!.^ l'^ ^''''' ^^''"^'^ ^'^^ ^^i« fouowiiiff bill of tixp^ . Amount of Assessment. ... ^ "ii i or taxes :— yw'rJv • '^^^ Town rate, three and eight-tentl^s mills on the dollar School Debenture By-Law, one and one-fourth niHs " * * Harbor By-Law, two and six-tenths mills KcdcMiiption of Debentures, one and four-tenths mil iV * * " School rate, five and one-tenth mills ^ High School rate, one and three-fourths Vnili; "" 1 otal taxes MlSCEF.LAXKOf^s EXEIiriSES— UNITS. 121 spcctivc ly «, 9 and 10 hours n, dny, and rcceivini.- the s^nno (iMily pny; after tlirec days each Avorks an iPoi .'d v more and the work is finislied in three davs nore . if the total^^m paid is ^114.05, how mueh of it should 4^1 ^ 220. Ih'inxn purchased /, of a mill property for MOCA 55 |nu Smith purel.ased J- of the same V4>em' a^ a rati ' 1./, her ; what did Smith's part cost him, and what fac ion ot tlic property remains unsold ? n action 221. Four men Avorkin^>- ci^ht liours a day take '^2 davs pave a road a quarter of a mile lon^ and' *U) ft br v' I . how many days will four men, two of whom w rk e 1 l t )K.urs and two ten hours a da>', take to pave a r ad 1575 yards lon^- and 8C.J ft. broad ? ^"^ i.-,m"' -i/V'^ ^^'"^ ^''i^'^ number the product of which with loOO will be a perfect square. iii'il' ™^'^^«^^'^«t number the product of which with 14175 MH 11 be a perfect cuhe. • '-u nirn I'i^ttm w/^'^'^'rll?''"'^^^'' *^'^ f^^'^^^^^^* of which with l.iio will be a perfect fourth power 225. If the volume of a brick (2 in. by 4 in. by 8 in ) be .al^n as the unit, what is the measure 'of the volume "^f a 220. If 27 is the measure of a cubic foot, what is tlio mt>a^ure of the unit in cubic inches ? 227. If tlu! area of a, posta^^e stamp (J in. by T in ) bo tho J.n8-lisli f,n,l ,„ i\h: Frenel, system, (iiven the „ encal value of any len.nh in one system, express iUnThe 22!). Compare, in wliolo numljers, the rates of speed of wo locomotives, one of win'el, travels ;W7? nU es^^^r 1 i Ii -s., ami the other 2(;2,'„. miles in 8 J ]„•«. . v|,," h' ',; ;;;|o,^;;a^as the tmit of nH.asnren.ent in olJr^^ 230. Find the len<^l,hofth measure both the distances 2 ft. I in. (■f'»iiS<'!^ J' in- '..hat will exactly '"^'"^ y"ft' long and 27 inches wide, express the unit of measurement in inches, in feet, and in yards. 239. If the unit is $4.8Gf, what will be the measure of $10.95? Of $48.6? Of $4.86? 240. Define the term^ unit und measure. The measure of a certain distance is 1760, when a yard is the unit of length ; what unit of length will make the measure of this distance 320 ? 241. The measure of the length of a piece of fencing is given as 100; when a foot is the unit the measure is 66 what was the unit in the first case ? ' be'^48%?''''lo'?'^ '"''^ ""^ '''''''' ^^'^^ ^^"^ measure of an acre What is the measure of an acre, Avhen a square link is taken as^ the unit ? A square inch ? 243. Find the quantity whose measure is 3." when the tmit is $1-085 ; also the quantity whose measure is 3-1416 when the unit is 7 ft. 4 in. 244. Find the least weight that can be exactiv weighed with either the oz. Avoir., or the oz. Troy as unit. ch toainilo ; inch on the measure of MlSCELLANEOrts KXEHOISES— UXfTS. ' I2a 1| g«ll«n.s, 'lulu '"*'' """^^-3 pints, 5 quarts, Lanas '^^Z:! SUZrr"^^ T"" '^y "- «'•«-" in".l., on t hi mTp tl,e Tow , "" Ti" '"' '^ ""'"' '" ■■'» inches in lengtht^nd u inlln';;; ti?'"," """""™^ ^^'V does it contain ? " "' ' ''"''' '""">' 'icrea linear measurement "' ^"'' *" ""■' of as the unit, tl,e sSnL is 'ioSp> «"?'!?'■ '''•'•■''^'=" Jupiter wl,e„ the cart,, "rssi«r;;iit.'"' the mass of Sueh a ^nit'irtLrrnf '"f* 'T' """ »^«- ^O""- What numC wilTiXesent n e ntt^'Vi^""'^ '""'^'^ ^ -^ ' '>5!) 'r'i,« „„,.!i ■ "-P'^''^'^'" '"<- mass of tlie moon ? bornl'tle „, iT'theTnt';;-" "T"'^'- ('•'-■"■"^« ™"«^) & t;n^. uiiiL, rne suns diameter m iii-jr, <. «, j -, '^ 'i.ameter of the sun when a mile is the unil ' ^ *" fooTi!; a'ith.reTi^i'i^s,'; .?'':;; ''t ^ '"■-'^■^ «- of the power of .in en*;,- 1. ., " ™"' ""'' "'« ■"ca'^urc .W ft. in 6 hon,.;! ^'"'' "'"' '^'"^ '''"**« "'000 tons through 3 days, where';'' W's'^o ?'"''• T' ' ^ ^™""'"'« ^^'O''^ in will It'take 3 wJm n and VUV ";'." "" ,""'*■ "°"' '™*? work whieh 12 n.en havc'half ri';isi:',;/?rio''£j; t"'' <"' li 124 ARITHMETIC. i r 256. akin^^tho rndiua as the unit, tl.(« circunif(;rencf3 inn'l.f^''^ V:^ ^''-i ^'V"" "^''"^^ revolutions will bc niudo in a -^ Vy^jfr ""^'"' ''•^' '' '''^'^''' '^''^'^^ ^''^^li^i^ i« ^ ft. 4 in. V -^7. It .)0 cents an hour be taken as tlu; unit, what num. ber wil represcmt tlie w.-i^cw of u nuin who receives $50 for worlving- 8 hours a day for 10 days ? 258. If a velocity of one foot in one second be t.'iken as the unit, what number will represent the speed of a man who can run 100 yds. in lo seconds ? 251). The number of square units in a, piece of carpet is Ji times the number of linear units in its h-ni-lh • if 104 uh'^^^L ""'"'V ^''"^^^"^ ^^ ^^i^^"'^ y''''^^> 'i"^l the units of measurenumt. honrw!^'i'"-n"i^''/'^^' '''''''^ "^^^^h '' man can do in an houi, what will b(, the measur(! of a piece of work which l?rTf '' r'?'^ "V^ '''''' ^"^^^ '' '^^'y' «^" ^^> l^o^^'-« ^'Hch ? Jbi. J he disance between the earth and the moon beiiiff as un-r' i^,^.'""''!:'^ ^^'i^'^^ ^'^^'^^'^'"^^ t« tl^- ^'^^'"th's radius nnXU 1 ' ''''^'''' ^''^"^ '"^'^'^^-'^ ""1^«' each of these knou^ nf H '"^' ''''''^^^ ^''^^ ^^"^^^"^'^ ^^^^'^"^''^^ ^^^l^^^t can be op\ ?T "" "'''^'^*' distance from the earth in miles? Jl.2 Using my walking-stick as the unit, I find the dis- tance between two telegraph poles to bo 50; I then ascer- tain the length of the walking-stick in inches and calculate tlie distance between the poles to be 114 ft. 7 in • find the r;;;^f ,l';r,i;^^ "' "'^^ ^^^'^^^ '^^-^ '^»»'- '-• - 203. A cLil)ic inch of water being the unit of weig-ht if heweight of a cubic inch of gold be 19-258 ; of silver, iy-474 ; and of a mixture of gold and silver, 14-723 each right to the third decimal place, find the bulks and weights ot gold and silver in one cubic inch of the mixture 264. The volume of water at the boiling point is 1-043 of Its vohime at freezing point. If a cubic foot of water at reezmg point weighs 62^ lbs., what will it weigh at boiling point ? • 5" au 2(15. Lead weighs 11-4 times as much as water, and platinum weighs 21 times as much as water ; what weie-ht 111 platinum will be equal in bulk to 56 lbs. of lead ? IWrsCELLANEOCrs EXEItCISEs- -UNira 126 the- tLk„r':/;:,;r>t' W' 'f '«.'?•••'"■'- «"-' grain. ^ ^ •'"*' "'■ "' ^*l"<-'li "I'isli oiio ft. 3 ill wci.ri.u 070 11. ""/^'^ ^ ^r- -^ J'l., < ft. T) 111., and 4 inucl, in four as D can in fi e d^u s K r i'' n' ''"'' ""^ mlier Hnish cutting "ol, .OS of •,.;,' '-^ '!n ' ," ''"" *°- lon^ would it take .^ to ;«Tol,?.:;".r "' "* *'>-^ "«- -i'*.of {rpa«f.ST,;cfl 'n°"' '^T',*'''^"''^''*' -"■ the^i<.|.?p::r.:Lzsi;r::iio;-'''' '^'•'" share"'of td.?""*""""""^ ""'•■« *"«" ^ ; what are Z greatest length of steel wire which when Ini,, J ./ " end, will just not brealc by Its own weight "^ ^ ^ """ > f'.i I' t I i fi ' i i:: ■A ■I -•1 II 1 : if 1 -j I I . [ 1 l" i 1. 1 1 i ' 1 ji 126 ARITHMETIC. 277. A vessel contains 120 gullcnis of wine ; 20 gallons are drawn tlierefroni and the vessel filled with water ; 15 gallons of this mixture are then drawn and the vessel again filled up with water. If this operation be performed SIX times, 20 and 15 gallons being drawn alternately, how much wme will the mixture contain ? 278. A metre equals 39-37 inches, a cubic inch of dis- tilled water weighs 252-458 grains, and a gallon of water weighs 10 lbs. Find the number of bushels, etc., in a hectolitre. 279. In how many ways can 1397 be separated into 3 different numbers of which the greatest common measure is i-Jii i •^ ^^^; ,^^1^'^^s of lead ore weighed 7 tons ; one portion of It yielded lead -^^^^ and silver 8 ounces per ton ; the re- maming portion yielded lead -i^^, and silver U ounces per ton. Ihe total yield of silver was 55 ounces \ what frac- tion of the whole mass was lead ? _ 281. In the Centigrade thermometer the freezing point IS zero and the boiling point 100^ ; in Fahrenheit's ther- mometer the freezing point is 32° and the boiling point Jlw ; what will the Centigrade thermometer mark when Fahrenheit's marks 65° ? What will Fahrenheit's mark when the Centigrade marks 4° ? What reading on the Centigrade will correspond to 0° F. ? 282. If the pure gold in a sovereign weighs 113-002 grains, while that in a 20-franc piece weighs 89-604 grains • find the least whole number of sovereigns that will be equal to a whole number of 20-franc pieces. 283. The driving-wheel of an engine is 27A ft in cir- cumference, and the forewheel 16 ft. Two particular spokes one m each wheel, are observed pointing vertically upwards ; how far will the engine travel before the same two spokes again point vertically upM^ards at the same time. How often will this happen in 9 miles ? waM^^V^^^^ ^^'^^' simultaneously point vertically down- 284. If 175 men and 240 boys do in 1330 days the same amount of work as 603 men and 1005 boys iA 350 days compare the average daily work done by each man w'itli Dnat aoiie by each boy. |! ' MISCELLANEOUS EXERCISES. 127 to contain only 2m By the Ac" o/lsf/^'V^T ^"<* declared to contain 99is „nK- • , * *'"^ '^"^''^l '«'"s a certain "fracK ,r/:e ill" SfofT """"^'"' °" supposi,^. the p-ice per ^^ ^'ZlZ^^"^^^- ^86. A coin consists of 11 Dart- P-nlrl ..nri i V 1 ,, the ailoy l«ing worth one-fiftSth ?f an qual'Tv ei'Kf will Lunv fill r'^'''' °^ ^"terandair be taken, the air ou' nHtv of »1i , "•'"' ''' "''"' 'P''"^ *« "'« ^vater. If" Kn h„ 1 ""^ mmersed in water to a depth of 3-' ft will be compressed to one-half its original volu no ■ if to a de^th of 64 ft., to one-third its original volume if tn It would be worth anW -pof.qi. « ♦^^^^"^t.icnangecl, .oM and Silver ^^Z^^^, T, ttpSr'Xet in 20 days , determine whSr'1 'J mo r'pro'Krct enjploy men or boys. If a piece of work done by men co t ^d Thf.T''' '!r """" ''°'^ «=««' if done byTovs? nnf ?• /""'"■■'' '""S'h, breadth and liei^ht of a ivct angular wooden cosed box nro is in ^'f' \oi <> icct- spectively andtheth'k!::s*;V% \oVrh ,f^ti:,eh■ '^ h". nd iwl""^*^' " ""f"^^ ^-^ "- -"l -hen d ■wo'l'^'^V'nd. ' '"'"^""■'^ *"" ^""^'"«^ ^'1"'-" ""'kH of 291. A ditch has to be made .^(io feet lone- 10 feet wide , 128 ARITHMETIC. 'i i^ ) ft > i 1 \ i ■ ■ t iu 1 i f : m m H 292. Into a gallon measure there are put 20000 grains of broken flint which is '2-:^ times as heavy as water, and 10000 grains of granite Avhich is 2-7 times as heavy. How many pints of water will noAv be required to fill the vessel ? 293. A fast train leaves a place A f()r another place B at the same momcDt that a, slow train leaves B for A. The fast train takes 2 hours for the journey and should meet the slow train f of the distance from B. The slow train is running at reduced speed, however, and they meet at \ of the distance from B. How much- behind time will the slow train be when it arrives at A ? 294. A train 88 yards long overtook a person walking along the Hue at the rate of 4 miles an hoar and passed him completely in 10 seconds ; it afterwards overtook another person and passed him in 9 seconds. At what rate per hour was this second person walking? 295. "If eight best variegiited silk scarfs, measuring each three cubits in breadth and eight in length, cost a hundred nishcas ; say quickly, merchant, if thou under- stand trade^ what a like scai-f, three and a, half cubits long and half a cul)it wide, will cost in terms of drammas, pannas, cacinis and cowry-shells?" Lila.vnti. (1 nishca == IG drammns =- 2i)^\ pannas, 1 panna -= 4 cacinis =- 80 cowry-shells.) 296. "If the hire of carts to convey thirty benches twelve fingers thick, thoisquare of four Avide aiid fourteen cubits long, a distance of one league be eight drammas, tell me, my friend, what should be the cart-hire for bring- ing fourteen benches^ which arc; four less in every dimen- sion, a distance of six leagues?'" Jiilavati. (1 cubit = 24 fingers.) 297. Having three separate parcels of powders weigh- ing respectively 84 lbs. 3 oz. ?>m gr. Troy ; 45 lbs. 10 oz. 252 gr. Troy; and 32 lbs. 7-232 oz. Avoirdupois ; how can I sub-divide them into parcels weighing each the same integral number of grains ? 298. The hour hand of a watch is y of an inch lono-, the minute hand | of an inch, and the second hand \ of an inch ; compare the linear speed of their points. MISCELLANEOUS EXERCLSES— -THEORY. 129 ^299. A sum of money is divided among A B and C as follows :-A receives a certain share, B one-third ind one-sixth, and the balance is divided equally amont them UnTs A '^ ^^^^^^^'"" '^"^'^^ half a^mf^S; heS t r?'^^v^'' T"', ^^^^%^'^ ^^"^*^' ^^^'^^^^^h and at a cott of ^n'. r''^'^"^^' ^'^ ^^'^^"« ^^^« decorated finishPrtni- L '^''^' P^^ ''1^'^'"^ >'^^^-^^' '^nd its ceiling ^?i? iw Mif'"'? '^^^^- ^^^^ ^'*" f^i' the whole wa? carpet at fl .40 per square yard ? „f a?^- m" "f *,'■'"'" ®^ y^"*^ '™S-' travelling at tlie rate 35 m, es an hour, meets a down train 88 yards onff at 12 DC ock, and passes it in G seconds. At 15 m n Tnd 6 seeo„ds past 12 tl,e np train meets a second d mvn train lo2 yards long-, and passes it also in seconds • it wW time will the second down train overtake ZflVstl' ' rf02. Supposing the cost of diffffinjr a trench to rtenpriri npon the depth to which it is snfk as^we as U o aSv ot earth taken out, and that the cost of diggTuff a uene^R fee broad ),y 8 feet deep is 20 cents per ya«;^vha would feet deeT? "'" "''"'^'^ ''° ^'""'^ '""S'. s'^Teet CSuo .303. A workman finds that he Ii.as to spend one-half of his income on food, tliree-twentieths on clothes thr!» twenty-«fths on rent, and he saves the rcnu,M; He" emigrates to a eountr.x- where food costs hatf as mufh clothes three times as much, and rent two and I half^^.« as much but wages are half as much a gain -find whether he IS better or worse off, the test being the eneth of ti,nl his .xvings would support him in tbe^countr^Th'e rt S 304. " In a case of ordinary division the dividend ond the divisor are alwavs simihir number? ..^H tlL ." is abstract." Explain this statement ' '"^ """'"'"' of-;^peS':idilt.™"""''^-"'"" '^ ™^'"-"-' - =' -- pea W "ubS^^lil^y "^"'"" ■" '■'''"■'^' - '« -- "f -. :l If: I I Iftll I IJil' ii 11 ]30 AIUTHMETIC. l?)l' Tr^l'"''^ ''^"^^«^' ^^««Pt 2 is an odd number ..O!!. 'I'lic sum of nny nuinbei- of even numbers i . 4+-/ ^^i iiuiuubi, ic.o.m the foliowinsoperations by sliort methods- o,r ,:",'< l-''! 8732:UH-lt;i; ;!l(;,S7x3!0(U8 315. I),VK e 3812 by 7, and tiie result by ndexniain how to got the correct rema indcr. ' ^^P'"^"" 31fi. riysubtractins-.t from lOtimes .7 show that 7 ' uni4?' ^'^''' ''''^'^'' "' ^^^ ^^™b^^ ^^^d its reciprocal is 320. In order that a fraction may be in its lowest tenr^ It IS necessary and sufficient tl,at the imm^ntor^n^ denominator be prime to each other ^^^i^eiatoi and I.sMuw'''''^*''''^ ^^">^ "^^^»^^^^i- i« divisible l)y four if the l.vst Uo h^.a,es on the rio-l.t are divisible by four • sni: V f ^'''"' the iollc,wi,,i>- sta,ten.ent:-?^4 is be mea sure oi th;M, iiiMii-iutude "'i--'^ ... _ . ^- '"^ ''"*^ "i^a- part of the uni;, m- i,!4'nitude iicli contains 5 times the sixth MISCELLANEOUS EXERCISES — THEORY. 131 number. . ■ numbers 5 is even ; ven, and I. by an odd lainder is 3r is odd. number, imber be y double 3r, wlien Id, when ni, when y an odd linder is r is odd. lethods- explain 3tracted on ly num- t'actions rocal is t terms or and r if the le mea- e sixth 323. Every square number is cither a multiple of 3 or 1 moi'e than a multiple of 3. ' 324. Shew that any number and its units' digit when divided by 2 or 5 give the same remainder. 325. Prove that no sciuare number can end in 2. 32(). Prove that no square number can end in 3. 327. Explain the difference between the operations involved in the following :— | of G and 0-2. 328. Find the value of 2-.!+ ^ H- 1 , J . 1 2 2x3 2x3x4 2 X 3 X 4 X 5 "^ 2 X 3 X iVbVQ 329. Explain the following method of adding the frac- tions in the previous problem : — 2-5 •16666 •04166 •00833 j00138 2^71805 330. Hpw many more terms of the series, in No. 328 must be taken to give the sum 2-71828 + ? ' 331. Knowing that ^12 = 1-4142 + , hnd ^2 to seven deci- mal places without going through the ordinary process of extracting the square root. 332. Describe the respective advantages of using vulgar and decimal fractions. 333. Divide 3654 by 2-03, explaining each step of the process. 334. If two numbers are to be multiplied together, the product of the first multiplied by the second is the same as the product of the second multiplied by the first. 335. If the two whole numbers nearest to half the square of any odd number be taken as the hvpothenuse and one side of a right angled triangle, show that the third side must be a whole number. 336. If a number be exactly divisible by 1 1 , the last digit in the quotient will be the same as the last digit in the lividend. i. . n. H'i I Hi: [\ 1, . ikLi. ft.,. 132 ARITHMETIC. 337. " The g-mitor part of the operations upon num- bers are perforniod, not by operating directly upon those numbers but upon tlie parts into which they can be de- composed. Ascertain how far this is true in the case of ordmary addition and multiplication. 338. In addition of num])ers the operation mig^it be performed by proceeding: from left to rig-ht. What would be the disadvantag-e of this, as compared Avith the ordin- ary method ? 339. Prove that the difference between two numbers is not altered if the same number be added to each. How It the same number be subtracted from each ? 31:0. Prove that "multiplication and division are mere modifications of addition and subtraction respectively." 3 U. Shew that the number of digits in any product can never exceed the sum of the number of digits in the factors of that product. 312. Prove that if the divisor and dividend be multi- plied by the same number the quotient remains the same \Vhat about the remainder? 343. If the dividend is increased bv a multiple of the divisor how is the (luotient affected ? ' How is the remain- der affected ? 344 How is the value of an improper fraction affected oy addnig the same number to both its terms ? 345. -'Division is ratlier to be regarded as the undoing of a imiltiplicntion." Explain this statement. ^ 246, The ditierence between any number and its square IS even. "^ ^ 317. The square of an even number is always divisible by 4. "^ 348. In oi-dinary multiplication the last partial product is always greater than the sum of all the other partial nro- ducts. ^ 349. Which of the digits may be the last digit of a s(|uare number? 350. Ivxamino hoAv far the following rule holds good •— " The truth of ail results in multiplication may be proved l)y using tlie multiplicand and muUiplier as the muUii^lier and multiplicand." MISCELLANEOUS EX ERCLSES— THEORY. 133 K ,1 *:iue Sid?' mv il't^ly" "^"^ ''^^™^' '"^^-^- "^ 352. Explain how a sufficient degree of accuraev ms,v be obtained in ti.e addition and subfmctio , of eSat S decimals to any required number of places wi lout co,f verting the decimals into fractions 303. From tlie fact tliat 10 square chains make an acre deduce the length of the link in inches ' 354. Prove the rale for the division of decimals Apiil v the contracted method to find ■95372843-44 736546 to eight places of decimals. '° 3.55. Shew that any proper fraction is increased bv adding the same number to both its terms. '"""''''"^ ''^ by "he dtn^iW '""'' *' '^"""^"' °^ ">« ""■"«■•''«»'• tiof nV ^Zi'^'f- ' '* "'='^«'^f "-y in th« addition and subtrao- d:;^miJurr'' '" '■"'"''•^ *"*' ^'■'"^""^ '« '••• -'--n 3.58. In what sense can i be said to be the value of -i » 3o I. It any number (not a multiple of 10) be divided bv 9 rond have no remainder then the sum of the hJtX7i„ the dividend and the last digit in the quotient v 11 fcio ofiO. In dividing any numbers bv 12 shew th.at there wdl never be more than two figures' between Ilie dech.Ta' point and the rejieating decimal uecimal 99^8 bvHMe ml;;.!.'!',' "'"'"'" ^^ ^'^-^Sm^^'^^ differs from 36- The fo Inw? *'"'",''' V>"' '" '""' ^'isbth »«'J"i«>s given todivideby ' ,,•''•. Multiply by 7, divide by 11, then bv ■' ■nid add one-eighth of one-thousandth of the resi It " Find e error in dividing 180 by 3-14159 by this rale "" hour he goT.."mne"'rH''H ' '^ "i™"*'^' '" '' 'l"^"-'<''' "^ '™ finl ; J "" ""•"''<''■ ''s to be divided bv two or more fr.ic- tions the result will be the same whether we divide by he fractions success! vely or by their product. ^ -ether the resnTi "T *'™'''.°n« "''« to be multiplied to- Sons'^ilay'bruk'en: ''" """ '" """'^^'^'- ''"'-■ '"« f™' H ' I ' i ! i' 1 I I I I i;! ii 134 ARITH^FETrc, fractions ,„«;• be token "^ '" whatever order the other fraction of equa value w i flenonnnator of any numerator and denom n Uor o f o ''"'i""'"f"«« "f «>« ■ HC'S The nmrtn,.v!;«-"''*^'™n fraction. pro<.uct^^"ta": mi'^nd'rr;^:'"^"'' *"''^-"« - *« square. '' '"'" "^ '"° ~'<' «l"-'c^ cannot be a perfect so,r„uTltipTo;?5. """'"■ '•"^•'^^ "y »»- -n 1 from an'"live,':£SiiTi,Tpr2.'ce'a"l'"' '"^P^?'°" -"<""- ins- decimal. ' " " ' P' «'"ce a terminated or a repeat- than'hf^um o;t\,e\Tll^"'^' """'^ '' -'-«>- greater which follow it ^'' expressed by all the digits wgdS.^!nz^rt----"^''^^-'^ 16| 48 5 „__ ^ 54,^^ = $54.03. ^75. Explain the follovvino- method nf p ^^ approximate value of |18.75 x^l.oTx l 06 x 1 06 -!!^ '^' 118.75 1 13 122.33 MISCELLANEOUS EXERCISES— THEORY. 135 376. Shew by means of /in oxaniple how an improper fraction is atteetoc by subtracting tile same number ftm Its numerator and denominntor. na?j/divrsion :-'' '"''"'"'''^ '''''^'''^ '' ^^^^^rtening ordi- 324) 432578 (1335 1085 1137 1658 38 378. Dividing the numerator of a fraction by any number IS the same in effect as multiplying the denominato" by it diJ. If a number measure each of two otiiers it wiUalso measure their sum and their difference. 380. If a number measure two others it will m .sure numbrs. ' ''' '''''''''''' '' ""''y "^^^^^iP^^« «f ^l^e«e 381. Shew how to convert a simple or a mixed periodic decimal into an ordinary fraction. penoaic Examples, -316, and -67185. 382. Define the terms -abstract" and "concrete" as applied to numbers. Is 6 x 3 = 18 a correct solution of the tre^nts'Ii^ir" '' ''^ -- ^' ^^- Postage-stan^^st 384 Reduce | and ,V to decimals and account for the resemblance between the repeating periods in the tv^ re! 385 What is meant by a -mean solar day?" How does the^l solar" year differ from the -civil" year I Explain fu ly the methods which have been made use of to correct the error arising from this difference 386 Describe the silver and the copper coinae-e of legTltid^^^^^^^^ ^""^ -^Pocti vely are sI4r ^nd So^ppe'r 387. Why should the fraction |f ^ be greater than | and less than | ? hi 136 ARITHMETIC. Ifil ;. y i' i i ; 1 1 ' !;■: i i :■ { ir - miriJdSir,:vT'''' '''"^'■^'''''' "^ ^' «— ^nts as|Ln';viKLr:nrL"rc;.s";rr!r? "'■ '«•? r""^-^'-^' • 393. Every common .n, Iti t^ . ? ' '"'"'"' °' *''« »"'ei'. 1-Ie of their I c. HI. "*^ '""* mmi),ers isamulti- ceSui. ?4s:.irreLrivtf?f '-rr"'"''- - quo;tnac:"twX\vl!;:,;^Sn^ r"'>: "'^^^ 397. " Tlic place of f ^ f "T**^? " *"' '""^ *36 + 9. principle forms to .'iB-!,f 1'. ?"'"';. *^"*'^^ »'"" '^b 399. Every prime number when div,Vipri i. n m, leave a remainder of either 1 or 5 ^^ ^ "^^^ ber W 9^^^T?^K '^'" ^"^^"^^'"^8' ^"^^'^^^^ of dividing a num b, 9 .-lake any nun^J^,,y ,,74965, for ^wS; T,. ., , 5374965 uivide the sum of the dii8 pn^tVl o ^^^^-^^^^^^d digit of the divi- MLSCELLANEOITS EXERCISES— THEORY. I37 40,i. Koduce to a decimal correct to fouVpiace. : i. x^ ^3 4 5 6 7 8 404. Kxplaiii tlie following moflmri nfLi • vious problem : ^ mctiioa of solving the pre- 8 7 + 4 ()H-5-5 5 + 5-75 4 + 5-375 3 + 4-0875 2 + 3-8437 l+2-il218 •!>804 405. By what arithmetical ouer-jtion ,„ a,, a ■ , pressed as a whole nan.be, •inrl., ill .'• , *; 'l""'"-'"' ex- recur in the decinml portion ' I i '""' "f "^'^"^ ^^■'" starting from the prop^' point ^ ''""" °''"'^'' '-^^ '" V., ""iZ 'r? '^'^'^''''^ ™''^' '« token ''"'■"' fmtli thrrsirirrtnr'Sir' '■^•'^^<' "'• -»- these fractions in successfnn n . , "f' "^'' ^™ multiply by 409. TO divide r8?,s^;\tr:c;Kr: ■ 38743295 F,V=f p. A.. 35221178 i«i|/; s^bt';;r^'tnf 5";:.?' i'r ^r"^ ^""^ '■'^•''■ under the second diffit su" t,V^ 7', *^ ''emainder 7 remainder under the thild ,if v '^°"' " *'"'• P'»eo tho mainderisplacedunde thelJ,' '"'Jl '" °" "«" ^ ■'e- and the 8 are resne"ti veil ?hl'"^!"'' '' '^''- ^he 35221 1 7 Give the reaSffortllrs meSr"' """" *'" '•^"^'"'"'-•' 4 m 138 AUITIfMrOTrC. .. I ! 11:; I I I I , , It I ■' JU. It a cub. 1,0 aiviacl l,y y th'e remaindeTis 'a i' inff'whfr,"?',',,? 'i' 'T«««"i«I. «iM.o„t actually divid- 4, 1 Tt- '"'""' '' '-'^'i'=*'>- divisible 1,N- HI) I hv.ii ■'"'"""""• '"'i* '>cen ascertained to l,e divisible by n the qu«t,ent may be obtained by subtriSion IL^ = o.M)00 dig-its and subtracting. ^* ^'^® ^^^^ ^^^^ 4!;-* fvT\ l^""^ no square number ends in 2, 3 7 or 8 4K). What form of vulo-ar fi-iPfmn «rin " ' ' 418 Explain the method of proving- results in n^niH «^i^^T\:i^-^^^^^^-^^^ ^^^^^^^ b/648 ;^^^''"^ ''' ''''''''''^ '''''''''' '' multiplying 3G7 307 648 293(1 23488 ^237816 .1.. .^ * 1 ^'^ ^^"^^ ^^'^ advantages and disadvantao-es of 676634634---. .28345345,--.. .1654321321..., PERCENTAGE. j .^,j cleciinal ? tt^ijujiuited or ;i circulalinjLr i '^- I ^.11. - qts. at J<;}.00 a hualicl •— 3. CO 1.80 .45 .^('..ori PROBLEMS AKISmo^FKOM BCSIKESS TRANS- PERCENTAGE S- l-'ind 1% of STOofjy 0/-44- ■.^r'^^f^^- ofii-SS. ' /» " ^'^''Oi 1% of §625; 1% Of tor'"' '- °'' «°«' ^'% «* ^««.,. 2% Of $725, 2% 5. Find 5% of §3,0 , 45^ „,■ ,,, ^ , ^^ ,^^^ ° ■ G. Write deci.„ally .-1- -^ i^ =^1 ^i ' • i 5iJ^TSr ^'iJtS'^^J'f ^^-l^!';!??^^ if it? i of it. !». Whlit percent, o f i.u- M, . il t-f."' '^^^^ "''i'? iofit?,^o^ty.o,rfit"l^'^»i^i"•jt;;;ofit^^ofity . 10. Whatfmotiunof lOOisiv l-^:', ''^ "■■'*'-'- of if? is C? WI.Mt f,-,iction of .4o is rt ^ '''' ''''''"*™ °i' ^00 i«'' w;i:vi:;;;.^:;:;;::,m'Hiv; ^^'-'----0^2 %lmotV^°''^""'^^^ ^-t%of200is0. What 10 (',:! 140 ARITHMETIC. 13. 7 is what % of 35? 12 is wheat % of 72 ? 95 is what % of 1900? 12A is what % of 225? « is what % ofl7|? 14. Find 12|% of 1728 men ; of 864 bushels. 15. Find 3^% of £175. 12s. Gd. ; of $144.16. 16. Find 1^V% of $265,000,000 ; of 50 guineas. 17. Find 33i% of 1260 marlcs ; of 172-80 francs. 18. $365 is what % of $5840? 19. 36 minutes is what % of 1 day ? 20. £3. 2s. 6d. is what % of £25 ? 21. Ten per cent, of a certain number is 13 ; find the number. 22. The number of boys in a school is 60 % of the num- ber of girls. The number of girls is 60 ; how many pupils are there in the school ? 23. The average attendance at a school this term is 225, which is an increase of 121 y^ on the average attend- ance of last term ; what was the average attendance last term ? 24. What number increased by 14 % of itself is equival- ent to 285 ? 25. What number diminished by 11^ % of itself is equivalent to 710? 26. On account of the increase of value of flour the price of bread is advanced 25 % of itself. Formerly ten loaves were sold for one dollar ; what number of loaves will now be given for fifty cents ? 27. A regiment lost 20 % of its men in a battle ; 10% of the remainder deserted, there then remained 360 men. How many men M^ere there originally in the regiment ? 28. The earnings of a mill for two years amounted to $6560 ; the earnings the 2nd year were 5% more than the earnings of the 1st year. Find the amount of the 2nd year's earnings. 29. Ice expands 10% in freezing ; how many cubic feet of ice will weigh 1 ton, given that 1 cub. ft. of water weighs 62-1- Bjs.? 30. A bankrupt pays 30 % of his debts ; the amount that a creditor receives is what per cent, of that which he loses ? PERCENTAGE. 141 o/ /o of 120% ^wh'If'*"'''?'' J"'"^''' foi-«4000, increased in value Qo'" '. "*' ^^^ *'"' increased value ? left each 'v^a.^^'inT"''' ""^ ^ "'' "'^ ^^"''■■y' "" 10 ; what are the numbers ? ' cent, has the volume .Wea^ed^'^ '''"""^ ' ^^ ^^"''' P'''- cent'-is^Fsm'?" ''* ^° ""■''' "''^" ^''' '>°- """^h ?«>• ?ostTl ^o i„ ' ?'"'"■ ■" ''"^'"^ <* '«'" and bat. which cost f 1.50; now much money had he ' 38. A metre is equal to S'J-37 inches ; a cubic metre i« how much per cent, more than a cubic ;ard ? '' of boVs £v lo Z "^^'f'' '"//"="<'"' *'^''««<''' 'he number fin^T • J *® iiimber of boys is 37i V of the whole • nnd the number of girls 2 /o "' i''"' wnoie , wht*!' y Th " ^"^ ^,'"' ^ P'^"'" °f ^^'«>-k in 1 day, B 12i »/ • ^orti^lat; arBil;:r-" *" "« ^-o ^^- -^-hi shar^ fn,"«t"nm° °T""^- ^°^ "'' * ™'"« «°"i 33^% of his Share foi I1200O ; what is the value of the mine ? time; tt„ V ''°*'' *,»00" ! ^0% of this sum was 40°/ of 3 times the value of the house tliat was built on the farm • what was the cost of tlic house ' ' amp^^'tZv^fJ"''' *"r P^'''^> ^° 'hat 10% of the part Sj; be li:'.""' '"''''"'' ^'''' """^ ^^^/o ofthe third 44. One-fifth is what per cent, of one-half v o^u .K°T' ^°°^ °''" '■'"•"• ■•'•"d n owns the remainder- C sells the farm for them and receives 4V of 'hTs",nn; d^rATece^rr"" ' "'^^' "'" ^"="'- «f 'he-selling pS iV I,: 'hi )■,• !f mt U'!; 1 ^1 m ' iilf! I Ijii fi; ii ' 142 ARITHMETIC. 46 J^^i-om a cask containing 12(5 piilons of wine, 2 ffal. ij qt. leaked away ; what per cent, was loit '-^ 47 How many pounds of bread can a baker make from . if' ,\^^'^^^^^'«= ^^^y« ^oods at m per pound Av, and sells at 15 per pound Troy ; liis ^^ain is wliat per cent, of cne cost r 49. Mr. Brown sold his farm for .915840, which was 107 less than he gave for it, and he gave 10% more than it was worth ; Avhat was the actunl value of the farm ^ 50. Four per cent, of beer is alcohol ; how much alcohol does a man swalloAv in one year, if lie drinks 3 i)in^s o*' beer a day ? i ^ w 51. Two percent, of a certain number, together with 37 of half of the number, makes up 21 ; what is the number ^ 52. inve per cent, of a certain number and 6°/ of twice that number, together make 175 ; find the former'number ■-' TRADE DISCOUNT. . Lri^"'^!?/'''!.*'?^''-^'^ '' quantity of goods amounting to |>bOO ; 10% of this amount having been throAvn otf how much did he pay for tlie goods ? ' 2. After a discount of 15% had been taken otf, a mer- chant paid ^850 for a bill of goods ; what was the amount or the discount ? 3. A merchant paid $170 for a bill of goods, after a dis- count of $30 had been taken off ; what was the rate of dis- count ? 4. A trader bought a lot of paper marked $5 per ream at a discount of 121% ; he received a further reduction of 2% for cash ; Avhat did the paper cost him per ream? 5. At what price mast an article which cost $12 be marked, in order that after a discount of 10% has been taken off, thei'c may l)e a gain of $.'} ? 0. An article sold for $4 ; a discount of 12'% had been given ; what was the marked price ? TRADE DISCOUNT. 143 7. A tradesman increases the price of an article which now sells for $3, so that after ft-iving- a discount of 207 he may still receive tlie former price ; lind the marked price of the article ? ^ mcUKca a A tradesman reduces tlie price of his ^-oods bv 10°/ • he allows a customer a further reduction of 10°/ off this 10. A merchant marks his goods at an advance of 33 1°/ on cost, and allows a discount of 5% for cash ; ijnd the cost of an article of which the cash price is $7 GO miK^^'f'''' ^'^^""^ ^--^^ ''^■^^'^' allowing- *a discount cf 10/ from tiie marked price of an article ; he marked his goods at an adv..nce of 40% on cost ; find the cost. 12. Uhat IS tlie cash value of 150 axes at $2.50 ner dozen, there being a discount of 10% for cash '^ 13. What is the marked price of° a book for which 30 cents IS paid, there being a discount of 331% ? ' •^^•;'^?io/*''^^ ^''^'''''•' ^■^'" ^^'^^^ C'-^ch ; foVthe one I re ceived 20% more than the cost, for the Jther 20% less thnn the cost. What was the gain or loss on the twoV '^''" 15. A lot of goods was marked 40% above cost- 30°/ discount was g-iven ; fiml the gain or loss per centof cost^ lo. V\ hat rate per cent, discount is equivalent to g-ivine' one^ialf of an ounce with each pound for good weight P^ nl^{ u' % ''''^'' Pf'" ''''''^- ^^^*«^o^"if' i« equivalent to giving- 18. A bookseller gives a discount of 10% for cash 'irul allows teachers a second discount of 5% on al carpiVce ' ^fj ^^' paid 83.42 for a book ; wiiat was t&S 19. A tradesman reduces the marked price of his p-oad^ by a certain per cent. He gives the sanie ra'e ZZt ?L'If,/'^ "'"'^ r^'' ^''' '''^- The cash pric^e snow jf| of the origmal^ marked price ; find the rati per cen? 20. A grocer mixes one quart of water with every two him t give r^ ' '^''^^'^^^' '''^^ '^'^' ^n^^We 144 t ii ARITHMETIC. il 21 What discount is given whon 15 articles at the re- duced price can be bought lor that sum which would pay lor 12 articles at the list price ? , ^ ^ ^ 22. A tradesman makes a gain of 10% of cost after giv- ing a discount of 10% ; what would be his gain per c? U of cost It he gave a discount of only 5% ? ^ woud ollnwi H "''^' '^° '^'^''' ^J^^^^i-^^te of discount welfcoiirt ?'"' '''"'" ^''''''"' '^^^^^^^^ if the measure 24. 20 and 10% off is equivalent to what single discount ? 2o. 10 and »^iiat % off is equivalent to 15% off v lb. The marked price of certain goods was reduced on account of damage by lire ; a further reduction of 107 is given for casli ; goods that were originally marked f 5 ;rer,^:aer '- ''■''■' -'^'^ ^<-^-^<-^^^ 27 A merchant, after reducing the marked price of an So ' «7 ^>;*^^^'^^ successive equal rates of discount, sold for dlscomil? ^'''''^ ^'''''^ '^^^' ''^^'^^ '''""^ ^^'^ ^'^^^ «f p-^it i1 /5' r''"''^ T'^f ^''' ^««^^« '^^% '^'^'ove cost and gave 111.% discount; what per cent, of the cost did he gam r* 29 Marking goods 5% above cost and giving a discount ot o/; ot marked price, is equivalent to Avhat loss percent. 30. An article was marked at a certain per cent, above cost ; the same rate of discount was allowed, giving a loss ot ^\- oi cost; what was the rate of discount'^ 31. An article which cost .^96 was sold for ^00 • it h'ld been marked at a certain advance on cost, but the same rate ot discount was given ; what was the marked price '^ 32. A merchant offered a reduction of 50 cents from the marked pnee of a hat ; he gave a further reduction of 45 cents for cash, tins being at the same rate as the former- what was the selling price of tlie hat ? ' 33. At what price must an article which cost $1 50 be in.'irked, so that there may be a gain of 25 cents after a liiseount ot 25 per cent, has been given v PROFIT AND LOSS. 145 pertnto^rif.f '' '"^^ ^^'^^^ m.e is equal to four 35. What ratfe of discount taken off" twice in succession IS equivalent to 20 and 25% off? succession PROFIT AND LOSS. . 4. A speculator invests $12000 and Iosps i^V .f i.- mvestment; iind how inuch^apitaire'hasTeft!''^ '' '^' a Le';.:\o^,S%r' "^^^ '^^^ '' '^'^ ''^^' -'^^ ^^21 leooo^'arM -^^h a capital of the frain tnthi ! '.¥ * '^ ^'''^ >^^^^^* ^^«d added 75°/ of tie gam to the capital ; the second vear he g-ained '>0°/ of this increased capital and added one-1 Sf^ of it "lo^^^^^^ capital ; the third year he gained 407 what V s the amount of gain in the third year^ fl '^; ^.^I^^'"^^^' ^^'^^^S-ht 2 iO sheep at f^S each • 5°/ of fl.^ ?otain 20kf '"^ "^^^^ ^^^^^^ ^^^ ^^ -"''^ re4ll!j? ^1 mo''t'°'' l^^^^d^'^d I'eams of paper were bought for fix irfai-iSS ;£;:^— rXS ati°o.a1n of'-wr'^r ,'" " ". \ -i what did the piano cost A? ^ ^ '^''''' ^ P^^^ 5 i ; I ! k I I ; HI l-l Wuu 146 AlUTJIMETIC 11. A bouij'ht ii'OOfls for fini'} mn i ,. , 14. A Avine merchant lo^o«! 9n°/ ^^-i • -. , by wh«t per cent, mli ^Zov^eth^ !^'" by leakage ; ion to gain 20% on his investment" " '""'' P<"' *'''^'- he sold fo, S4^!nVrilf ^T" ''^""^^ «■«» ^'ortb SIOOO • wbat^ffonlltlt be'"f/;'r^,r ny'^i*' '^ ^'^ ^^ «*-^'5. *- $4.76 «mnihrsVd%??S^^^^^ ^^^'^^^^^ '^ -^^^ for article is sold for $4 25 ' ^^'^^ ^''^^S-ain % when the 20 Bought wheat and sold so as to e-ain loio/ vested the whole snm m-.,! >., i iV ^^^ ^-i/o ; I'ein- invested andlo " '" '/ «„,'"? ''!' f ">« '••^'^ »f g-^i" ; I'e- -nainde. to-p^j '',1 1^^:^ ^,^,-- "0 seii t,i .uust the awles be sold tlv ttho '''^•™'? I''"" <"'"'• «f^««' S-ain ? "''" "'^"' "'»y bo neitlie.' loss nor 23. A dealer sold two hor.sps fni- «ioa „ i arain of 20°/ ^ ^uniu^ on o%, I i^^ay make a PIIOFIT AND LOSS. 147 2o. Allien milk is sold at tlie rate of IG quarts for ono doUar, there is a gain of 25% ; wliat would Z it ^dn y find ihe |a?n %r "'' ^"'^ "^"^^^^ ^ «*" ^^^ -Uing prioe, 27. If a tradesmanmarks his goods 407 above costinrl 29. Whisky is bought at $1.25 per gallon • how mnr.i. 31. How much per cent, above cost must a man in^rt his goods, in order to take off' '^07 'in^l «hii ,., i ^ of 207 ? /° ^^^ make a profit 32. What per cent, is lost by markinp-crnod^ i->i°/ ..i ^ cost and then giving a discount of 1^^^/ "^''^' ^'^^ ^^^^^^^ 33. A merchant lost 6i% by markinghis «-oods at n o^v tarn per cent, advance on cost, and then giving the snm J per cent, discount ; what was the rate of Count ""' 34. A merchant lost llJ7bv ^^^<^vhh^.^ i • \ cemin per cent. >^y^rJo^X^^rin^%^X^ cent, discount , what was the rati ofclisoo«,rt V '^ ' nnri „Ti ""y'^'^y ''">•« P'ipore at tlie rate of 8 for 5 cents make • " '' ' "''"'' ''"''' ' ""'" r™«' P-' eent. dorhe itbl;om:"n':r;rorof -^"f :?tL.'"''"" ■""'=" ■"-' to the agent whoseUsthe1,ofsf '/'"'' P"^'°^' *^" '"'"'■"■^ '^stentl" tov'nfT"'"! ^r''? '!"''•'' *** "«' >•■'''« »*■ 100 for -0 cents ; 10// of her stock s lost hv deoav • whnf «/ ,i she gain by selling 5 for 3 cents "^ ' ^ ' "' ^^ '^'^'^ X • iii ! 148 ''•' i f U {■: f-IJ- ■: ARITHMETIC. 3a. A sold ail article to B, B sold to and C to D : each made a gram of 5% ; C's gain was |5 more than A'b ; what did D pay for tlie article ? 40 When potatoes were sold for 120% of cost, the gain on bushed? ^^^ ^ ''''''^' ' ''^''''^ ^''^ ^^'"^ '''"'"^ i"*'^^^ P^^' 41. A drover sold 5 cows for $60 each; on one he gained } ■ko^'^'l'''''''?^' ^''^"^ ""^^^^^^ 2'^^% 5 ^^1 the remaining two he lost 33^// ; did he g-aiii or lose ? What per cent. ? 42. How many bushels of corn at 40 cents a bushel must be mixed with 100 bushels of oats at 50 cents a bushel to make a gain of 10% when the mixture is sold at 50 cents a bushel :> 43. If 3 articles are sold for the amount that 4 cost what IS the gain per cent ? ' 44.^ A and B invested equal sums of money in business • A gamed 20% on his money, B gained i^lOO ; B's money was then double A's ; what sum did each invest ? 45. A merchant mixes a lbs. of one kind of tea worth b cents per lb., with c lbs. worth d cents per lb.; find his gain per cent, in selling the mixture at e cents per lb 46. A grocer sold goods for $16.10, gaining 15% ; if he had sold them for |20, what w^oald have been his' rate of gam ? 47. A merchant bought 5 hogsheads (63 gallons each) of molasses at 25 cents a gallon. He paid |8.20 for freight and ^1.80 for cartage. He loses 10% of the molasses by leakage ; 6% of all sales cannot be collected ; for how much per gallon lunst he sell to make a net gain of 40 per cent.? Ko/^' T^ person buys an article and sells it so as to gain 5%. If he had bought it at 5% less, and sold for $1 less he would have gained 10%. Find the cost price. 49. A tradesman's prices are 20% above cost price • he sells at a profit of 8%. What discount did he allow ? ' 60. Smith having lost 25% of his capital, is now worth as much as Jones who has just gained 20% of his capital. Jones' original capital was $9600; what is Smith's present cai^ital ? D ; each what 'a • B gain on jrice per e g-ained g: two he i tiel must iishel, to 50 cents 4 cost, ashless ; ; money worth h And his f lb. ; if he rate of each) of freight 5ses by or how n of 40 to gain |1 less, ce ; he V? worth lapital. )resent COMMISSION. COMMISSION. 149 ti >mn ^^"^"^'■^^^*»!^ iucrcliant sold a cargo of flour for ^12000 on a commission of 2% ; what was his commission ? fn.r^twrT'"'."'/'''' "lerchant sold a consignment of apples 101 ji?l«oU; what was his commission, reckoned at 24-°/ ^ 3. An agent sold 15000 bushels of potatoes a,t 40 cents ToL I^ni ' l^^^^«"^"»i««io" ^vas 21%; what sum did he send to his employer? 4. An agent received a consignment of wheat which he sold on a commission of 1|% ; the amount sent to his employer was eciuivalent to ()5^ cents a bushel ; at what price Avas the wheat sold ? ' 5. A commission agent sold, on a commission of 11°/ a cargo of 1200 bushels of corn at 45 cents a bushel ; what was his commission ? ' • ^'J^ ''^^''^'^ transmitted to his employer $1568, retain- ing I.-32 as his commission for effecting a sale ; at what rate was the commission calculated ? 7. A commission merchant retained $40 from the pro ceeds of sale of 50001bs. of butter at 20 cents per pound • what rate of commission did he charge ? ' 8. An agent sold a consignment of goods for $5200 and sent $5096 to his employer ; what rate of commission did he charge ? 9._An agent's commission Avas 4i% of the amount remitted to the consignor of the goods ; what would be his commission for selling goods to the amount of $10000 '-^ 10 The price received for goods sold by a commission merchant was 105% of the amount sent to the consignor- the commission was $45 ; what did the goods sell for '^ ' ^\' ^^ '^g^ent's commission was $50, for selling some land, at $125 per acre; how many acres were sold* at that Pi'ice, the commission being reckoned at J% ? 12. An agent charged $75 for selling a house for $10000 ; what rate of commission was charged ? ISW-A commission merchant, in selling, reckons liis commission ill I. li ^ I I'l: ; f ' 1-1 " IJ !!;; . if 150 ARITHMETIC, 13. An jiH-t^nt Tmys for his cniploycj- lOOO l)uslu',ls of whciit at «)5 cents a l)Usli(!l ; wlint is' the; ii^c^nt's conimis- sion at tlic rale of 2.V/, ? II. ]IoAv much money must be sent to an a^^ent tliat he may purchasi; a liorse for me at .irlOO, and liave his com- mission of 2%? 15. A mercliant sent $i>25 to an a^rent in Atlanta, re- questing- liini to purcliase cotton, after retaining- liis com- mission of 5% of the sum paid by him for tlie cotton • wliat was tlie commission? ' ' 16. $1750 includes the price paid by an ag-ent for cer- tain goods, and his commission of 2%; what sum was paid by the ag-ent for the g-oods ? 17. An ag-ent received i^^'AfuO to be; used in buying- sugar at 5 cents a pound, after retaining- his commission of 2% of the amount i)aid by him for the sug-ar ; what was his commission ? 18. A commission ag-(;nt retained |80 out of $2480 sent to him for investment ; find his rate of commission. li). 'An ag-ent charg-ed me $1.15 for a l)ook which cost him !)5 cents ; what rate of commission did he charg-e? 20. Sent $122 1 to an agent, requesting- him to buy poi-k for me at $1.75 per cwt. ; he charged 2% comm'ission ; how many lbs. of pork did he send me ? 21. An a.g-ent boug-ht 12000 bushels of wheat at 80 cents, and charg-es 1|% commission ; how much money must his employer remit him ? 22. A commission agent received a consignment of 500bbls. of apples; he sold at $2.50 per bbl., and after deducting- the proper commissions, invested the balance in sugar at 4c. per lb. He charged 2 % commission on sales, and Ih % on purchases. What was his total com- mission, and how much sugar did he send to his employer? 23. An agent sold on a commission of | % a cargo of 1200 tons of coal, at $4.75 per ton; he invested the net proceeds on a commission of 1 % in lumber at $18 per M. How many feet of lumber did he buy? 24. A broker receives $12000 to invest in pork, at 4 cents a pound ; he charges 2 % commission. Hoav much l)ork does he buy ? COMMISSION. 151 2(J00 l,l>H. ot flour, which he sold at .$8 a ba^Tol on a •W-ntoiitot the proceeds, amounted to .«ir)0- lie Iw iwlit co«on at 15 cents a ponn.l with the net ,„-o„e'ed; Si coSolid^r'^yr '"'• ""■^"'^'- ''"''■ ■""">■ p--"* "'■ 26. A consio-nlnent of g-oods wan sold for &!l-^5()(). tho 112150 What rate of com mission was chai-ffcd''^ 27 An agent receives $(1800, with i.istructi(ms V> invest iy iM n. vin''"-" ^' r"^^' ^'^^^^"""^^ ^>^^^ commission "t - /^ and paying- in advance the freight at 2()c. per cwt How much sugar does he buy ? ^ ^• 28. An agent receives 1500 hams, .Tveram' weight 25 pounds, which he sells at 10c. a j.ound ; he pays fre t 20 cents per cwt., and charges a conimission of !> / on snFe He 1 ins ructed to buy tea at 45 cents a pound/to p.vpav the Height on the tea (20 cents per cwt ), an( ret , in his 2y%nt\^!^''''f'^'^'' -^- commission on sales and i?:' K ^ ^'^^ amount tho agent receives. SO. An agent charges 2 % commission on snles and •>■} 7 for guaranteeing payment ; he receive.I altogether fAo nhat was the amount of the sr les ? 31. An agent charges 2 % commission on sales and -y-^ 7 for guaranteeing payment ; Ids commission for selli,ig"4s 140 less than the guaranty commission. ^V\u^t w.s The amount of the sales ? >Mi.it an,.^ the 32. Sold cotton on commission of 4/, invested the net sC wS ^SS''yr/l^ °^ connni.ionf the tot:',' co' nn"^! sion nas h.i.iO. hind the value of tlio cotton. ^ionotHV^''Tft^!^Y ?"">;'""","«"' «f «Pples on a. eon.n.is- sion ot .{ 4 After deducting his eonnnission and rewi-y- .n«- a sutnejent sum to pay tl,e f,-eisl,t at M ee, t pc r ,nn he bought fhmr nf: *■> wr> ..,. „" . . ^ ^\ <^ »vi,.^ he ho„,,;t fiour at feHO p;.;;. on a e;m;r;:,f ^^ ine total rninTrnQsiiVkn -.,^.,0, <:^t"y ti- . 1 ., .4 The total commission was SHP,. bought. '=/ Find the amount of 'Hour 162 ARITHMETIC. HI h i ( i n 34. An agent rcccivod n consi^ninont of wheat, which he sold charging 2 % coniniisHion. AVith the net p ,cee Is after d(.lucting his connniHsion at U %, and p vn'/vW fre.^^ht at 25 cents per cwt., he bou^nft suf,^"r at V^cents a pound The agent's total commission was 170 * Fhid the number of pounds of sugar bought. offy ^wil)V?h'f^''"/"''''^',"V'"^^ ^'^^"^ ^'' '^' commission of I V f^ n 1 "'^^ ^'•''''''^' ^"^'^^^ ^^'-^ «" ^ commission the tea commission was |50. Find the cost of ft ^^^^^'« ^y p^y^^^ « p^^, mmm of .^24 ; the rate was .|-% a year ; hnd the value of the house, .| of its value being insured 11. A premium of $37.50 was paid for a two-year polic^^ on a dAvellmg worth $0000 ; the rate was i% of the policv lor each year ; what fraction of the value was insured ^ 12. An insurance company charged $18.45 for insurin«>- a house worth $2400, for one year ; find the rate per cent" «imnn ' "'"'"''^^'f company took a risk for one year of fn?^ ?•?" '^ ^^'.^''^^''^^^^^ ^^orth $15500, at f% ; it covered minnwi H 'If r T^^''^'' ^^^^P^''">^ '^^ |% ; how much pre- mium did the hrst company receive above that which it paid the second ? 14. (a.) J low much is paid to secure a policy of $100 for one year at 1% ? Ans. $|. ^ ^-iwior _(b.) How much do(>s the owner receive above the pre- mium paid, in case of loss ? Ans. $1)9^. (c.) What must be the value of goods insured for $100 at^/^, so that the OAvner may suffer no loss in case the goods are desti-oyed ? Ans. $1)9 j. . Vo; *'^^: how much must a cargo woi-th $7940 be insured ' //' I? '^'''"'■'' ''''^'^''*' "^ h)ss if the ca,rgo is lost^ 10. 1^ or what sum should a houses worth $3905 ho in^vvM to cover the value of the house, and the cost of the policy at ;^ / r* ^ -y 5 8 INSUIIANCE. 155 fn n!' ^'^5,^^''^^""^ '^^^* ^ ^<^ssel worth 115800 be insured 19. A company took' a risk of $40000 at U 7 re-insurod 40 per cent, of it at 2 % and 25 per cent, at' 2% V Whit of1?\f I'T^^'^o^ ''lyr^ '^* li%' i-e-insured 50% r!/o r^ *./°' '^ '^^ % ^^ ^^^^ remainder at 1-1 y. What carried? ' ''"^^""^ ^'"^^^^^ ^^ ^^^ amount {"f ril It ^ 21. A block of buildings worth $1000000 was insnrod in company No. I. for $25,000 at 11 7 rcommnv No. II. for $40,000 at 1 7 in companV' No "" nP^?^ fi W a. I %^ Find thel^^elniur'^ail if ea "ca^/ If the block be damaged to the extent of $100 000 whS amoun of loss will be borne by each company p' .Jo '; ?;^^'^ '^ '""^^^'^^ foi' I of its value, the furniture foi t of Its value. The rate in both cases is r/ Tl e house IS worth 5 t^mes as much as the furniture ? the total premmni pa.d is $12.(;o. Find the value of the house 2d. Ihe premium on a vessel and its cargo is Sl^n • the rate on the vessel is |% and on the cargo « y The value of .he cargo is double that of the vesef and each s insured for f of its v^lue. Find the value o? the car^^^^ " 24. A vessel was msured for $20000 at | 7 in one com pany and for $25000 in another at r/. What rato "• premium is paid on the whole insurance ? carc.o lortirinooo h ""'"'' ^. ^"^'"^ ''''''^ ^^^^OO and cai^o woi th ^2 1000 be insured at f and ^ 7 respectivelv to cover the total value and the prlmiunis paid ?^ ^' of it; yZe'^Vf '""^ """'''t'r ^' ^«^^' "^^^^'^d for ^ or Its value at 2} ^ paying $75 premium At wh'.t price per barrel must he sell to gain 25 7 of cost as we1 as premium p^iid ? »= /o "i ^osr, as aa ojj i I i 11 ; f r I ; 156 ARITHMETIC. •^^" ^I¥f/^^ *^'^ "^^^^^^ ^^'^ ^^^'^^^ if the insurance pre- mium of 4 /on I of its value, including 50 cents for the policy, equals $21.50? S^rS^' 'i"'r ^"^^^ "" ^T^""' """"'^^"g" ^2500, upon a lot worth ^500 ; the house was burned and the insurance company paid the full amount of the policy, 4 of the value afr 4 7 . Joui'^iriossT "^' '^^' ''''' '^^^^^ ^^^^ *^^ --'« befnJred If^p'/^^ yacht is $2940 ; for what sum must it be insured at 2%, to cover, in case of loss, the value of the yacht and the premium paid ? 30. A house worth $2962.50 is insured for $3000 to cover value and premium paid ; what was the premium paid ? 31 A vessel worth $4925 is insured for $4000 which sum includes 4 of the value of the vessel, and the premium paid ; what was the rate of insurance ? r ^ um 32 A carg-o worth $22125 is insured for $15000, to cover f of the value, the premium and $100 besides ; what was the rate of insurance ? ' "'''' '^^^ -> nf/i ^^^7^"y,^^«- }' insured a building- and its stock for No rrZnAi' ^^'Tl^ATf' They reinsured in Company fi,?; "^i i^'^ ''f'' ""^ ^^^ 5 building and stock being de- ?L «!:«. ^fr^V "" ''''^"? company lost $49000 less than the hrst. What amount did the owners lo^e ? 34. My house is valued at onc-lialf more *han my bro- ther's ; my house is insured at r/ on # of its value mlr brother's at |% on | of its value, l^y $^2 ml'^irmiZ than my brother ; find the value of each house 35 A shipment of flour was insured at 47, to cover A of the value and the premium ; the premium was $15 ; hnd the value of the flour. ' 36. A drover is taking a herd of 400 cattle from Quebec to Liverpool ; the average cos* of the cattle Avas $45 ; for what sum must he have the cattle insured at ir/, to cover m case of loss the value of the cattle, the prlin ium pIS' and the cost of his passage, $84 ? ^ ' s^t%} 7n^lf\^'''''^ ""^^''^ 7^^''^ ^^ ^ ^^^^^^ "^^^«t ^« insured at 6/, so that in case of loss the owner may receive ^ of the value, m addition to the premium paid ? * TAXES. 157 TAXES. 1. Every citizen of Borden is called upon to pay for public use 2 % of the value of his property. What^tax aoes Jones pay whose property is worth ^5000 ^ ^•^m^^r^'^r^ f "^^'^^ '^^^oi'ding to the assessor, wl ??V .? '■''^^ ^^ l^^'^*^^'' ^' ^i ^^^t« «n the dollar What IS the amount of my taxes ? 3. Find the tax on property assessed at 112000, when the rate of taxation is 2 cents on the dollar '-^ 4. When the rate of taxation is 15 mills on the dollar wnat IS the tax on property assessed at $2000 '-^ «7?n nm ^^!!^/^^^«^««ed value of the property in a town is 1750,000. What tax will be raised when the rate is v'l mills on the dollar? 2 of ti to?i'oooTh'? ^""l^^^ '''P'"'"' ^^ ^^^ y'^'^ '^'^ «^"^^ ot ^1,500,000; the taxable property of the corporation I** ' assessed at 135 million dollL."^ fL the ratronaxat^on /, ine property of a village is assessed at $800 000- the rate of taxation is 18 mills on the dollar ; it cost's 2 7 of the tax for collection. Find the net amount received by the village. 8. A town requires $19600 to meet expenses for the r^r/W f^P''^..^^° ^'' collection, What"^ iiust be the rate, it the taxable property is assessed at $1,200 000^ 9. As income is $1200 annually, of which $400 is ex- mefs4T? '''''''''''"* ^^'''' ''''' "^^"^ ^'' ^'^^ ""^^'^ '^' ^1 im ^^'^'''°^ '''T"''' ^""'^ ^^^' "" "'^'^^ ^""y w^iose income is H500 a year, the rate being 16 mills on the dollar ; $500 being exempted from taxation ? $2000 a year $600 of which is exempted, the rate being- 15 mills on the dollar ? ^ 12. Mysalaryis$1500; my net income is $1482 40 after ^^7o"^r "T"^^ *^^ ^^ ''^^ ^^'^'^■^ *'4^- What is the rate ? \fr\ ''''^ '''T""''^ "'^'^"^^ ^« -'^ % of iHS capital ; he pays $25 taxes, at the rate of If % on income. What is his capitax . u iiVtV I. : ! i , I \-4 !■! I i;.' f ( I li, i 158 ARITHMETIC. %T xLe evt n^t^ "" '^» o^^-- «00- What was the rate ? which w2 ZCal f"^*™"''"? '^ bridge was $10000, incomc^'T. 'iff Co' "'"''' P''^ '■•'" °" "" °^'"- ^^^ ■' iiioumc;^ ie._fc, ,jL.n .^1000 are exempted from taxation S^/^*^^;;^^^^^'^^ion is 15 mills on the dollai^ How m?ch better oft is man whose income is .$i)95 tlmn a man whose income is ,^1000 '^ assessed Wn"/!^^° '' '"^'"'^ •°'\''^ ^"'••'*?« °f ^^l'i«h ''>« assessed value ot the property is .?255,000. What is the tax on property value ai$1800 9 "'mi, is tne 1.8. A tax .,f $15000 is, levied on a town the assessed valuation Leing- $030,000. What tax doe^ a nfaTmv wliose njeome is $1300, $400 of which is exempted? ''^ IJ. 1 paid $24 iiieome tax, ¥400 of mv income beino^ exempted; the rate wa. IB mills on the doS mm was my income? ^^^un. >vnat 20. Smith bought a house for *♦;()()() • it is assessed for 2 aouai he house is insured for f of its value at 1 7 Tf bmith had loaned liis money he mfght liave received^^BOO interest on it for the year. W]u.,t niontllT In was Smith really paying for his house ? 21 What sum must he assessed on a scliool district fn build a school house worth ^.700, and pay 5 % t colic- • 22 A tax of j<4500 is levied on a village, tlie assessed valuation being $180,000. What tax doe a mm pw Whose income is $1350, $400 being exempted fiomtaS DUTIES AND CUSTOMS. CO him at $2.40 ; he pays 20 cents postage, 1.5 y ad vnlo What d".^' r' r""'' ' ^"'^^ '''' ^^ -^ ^^'« whi:"" n hat docs he charge me for tlie book V i DUTIES AND CUSTOMS. -.59 ^\ |^,thcro wero 110 duty in the previous i.rol.leu, wl.Mt would the dealer char^^e L for the book so s o , l!^ tlie same rate or g,m? What would be the cl ni^e o make the same amount of gain ? ^ nf%7i!''V\^^''r ^^^^^y P'^i^^ on an imported book, invoiced at j?D the duty being- 15 per cent, of value ? 4 1 import a piano on which there is a specific dutv of loOand an ad valorem dutv of 20 per cen • I tl?\n geUier ||3r t^je pi.UK, e;.,o. ^ What 4s t^lcll-nvoi^^,^^ G A\nv;/ ; 1 .' ' ''' ^'"''^"' ''^' ^'' ^''' ^^^"<^- ^'^l valorem. relTin^VtoV,'r^.^f '^ ;^""''' ''"'" ^'^ ^^^^^>' tl^^ ^^ealer bciiini^ at J.) /^ alK)V(' total cost ? 7 iMnd tiie expoit duty on a pine loo- of uniform section lielen^-tli bein,- 30 ft. and tlie diameter 2 1^11 n Ze rate of duty being- $1.50 per cord ' 8. Find tlie export duty on a stick of timber 20 ft long- foOQ /'f '"V ''■ '^'''^' *'^^ ''''"^ ^f ^lnt^' being 12 S lOOQ feet board measure. ' ^ ^ 9. Tlie duty paid on a consig-ninent of 50 pounds of manufactured tobacco was .fl7?50 ; the dutv on sudi tne Aalue of the tol)acco as sliown in the invoice. 40-*n .1 ' '"1 ^'I'"""''^ '"-'"'•^ «^^ll« ^^t '^n advance of 40 /, on the cost, laid down in his sto.-e. I pa v him ?^-3oi for an orpn, on which he paid a specific du v of $80 -ind 11. Giving- tlie donlor tlie same rate of proHt hv Low much would tl,e price be reduced i„ tlic preceding- proWom if there were no duty ? " -^'^^"^^"^ 12. A g-rocer imported 150 cases of port wine 24 bottks ne paid an ad valorem dutv of 20°/. The froixHif ..nr $4384 ; what was the invoice price per bottle ^ 13. If goods invoiced at $1200 cost $1800 when laid down m warehouse, the cartage and freight amouming o ^75, what was the rate of duty? uuum^ lu 160 ARITIIMETrc. ! i I: 14 The duty on surg-ical instrument eases is 357 • that slrn'r^'-'^^-'^^' -^%; tl^-^l^^y paid on a case of in, St union s invoiced at $^30 was $7.50 ; iind the invoice price of the instruments alone. 15. The duty on 1000 boxes of raisins, each containing lo pounds, was .-^2.0 ; the raisins Avere invoiced at 8c. pel b.; the specihc duty being- 1 cent a pound, determine the ad valorem duty. 16. The duty on imported axes is $2 per dozen and 10°/ ad valorem. The whole duty paid on a lot of axes was $561 he specihc duty being- $24 more than the ad valorem, l^md the number of axes imported. 17 The duty on a bale of canton fiannel was $3.75, the specihc duty being 1 cent per square yard, and the ad va orem 15% ; hnd the v> Idth of the fiannel, given that the bale contained 100 yards, invoiced at 20 cents 18 The duty on wine cont.ining 20% or less of spirits Ls 2o cents per gallon, and 30% ad valorem : for every degree above 26%, 3 cents more per gallon. A man wishes to import 100 gallons of wine 30% strong, bright at $2 per gallon; he orders enough water to be mixed with It to reduce the strength to 20%. Does he gain or lose m the amount of duty paid, the total value of the wine being reduced 10% by the mixing? Does he gain or lose on the whole ? 19. The duty on imported window shade rollers is 30"/ ad valorem ; on the shades 5 cents per square yard and lo% ad valorem. Each shade is Avorth twice the value of the roller. The duty paid on a dozen rollers and shades '^nTif f' ^^ ^''f' '''^'' ^'^••^^5 the width of each shade was 4 ft. G m. ; find the length. 20. The duty on rubber-lined cotton fire hose is 5 cents per pound and 15% ad valorem. The dutv on 100 feet of hose, invoiced at 20 cents per foot, was $15.50 ; find its weight per foot. STOCKS AND INVESTMENTS. 1. Find the value of 40 shares r?100 each) of bnnk sto^^-- at $80 per filiare. STOCKS AND INVESTMENXa 161 2. rind the value of 75 shares of railroad >lr eaeh. stocit at $125 find the cost of 3. Merchants' Banlc Stock sells at 140 ; 15 shares. 4. Find the cost of $4500 stock at 90. 5. Find the cost of $2700 stock at 108^. 6. I instruct a broker to purchase for me |5G00 insur- ance stock which is selling at 851 ; the broker charges i-7 of par value of stock for commission ; how much must I give the broker to pay for stock and commission ? 7. Sold through a broker at -i% commission, 80 shares at 92. What do I receive for the stock ? 8. Purchased $8000 stock quoted at 89X ; what did it cost, brokerage |% ? 9. How much stock, at 80|, may be bought for $G450? 10. How much stock, at 80|, may be bought for $6460 brokerage i ? ' 11. How much stock must be sold at 117|, to produce $4710? * F^^^uoe 12. How much stock must be sold to prodtice $4710 the stock being quoted at 117|, brokerage i ? ' 13. Find the income received from $4000 stock, paying an annual dividend of 8%. 14. Find the income received from $8450 stock, paving an annual dividend of 7%. ^ ^ & 15. Invested $9100 in bank stock at 90|, brokerage A and sold out at 92, brokerage ^ ; what did I gain ? 16. Bought 86 shares, quoted at.96|, and sold when the stock had fallen to 95 ; how nmch did I lose, brokerage I each way ? ^ 17. What rate per cent, do I gain on my money by in- vesting in stock at 95 which pays a 5% annual dividend ? 18. What rate per cent, do I receive on my money by investing in stock at 94|, brokerage I paying an annual dividend of 5% ? -"*=.. 19. Find the rate of d^'vidend paid by stock, when a man who owns $24000 of it receives 81920. 20. How much 6°/ stock must be bouglit to give an an- 'Jivil income of $210? Wlvdt will it cost at 75, br(>l> ei"a; ;ei? El 162 ARITHMETIC. i'i ! . i I' I ' rill t niii.j 21. What is the price of a 7°/ ct„„i, . • , the money invested v * ''^ "»<''' "'''"='' I^iys 5% on r^^^^l^^:!^ 7' '''^^' "">-« 4|% on the cefdsi^?ernri i^^ek^n'^if "- '- in income, b.'okemge eihway 1 *' '"' '"" '""""^"on .^^- A man cleoi-eases liis inconi,. «jka i of 3% stock at (17 an.l ii.v.sth " in 4 / 7 '"^"'"«; °»'' amount of 4 % stock .li,l bl'lmyl" ^ "' P'"'" ^^'«" 4%^tocn"m,t7is''itlr''''f^« ■■^"" "'™^« i" Wllat amount of 1 7 tocl" Z he'sHI ''r?'"'" ''^ ®''«0- 27. Bouglit 64800 stoci!. 75 A ' ',"f «'?«" i ^^ share, mus'i I sell it to gai.f slw ,^* whatpnce, per $100 thf^^ntrHf M^aiif a ^tr "' ^^ °" at 120. His nexf li'.lf ^/^n, \.\, ,^^ ^^^ <^''« s^-^me stock the amount'oTthl flmCi%''°''^°!^ "' °"'-- "-' "^r H% on his moneT' th^oU,: jfaW^g'TyT'' ""'^ '-^^ ii-4o\rt:ri,t^r?;iTai" ?J ?r -" ''' »»<> the latter to the former ^sum%nffi-V'? ''■'">'*fe''« from comes from the diff",.6nt ™k "thrsi; nf 1^^' IT amount of money transferred v ' '^ "''^ or> the^preference st'o^^ ^^r::^:^^^^:^^ STOCKS AND INVESTiMENTs. 163 -ys 6% on 1% on the . and the It is the the pro- Iteration lin^*- on!. . Wlmt vests in y $480. •er $100 4% on le stock hat was at after Y make 90, and •s from the in- is the 4,000,- it G7, be the %, the I stock d can whole 3? ->. . 1 >u ^^'^l^l'/^'' the more profitable investment, 6°/ stock at 128, or 5 % stock at 1)!), brokcra«-e • ? o4. Tl.e whole stock of a company is $1,000,000, the net p«in IS sufhcient to pay G % on the whole amonnt. The ;Hv1!;;!^^7'^h-^ /^ :^ ordinary shareholders, the balance gun- 8 / on the preference stock. What was the amonnt ot the preference stock ? 35. A company with a capital of $200,000 paid 8 7 dividend to Its shareholders. Afterwards new slock was InUrif'^'^'or^ wl^' ^'''''' '^^"^'""^ "^' ^^'i". tl^« company paid only o //. What was the amonnt of the new stock onf mV ;i"^'^^V-'™l' ^''T "^ '^^ '^'^^^ '^'^ 75; he sells out at 80 and invests ^ of the proceeds in U % stock at .)b, and the remainder in 5 % stock at par. Find Ids in- come from the latter investments. .f/o/Yfl'/pll'^m ?*• '' ^'''""'•^ '''' ^"^"^y ^'y investing- in a 21 % stock at 00? 38 I sold some stock at a discount of 10% and made U w,"'^ "'""''"^ ' ''* ^""^''^^ ^•'^^*' «^' cliscount did J l,uy P 39. ^ hen money is worth 4%, what ought to be the price ot consols Avhich pay 3% ? !S? ,"• t'%f ^•^^^ '-^tll^i brokerage in eachcJsei; fbi^Mrs"^^ "^^^ What does the broker receile 41. A person owns $15,000 bank stock paying 57, which he sells and mvests the proceeds in 0% stock at 120 Ms ir:z ^:i:zr' '' ''' ^ ^-^ ^^- ^^^- ^^ -^^^^ 42 What sum invested in a 5% stock at 115 will yield a net mcome of $17711 after paying- an income tax of 5 mills on the dollar on all over ^400 ^ h^lJt^^^^f^"''^ "^ Por annum, what sum should fn i^f -^ i 'V ^^ ^'^'"^' ^''^''''''^ ""'^''''''^ interest at 6°/, to be paid off at pur at the end of 5 years ^ • anf ;o?d''n.Tf f.'"'^^'' f ^-% ^^i^^'^^^nt' "^^st be bought, and sold at 8% discount, to make a clear g-ain of $100 broke rag-e each way l^ ^ ' 45. What money invested I list be the price of 5% stock to gain 8' on ) .1 I h i 164 ARITHMETIC. ^(^. A mail mvostort a certain sum of monov h, .. ao/ stock af, UA)L brokom.-e ' ■iihI l... if / , '^ "^ '^ ^/^ 57 stock <)«»7 K.. "^ r ' ' '''''^ '^^ "^"^'l* >»(>»*(3 in a ki.rofs'ock? '^ "^^ ""'^ "^'"" ^^"^ '^^ invest in each C%*L^'afl^lf '^"" "^^^^^"^^^'^' ^% «^-k -t 113, or invt tod !lf!^ ^ rnum'^ -^,^f ^0, and the proceeds An A - ° i annum hnd the chance in incomp 4!). A person invests £->')(\o i<». ;„ „ ' ^^'^ "^ income. bv the tr'ms'iPtinn9 ns i ^' " "'i'' <^ioes ne gain or lose sians |.J [Brokerage on Consols |, on Rus- SIMPLE INTEREST. fo/onJ fif l!owTn^'".' if ''^^^^'^ ^^^' '^'^ ^^« «f "^oney of tim I ' "^^^">' '^^^^'^'"^ should be paid for the use of 1100 for one year ? For the use of ^innf^V f. For thp n«n r^F ^rnr^ ^ ^^ *'^^^ i^^ OWG year ? j^ oi rne use of 5!^oOO for one year ? j "■^ • 2. If 6% of the sum lent is charged for the use of moTiP.. SV^O ?o"'' '"" "^"J^^ ^^^"^^'« «^«^^^d be pIm fol he us^ of 1450 for one year ? For the use of $275 for one fear ^ For the use of $242.75 for one year ? "^ 3. If 4% of the sum lent is charged for the use of monp.r 4. If (,% of the sum ^-nt is charged for the use of mnnoA- of'iToo'f"'' ''^T ''T> ^^"^''-^ should be a d for trise oL tM?d of ''''-^''l^% ^ y^'^^ For the use of $100 for 5 V fi°/ . r""' • « ^.'\ '^^" ^'" "^' '"100 for 2 months ? 5 At 6% a year find the interest on $800 for 2 ve-irs Id 6 months - on sKirio +>.». 7q .i . .1.^ i"^ *" years and 6 months • on $450 for 73 and 14fi days. CiHys ; on 1 720 for 3 years SIMPLE INTEREST. 166 in a 6% >!•2G.50 (d) $1827. CO (e) $925.40 (/) $1000 u 91 " 2 at 4% per annum. << ^1° 4 (flf) $1263.80 " 3 ih and 4 mos. " 5 '* '< 4^ . " r>5 days ''' 5% « - " 73 "" " loy < (^) £1800 12s. (id. for 2 years and 140 days at 57 per annum. *^ /° ^ r.^;. ^?? ^^^^/T ''"'' r^' ^'^ ^'% 5 ^vliatsum should be returned to me at the end of the year ? 9. Find the amount of 81200 loaned for three years at 4/, per annum. ' 10 Find the amounts in the different examples in num- ber 7 above, using 8% in each case. 11. Find the inter .St on $1400 loaned on May 1st 1890 and returned Nov. 14th, 1890, at 8% per annum. ' Oct 28rh is'm Tf^T'"'' ^^'^^ ^'""' '^^"' 1^^' 1^89, to uct. j«tn, 1890, at 6/^ per annum. 13. Find the amount of $1275 from July 4th 1888 to Jan. 15th, 189U, at 8% per annum. ^ ' ' 14. Bought a farm for $5,500 to be paid in G months with iirterest at 4% ; find the amount of the payment ^ 16. The interest on $550 for 2 years is $44 : iind the interest on $100 for 1 year at the same rate ^rl^iJ^'r^lT"' S'' ^^^^^I""' ^ ^^^'' '^^^d 146 days is $58.80 ; find the interest on $650 for 2 years at the same rate. .ILF'^^V'' ^''^ ^'''^'' P®^' ^^"^'•' ^^'^^^^ the interest on cl440 for 1 1 years is 72 ? 18 Find the rate per cent, per annum when $400 n mounts to --US in 3 years. 19. $1200 amount: "to $1290 m 2 years and 6 months • nnd the rate per cent, per annum. ' 20. $1500 amounts to $1515 from Oct. 28th to Jan 9th- find the rate per cent, per annum. 21. In how many years will $500 amount to $600 at 4/ ? i- i tfr I 166 - 1 , • 1 i i 1 i ^ J 1 : i i : ' i .1 m i'l m\ AI'ITHMETIC. ^2. In what time will $l2r,0 jjinounl, to UlMO at .HV v -'4. In w mt time will mi.W doalAv. iisclfnt 47? at 3% ? ' "'^ '"""^ '""'^ '''"-^^ '""' "'' ^'^^'"^y '^^'^^W^* itself tholm'le^t v"' ' '"^"*'' *'" ^^^^^'^^^ ^« w].nt fraction of thepri^cii.l^''^'"'''' ''' ''''''''' ^' ^^""' ^'•^^^^•«" ^f ,.tf in ^otilowh^'r- ""'"" ''^^ "'"'"^ " ^'^^" ^^^^^- (r^) At 7.',/,; for 4 years. {h.) At (;|% for L> years. (c.) At ;-5^/% for (; ycuvH. {(I) At 5% for 20Vears. (e.) At 8% for G nionths. (/•) ^t ()■% for 4 months. .,. ^ (.(/•) At 10% for 2l>2 days. -.». Ill one year the interest is .\, of the sum lent- fn ""'Z^^yr^^' -''^ ^'^^ i^^^-'-e^t ecfual the sum lentV su At J/ per juimini, in 1k)\v mmiv veavs will fi.o ,•„ teresteciualthesHniicuf,.' * yeaii, ivill the m- 6i%? ' Am%f''Ar-'% ? ™" "'' '"»■>«>■ ' ' At'Tif ? " '"'" "'■ """"'y t'''^'"«ite<'lf.-« of "^^i"o7'f "i"' "''"' ''™*"'' "^^ *-" "■ '"""th. pays tlie taxes what is the value of tjie house ? money invested ; 34. What montlily rent will i.^v the taxes it 1°/ nf 36 Tlie interest on *400 for 1 year at a certain n,t;e together witl, the interest on $500 'for the same thne fj double that rate, amounts to $28 ; find the rates 37 The nitorest ou $300 for 2 years at a certain i-ite Miat laie, is Jj^lU.) ; hnd the rates. 'M HIMPf.E INTEUEST. 167 rate, is ^25.25 ; f1n!f th^ rll^t '" ^^'"''^ ""' ^ '''''"' •|!». FHuUlHwimount of .^00 iu 2 ycu.vs at 4 /. 0. J) .at siini .nn.m.ts to ^540 in 2 years at 4 / ? ^^iL , innnts to maJhl'''^ """' ''^P'^^it^''^ i» ^ T^-'-.k now at 4 7 will in 'f montlks, amount to sBl.'j;);) ? •* /o ""ii, m 43. What sum deposited at 6 "/will in -^fwi-.x- . to cnono-i, tn ...i,r fi... 4. /° ' " ^l.Mlays, amount tu enouft-ii to pay the taxes on a hulldimr worth ^'lODOn taxed at If) mills on the dolh.r? *.^0,000, wiu amount to ^i^i) ^ one-half of tlie time ^ 4b. A pei-son ))oi-rowed monev for 'it vonw .,<■ w cert and repaid prineip.l nnd int^^^ ^h^S-^O lIZ mueh was borrowed V si \Mrn .. .>_u. How tiply theeum by one-half of tlic lininhcr of „„ , l^^' ^.n p' romove tl,o aeeh.ul point two ,.,,,cc.s to the'lLf""' " tpi; 'n best offer money being" worth 8,1 per cent. ? •>t>. If I borrow $1200 for 3 ve'or^ -if f\y ,,M-fi. ^i h iH n oU, '^'' P'^r^P''^' ^"^^^ t^^- ^«^t year, liow niuc 'i Shall I have to pay at the end of the 3 years v 01. A man 0Hffa,o-ed in Imsim.ss was m.-i kino- 1 r.o/ ,>.,^. year on his capital of i^lDOOO ; Ik. L^ve i IK bn i 1 a i^acr fiee of 10"/ of his ,...niM 1 . ^ , ""'^nK^ss at 00/ , . ^ /o "'^ ^"^ capital, and loane< his monev at 8/ ; what amount of income did he lose yearly ? ■ f r ■ i! '': i F If I i n I Mi ; '■ t !; 168 ARITHMETIC. 52. A dealer bought 12000 worth of flour on 6 months' credit, and sold it immediately for 12^% advance. If from the proceeds he deposited in the bank sufficient money to amount to the $2000 at the end of the six months, rate 5%, what sura had he left ? 53. What does a dealer gain by buying goods for $2500 at 6 months' credit, and selling immediately for $2800 banks paying 6% on deposits ? 54. The interest on a sum of money amounts to X of the sum in 7| years ; find the rate per cent. 55. The interest on $5000 in a certain time amounts to | the interest on $100,000 for 1 month at 6% ; find the time 56. In what time will $l.38.i amount °to $1.66|, at 5°/ per annum? "^ ^° BANK DISCOUNT. 1. I have in my possession the written promise of John Jones to pay me, or any person I may name, $1200 at the end of 90 days ; the Bank of Commerce gives me $1200 less the interest on it for 93 days at 6%, for my claim against Jones. How much do I receive ? 2. $500^. Toronto, June 1st, 1890. i^our months after date I promise to pay John Jones, or order, the sum of Five Hundred -P^«^ Dollars, at Dominion Bank here. Value received. Henry Smith. {a) On what date Avill this note be paid by Smith ? {h) If Jones sells this note to a bank on June 15th, how many days' interest will the bank deduct from the amount of it ? (c) How much will Jones receive if the bank discounts at 0% ? {dj What is the discount ? Find (a) the day of maturity, (h) the time between day of discount and day of maturity, (c) the discount (d) the proceeds of the following notes : 3. $1000/'J>y. Hamilton, March 1st, 1890. Six months after date I promise to pjiy Oliver Bland or order, the sum of One Thousand /'^'^ l)()Ib)rs,at my office here. Value received. John Smith, Discounted, June 4th, 1890, at 5%. BANK DISCOUNT. 169 4 'fi^^'ifi s SIX ^0^ -Dollars, at Imperial Bank hore Voi^^ • I Discounted immediately at 67 Value received. 5. $1250 ^ p ^/°' ^- Brown. Fifty ,,VDoU..., at *::X.?Srh'eI""ilC;lr Discounted Feb. 1st, ISiJO, at 6.|%. ^"'"""^ ^''^^™''^^ 6. I»640fyjj. Baerie, Ont Mnt, -ru.^ mco Hundred and Forty --^^ Dd ars Vni„« thousand Six ./ 10 J-'"iiais. Value received. Discounted July 2nd, 1888, at 8°/. ^''^''^' ^^^^^^^. • . ^w/Djj^jj^, Grimsby, Ont. 7>r uf ^oon Ninety days from date I promise to p i\^WmHm R^H ' or order, the sum of Two Thonsnnrl i;'/^ ^^ Vr ' ^^ -l^^^i'ker, Sixty-nine -00 Doll'^r^ vt.,? .^"^7®'' Hundred and •^ ^""^ 100 -L'uiicirs. Value received. Discounted Dec. 24th, at 6% •^''''^"''' ^'^'"''"- Dollars. Value received ^ ''"^ Tr^Z'^r"^' ^'^^ Discounted June 4th, 181)0 at 87 •^'''''^'• 0. $4000^oo_ ' /°* ^^'^^^iTJo- Toronto. /Vo?? i>o/A rwon Q. ^^-'^^i'cKT New York Tlf//,/ a/;, /oon Six months after date I promise u l^T^i ' i^'^?' or order, the sum of One ThousTnd T^ IT''^ ^?'^^^' Thirty-four and ^"'' Do n-/ Tt- 1 ^^^ "^"^ Hundred and Value recoiwd "' ' ^'''"^ ^^^/^"^'^ ^^^^k here. Discounted June 4th, 181»0, at G%. '"'"*'''''' ^'"''^''' 170 M i • • ■ i''l I > i': ARITHMETIC. ot i 01 onto here. Value received. Hexry Scott. Discounted immediately at 8%. it ^^'^^'l^' . . Whitby, Feb. 3rd, 1888. ]^our months after date, Ave promise to pay to tlie order of Charles Beemer the sum of Five Hundred' and Seventy- six and y^v J>ollars, nt our office here. Value received. p.. . 1 nr . ^- WiLMOT & Co. Discounted Mar. 1st, 1888, at 8 7 tL;^''^'^- . . i^^KXs, i.W. 6Y/,, AS87. Th oe months after date, I promise to pav Samuel Cole, ordei-^ the suni oi Four Hundred and Ei.^^htv Dollars / at the Standard Bank here, with interest at 5 y Value n; ^ 1 1. T .o , IriuMAs Johnson. Discounted Fob. 18th, 1887, at () %. 14. $20U0^'^. T()ROA^;o, Mew. 4th, 1889. Sixty days attn- date, I promise to pay to tlie order of Henry (^mliara the .um of Two Thousand Dollars, at the Imperial Bank l^re, Avith interest at ('>%. Value received. TV , . Alexander i\lcCuAiG. Discounted immediate! \'^ at 8 7. 1^- e4.^00^^. "Port Hope, Auq. 2Wi, 1881. Ninety days after date, I promise to pay Thomas Scott or order, the sum of Four Thousand Two Hundred Dollars' at the Bank of lAlontreal here, Avith interest at 7 7 V-ilno ^"T^'''\.. , o. George Ielly. Discounted Sept. 1st, at 8 %. Ij;. When must notes dated and draAvn as follows be paid : — (a) Jan. SOtli, at 1 month ? (h) Jan. 2Uth, at 1 month '? (c) Jan. 28th, 1888, at 1 month ? (d) Dec. 31st, at 2 months? (e) Nov. 2!)tli, at 3 months? 17. The iut'M-est on any sum for 73 days, at 5 7 is what fraction of tlui sum lent ? " ^''' ^ 18. The inicicst on a note for '.ir) day.', at G 7, is what, ii'actioii oi the cimoui:':; of the uole ? lUNK DISCOUNT. 171 10 A bank discounts a 92-diiy note at 7. Find wh-il 'Io^'a 'I'' f''?''''' '' ^^"^'^^^ '"^^ ^'^'''- orie note ^U. A bank clmrg-os what fraction of the face of ;i note when discounted 73 days l)efore it matures, at 10 yV' 21. AVhat fraction of the face value of a note does one befo^ritird^fe?"' "'"' '^^^"^^^^ '' '' ^%' '^ ^^ 22. A note is discounted 60 days before due at 8 °/ and piXe |!s?i;f ' '"^ ^^^^eounted immediately at 8 %, to 25th "at?!?.'.!'' ^''/v '"'" '''''''' '^ ^ ''"^'^ "^^^de May ducelkVos ? ' ^^^«^«"^^ted June 8rd at f, %,'to pro^ 25. A note for S730 was discounted 45 davs before it d!"r ''"'^""'^ ^'''•'^- ^^^^^^ -^^ the rl^^e of ^lOOo'^TqT'' Tf'^'^'^- ^"''" ^'^ ^''"^^^ ^-^^^ for a note of lls^ounl? '^^'^''^^'*' '' ''^'' ^^''"' ^^^'^^'^ was the rate of Aug- 1st, and was discounted on June 4th, was ^-^0 80 - hnd the rate of discount. ^-^..iu , 28, The discount on a note of .^14G0, discount(Hl (0 davs count '™ ^^^'""^^ '^^'' '^^'^ ^'^' ^"^^ ^'^^ '''^^ ^^^^ i«Qn "^^'''' r?'''? '"^"^ ''''''* ''^ "^^'^ ^^c ^^I'^^wn on June 1st June l!i^:^ • ^ ^'^^^"' ^^^'^'^^ ^^^^^^^ '' - discounted on June 14th, at 8/ the proceeds will amount to ^858.G0? 07 yi^ldediM^s'rs'' 1^"^'"' '" ^^>^ '"'^'^ ''''' ^' due? «^1'1-1«.18. ^Mien was the note nominally Dec'^th^'l'^^lr^T «"/"'"' ^'^^'^- ^'-' discounted on Dec. 20th, 1887, at 8%, ana yielded 1289.12. On wh^it date was the note draAvn ? v^n A\nat 32. A sixty-day note, for $1200, with interest at 67 is d^^med on the day it is made at G%. Find thet-" 12 Ill ■' ; 1 1 1 'I : |, i i 1 r]:ii ^1 in , U j ! P \i,l i 1 .i 172 ARITHMETIC. PARTIAL PAYMENTS. 13500^ m On dPnnnri T • • ^^^ONTo, Mm'ck Ist, 1888 received, with interest at 5 percent ^tatunL"'' '''"' On this note the following payn.ents weTnucfe™™ May 13th, 1888, ^.WO. ' the ■inte^rirdu™""'^ '" "^^ '"^^■"''^"' » S«P' «*, exceed paymenltXT'eth'. ^"°" °"^ ^^^'^ ''«- ™«>-^ »"« $5^^r '»""'> ^'^^ -e«t due on Sept. exeeod'^hc'rtert'at? '"" """""' "^ "^ '™ P'-^^^-'^ 12. How much does Cole n\vr» ^o.^^f ^,> ^ -i-oa v.oit:i o>vL bcott ou dan,. 1st, 1890? PARTIAL PAYMENTS. J7«j $400^. On demand, I promise to n.x ir-fr' '^''''- ^'*> ^^•9^- the sum of Aur iZdn- 1 h 1 '^^r'''' ^^'^^' ov (Mi^r, with interest at 7 r^^ ^^^?'^^« I'^eeived On this note thei^io.^!;:;:^:!,,, .J^";::^^^- March 15th, 1890, ^20. ' July lOtli, t. ,«!(• note foTllOoi r^fnSritt""^ ?"''■ ™-'' '''^■"-''' 8% : June 1st 1888 *qm . V 7 . ' l^e^^nng- interest at 18«'^, -^100; June t't 1889' ^4fio If''''' '^J^' '^^^"^ 1«^' June 1st, 1890 v ' ' ^^^^' ^-^^^^ "i^^ch was due on eaiit-id^i::^:;,^ -.Siut^ '^??'. ^^^^-^ ^^^^ balanee, witJi tiie prMe<^f 'f ^^^ prineipal at any ti^^ ^ , ^^r^^^f f "^^ ^'''^K '' ^1- date. Tlie transaction Z,Zt^^^^^^ !f?" ^'^''^ "P ^^ Sept. 1st, 1889 he ^-ud W t t ^'"'^ ^'^' ^^^•*- On lt>. On a flcniand note of 0Hi iq«7 1800; July 2nd, 1887, $000 ; July 2nd 1888 ^^uw' ir ' mueh would pay off the mortgage on Jan 2nd ?880 7 Soo ^?r bearing interest at 5 % ? ' ^^^^^ '^^^ ^Te'year after date, for vahrrec^i;'e? '^'^• pay Mack Jones, or order the sum afrl rri ^ '''"'''' ^'' lars, at Molson's'bank heve " "^ ^, ^^'^^"^ ^'-'■ . On Jan. 7th, 1889, Jones paid on this note 4 'no mg to pay interest at 8 V per annmn ,,..1 , ' ''^■''^''■ balance. lie made the iblliwin: p' "n^ he pa d the 1889, SaOO; June 7th, 1889, $20o/ ' ^' ^ ^^'''^ ^'^' 18. How much remained due on Dec 7tJi l8Mov n, kon time by months). ' ^^^'^ ' (^^ec- ^''1 ^^J^J^B j||HB| ^^N ^HhUI,' ^J fli '^f a III !H liii 174 AIIITHMETIC. r.o ^1 oV'!'''^'''^'^ ^^'^^^ ^" '^^^^c 3^tl., 1888, a^reein^ to pay 7| // interest on the i)i'iiicipal, reserving- the riiiht to pay on any part of tlie principal when tlie interest is paid up to date. On Sept. nth, 1888, I paid $200; onJuneSOtli, 188J, I paid J.150; what payment made on Jan 31st, 1891, cancelled the debt ? ' liiii^ i' • EQUATION OF PAYMENTS. 1. The interest on $100 for 5 days equals the interest on Avhat sum for 1 day ? • 5' ^"^ ^^'¥* ^^"^ ^^^""^ ^'^'^ interest for 1 day equal the interest on $50 for 10 da>'s ? ^ ^ 3. The interest on $125 for 4 days equals the interest on what sum for 1 day ? 4. The interest on $500 for 1 da>' ecpials the interest on $50 for how many days ? 1000 fS'i dTvT '^''^^'' '''' "' '^'^ ^'^ '''''''' '' '^'' ^^^ «f' 19^0 ^1 i:^l ^^^^' "^^ '' ^'' '' ''^'''' '^ ^^- -- Of ^ 7. How many days' interest on $50 is equal to 12 days' interest on $62.50? ^ 8. Jones loans me $200 for 4 months ; for how many mont.is should I loan him $160 to balance the favour ^ nf IrmT T''^^ Tf^'^^ ^'^ ^^^^^^^ ^^ ^^^^^1 t« the use ot !it^bOO for 5 months ? 10. Smith loaned me $300 for 4 months, .'*500 for 3 months and $450 for 2 months ; how mfich money loaned hniith for 1 month would balance the favour ? 11. I owt $400, due in months, and $100, due in 11 cTebtt' ' '''^^'' "'''^^ ^ ^'''^' '^^^^ '''''^ equitably cancel the 12. I owe $500, due 6 months a-o ; $800, due It months ai'o; how many montlis' interest should I pay on ($800 + $500), in addition to paying- $1300, to cancel my indebtedness ? '' EQUATION OF PAYSrEXT.S. 175 13. Bougiit from Morton & Co. iroods on fnlinw;,,^. So' .l^^J^^ P^^^^'-^blc in 40 days ; at what time might the H900 be paid in one payment ? ^moJ^^V^a S^l^^^^i^"^^^ of payment of the foHowin^^ : S days. '^' ' ^ "^ '^^" "^ '^^^ ^^^^>'^ ' ^^'^^ '^''^ ^" nn\h\^T^'^- ^'^''' ^^^''^^" ^^ ^^^^•' ^««^^« amounting to 12400 ^800 dno ';^n^, '"''''' '' ^-'^^^ '^'^'^ ^1-^00 due in 10 days days, .1?(,00 due in 12 days, and $900 in 22 days pa>s ig- of It in 3 montlis, | of it in 5 months ; Avlien does the remainder become due; ? 18. A merchant bouglit on Jan. 2nd, goods on the fol- credit ? ' average term of 19. Bouglit from A. White & Co. on Juno ^rd D-nnri« o. M^w. : 11.00 .ish, ^2400 on So'd^ys "^^^i ^t^? I settled by paying .^4000 cash and giving my note for the balance. Find the time of the note • ^ 20. A debt of 1,5000 is due in 40 days; $2000 is mid 1. days beft,re the debt is due, and .1?1500, 2 dav^ befo e the debt IS due ; when should the balance be paid ? tinu^^fny!.!^'^^'"''^'';^'''" ^f «^'«^^it' ^^nd the equated tunc of pavn en from June 1st, of $400 due in 30 days kWO due in 40 days, and ^'500 due in GO uays ^ n.i^u ''""''^r''''''^' ^"""^"^'^ ^'^^'^^^ ''^'0"i the wholesale house Si ioo '%V ^\'''' ;^''^' ^^^^ «'^ ''^ '^^y^ credit ; June 1 5 I ' 23. Bought merchandise from Macdonald & Co as fol time ^ ^ ' ^ ^'' ^^^ '^''^''' ^'^"^1 ^^"^ cqn^ted ^lon , iiMUOO, iluo in .todays ; Hud the wpiatcd time. u fi n f S - ti I i i III! , ! Ill'' 176 AUITHMETIC of ]).'iyiiiGiit. ^ "^^^ ^"^ 0(iimtecl time 2(1 Find tlie c ^^'^^'^^ 90 w. .. , ^ "^'^^'^^^^'^^ ^% Pf'i' annum? Ham. Smith. q^ 1890. Jan. 5 To indso i Jan. 20 1, I Feb, 1' „ , Feb. 15 „ H .«*840 900 750' i 800 1890. Feb. 1 By Cash. I Feb. 20 „ „ $1500 500 ju^ iS'TCtn'' •' ''" i^^-^^^^^^nee thiT;;;;;^;;^;;; .muc ibt, l«!)0. allowing mterest at 6% ? COMPOUND INTEREST. 177 ic-half is ted time s :— account Cr. $1000 600 a lance a lance on all Cr. $1600 500 It on COMPOUND INTEREST. 1. I deposit in the Standard Bank $100, on which I am to receive interest at the rate of 4% per annum. How niucli is there to my credit at the end of 1 year ? 2. If I leave the deposit and interest in the bank on what sum should I receive interest during the second year ^ 3. How much is there to niv credit at the end of 2 years ? 4. On what sum should I receive interest during the third year ? ^ 5. How much is there to my credit at the end of 3 years ? 6. To what sum will $100 amount if left on deposit in the Standard Bank for 3 years, interest calculated yearly at 4% ? * J J 7. What is the amount of $1 in 3 years, at 47 per an- mim, interest calculated yearly ? " 8. What is the amount of $100 in 4 years, at 57 per an- num, interest calculated yearly ? 9. What is the amount of $150 in 4 years, at 57 per an- num, interest calculated yearly ? ^ 10. What is the compound interest on $150 in 4 years at 5% per annum, interest calculated yearly ? ' 11. Find the compound interest on $875.25 in 3 years at 4% per animm, interest calculated yearly. ' 12. Fnid the compound interest on $1250 in 3 years, at 5/ per annum, interest calculated yearly. 13. By what fraction must any given sum of money be multiplied to g-ivc the amount at compound interest for 3 years, at 5% per annum, interest calculated yearly ^ 14. By what fraction must $525.35 be multiplied to give the amount at compound interest, for 4 years, at 47 per annum, interest calculated yearly? 15. John Smith deposits $100 in a Savings' Bank at the beginning of each year, making the first deposit Jan. 1st, 188;). How much will there be to his credit Jan. 1st, 189l' the bank paying 4% per annum, calculated yearly ? 16. What Hum of money will give $150 interest in 3 years, at 4% per annum, compounded yearly? Ill \^ ■ i 1 ' w i 1 1 i 3 M ' ■ 1 ' f i ; 1 i ; ; 178 AllITHMETrC. 17. Find the amount accuiiinlatod af. the pnH nf j ,^« tercst, cHlcuIatt'd yearl\'. "^ ' ^^ compound m- 18. ^Vlw,t is the'diinuence between the nimnlo ...i ,^ -np^md interest on mi, lor a yea.;:!;? r^l^^m^illluS intlLt t ^l^t:^^- ^~: 5-r ^-^;^ ^«- compound yearly, is .f iQo. j.i,,a the sum ' ^' compounded 20. AVhnt sum of money loaned nf- t°/ r^«,, „ pounded y(,M.lv will nmrmnMn i ^^ ^^^ annum, com- -:►[ I/- I .. • ' amount in 4 years to ,^1200? 21. iMnd the amount of $1200 in •> venrs nfry^ -4. l^nd the into-est on .^i "in 2 Wrs iPd 7^ .lo 5% per annum, c.leulatcd yearl.N^ ^lays, at -i). A\ hat su]u of monev" wiir\'ioM c:iaa • . .0% peV::;Cn,f tile';::?:;:: trrr "^ ^^^^'^ \Vh^^i^ t^. ^1.^ Jiiv.i« d;5e in tnc last 4 years is 1.^<)9^ VV hat IS the pi-esont po])uhition ? ^ '^'^' }a) \V\J\ f';V'"^^^"'^' ''''''"^^ ^'^^ .yearly, (a) n h.,t L, the amount of $100 at the end of 1 year? (I>) A^'hat is the amount of |1 at the end of l^^trf '^^• (0 M'hat is t.e amount of 1109.521^^.^/13 Alls. $16!).r>2xl.05, or (d) AVhat is the amount of $A at the (^'of i yf^ p (?.! .^"^^1!;^^' '' ^^-)^- '1% per annum, interest adclecl yearly v "''' ^'^'''' '^^ ^. What is the amount of ^i at the ^ ^^ ' ^'•^'^'• H/o per annum, ,,dderl vearJr v a^ ' ^^'"'''''' ^<= 30. What is the amount of ^t^r.n .. „ '^"'- '^l I "O'^ -■>)"'. at 3% per annum, adS .' .f't^ '^\ '^ ^,'" /»^" ^ y^'-^^^, at 4% per anniiii,, added ve.irlv v "«''-'5- wr 3 years, amounlof^iiriii'"' compounded yearly, find the 35. Find the ainoimtof iBl in 2 venr<. ^nri o 6% per annum, payable ve'.rlvv \ f"'^ '^ "lonths, at 36 Find ^<- i-/ payjbh3 half yearly, amounts lo 1^2^ 4;; ^'' '""""^' ».82. Find the s4m^ '""""' ^^"^r-^^"^^-! yearly, is 4t^r;?=reo^!:^^^^ second is $1, the d ffc^^ee h^^eflho" w ''^^^' ^^" "^^ Beeond year and that of th^th i .1 ye^^^^ \^« rate per cent, per annum '" " '^ > ^ '' » ^^ *l -O"*. Find the 42. What rate per IS equivalent to 3 yearly ? ••cent. j)or;!i!num v ompouiKJed Vcai'lv o/o per half year, compounded 'h •ilf -Jfr -^ A #. %,<^ .0^, \^^ IMAGE EVALUATION TEST TARGET (MT-3) Y A c& Q^ :/. 1.0 I.I 1^ 1^ = 2.0 L25 i 1.4 1.6 Photographic Sciences Corporation 23 WEST MAIN STREET WEBSTER, NY. M580 ',716) «72-4503 ? <>.' 4io W4a 4. fe r: i.i ^ Nil! > ■ n J II 180 ARITHMFTIC. 4H What rate per cent, per hcilf year, compotinded half yearly ? '' "^^'"^''^^"^^ '^ ^'% P^^' •'^^"i^m, compounded 44. A sum of money at compound interest, added yearly amounts to .^l-.»600 in 2 years, and to .^178506.4 in 4 ^^uL ''^ "'" '''''' ^^^^""^'>' ^^^^^^ ^h« ^-^^'^^ P^i- cent, per III Ilium, fn,.^'l"h5c^*''^ /' ^'''' co/»i;r)^/«f/ interest on a certain sum or the second year, .ii?l 1.(325 is that for the tliird year i'lnd the rate and the sum. ^ ^rJf: J^lri?^"'"'^ ?" '' '''''' ^f i^^«"^>' f«i' one year, is ^oO ; tlie difference between the compound intei-est of the second year and that of the third is $2-08 ; the difference between that of the third and fourth yekrs is $2. 16 3' Fmd the rate and tlie sum. ^ i ^ is 0- ibd.. 47. The compound interest on a sum of money for 4 years IS .^254.78784. The difference between the^nterest of the hrst year and that of the third is i^4-81)fi • the dif terence in tlie second and fourth yeai's is i^5-00184 Find the rate and tiie sum. 48. The coinj^ound interest on a sum of money for 4 y^irs, reckoned yearU^ is -^-V^^ of the sum. Find the 4'.). The difference between tlH>, interest at 107 per an- num, added yearly, and that added half yearly, fbr two years, is So.VOdi. Find the principal. ^' dO. Find approximately tlic amount of .^1000 in 10 years, at 7% per annum, compounded yearly. PRESENT WORTH AND TRUE DISCOUNT. 1. Kind the amount of 8100 in C, months, at 87 per annum. ' . ^^/A/ o ^A I 4- , , . ^ns. $104. ^. W hat sum place.l ,„ a bank now, at 8% per annum wiI.nnonntto.KMin.;monthsy ^ Ans.^a .->. Smith owes Jones .^104, due in G months. What r.iin ].aid now will cancel the debt, money beiuff worth ^V'i'-''".'h"' ... Ans. .QlOO 4. i-ind the aiiioiint of ^I(K) in. months at 8V ,,or -^n num. ' /' ^ ./"." Ans. 8100. PRESENT WORTH AND TRUE DISCOUNT. 181 5 What Slim placed at interest now, at 8% per annum will amount to $636 in 9 months ? Ans ^600 6. Find the present worth of 8636, due in 9 months* money being worth 8% pc>r annum. Ans. ^ew.' 7. Piiid the present worth of ,$800, due in 8 months money hemff worth 6% per annum. 8. Find the present value of 81, due in 6 months, money being wortli 8% per annum. ' ^ 9. Find the present value of $1, due in 1 year and 8 months, money being worth 6% per annum, compounded J c a 1 1 \ . 10. Find the present worth of 88000, duo in '-> years banks paying 4% per annum, interest calculated Vearh-' 11. What sum should be deducted from 8l-?00 due in'6 months, if the debt is paid off now, money being 'worth 47 per annum ? -^ & " ^/o 12. Find thc*true discount of 81350, due in 9 months money being worth 5% per annum. 13. Find the true discount of fa) J485.50, due in 116 days, at 71% per annum. " " 1 year and 5 "months, at 6% per annum. " •• 2 years, at 4% per annum, com- pounded yearly. " 3 y(;ars and 219 days, at 5% per ^ annum, compounded yearly^ " T) years, at 5% per anniun* com- pounded N'carly. 14. A former pays a yearly rental of 8400. What sum paifi now would be equivalent to the next three years' rental, money being worth 5% per annum ? '^^}'^^---f'>'^nks pay coinpoimd interest on deposits. fn i'lm -^ «^"i; P^iicl cit the end of 2 years, is ecpiivalent to is400 paid at the end of each year, for three years money being worth 4% per annum y ^ ^900* .il"'-'"'/^ ^^''^' ^^ wi^^ - >'''''' ' ^^^^' '^''^ "^ ^ vears ; HI w ro/ '^ ^''''"'' ^^^''^^' '^"^^ 1''^^^^ ''^''' ^^'^i^'^l "eance the debt, 5% per annum ? 17. The rent of a house is 825 jwr fa) What is the interest on $b ? Ans. 6a. (hj What is the amount of 6b? Ans. 6(a + b). (cj What is the true discount of 6(a + b)? Ans. 6a. (dj What fraction is the true discount of the amount ? a An. a + b' 22. If the interest for a certain time and rate is— of the b principal, llnd what fraction the true discount is of the amount for the same time and rate. , ^ Ans. a + b* 23. Find what fraction tlte true discount is of a debt, due in 3 years^ at 5% per annum, simple interest. 24. Fiftd wliat fraction the true discount is of the amount, due in 2 years, at 4% per annum, compounded half yearly. 2."). Jiy how inuch does the aniouut exceed the present imrth ?' 2(). By how much does the interest on tlie amount ex- ceed the interest on the present worth? 27. The ditto rence between the interest and the true discount on the same sum, for 2 years at 5% per annum compounded yearly, is 68.20. Find the sum. - ' 28. Tiie ti'ue discount of a cx^i'tain sum, <\\u\ in 2.', years at 4% ]H'r annum^ compounded half yearly, is ' 63r)0' AVhat is the sum? Ill PRESENT WORTH AND TRUE DISCOUNT. 183 -•>. Iho (M)mi,(Hi,„l intorost on n certuin sum, f,.,- a ccr- Uiii time and mte, is |250 ; the true diseount for the same tune and rate is $240. Find the sum 1ho?;M^r'^'^''?"''^^''^'''''^''^■^^'' compound intei-est and me In. '''" ' ""f "'^"^^ ^""^' ^^^^ '^'''^ ^"'^^^ ^^»^l ^^t the Stl^'S.Il"^ ^^ '^^ ^'^^^^■^^^- ^^- time is 2 years. 31. The true discount for .^ yenrs is ff?oo of the com- pound interest for the same time. Find hi'rate per ceTt per annum, compounded yearly. ^ F^.ttJ''^ •"'''' discount of 8270, for G months, is $30. J^Jiid tlie discount on tlie same sum for 1 year, interest compounded lialf >^iarlv. ^ ' niccicbt 3:-J The true discount of 821.S, for 1 \'ear, is «^51. Find pouncied jialt yearly, .J^'/^'"" compound interest on a certain sum for the second yenv is $49.92, and for the third year $51.91G8 Jund tlie sum and the rate per cent., compounded yearly" o;).^ Uie difference between the compound interest on'a certain sum, for the first and second years, is $2.40 ; the difference for the second and third years is 82 -490 Find the sum nnd the rate per cent., compounded yearly. ob._ Ihe difference between the compound interest on a certain suni, for the first and fourth years, is $12.<;i. J ind the difference between the interest for the third year and^that of the seyenth, at 6%, compounded yearly. .rf^' vino/ '? ^^'^''^ Of money whose true discount for one year, at 10/, is greater by $3W«, than the sum of the true discounts ot one-half of it at 8%, and the other half at l->7 ror one year. ' ~^°^ 38. Boug-ht a farm for $10000, payable one-half cash the remainder in 1 year, with interest at 67. I sell ini- me^nitely for $12000, payable in 3 months, with interest .tt 4/^. \\ iiat IS my preseid gain, money being worth 57 per annum ? ^ & ^ '^•i • /o 39. Show that the true discount is the present worth of the interest, for the same time and rate 40. A town borrows $1 2000, to be repaid, principal and interest, m 4 equal annual payments. Find the annual payment, money being worth 6% per annum. \i- M, 'kp ^f ..4 Ilj u S ; M I 1 ■ ll II' ; ' ', i ij', ii:iiil i HI! ill 1^; If'^Ml^ 184 AllITHMETIC. of^monev '"'^';'''=?«« ''"'^veen the present worth of a sum .,,^ t." "* .'^-'i ^' i*^-- C'5,10 the rlim'reiice in the o>mP nf '^ ANNUITIES. est payable vcii-lv w i V; ?! ^ P^^ cinnum, inter- pciNciuiL yuiii>, AMii yield a,! amiiial income of ^3^iO'>> .400 Tl V!- ~ "";- oT,f p;rreS Stv of holntion.-Tho value of a perpetuity of $400 is $8000. 1 he present worth of $8000 due in 4 years is $8000 x (i-»)i Ans «.J;i ^^'''"'1 '^ *'"' P'esent value of a perpetual niinmYv «f S.:t'':f4i-rtr ? •-^^^'^ =' '*^ -^ - «^-" thffii^Vl'i'V"'*'"'"' ?'"'' °f '' perpetual annuity of $200 fated at ^^xtlX '" "^ '''^'' ''' *'- -"1 «f •' > A eS i.J! ill. ANNTTTIES. 185 yeai-ly. to lun 4 years; aaculated iit 5%, Plc*11r-m '"^ '"" '"'■"""'^'^ °*' «- -«■'« - exam. thif sum '""Tn "•'" ™''' Y ''''^''"™ ^ '000 a year ; of system by paying tl.^^s''^" a '!'",„ ty for'To'^? money being worth 5%, payable ^•,.a.r™?^ ^° ^''''"' npv?'.- '"'""''y. P'iil »t tlic 0,1(1 of each year for the ^^'p:^::s^S^eVS;,^^-^--" venr ^JT '^"\^^" "i«^^^'>^ deposited at tl.o end of each purchase a perpetual ai^.uSry o'kt aef:.r:'d''> v'e^rf money being worth 0%, yearly? ^^1^.11 ea ^ jcais, viv* +wf "''"' of money, deposited at the end of eacli est at 6%, payable >-earlv/ i ul to Am 5 v^'i,^ v^'"f '": sum should they sell, money be n«. ort^ ^^ve^rlv v''"' Present value=S720^ I'i + mr + m'+ik' tt^"A I- fli'OOOx (»';)«. ■ ^^'■^ i^" iwi;^ ' ^i i!,i : i; ■' , i ; , i i ■ isn ARITHMETIC. i-iii: 21. A vilhi^o olloivs for snlo dclK'nliii-cs, to niii 5 years bearing- intorcsl, at O/ per .'iiimiiii. At what rate; sliouNl they sell, money being worth 4% per aiuiuiii, pa\able yearly :> Value of 8100 debenture 22. A man agrees to pay for a farm $1(100 a, year for 5 years. What sum paid now would be eciuivalent to this price, money being worth 4% per annum, payable yearly? "^ 23. Find the present value of an annuity of $200 pay- able for 12 years, the first payment to be made at the end of 2 years, money at 3% per annum, yearly. 24. Find the present value of an annuitv of $400, pay- able half yearly, the first payment to l)e inade at the eiid of 1 year, money being worth 4% per annum, compounded half yearly. 25. A mortgage of $5000, bearing interest, at 07 per annum, payable yearl>-, has 10 ^'ears to run. Find its present value, money being worth 5% per annum, i)av- able half yearly. ' ^ ' 2G. A mortgage of $4000, bearing interest at 517 per annum, payable half yearly, has 5 years and 3 mo'iuiis to run, the next payment of interest being due in 3 montiis FiiKl its present value, money being Avorth 5°/ ])er annum pRyaMo half yearly. 27. A town borrows $12000, to be repaid in 20 equal annual instalments. Find the amount of the annual pay- ment, calculated at 5% per annum. 28. Find the present value of an annuity of $1000, to run 15 years, the first pa>'ment to be made at the end of 1 year, calculated at 4% i)er annum. 29. Find the present value of an annuity of $(;00, to run 19 years, the first pa>'inent to lie made at the end of 5 years, calculated at 5% per annum. 30. Plnd the pr(>sent value of a pei'jjetual annuity of $150, to begin in 15 years, money being worth 4V/' per annum. " '^ M! m ANNTTITIES. r» yviiVH, ; .should J)avtible 187 oi\r, for aleiit to l>ava])le )0, pay- < tlie end 30, pay- tlie end )0uudeil 0% per 'liul its 11, pay- Milhs to uoiiths. uinuiu, ) equal il pay- 000, to end of to run d of 5 lity of % per A iii()rto-..,u-,, ,)]• $2m\ boarin-- intcuvst at 87 per I, payal)l(! Imlf yearly, has lour years to run. Find 81. A ni( annum, ts present value, calculated at G% per annum, payable half yearly. ' i .^ 32. Find the amount accumulated at the end of 15 years, l)y a, pei-son who deposits in a bank at the beg-in- ning-of each year the sum of !?200, the bank payinir 47 interest, compounded half yearlj^ 33 A farm bears a morto-.,o.o of mm, at 8 %, interest payable half yearly; the mort-as'c has 5 years to run. What sum paid now would be equivalent to' reducing- the interest on the mort^a^v, to 5%, money beinff worth 47 per annum, payable half yearly? ^° 34. A father dying- left four sons. lie bequeathed a pei-petuity which he owned in the following- manner : the eldest was to have the income for the first year ; the next m ag-e, the second year, and so on. Find tiie comparative values of the sons' shares, money worth 4%, added yearly. 35. A man pays $150 yearly, for 15 years, for 'an en- dowment policy of $2500. Find the accumulated value of LP'^^y^n ^^^^^' I'^^^^^i^^i^S' ^^lO'iey Avorth 0% per annum. ^■ih. A\ hat annual deposit, for 15 years, at 57 per an- num, calculated yearly, avUI amount to i^5000y '^ 37. A man pays $240 yearly, for 15 years, for an en- dowment policy of $4000. Reckoning- money Avorth G7 per annum, payable yearly, how much is he "paving- each year for the l]fe risky " 38. What is tlie present value of a mortg-ag-e of ?55000 bearing- interest at 6%, payable yearly, having- 5 years and 6 months to run, the next payment of interest being- due m 6 months, money being worth 5% per annum, pay- able yearly ? » i^ j 39. Money is worth 4% per annum, payable yearly. At what price should City of Toronto Bonds sell, bearino- in- terestat 5%, payable yearly, and having 6 years to 'run? 40. A man buys a piano, g-iving his note' for $350 re- newable at the end of 3 months, less the quarterly instal- nient of $2o ; for the first 9 months no interest is charged • after that time, interest is paid each quarter, at the rate of 6%. What cash price is equivalent to this, money be- ing Avorth 5% per annum? "^ 13 188 \ **".' 1 1 ^: :;* / 1 1 I- ARITHMETIC. i'i ; H: l-ay each yoa. L^^llH^^^^J^Z to'tt S)"" t'" oniiiff ,ii.)noy worth (;;/ i».ranmi,„v *'**"' '■''"''- Oder a ten-ycar ei low ,,.„/??■' *"'''« same man thev miumsincach case boin." i,,i ..V h r^i^^; ^^''' Vr^ l.ad tak.n an onLZl'nt policy ?"" "" '"^"'^ '""" «' '"^ beSr off !i^;:«';i^'ift°h:v."i' '"i"'" ■^«'"'»- •■"- "'"«" each year, for 15 vc-ir/nt r'/ „ '^'^ "'« '»%'iiimnff of ;;-yeaHy, w„, aS. fo^lio^; r^dTtK;! in"a^87"J;:^,frr''} ^ f'"' "^ *-'"«'0' drawing fund of S75fj8'„'^;^eL V ^''"''y' '"^ P^'<» ^^ V =' «inkin| inttL'"a?57 'rL^S™ ^-I'Vlt"^' f^^OOOOO, drawing $10000 a year? """' '"' P'"^ «» ^ya sinljing fund of , I 1 l'i[||; S S " ' ■; i ' f i -P 1 H: i r .) * i .: 11 PARTNERSHIP. i5Ltrthre,S"ofVn:^:f:r'2vtf ^' "■'"'J '^■•"'^•""^ $2500 How Should thisVa'be'a^ "'"' ' ^"" "' should this gafn bcf/ii^ded? ^ '" '"^ '"''"*^- ""^ PARTNERSHIP 189 3. A and B (iii^r-i^ed in tlu; lumber trado, witli a joint capital of $12U0(); at the end (4"a year A's ^^aln aniount(3d to !i,400 and B's to .*^800. J low much cai>ital did A put in the business? ^ 4. Sykes and Smith formed a partnei'ship, with a ioint capital of ^.S00O,-Sykes K, receive $1200 a year for man- aging tlie business ; the total gain for tlie velir was ^'^200 ot whicii hmith receives -li^loOO. AVliat amount of capital aid bykes invest? 5. A, Band]) forma pa rtnersliip; tlieir respective shares liH!^ r""'" ^ ?'""^' ''"'' *-^^^' ^'^''^^^O'^'i^l -^'X^JOO ; A invested *4000 less than B. How much did \) invest^ 6. A invested .1i^4r>00, for 2 months, in a certain business : B invested !B4000, for 3 months. If tlie ga in is divided in proportion to the use of each man's investment, what amount should A receive out of a total gain of $2800'^ 7. A invested $1G00, for 3 montlis ; B IllOO. for 2 ^2400. Find each man's sliare. - 8. A invested $2 100 hi a business for G montlis, acting asmanager for that time on a yearly salary of $1200- ?^ivested $13000 for 4 months, and during that time re- ceived ^350 as oookkeeperjC invested $4000 for l-> months, and acted as manager and bookkeeper wiien A and B were not in the business ; the total gain for the year was $8G40. What was C's share ? 9. At the beginning of a year A, B and C enter into a partnership, each contril)Uting $4000. At the end of 4 months A withdraws one-half of his investment, and at the end of b months B ^^'i(;hdl•aws f of his. The gain for the year is $()000. Find C's share: 10. A, B and C entered into partnership, contributinir S i?:^f il^r^ ^^^T' ;^-^^^' ""'"'^ ^-'^00 ; tl^^i^ ^^^ins were $1120, $880, and $1200 respectively. If B's capital was in trade 2 months longer than A's, for what time Avas each man s money in the business ? • li" ^' ^ ^'^^ ^ formed a partnership, their money being in the business for 2 montlis, 21- months and 4 months re- spectively ; their gains were $600, $500 and $800 respeot. ively ; A's investment was $3000. Find B's and C's '(}' i( U I Hi 190 ARITHMI-rnc. III'! fir ^ 1-. 1 lardy . -111(1 JiMU's arc in I'arfnorsliip, Harclv having nivostcd $12000 and Jones $15000; Hanly acts as man- ager ou a yearly salary of $2100, the salary to be reduced in pi-oportion if the capital is reduced; at the end of 4 months Hardy tnkes $3000 out of the business, and at the end ot b inonths Jones t.-ikes out $1000 ; the total o-ajn for the year was $0000. How much of this does Hardy re- 13. Lock, Smitli and Knight formed a, tradiuL'-coniiunv Lock put in $2500 for 10 months, Smith $2300 for ii months, and Knight manag-ed the business for 12 months his services bciiio- considered equal to a cnpital of $->OOo' they gaiiK.l $2!»72. What sum should each man receive? 14. B and C formed a partnership to dig a trench • Ii furnished 100 workmen for 40 days, C 120 workmen for 30 days ; the\' received $12000 for the work. What was the share of each y 15. A owns I of a vessel, B i and C the remainder ; the vessel IS insured for f of its value ; the vessel is lost, and A, alter receiving his share of the insurance, finds that he has lost $1000. What did B and C lose respectively ? IC. At the end of a year, from the commencement of their ^business. Smith, Jones and Cook, after "takino- stock," find the amount of goods on hand to be $40000"- cash on hand, $22000; del)ts due them, $25000; amount of their indebtedness, $47000. Make a statement of resources and liabilities, showino- net capital and gain. Find each partner's share of tiie gain. Smith having put in the business $8000, Jones $9000 and Cook $3000. 17. A, B and C form a partnership, with capitals of $7500, $15000 and $22500, respectively. A divaws out at the end of each year $750 ; B, $1200, and C, $1350. At the end of 5 years their capital is $42900. How much of it belongs to B ? ios Good of Hamilton, the sum of four thousa.- • dollars. O>o I bu vs from a, Toronto bank an order on a Hamilton bank direet- in^ the latter to pay Good moO. How nuich must Sr pay, siich ord.rs scdlin^^ ,,,t |-% premium ? ^ for'^^Z'^'-'VlTI' '*^ 1'^^''^ foi-''^^iS-l't'-i« for 1500 nancs at o'lb francs for M. JdcZT^V^-"" ^'^''^ PfV>f exchange between Canada and Great Britam was i!?4.44^- for £1 ; the lec-al par at present is $4.86.| for £1. The present value «t per cent increase on the old value ? ^ 10. Find the cost of a sixt>-day bill on London for £8000 exchange being- quoted at 8|. ' 192 n '• ARITHMETTC. of exchange is £^J* of 14.44^^ for £1. excLSiny^'.f ' clemand-bill on London for £1500, ^'>!m whf """'?"*' ''^ clemand-bill can be bought for 'IrilUO, when exchange is at 8 '-> ^ 13. When $7300 is paid in Toronto for n bill of ..^r change on Livernool for ^^if^nn i.^ • , ^ *^* ®^' s xjivcipooi 101 d^l500, how IS exchano-e auoted '<> 14. A merchant in Winnipeg owes $4000 in New York- ^^^■:fy7 ^'''^ '' ^ % Pi-emium, but excl Inge on ^y Shir^ ^""^'^ '^^ ^^'"^^ ^^^^^^^« «^ ^ew York j /o pieminm Compare tlie cost of a bill on New Yorh 'hrdebr^' ''"' ^'^"^ ^^""^-^^ ^^^^'^^^^ wh'Ll^ldp'y chlsVforlor?,'^"^' ^"i ^"^ ^^^'^^ f«^ ^2700 was pur- cnased toi .f2G73 ; what was the course of exchange ^ fraYes'fo?\^ori^'^''^ ^^""^'^^ ^'" ^^^^'^^ 294,000 Thk inon • T • ^'f^^' '^" ^^^^"^^'^1 "^tercst of 12)20 Phis loan IS transmitted through London when exchange n London is quoted at 25-30 francs, and sterling exchange IS ^. Jind what rate of interest the coinpan-/ mvs on the money actually received. company pays on 18. The value of an ounce of the gold of which sovor n 1837 hxed m fineness at 900 thousancUhf pure • the mm pnoe of English standard gold (22 caratTflne) is tltl^r .1 - V ?'''' """''«• ^''■"'» '■!'««« fi-cte show tl e tiat of the statement: "By the new par of cxohan"e sterlmg money is worth 9 ' pei- cent, more' tha^by tile cfd EXCHANGE. 193 20. A Glasgow merchant ships to his Montreal agents for sale goods for which he pays £616 in Glasgow; he pays an ad valorem duty of 12 % upon the goods, and a commission of 7 % to his agents for their services. The goods rccolize in Montreal $7800. Find the merchant's net gain, a pound sterling being equal to |4.86. 21. A bank in Toronto remits $10000 to Liverpool as follows : First to Paris, at 6-40 francs per $1 ; thence to Hamburg,_at 185 francs per 100 marcs; thence to Amster- dam, at 171 stivers per marc; thence to Liverpool, at 220 stivers per pound sterling ; how much sterling money will he have in bank at Liverpool, and what will be his gain over direct exchange at 10 % premium ? 22. A commission agent sold goods to the amount of $12500, charging a commission of 21 % ; with the net proceeds he bought a draft at | % discount. Find the face of the draft. 23. When exchange at New York on Paris is 5-16 francs per $1, and at Paris on Hamburg 2 -121 francs per marc banco, what will be the arbitrated price in New York of 11520 marcs banco of Hamburg ? 24. A Toronto merchant owes 1800 francs in Paris, he buys a draft on London when sterling exchange is at 8 • exchange between London and Paris 25-20 francs per £1, What does he pay for the draft ? 25. The par of exchange between Paris and London is 25-2215 francs for £1, and between St. Petersburg and Lon- don 38-177 pence for 1 rouble. Find the par of exchange between Paris and St. Petersburg. 26. The par of exchange between London and New York is $4-866564 for £1, and between London and Am- sterdam 12-1071 florins for £1. Find the par of exchange between New York and Amsterdam. i^^H H ^H') *< 9 K" 1 ^ d ■' r •< '^ 'Kf <■ i m B^'i - 1 m m'^^'im mMMm Kfnm ^9 'iii ' ^1 'I ¥ i I I Ni ij H m !:• Hi J!.;! ft ' '■• flli!!! I ! llJill i I mlillli;:!.: 194 AlUTHMETIG. PROBLEMS IN MENSURATION. I^r convenience of reforencc the rules fnv ^-hf • • measurement of surface., and thT vo u nt o^S? '^' liere expressed genenilly ; ^"^^imes ot solids, are 1. AVJiere a is the measure of the side 'mrl -h r.f ^u . n/3 (s— a) (s— b) (s— c) where 2s = a + b + p on&S-rlre ll f " ^" ^ "— .ti^ntrto that "xe^xsare of the are. of the curved laS^ •ie":^''' "" I'ln^ nicasuie 01 tne circumfereiiopnf fi./:i i.^ ^ ^i leno-f], .i,wi , -i/i '^^® ^^^® measures oftheheie'hf ^- U here h is the measure of tlie hei>hf- nf o ^ t ^ •md A is the measure of the nro-, if ?^i ^. ^y^^^^er, is mea3ured by h x a. ^' ''^ '^' ^^^^^' ^^^ volume ^. Where h is the measure of the heie-hf r^f o ^«- A the measure of the base tbp ".i,!^ .^^'^ P^^^"^ ^na Ji X A. ' *^'® volume is measured by 8. UHiere h 'n the measure of the heie-ht nf . ^ volume is measured by ^ x h x 4 ^^^®' ^^^ cif5r^S^'::-±;'^^i---t-e„fa . • ™« "'^''^''i-o of the „,rea of a oircle is /r= .^^ 1-. Hk. n.c.ure „f „„. area of the surface of'a sphere Vi. The measure of the volume ofn sphere is :' r: --^- Pl!OHI,£MS IN MEXSUBATION. 195 14. The measure of the area of a trapezuim (4 sided figure, having two parallel side.), is !^ ti' . h, where a and in'!:^7liosf7zL':'J::;:i>:' <"-■'- ^^^ '-p-«™ into two 1. A ladder, 30 feet long-, stands niu'ijrht asainst -i wall Imd how tar the bottom of the ladder nLt be pu iec7o t' to lower tlie top C feet. puiiea out dow A fre^h': f *■'"' '™^' '"" r'='^«' ^" ^^ to re^'ch ^ win- dow i4 feet hiffh on one side of a street and from Vi 1 same spot it will reach a window 32 feet hth on the oth r Bide 01 the street. Find the breadth of the sf^et .i. J he radius of a circle is 26 inches • the Dernendic, .'5. The radius of a circle is 4 feet ; from a point 7 feet VVhcit distance will be saved by rutting- rig-lit across in the direction of the diagonal ? iJ^nwicioss m 7. A g-as-jet is 11 feet above the T)avement TTmv i'nr must a man, who is 6 feet 10 inches hi^rsmd^-Ii I so as to cast a shadow 7 ft. long ? of thnw'^;Ji" 'r'^^'^'V*^^ '^*'^^ "^^'^h^ ^^'«^^^«t «q^^^^i-e stick ence i'l2l!i:t.' ""^ '^ "^^ '^^"^ ^^ ^^^ ^^^^ -— fe- 25 m. at the other, and 16 feet long-? 10. A cubic foot of gold is extended by hammering so as to cover an area of 8 acres. Find. (>orrect to 7 places of decimals, its thickness in a, decimal of an inch I 1 1 h I f M II 14 IV 4\^ ^ 196 ARITHMETIC. hill iv^Ti ' ^t ^T ^^"^' 1^ f^^'<^ ^road, and 7 feet offh; fl f ^""^^^ of string- will reach from any corner of the floor to the farthest corner of the ceiling- P breadth' of f'vn ?.'f^'"'' of paving a road of the uniform Dieaatn of 4 >ards around tlie inside of a rectane-ular TeTdth 5f '":;;i ' T ^^"^^i^ '' ^'^^'^^^ '^ ^^ y^^'--^ ^^^^r^'^' '''' '''' '' P^'^^"^ ^---- ^^S -"ts per peJimeTern??/^'''^-^^^' ^« 2 feet. Find the whole poiimeter of its semicircle Finditltdl^s'^' ^"■'"''"* '''' '''^''''''' is 80 inches. 15. Find in feet, to three decimal places, the side of q square containing 2f acres. fni^' ^u"^ *T ^^'^^^ ^^" '^^ isosceles triangle measure 65 feet each, and the base is 50 feet. What is^the altitude ? }J n "^ '^^'\^f ^^^^^ contains |- of an acre. Determine the Ien|th of a side of the field, correct to the nearest inch liiih on tbo "i '"^f ^^1^8- from the top of a precipice 1.30 feet high on t]H3 bank of a river, to the opposite side, is 380 feet long. How wide is the river ? 19. The end of a round stick of timber is 8 feet in diame er. What AviU be the size of the largest s^ua^e stick that can be hewn from it '■> square con^.'i^^ f;;f ^^fe^ular piece of ground is GO yards long and oon.,ainb |- of an acre. It contains a grass plot bordered by a walk 6 feet wide. Find the area of thc^>lot 9^ mo . M ^^l"'l^«;*^''l.ci™mference of the earth is about • ^^Vr;^?^^ ,^^^^ ^^^^y persons can stand in a room measur- ing 1.5 feet by 9 feet, supposing each person to require a space 27 inches by 18 inches. 4uii«a 2.3. The sides of a quadrilateral field are 20, 30 25 3'> chains respectively, and the dingonal joining the fi'rst knd third corm^s is 40 chains. Find its area in square yaMs 2L A ladder standing upright beside a wall 50 feet [ugh, just reaches the top. IL,w far may the foot of the M Iwh '" nf Trf ^^""^ ''^^ ''^'^^' ^^'^^ still reach within 11 inches of the top ? PROBLEMS IN MENSURATION. 197 25. Find the length of a ladder and the width nf th„ street If when one end of the ladder is pLZfo-ainsf^ and when it is turned and placed against t w^l Z. 1' opposite side it reaches a height oTlSfletTnd fhr^ with its first position, a right nfgle '' '""^ '"™^' tivety.^' Fhidi:^.!*''"""'*' ''" ''' ''' '' f-' -P- 27. The diameter of ca carriae-e wheel is 80 h^ohn^ . «« i how many turns the wheel males in trlvenlg one nik . 28. A road runs around a circular shrubbery • the Zp,'- Sro7?h:;:r ''^"^ ^-'^ theinner420Ak Sh^ ; B^ iTi.r :p",iS^s:;K"sr %r\ lene-thenerl hv 1 foof Q ,• /'"'^V*"^8 ^•^>'-' sq. yds. is to be 36 The diagonals of a rhombus are SO nnri ^jk ^„ . spectively. Find its area. """^ ^^ ^^^^ ''^- 198 AUITHMETIC. h ? . I '! I 30. Find the volume of a frustum of a cone, the radii of whose ends are 7 ft. and 12 ft. 4 in. resjjectively, and whose altitude ^"s 9 feet. 40. Find the dimensions of a rectangle containing 240 sq. ft., if its length is to its Avidth as 3 to 2. 41. Suppose that the planet Mercury describes in 88 days a circle nround the sun^ of which the radius is 37000000 miles ; find the number of miles described by the planet in one second. 12. A ladder 42 feet long, placed with its foot 24 feet from a wall, reaches within 3 feet of the top. IIow near the wall must the foot of the ladder be brought that it may reacli the to]) ? 43. An electric light is 18 feet above the ground. What will be the lengtli of the shadow of a man (> feet in height, if he stands 10 feet from the post on which the light is placed ? 44. One side of a quadrilateral field measures 25 rods, the side opposite and parallel to it measures 3(') rods^ and the distance between the two sides is 12 rods. Find the area. 45. Which requires the most fence, a circular field 15 rods in diameter, or a square one whose side is 14 rods ? 46. The depth of water is 7 feet in a circular cistern, the circumference of whoso base is 20 feet. Find the depth of the same quantity of Avater in another cistern, the peri- meter of whose square base is 20 feet. 47. The parallel chords of a circular zone are 12 and 16, and its breaxlth 14. Wliat is the diameter of the circle ? 48. A rectangular garden, is to be cut from a rectangu- lar field so as to contain f of an acre ; one side of the field is taken for one side of the plot, and measures 2| chains. Find the length of the other side. 49. The length of one of the diagonals of a quadrilateral is 40 feet, and the lengths of the perpendiculars on this diagonal from the opposite angles of the quadrilateral are 12 ft. 4 in., and 9-25 ft. respectively. Find the area of the quadrilateral. 50. Find the diagonal of a scpiare whose area is 14 s(]uare inches. proht.ems m me.vsuuation. 199 eaeh;ile''i:'"80 "'' " •■''<:»^''«'"'- I'-'vinjf 8 squares alon-- f„ , -.i; ,'i;'"*''"''"' "'■•'' is ■*« ™lsin length and SO v<1, m;^ Klth. F.„d in feet the side of a sq„a.^ fie'd'rfe <■ '^i hi^lt' ^VhTi'^.r'*^ r!"°"^' 1« *■"«' ^'1e, and 12 feet fn tit ^^^'"V "'« distance from one of the lower corned to tlie opposite upper corner ? oointis 55. What is the volume of -i fmon,™ „<■ iTr>j/i +i.^..-i » *y'""^^ 01 .1 iiustum or a souare nvr. "irwh-it is';f"f ""r '^"""^'-'^ -oi'utfon '?' '"" " .X. Hhatistlie leng-th of a side of the srcatest enh^ 4 in ^.'''f^fiS-ti'ar court measures 21 ft. C in, by 1.3 ft squa;.e'fle"d 0, tai U^'slx '::;:r"nd"?hr"""'"- "'■ '' is in "th^ f ^. !■"'"''"" "^ ^''"'°"' of ^™'er in a tan k " hieli of^c|r::i,^-!^— -- -- - Penmete. block oeeupii LUsq.'nchef "^ "' '""^' '"PP"^'"§^ *'''''' tlie d\^meIc'"c" v.fv'''r ''^'''''"'" ^"' '^ «'^'«™ «0"toi". deep ? '" ^'''''' '^ ** ''«'^«' ^"d which is 4 feet \ f! 200 ARITHMETIC. '?' J ■ j|d i. li' i: w ■ : ': '{'■ iiji f ■ '. ll ir. i e h I. ■ i';J li- Ilif H- 65. How fast must the water rise in a well whose di. ameter is 7 feet, so tiiat it may remain the same depth when a pumj) is em^Jtying it at tlie rate of j- of a ton of water per lioiir ? 66. If we eonsider the earth a sphere whose diameter is 8000 miles, iind tlie scale in miles per inch hy which its surface must ])e represented on a globe whose diameter is 1 foot. 07. Find the numlier of cords of wood in a cylindrical stick of timber, the length being 40 feet and the circum- ference 22 feet. 68. Find the area of a triangle whose sides are 41*26, 39-4 and 21) -2 yards respectively. 69. Find the area of the uniform circular walk 2 yards wide, surrounding a circular pond which contains 2J acres. 70. Find the whole surface of a hemispherical bowl whose inner diameter is 4 inches, and outer 6 inches. 71. Compare the volumes of a right circular cone and a cylinder of the same altitude and base. 72. A tree breaks off'a certain distance from the ground, and has its top resting on the ground, so that the two parts of the tree and the ground form a right angled triangle. The height of the stump is 12 feet, and the dis- tance of the top from the butt is 34 feet. Find the total length of the tree. 73. A circular pond which is 4 miles in diameter has a driveway around it. A man wishing to reach a point directly across the pond from where he stands, can drive at the rate of ten miles an hour and row at the rate of 6 miles an hour. Which is the speedier way ? 74. A rectangular field containing 3 acres, is surrounded by a road of the uniform width of 66 feet, the total area of the road being 3 acres 36 sq. rods. Find the length and width of the field. 75. A side of a square field is 36 rods. Find a side of a square field 3 times as large as it. 76. Iron being nearly 8 times as heavy as an equal volume of water, find the weight of a solid sphere of iron whose radius is 3 inches. PROBLEMS IN MENSURATION. 201 77. Find tlie cdfro of a cube which is 5 times the volume of a cube wiiose edge is 2 inches. voiumc ,. /^; I^'^""^- ^'^^'^ ^*'^ ^^''^ ^« climinislied by I of itself Bv wliat miction of itself is the volume diminished" ^ in theS^fVto r''"'" ''*"' 'P'"'^^ whose radii are zone'is'^(!?frr'''7^^ '^?^^ '' ^''^ ^'' ' ^^'« '>»-«'^dth of a Find the otht: '"' '^" ^^' ^'^ ^'^^''-^"^^ ^^^^^-^^ ''^ f-^^. 81. A house is 60 feet long and 25 feet broad Tr, o wbi; ^- '' *^'^ ^^Perfic'"'-^! area of the outside of a box whose dimensions are 9, 10, 7| feet respectively ? i. qoi T^^^^'!^^ ^f \ triangular field, containing 1 acre IS 90J yards m length. What is the altitude ? ^ ' «5. A room is 24 feet by 20 feet; in the central nnrf i« a carpet which measures 21 feet by 17 feet FM the co o^TL how many square flower beds 4 feet to the side can be arranged m a square plot whose side is 7 feet ' 4u incnes. J^ md the difference in their areas, if the sides iJ^ £,f *a"^le are in the ratio 1 to 3. ' heavy as' thnt^nf '^ "'"'' ^i ^''?^ *^ ^^ ^^^^* ^-7 times as neavy as that of an equal volume of water Find thp weight of a rectangular box, without a lid, full of water f the outer dimensions of the box are 4, 5 6 feet 7esDec lo^'r^u^ '^'^ "*^^ ^^ 1 "^^h i^ thickness. ' ^ ' Find fhP nnT' V^* '\ '"^"^^'^ ^^^^' ""^ ^12 an acre, is $132.24 yard. ^''"'"^ ^ ^'""^ '^^'^^^^ ^^ at 35 cents t 90 The sides of a triangle are 13, 14 and 15 feet re 202 AIUTHMKTIC. •i" 1 ■ , ... ' '1 « f 1' , I 92. Find the cost of pnviii^- a nmd ol'tlHi mi i form breadth of 4 yards around tin; inside ol' a reetan^niiar piece of ground, tiici length of wliich is 85 yards and the breadtli 50 yards, tlnj cost of paving ])eing 2U cents a square yard. •J.'}. How many plnnks, 2 inclies tliiclv, can l)e sawed from a log 10 feet in circumference, allowing I of an inch for Gcach saw cut, and 2 slabs, each at least 5 inches thick, to 1)0 cast aside ? l)t. An elm tree partinlly broke off, and fell with its top on the ground. If th(5 height of the brenk is 15 feet above the ground, and tlui distance from the foot of the tree to the point where the top reaches the ground is (W; feet ; what was the entire height of the tree before it fell ? •.>5. How many square yards of canvas Avill be required to make a conical tent \) feet high nnd having a base of 4 feet radius, no allowance being made for seams? 06. How many gallons of water will a circular vat con- tain that measures 12 feet across the bottom, 15 feet across the top, and 6 feet deep? 97. How many yards of paper 27 inches wide are re- quired to paper the wnlls of a room 18 feet long, 12 feet wide and 1 1 feet high ? 98. A square and a rectangular field have the same perimeter, 100 yards. The length of the rectangular field IS 4 times its width. Which contains the greater area, and how much ? 99. If it costs $448 to fence a square field at $4.40 a rod, what would it cost to fence the same amount of land in the form of a rectangle whose sides are in the ratio of 9 to 10? 100. Find the width of a rectangular field which con- tains 12 acres, and which measures 24 chains in length. 101. The radius of a circle is 8 feet. Find the circum- ference of another circle of I the area. 102. The area of a triangle is 3i square metres, and its altitude 2| linear metres. What is the length of- the base in feet ? 103. The sides of a triangle measure 14, 12, 9 chains, respectively. W^hat is its area in square yards ? iMtonr.KMs in MEN.sLiuArroN. 203 area of the oc^ou y*"'''" "^^°" '** "'"'••'')• ^hat is the ner on the top L , t " v " '" "'" "'''"'""' "I'P"^"'^ «"" circunif,.,.' „ce. * ""-' '''''™ '» •^•'' '■>^'"^- J"'"" the outer pattern 'cZ-y??; '?[;!' '"' P'^""'.'" ""="- -W«. -it" a wire costs 3 cents al'mi ' 'J ^'arlicd wire fence, if the ice" inel T'tJ ;,":;r'"\f"i T'"'' '^ ^T™ "^^ -"" wide, and 8 fc^t d ".ff '^'"^ "" "'"'"' *° f"" '«"?■ ^2 ftet widf aromf,,'!','' ''°'! f, ^"^'"'^ ^i«' asphalt a wallc 6 feet yds and ?^ff ^f ',""''' '""«^' '■■'^^ '^^ »^«™ffe width of 7 contain 1 /'If,'"'''' '""/•'^° windows, 90 of which severally anSt he mn^Sr,:' T"' '''"■"-" ? '"^"^'^ "^ ^^ inched each mn„T- 5 .^mrtows seyei-ally contain 12 panes ^a rtbe wlXVt">J"" Y '■* ""■'"^- '"'"" *'-■ -^ of 1 1 li p- J !. "■* --' '^onts per square foot. iio. J'lna thesidoof asf|iiiii-e which is poih\-t1(.„' ,-, area 10 a eircio i„i,^.„ „• /, ' "'\" "> equivalent in circle, whose eireumference is 55 inches. 14 m> m I 204 A KITH. ME nc IK). A square field eontains 121)0 wq. yds. Find the area of a reetan^nilar Held wliost^ leiift-th is double its breadth, ai»d which has a ijerimeter equal to that of the square lield. 117. If the pressure of 15 pounds to tlu^ square inch be applied to a circular plate 2 feet in diameter, find the ,otal j)ressurc. 118. The inner diameter of a circular Imildinj*- is 112 feet, and the thickness of tlu; wall is 22 inch(!s. Find how many scjuare feet of ground the base occupies. 111). Find the radius of a circle which is e(piivaleiib in area to a siiuare, the side of which is 40 inches. 120. Find the area in acres of a qujidrilateral, whose dia^-onal is 80 chains, and the perpenuvular from the opposite ang-les to it 29 chains and 28 chains respectively. 121. Find the area of the sector of a circle whose radius is 60 yards, the arc of the sector being- 280 yards in length. 122. What is the volume of a sphere whose surface is 616 sciuare inches? 123. How many pails of water may be contained in a cylindrical cistern 4 feet deep, having a diameter of 6 feet, if each pall contains 12 quarts ? 124. Find the difference in the perimeters of two fields, each containing 2^ acres, one being square and the other rectangular, its length being 4 times its width. 125. A gardener wishes to lay out a flower bed. He first lays out a square, 4 feet to the edge, then with a radius of 2 feet, and the 4 corners as centres, he describes 4 equal circles. The part of the square enclosed by the arcs of the circles he makes into a flower bed. Find its area and its perimeter. 126. A string is wound around a squir^ block, edge 2 inches. Supposing the tiiag to be unwound in the plane in which it is now situated, and always kept stretched find the length of the curve, which may be traced by the extremity of the string in one complete revolution. 127. In the preceding examj^le, find the length in two complete revolutions. 128. Also, in 21 revolutions. •MtoHLKMS IN MENSIJUATION. 205 l-i>, Three vnni\\ clrchN of ivwiin^ u i* ^ on., another exteVa.! V Fi,", , ' ' 1 fli. H,"""'' '""'''' cloHca by .1,0 arcs boUvoon th' ' ,, , ^' ^ „ " »'«« "'- H rope 4oi:.uo;',i;'::;;:,r" '' ™"' "''"^" ■■' '«""-• -''" 133. > 111(1 the cost of cnrnptiTur ^.v;fi...,V • • " out*i„y ;:i™,:;',: "" <■•'''•'''=' "■"•"%' '«"g".wise, and ii;';.: 134. The area of a sector of ;i circle is ooq „^ ^. ,, Il;^.^n,a,,v .,„a.-o .cct of tin' c.oc, t„c X^^'o^tJ,' ^S in.!™;r'^ ff^^rcleno.- ]»ys out a flower bed in tlie following "e sidr. , i'--'' T =' '""•■"■"■' ^^'"«»<' «W« i« « foot on .vhoicpiot enclosed, ai.o, iind'tb-e p^e,'!; e of 'f "' ^"^ 137. The sides of a triaii«-le are 1". ii i r, ; ' , are tsei^^::^ ^n:s^^<;:;s'r ^^?;i^?^^' ?-^- the space enclosed by it ' ""' Pennieter and the zone between the chords.' "'"' ^'■'''' "'' .■I h it h tf 1^ ! m m t S ' ti 206 ARITHMETIC. unlfom1.?c^^^^^^^^^ % surrounded by a road of unitoim bieadth, the inner side of the road measuriuir 3bOyds. m circumference, and the outer side 490 Whif quantity of ground does the road cover 5> * 1 U If the diameter of the eartli is 8000 miles wh-it "> ''S?h7' t^'""'7'^^^' ^'^ ^^^^ times'sbulir^' 11^. i^ind the volume of the greatest sphere wliich can 1^ ^^t^c^^'^f '^"'^^ ^^''^^^^^ ^^^^^^^ ^^-'- 1 ^.i. h md the solid contents of a rubber ball 4 inches in tern in ■ ho.^r^ '^^ 11'!"' circumference, can liil a cis- tcin in . liouis, m what time can a pipe, 4^ inches in cir cumference, iill it? i t , -^^ 'Ji'-ncs m cii- tie outside Of a square court, containing 11)6 so yds at 20 cents a scjuare foot. ^ ^ ^ ' '^^ 14(>. The radius of a circle is 12 feet- twn r^avoii^i chords are drawn o„ oppcsite »Hes of .h cen ^ o'^fe "b Se"y do'" ''.""? h" -^ "*-"<' "'■ «"• -'^l tl.e' other an '/.''^ / "«'fl is "'n thousand times as larjrc as the nln>i whrch has been ma0. The area of a sector is 115 so. in the -wv^ nf .-ho CH^le is 275 sq. in. Find the arc of The sect'?r. '''" ^ l.)l. What IS the depth of a gallon pail that is r> inches .'.cross, the sides being U].right? 152 A circle and an e({uilatcral triangle have ecRuil perimeters, each being 3G inches ; find the di.Terenci of their areas. 153. PROBLEMS LV MEXSUIIATION. 207 Stands c^^r^i^jiL^;;;?::,^^^^ -f^ which the space occupied by The moa TfL f ''''' •' '^''' ''''''' ^*' of it is 71)6 yards ^ ' "" "''"'^' cn-cumference 10b. Also in 4.', north Jatitude. are draM-n off. "^^ ^* -^^ gallons n.aterh.1. ' ' *''" """"'^«'' "* ^"W" "iclies of the ofaneq„i,.tenatriJ,;,e;.id si K:";7h'"f,,';°'-"';'- then wound around the ti-i-ino-ln k„ , "'""*>' "* unwound. Find the area of th >T, ' '^, ''"'P' *''"'- "1"^" and the line i, ikcdomhvf .""*="'"''''>■'''« ^'"'"^^ one con,plete rc"ol«ion ^ ' """ttached extren,ity ir, tenodtJr^C^ro trSXd fo*'r .r ^^■"'^" '^ '■- of a side, find the 1^;! '' '" "'" ""''<1'« i'""'' What is its whoIe"s„pe,,ici;.l a°;,,V™' ' '^'''' ---P-tivel.. of the^o i-^dii if th f . f , '!• ^ '"" «'■« "le len'^f.hs tl.a:n t,,e in- ;^lf ;,!',r;^;,;:-t^:,',-f - ^'^ -vomtions more 208 ARITHMETIC, h: ! 1<;<>. At what (listMiicc from the top must a cone, 14 in- tiie tAvo jjarts may be equal ? 1 07. A vessel in the form of a right circular cylinder yJ^J^'VI^^"'^'"''^ "^ ^ ^^'^'^"^' ^^"^^ ^^^« ^^P^l^ of the e^ ^^.^d ;;: SI '' ''' '"^"'' '' ^^ '^^"^^^^ ^^ ^^^^ 1G8 Find the number of cubic yards of earth dug" out to make a tunnel 80 yards long, whose section is Tsem circl.-, witli a radius of 10 feet. to..^*!!; /i'^'p "'''"•^' P'''"'^' ^^ '^'^'"^^'' t of '^" "ich in dianfe- tei and i of an jncli thick, can be coined from material in the lorni of a cube, .vhose edge is 3 inches ? ^yhU^; l''f '^'""^ 'f .^ ."^^ '' ^"'''^^^^^ m^ of 11 inch bore, ^vhich IS I- inch thick and 20 feet long, at" 8 cents pei' pound, supposing that a cubic foot of lead weighs 11500 171 The altitude of an equilateral triangle is 14 feet Find the length of a side and the area of the triangle. 172. A ladder, wliose foot rests in a given positfon iust about its footjust reaches a window on the other side If the two positions of the ladder be at right ang-les to enel, other, and the heights of the windows be 36 and 27 fee IheSl?' '"'^ ''' ''^^'' ''''''' ''''''' '''^ ''''' ^^"^"^^' 173 The driving wheel of a locomotive-engine of diameter 7 feet, makes li revolutions in 1 second ^^nd fio\lt ^7^7 "'?"' '^ '''"'" ^^^'^ necessary to make a enTJ r .!''^^ ''"^ ^- '^^^^^'^ described about it, when eacli side of the square is 4 inches ? 1*5. The large hand of a clock is 11 feet long. How many yards per day does its extremitv travel '^ 1/0. l^indtiic internal d<>pth of a cubical box which will exactly hold 3 gallons. ' ^"^^^ 177. A soyei-eign is ^ of an inch in diameter, and J- of ind foi'iiK'd int I cube, find an iH\i>;v of the cul )e PROBLEMS IN MENSURATION. 209 178. Find how ]n/uiy ^-mIIohs a,ro contained in a vessel which ,s in the form of a ri^ht circular cone, the Iv' Has of the Imse being 8 feet, and the shmt side 12 feet 17.). 1 lie g-reat pyramid of Eg-ypt was 481 feet in heie-ht ^""7^-; "f '^'^-^^^-^ - -mare 764 ffe t7n length. J^md its volume in cubic yards 180 A solid is in the form of a right circular cylirder with hemispherical ends; the extreme length is 42 i'e i,^ and the diameter is 5 inches. Find the volume 181. i^iiid the weight of gunpowder required to fill -, hollow sphere 9 inches in diameter, supposing That 30 cubic inches of gunpowder weigh one pound inche^; .^!''^ f'^f'^f Jhe base of a cylindrical vessel is 14 inches a block of stone is placed in the vessel and is covered with water; on removing the block the e^l of the water sinks 4 inches. Find the weight of the block o :r;.:^r?;'" '' '- ' ^^-^^ ^^ ^--^ - - ^^' 183. A Avater-wheel, whose diameter is 14 feet makes 50 revolutions per minute. Find, approximate! v the number of miles per hour traversed by V point oi the cr cumference of the wheel. i^ uu cue cii- 184 A locomotive, running at the rate of 35 miles ner LToAd l' ^fr;"^i"'"'^ which makes 4 v.^olution: if" second. ]< md the diameter of the wheel Alexn'.^-" 'J'^'r ""^ ^^''}l'^y'' P^"^^^'' ^^^^^ch is situated near Alexandria, m Lgypt, is a single stone of granite Tl o height is .)0 feet, the diameter a"t one end is 60 inches ^d at the other end 7 ft. 6 in. Find the volume. ' '^ 18b. A cask full of water weighs 480 lbs. ; the cask vh^n eanpty weighs 31 lbs. Find "the number of galTo ^ the cask will hold. b'^iJons 187. Find the area of the whole surface of a pyramid on a square ba.e, having its other foces equal ; e^ihl^ o'^' _ 188. The surface of a sphere is equal to that of a rio-hf circular cylinder ; the radius of thJ base of the cylinder k^ * I —t- iii^l sBsmmmm 210 ARTTHMETTC. 18!). The volume of a spliere is cciual to that of the ri-ht circuhir cylinder in the preceding- example. Find the surface of tlie si)here. li)0. A square enclosure has a side 40 feet in length, m front of It, and at a distance of 40 feet from each of the two nearest corners, a cow is tethered. If the tether rope 1=. 100 feet in ength, and tlie cow is unable to enter the enclosure hnd the area of the ground from which the cow will be able to procure the grass. 191. How many times larger than the earth is the sun If they are considered splieres, the radius of tlie earth bein^ 3950 miles, and that of tlie sun 441,500 miles '^ 192. How many gallons of wine are contained in a cask weighing 450 pounds, if the cask when empty weighs 20 lbs., and a cubic foot of wine 995 ounces? " 193. If the earth be regarded as a sphere 7912 miles m diameter, and the moon a sphere 2160 miles in diame- ter, hoAv many times the bulk of tlie moon is the earth? F.- f ;/ f T*''' ^T*'*- '^"'- ^^"^''' '''''^ 13 ft. 4 in. Avide. i^ md the cost of covermg it with sheet lead -i of an inch thick, suppo^^mg that a cubic inch of lead weighs 6-5 ounces, and that 1 pound of it costs 7 cents tJX^' VT ^^^i^l^^^V"!''"''^'^^ ^^* planking It inches thick. If the external dimensions be 3 ft. 4 in 2"'ft 6 in and 1 ft. 8 in, respectively, how many square feet of plank- ing does it contain ? ^ 196. A telegraph wire is 70 kilometres long, and 2-1- millimetres in diameter. I^ind the volume in cubic deci- metres. 197. The top of a flag staff being broken off by a blast ot wind, struck the ground at a distance of 15 feet from the foot of the pole, Find the height of the wlioh^ flag-staff supposing the length of the broken piece to be 39 feet 198. The area of the coal field of Soutli Wales is 1000 square miles, and the average tliickness of the coal is 60 feet. If a cubic yard of coal weigh 2200 pounds, and tlie annual consumption of coal in Great Britain be 70 000 000 tons, find the number of years for which this cial field alone would supply Great Britain with coal at the present rate of consumption. PROBLEMS IX MENSITRATION. 211 5 Sufof u,t ."r ;:s:l %' 'r;' -'-; "•"->• »-. in being 12 ft. 4 in dcon ,nfl ifl. , •"■ ""'"''■' «'e stream ameter, each payh^te six nS r/^' "' "'"'"'' « -J'" agree to grind f]o,vn h „• ' ' ™ *'^'' expense. Tlie v ameter, uselo,™ flm] thf -^ *''" •""'""''' ^ "'«''« i" di- their lowe'S,S:tthrs™r'''f'''- T 'r"' ^'•°»"''' -i* of the two cxtitme pole "re ^'^^l^T' '"'''« "^'-"^t. the former is 20 tvTZl Lfie^f ^f ^'"'^- '^''« '°P "f tlie top of the inW lie no e -i" * f ^'''^f -^ ''''«'' *■'•«>» 12 feet distant t Vo the M,i'' 1 ''° ""''"'« I'ole stands horizontally. W "t tS"' h "i''^"""' '"'"• '"^asured the distance between the to' ''f^*^ ''?'' ""*"<^ I»'«. ''"d totai;„;i?;^!,r^ea mt bo^?f '' •'"^ '" "'*'• "^-' -■'>' answer in inches ? '^ '"""'•' ^'"^*- expressing tlie 203. A vessel when empty weio-hs 1-Ti i-ii., and when full of water we o-l,r7 q i -T '^''«STainmes, the capacity of the vessoHn fii • ! '^.■'<'S-'-'"»»ies- Find 204. If imn k 7 7^^ ' '" f °^"<> decimetres. water, fin Uhe r JdluTrfV r4?""^'/' t" <^"""' ™'"™<^ ^r 205: Find the n "mro of " 1°"'"^ ''*'":'<^"' »'">*• ".easuresofwhose"sMe":r:;:ta:d^S <"• ^ »'--"^'«. «'« t-"s pntp fo!: mf,d!;"|','pij- "a tv™'*'! ""'"-•'^ "^-p. -"- lost in drying and o^ f-fnV ^ P<"'-centage of the pulp is 2 feet 6 inche?'wwe L^ 004 ■''T'' 7™° y«'rds lol.g, What per cent, of 'J^'e X '"t lojtt .^liy^ ? "^'^»^^- >«,jm \ I . It 212 ARITHMETIC. GENERAL PROBLEMS. 1. A can do ns much work in 9 days as B can do in 10, hut A works only 10 days for every 11 B works. How should 5;\)1).70 be divided for Avork, under tliese conditions ? 2. A man insures a house, worth $4000, for | of its value^ at 2% premium. If the liouse be destroyed, find tlie total loss sustained by the owner after one premium has l)een paid. o. AVIiat pi'incipal will amount, at simple interest, to $373.75 in 2h years at G% per annum ? 4. Three men form a partnership, contributing* $3200, $4000, and $1500, respectively. How should a gain of I52G.50 be divided ? 5. I mix 3 lbs. of tea, worth 40 cents a lb., with 5 lbs. of tea, worth 48 cents a 11). At Avhat price per lb. should I sell the mixture to make a clear gain of 8% ? (). If 3% stock is selling at 84, what should be the price of 7% stock, to realize the same interest on the money in- vested ? 7. Find an agent's charges, who sells 3500 bushels of wheat, at 85 cents a bushel, on a commission of 4J%. 8. A merchant buys 4000 yards of carpet in England, at 4s. Gd. per yard. Find its value in Canadian currency, exchange being £1 = $4.87. 9. Find the cost of painting a floor, IG ft. in length and 12 ft. 3 in. in width, at 18 cents per square yard. 10. Divide $201 between A and B, so that A may have $12 more than half of what B gets. 11. Two men start together to go in the same direction around a circular track. The first goes 5 yards while the second goes 4^. Where will they first be together? 12. I sold a book for 78 cents, gaining 12i%. Find the cost. 13. When, first after 4 o'clock, will the minute hand be midway between the figure IV and the hour hand ? 14. Two men working together can perform a piece of work in 18 days. If tlu; job is worth $12G, and one of the men works 5 days less than the othei', how should the money be divided ? OENERAL PROBLEMS. glS turc worth 57^" eent'apouml. '"'"''''' '" '' '» '»™ -""x- cents at 85. t^scjiig .ssoioo in the U per annum. *^""' ""'"'^y being worth 8% per fl. Find the cost of .^.wh^ a fi^d 40^1' ';""" ' rods wide, witli wheat at 70 eonf. i v. ?''' '°"8' '""t" •''•'^ bushels to sow an acre *'""'""' '^ " ^k"* U Sule^';,;': lil^t^'^^J'- - *-* A may have Cin lOciays, and C tnd 1' i,a2 d Tvs Pi'f'''^?' ^ '""• 20%Vand on ihfo'ttrTlost'C '"h ' °" *7"« rS'^i"«l or lose on ti,e w^hole trlntetloiv ''"" "'"'^" ""' ' 8-" to ^152 07'i:;"2 ™arf' ""'^'■^^' '^ ^'""'^ ^l^^ win amonnt ^^^^27. Reduce -714285 to a vulgar fraction in its lowest are A and B worth^'^Ile^tu'SyV^'"'^" " *^^«*- ^^''-t p.-«ee^'^i;:\:i^:;;tl,^ 'aftcH^'r^ «B^2, «s the i"^^'.v!:^i;l^^: -^^.;;[^ -1^ - -. -a. .te Of 214 ARITHMETIC. Mr i. r -I; rfi'fii; I'v ^,5T^'* '"'''"■' ^^ ^^^- ^^ ^®^' ^^'^''^'^ ^'^ cents a pound, with ^U ll)s. of tea, worth 35 cents a pound. At what price per 11). must he sell the mixture so as to iraln 207 on his outlay ? ^° OA^; ^^^^^' "^^^^^ ^^^P*'^ ^*^ wasted in carpeting a room, JOtt long and lift, wide, with carpet (running length- wise) f y^l- "^^ide, and having a complete pattern every 8 It r "^ 3 A can do twice as much work in a day as B, but he works only ;| of the time. How should $22, which they receive for their work, be divided ? 4 When, first after 3 o'clock, will the hour hand of a clock be midway between the figure III and the minute hand .^ 5, A and B agree to pay their travelling expenses in ?S gj;?!'''''^^^" ^^' 2 to •^- -^ P'^ys on the whole $104, and B ^20(>. What has one to pay the other to settle the ac- count ? in?; 7^^of ^ii'^« of the selling price of certain goods is lU/ less than cost. Find the gain per cent, at which the goods were sold. 7. In what time will 8212 amount to $245.39 at 7°/ i)er annum ? > /o t 8 Find the sum i-ealized on the sale of $3000 stock at 8G, brokerage at J %. 9. Divide $4669 among 3 men, in the proportion of 5 7, and 11. ' l^- /"^^ the true discount off $422.50 due in 9 months ^^ *->/o per annum. 11. What is the cost of insuring property, worth $700, at 2/^, so that in event of loss the owner may receive back the value of the property and the premium "paid ^ 12. Divide $31.50 among A, B, and C, so that B may have one-third as much again as A, and C one-fourth as inuch again as A and B together. 13. Find the cost of fencing a 2 A acre square field at 50 cents a rod. 14. Find the net income of a man whose total income is moo, on $500 of which he pays 18 mills on the dollar taxation. GENERAL ruoiU^EMS. 215 16. Simplify -3^-^ X -002 X 1 -8 •0001) X -038"" at 5y;!:;!l|;;;r'r'' ^"" — " «« «^63.17J- i„ 4 yea.. broke,-a;;7%T" "'"^' " "'^" I^''^" f"' »4000 stock at 87, mfjaf.vst:rk''S?4':'-^r ^''^ "°"-™'"''»- -1. 1 insure a house for =} of its value '.f '-W m^h if .t them l1 1* 'rf eosf T^f--*'''' '""' "f'«'- '^««l""*^ them for $48 each How n t^-'."/; «'«ek oaeh, he sold transaction ? ^""^ "'"'"' ""' "« S^'"" °'' l«s« by the Wife; :it,^ tt^r rl 3^tftj?bl;s,fc° "IJit"'""" '^"" '^'^ bushels of oats to sow one acre ' " " ''"''""■'^'* ^ wMt3Mfr:;rtrrot!lr- ^n*^"- r'-'" ^^ -^-'^ ^^ "'•. must he « . i fi • "^'''^ ^ "^- ^* wliat price per lb If p."","'« "iixture in order to ™in 107 * long U ft wMe°:ncfl?fr M^ '^'=- """^ "'■'^" ™°"' ""• dows 31 w K ft V '"f ^"^'^' '" ''''"ch are two win- inTes',^i^(r^-'rsT;r' ' "' '''■• ^"" p^p- ^^^ at '.fmSeS'" ^.^ ^i/Jlf ^';- -e «rst, put out 3 vears of f^°/ ^ / T^, ■^^"''^^ ^^ 'Vc^ equals the second for 97 A t f ' ^'i"'^''' *'^® ^hi^d for 4 years at 4°/ 27. A bankrupt owes A $4000 nnVl R ^Tm' i i- assets arc ?921.25. How slSn'Lelt'd'ili; "',''" bushel, and sold it at 77 eents a^ushel. lose on every dollar he paid ? How much did he iSXti. 216 ARITHMETIC. 1. A and B can do a piece of work in 5 days, A could do it alono in 8 days. How long wcMikl it" take B to finish the work if both togetlier liad w^orked at it for o days ? 2. A grocer who throws off 5% for cash, sold the follow- ing : 3 lbs. raisins, at 8 cents a ponnd ; 4J lbs. tea., at 50 cents a pound ; 4 cans of salmon, at i\'> cents a can ; 8 lbs. of sugar, at ill cents a pound ; 5i lbs. of butter, at 20 cents a pound. Find his cash receipts. 8. I sold a horse for $UyH, and lost 15% by the transac- tion. Find the cost price of the horse. 4. A and B engage in trade ; A puts into the business $400 for 6 months, and B puts in .^300 ibr 7 months. How should a net gain of |450 be divided ? 5. Find the income arising from investing $3370 in the 3 per cents, at 72. 6. A sold $2500 worth of goods to B, at a, profit of 6%, and B sold them to C, at a loss of 5%. Find what C paid for the goods. 7. A certain sum of money in 3 years at 4% per an- num amounts to $33(;. Find what it will amount to in 4 years at the same rate. 8. Find the cost of gravelling a walk, I yard in Avidth, around the inside of a 2| acre square field, at 15 cents a square yard. 9. A, B, and C do a piece of work, and are paid $42 for it. The money is divided according to the efficiency and time each worked ; A's efficiency is to B's as 2 to 3, B's to C's as 4 to 5 ; A w^orked 6 days, B 7 days, and C 8 days. How should the money be divided ? 10. Divide $369 between A and B, so that A may have $4.20 more than 14% of B's share. 11. A merchant sold an article at a loss of 8%, but had he sold it for $1.05 more, he would have gained 7%. What price did he sell it at ? 12. A man divided a farm among 3 sons ; to the first he gave 110 acres, to the second ^ of the whole, and to the third If as much as to both the others. How many acres did the farm contain? (iKNERAL I'HOHLEM.S. 217 14% of the cost ? "'"^ " '•"' ''■■"' ''« ff'"'ne'l .veil; a^;ti';,Tt,rcS sateif r '- ' lat on What had it at th. h,.^;;::™;/^ t hf t/n c' .' '"''"• of mro 9 Zn hfrc'jori: T "'"•'"/-■ '° W '' ^'"" 3615 iw.4'u;.s^^^^^^^^ Required his loss per cent '^ "■" ""' ?''"=«• w|p:^^^xs^^i-=?v-*-- j— t^to IMS in ail, how .any has'h'e T::^ ItZ . 24. A and B start toe-etlicr to trnv^i i^ y-i tion. A travels at the^rate o 7 Ss pe hour '".fr?" ways remain I as iar ahe^id nf R „„ n u ' ""'' "'" What is B's rat! of traveling? '"' ^ ''•"' *''''^^"«<'. 25. A speculator received a stoolc dividend of ^v „i- ,, increased the number of his shares to I'W^lOm n many shares had he at first ? ^''^ •'• "'"^ 218 AlUTIIMETIC II if 1. A (Iruff^'ist «-Jvos a Ih. troy of ccrtnln pxMls instead of a II). avoirdupois. Wliat iH Lis pu'n nud tlu. cuHtonier'g loss por cent. ? 2. Find tlio cost of carpotin^- a room 20 ft. lonfi-. 17 ft. r. in. m widtli, witli cnrpct 2 ft. wide, at ll.f.O a yard, tlic carpet running, lenfrtlnviso, and witl.oiit a pattern. .J. lund tlie cost of fencing- a, farm in the form of a rcctan«-le, avIioso sides are in tlie ratio of 4 to 5, and which contains 800 aeres, at U.'Mh a rod. 4. A and l| work ('> and'? lioiirs respectively for one- day and receive the same wa«-es ; on tlie second' day thev work 7 and 8 hours respectively; they receive forthei' whole work .^18. 10. How should it he divided ? o Divido 1248 earnin^^s between A and li, when A works only 4 hrs to every ('. hrs. B works, but is able to do as much in 5 days as Ji cm do in 8 days <;. A ffrocer, by selling- 5 lbs. of tea for a'certain price, piined 12% ; afterwards he increased the T)rice. ffivinff only U lbs. for the sann. money. What per cent, did he make at the mcreased price ? 7. A merchant lost 25% of certain perishable ^wls and sold the remninder at a g-ain of 30%. What -Avas his ffain or loss per cent. ? /o ^ ma 8. Simplify - 0004 X -O OOni x -O;] •0008 X ;{^i !>. A builder ])ays 3 times as much for material as for abor ; had he paid 10% more for material, and 6% less for labor, his contract would have cost him m'>:^7 92 What was his contract price ? * * loTci Js """^ ^'''' ^^"^^'' ""^ " '''^'' ""^ '' '*^"'"''' ^'''^'^ containing 11. T invest ecpial sums of juoney in the 4 per cents at 87, and m the 5 per cents at 102, and realize $27 a year mor9 income from the latter than from the former. How much do T invest in each ? 12. The hour, minute, and second hands of a watch re- vove around the same centiv. AVhen, first after 7 o'clock will the minute lutnd be midway between the othel- I!' i OEN'EUAL I'HoiU.EMs. 210 "months. Ilow shou n^^iT^../ i ' ^T " ^^'^^ ^^' -^ them ? ^ •^^^- I^'^'»ts be (lividod between or'itA:^:^:!:^:;^^^^ w so that m ..0 Pi-eniium of 2°/ mic fbi. bL, ''"'"?,?f the J,ouse .ml the vaiiced? "^'^'^^t^s he charging for the money al con^ii!^;:;;'i5:i;::;f^'' ^^ ^^^^ ^^^-^^"^^^ of a.,uare'fieia whi; t^:^:Ti:]^:^:^ ---y ^^ 3% stoek at 72. 18. A merchant imnnrf ^''^^ "'"^"^^ invested ? them for $21)25, losing 107 t ere ^ ^ f"^ "^ ^^^^ voice price? ^ ^° tneieby. What was the in- th 'm .u aZr l^"t:":X^ " '»' "'• ^oocU, sens J of "■ainder at the increase 1 .^^ "' ''y selling, tlie re- whole transaction^ " '"" "''J' «^"i» «% on the b«fo;c^t&rir:rt': "ret ,''■''' '^v'^ ''■^"''"■-. on the dollar How n,„^ ' ° ""^ '™''"' 0"'y 50 cents 2. A,,,,- n"?ci,ri^s«r or^"''^''='>-^ *!• 2J a gallon, 12 gallons worth ^t9^/^ ^r'"" "'«'■* gallons worth !{!1.50 a inllon ,,Mti, ic ?, '^ ^■''"°"' '"'"'' " sells the mixture at $fa ^ '„o, f ,?•':? '""^ "^ '^•«"-'-. and 22. A bov engao-erultlf . f '^ '"' S'"'" l*'' cent, suit of clothes. '^HeTo.ryei t h''"'^"' f ^''" ''''■' ^^'^ ■•^™l a entitled to $25 Z the suft Wif "."^ °' " ™""*^' ="><3 '^ snit? " ^'"*' ^^'lat was the cost of the andVtKatUt ™ 'ri'^r*' ^'^^'"'^ ^-" '---"• come is I865..50, «nd hfsTtali Zr '^" ' '"'^ "«' "'' ^4:. What time botu-A^-, /• "^^"'^le. and minute hands of a watch he'toL^herf "'" '''' '^°"'- 15 ri 220 ARITHMETIC. 1. A broker receives $25 for investing' $4325 in stoclc worth 8(;. What rate did the broker charge? 2. Find the perimeter of a square field containing lOacres. 3. A merchant sold 270 yards of cloth at a profit of 15%, and 398 yards at a profit of 7%, and found that had he sold it at a uniform profit of 11%, he would have real- ized $4.88 more than he did. What was the cost price of the cloth ? 4. A man sold a horse at 20% profit. If the horse had cost him $10 more, and had sold for the same amount as before, he would have lost 5%. What was the cost ? 5. What is the difference between 40% discount, and 20, 10, and 10% discount? G. How much will I save annually by investing $7950 in the 3%'s, at 75, instead of in the 4%'s', at lOG ? 7. A merchant who gives a discount of 5% for cash, sold the foUownig goods : 14 lbs. of sugar, at 6^ cents a pound ; 7 lbs. of tea, at 35 cents a pound ; 3 brooms, at 25 cents each ; 20 yards of cotton, at 8 cents a yard ; 4 papers of pins, at 5 cents each, and 3 spools, at 3 cents each. He received in payment 5 dozen eggs, at 12 cents a dozen, 2 pounds of butter, at 20 cents a pound, $1.90 in cash, and booked the balance. How much did he book ? 8. The trae discount off $130 for a certain time is $10. What would it be for double that time ? 9. A merchant marks his goods at a profit of 30%^ but throws off 20% of this price in selling. At what advance on cost does he sell them ? 10. A and B run a 300 yard race ; A runs 12 yards while B runs 10, but after A has run one-half the distance, he runs only 8 yards in the time in which he formerly ran 12 yards, while B continues his original rate throughout the race. Which wins, and by how much ? 11. Find the difference betAveen the simple and the com- pound interest on $400 for 3 years at 7 % per annum. 12. Find the diflFerence between the true and the bank discount off $620 due in 3 years, at 8 % per annum. 13. A teacher pays $5.40 taxes. What is his total salary, if $400 of it is exempt from taxation, and a 2 % rate "is levied on the remainder ? 25 in stock GENERAL PROBLEMS. tiiiff $7950 221 1«. a": f lS'--"^e cloe^tj"- /;"-"- '- «0 cents '''-■ te:i^r;';;!.;i;f- r '- '- -- ^r n« cost 22. A wholesale dealer \nM ','""''' >''"•^' "''t 57 Profif t„ ^Hin or loss percemr' '""" *" -^'^ '"-• l-iKavera';; zt^F'^ 'Hri:--^,//; ,1 f r b ., ,o da.s ^"^^^^Z^rT^^ - "t^2 'da^ri" '^^ 6J%;impL tat-esf S,;'' ;' f "?"" *^300 for 3 vears ,f teres f, P '^^ ^"'^n to leml it •)(- ««/ >^ais at ^6. What rate Of discount is 40,10 .nd JO V ' ■'''''^"d 10% equal to? 222 ARITHMETIC, - ! ; 11" i 5. stock at 71) I 1. A (Irtig'g'ist marks his soda at 50% profit, but by a mistake sells a pound by avoirdupois weight. What per- ceiitag-e does he really gain on the pound ? 2. Bought land at i^40 an acre. How much must I ask an acre that I may abate 25% from my asking price, and still make 30 % on the purchase money ? 3. Find the cost of planting a hedge around a square field, containing 2| acres^ at 12J cents a yard. 4. A commission merchant charges .$53.00 for buying goods on a commission of 2%. What sum must his em- ployer remit to him to cover his charges and to buy the goods ? A In-oker charges -J% commission for selling $4230 What are'^the net proceeds of the sale ? G. A man insures property worth $7140, so that in case of loss he may recover the value of the property and the premium paid for insuring, which was at 2%. What was the premium ])aid for insuring ? 7. A and B invest money in business in the ratio of 5 to 7. If they gain $1005.51, how should it be divided ? 8. If a piece of silk cost 80 cents a yard, at what price must it be marked that the merchant may sell it at 10% less than the marked price, and still make 20% profit ? 9. When, first after 7 o'clock, will the hour hand be midway between the figure V and the minute hand ? 10. A farmer sold his farm for $5300, and considered that he had gained a certain amount, but a note for $80, which he had accepted in part payment, proved worthless, and as a result he gained only | as much as he had ex- pected. Find what he considered the ftirm worth. 11. Find the compound interest on $39 for 16 years, at 4 % per annum. 12. Find the sides of a rectangular field^ containing 3| acres, if the length is to the breadth as 3 to 2. 13. I sell $3500 stock at G5i ; find my receipts from the sale. 14. A speculator paid $1400 for two lots, the price ol one being 40 % that of the other. He sold the cheaper lot at a gain of 50 %, and the dearer one at a loss of 30 %. Find his gain or loss on the whole transaction. (iENEHAL PIMJBLEMS. 223 shaUhepl,acedtothee,lcli offh^i' "'^ ""' '°'*'l Profits remaming p,ofits he d vWed jn n, n"" '™""»»y, and the invested l,y each. At tl e end nf '^™P""""' '» 'he capital profits was §67o. Find he .n '" ^"""'" ^^'^ «h«re of the on the entire capital P°'-««»'«?c of proHt realized f'e pays the men S 4oM? "'"",'""" "^ ""=" ='"'l 8 boys • The average r^ri ^ he p' d"^ ™^ tf '"'" ^'^<^-^'-^^. many men were employed ? " ''™'" '"^ "•''>'• HoW l.l. Divide $3!) hetWeeii'A nnri « shai-e may e,|ual 177 nful ' **" *='■« ^% of A's i.-wKss^o^r^^"''^*-")— " tMrd at aSS" V;;r'"J|v';;;;;fl" ■•' '»' «'• *,-o^nHsli(3 invested? PM I i*, 224 ARITHMETIC is ; 1. A and B made a joint stock of 81300, by which they gained $715, of which A had for his share 8275 more than B. What did each contrilnite of the stock ? 2. Find the perimeter of a rectang-ular field, whose sides are in the proportion of 2 to 3, and wliich contains 3 acres, 120 perches. 3. What per cent, of the cost price of an article is the selling price, if the marked price is 20 % advance on cost, and the selling price 20 % less than the marked price ? 4. A l)roker invests $2290 in stock at 85 J, and chare-es -g % brokerage. Find his In-okerage. 5. A merchant bought 100 barrels of flour, part at $7 a barrel, and the remainder at $5 a barrel ; ])y selling the former at 15% gain, and the latter at 14 % loss, he just cleared himself on the transaction. How many barrels of each did he buy? 0. I bought a hind and a fore quarter of beef, weighing together 252 pounds ; I paid 7 1^ cents a pound for the hind quarter, and ^ cents a pound for the fore quarter, and found that I had paid 17'- cents on the whole more than if I had bought l)oth quarters at ^ cents a pound. Find the weight of each quarter. 7. If a de])t, after a reduction of 3%, becomes $1008.80, what Avould it become after a reduction of 4% ? 8. Divide $700 into two jjarts, such that the simple in- terest on one part for 4 years at 5% per annum, may be equal to the simi)le interest on the other part for 2h years at t)% per annum. 9. The difference between the annual income derived from a certain sum invested in 81% stock at 187|, and that from an equal sum invested in 6% stock at 134 is $10.00. What is the amount invested in either kind of stock ? 10. A grain merchant sold 435 bushels of wheat at a profit of 13%, and 325 Imshels at a profit of 11%, and realized $1.10 more than he Avould have realized had he sold it all at a uniform profit of 12%. What was the price per bushel he paid for the wheat? ' GENERAL PROBLEMS. 225 |;,talo»sof30%,u„d the cheaper one at a i.inT^y' in.cl„.j.„inorlo»spcr cent, on the M-MeCr/c^cd- 1-'. A mei-eliant at one time askcrl '>v/ "^ "'"^""^^"on. for an- article, but afterwarcLo cU fori'/ ' ''rr* this price. Find his loss per cent. '' '"' "Vo moie than 13. Twomen ranaround a circnl-ii- f-i.n„r, inrv , . PJ^^edthat they start togethe.^ trrnT'^re an"had [irtflJtf TlZ^Z^ -""'-•.f ^^^- quired 22 cents more to emul im' o if™ .Te "'''^'" ''t ceuts ^tch. How n.any beg,.«.s'";re?rere'?" "'^^"'^ ' am'o„ntSn"mon7 '"''T' %' ^""P'« '"'--^^^ «34r,.«0. Find tlte s'um ." d rl^""' '" ' "'"""'^ -»»- '» 17 A liimberman sold 3(i840 feet of lumboi- ^t- 'ko, ,., per M, and gained -'Sy. Howmueli wnnlri hi i H*'^^ had he sold it f„r $17 per M 9 '"''''' *^*'"^'* dLl'til'^Their SlS lr5%™',";f^"'./" "^ -"« .•espectively. It' tl.t'tore 1 'iir "ular S^so^To?™^ for"m.^:rtij;ts."H:rh:s'5rnr-^^^ material and 77 more for I'lhor tL ^ ^ "'""''^ ^^^ cost .110144. Wlu't 4s7ts :lo.t ? ^'''''' '"'"'''^ ^^^^^ 20. A merchant's wholesale Drioe i^ 1^°/ o.i each kind did he sell V " '' ^"^'"> '^'^^^-^'J^ of f 11 1.1 u . '■. ! li I li:'. m 226 ARITHMETIC 1. 1 invest $4000 in stock at 112. Find the amount of the stock I receive. 2. ThejointcapitalofA, B, C, andDis$3150; A invests $2 for every $3 B invests, B U for C's $5, C 80 for D's $7. Required tlic amount invested by eacli. 3. Divide 99 into two parts, such that 5 times one part may be equal to G times tlie other. 4. A, B^ and C work 6, 7, and 8 hours a day, respec- tively, for 2 days, and receive the same daily wages. They finish the work in 3 days more by each working one hour a day extra. If $91.30 is paid for the work, what should each man receive ? 5. An article sold, at a loss of 25%, for 85.r»2|. What Avould be the gain or loss per cent, if it sold for 17.00? 6. A speculator sells $3650 of the Merchants' Bank stock at 142, which yields a semi-annual dividend of 3i%, and invests the proceeds in Bank of Hamilton stock at 146, which pays 4% semi-annual dividends. Find the change in his annual income. 7. A merchant knows neither the weight nor prime cost of a caddy of tea. He recollects that if he had sold the whole quantity at 70 cents a pound he would have gained $7.00, and if he had sold it at 50 cents a pound he would have lost 83.00. What was the weight and prime cost per pound of tlie tea? 8. A and B had the same amouiit of money, A lost 39% of his in speculation, and B gained 45% of his in another speculation ; both together now have $164.80. What sum had each at first ? 9. An army, in a defeat, loses one-quarter of its number in killed and Avounded, and 4000 prisoners ; it is rein- forced by 2500 men, but retreats, losing one-fifth of its number in doing so ; there remain 6000 men. What was the original force ? 10. A person buys a lot of land at $40 an acre, and by selling it in allotments finds the value increased three- fold, so that he clears $200, and retains 25 acres for him- self. How many acres did he buy? 11. Find the compound interest on $48 for 8 years at ' per r.::::iiin^ interest calculated half yearly. (JENERAL PROBLEMS. 227 T)eJ il^d^'.fp^^"^' '"xx^'^r ^""'^ «^ P^^^« i« 5-l« francs Wo'wh^twinT.?'' ^^'f^'^'^'S 2-121 francs per marc 9fi?sn J T^^ ^^f''® ''''^'^^'''^*^^^ P»'ice in New York of 26880 marc banco of Hamburg'- '-> ^ "a ^ ur 15th' ^T'lfl^'^H ^'' "'? faee value of a note made May 7 ^K^J^^''''^^''' '^"^^ discounted June 27th at 8°/ tc) produce |5G0 ? . ' /o' ^" in etclf c^s^'ift '""^5'' .'^? T"' "^ P°" ^""«' 3« battles hemW nnti ^f' 5% had been allowed for breakage, he paid an ad valorem duty of 20/.: ; the troi«-ht and cart WhaHartlfe"'"''-*'"' '™'' "" ''-'">"' -«' "-« *m3 "o." vv nat was the invoice price per bottle » $969b'^Thrv„r°?f '■"f.'''''':'^ l>y a village for taxes is WbJO The rate of taxation is 17 mills on the dollar and amoun Ttl? ""''^"'' 'K'' '"" '<"'" '^^^ What s the amount ot the assessment ? 16. A merchant had 300 barrels of flour insured for 2 of Its value at 3%, paying §;!(; premium. At whS prL^ as wen'a Vr^' ""•'*'" " '°o^'''" ^5% "^ "^ p'ime^eoS as M^eli as tlie premmm paid ? .,e"draV„Tn:l^rvt:/^;-l-^--- rxtu'rers ^'■^"' ""'^^- "°- "-" -'- ^"" whit ™,r een? S' ^"^'i' ''"'^. '"^" "<'% '•f' *« remainder, wnat per cent, of my salary is unexpended ? be/n<; a, t^^^ ""? "'"' u°'" *^^-'"' "^« P'*" ?«>• b»«hel of the l':'ti,Vmidnigh;;"™ « "' ''' «"'^ p'^^' -- ^^ « fnr^lf'tt'^m '"J^";!^''' -^ ''^l? '"'^^^' '^" a yard measure 4 inch too^h^rf i • '"I'^'S it he u,ed f 124.80. Find thl co^J'p'^fce of h'e cToth .nd tf '" ""'"^ the merchant sained hyl,is diJ,ono.sTv ' '■"""""' tion A^n^ sS'Tr?t'eo?V" T'™' "' "^ «»™« ""•««- remains one half t f , nh ",""? ^iT'' '"""' ""'' always What is B's rate of u-avtllinl'f °' ^ ''^ « ^as travelled. covers as mnch ?n 1 leap as she does ?nt 'tin'' ""' '^'^ leaps wm the honnd tak^ before he cSch" s the hTe™'"^ Will the hour hand ^'^.^ ^^^l^J ^:^- 230 AIUTHMETIC. n^i: If I h i f ' 'mil ^ If* m n : i ; 1' i ; 1 1 i 1 i i and l<«i„,. 15% on tht su,4 " H s to a ^'4 beit%63 50' hind the cost of each commodity ^ * •^"' «3^«5rsh™'lnd''4?"''/'^''',™'^ "'■ «%• '^^ «>«"• hart 4. A nuwcliant mai-kod his floods at an advance of 20°/ JlSlnirtrpt'cr" ^'^^ '''^^ '"-' ">'« ^Hef ^i (1) half-yeari'v.^i; year?/: ^ """■ ""'''''^' '^""^""'"''» (y. What will it cost to ffnivcl a Avillr o ,.„^^ -^ as interest How much - respective! V. *= cuo x^, ^^, inve^Nl'Tn ff''-";?''?; 'li''''*' money will a man make, who d^Mends^ ""'' ''""™"' ^'"'^'^ ''' lOS' ^^ P'^yingSX 10%taTen'offftli;;i:r"'=" ''''"'^''" ^'^'^ '"--"'' »i 13. A gentleman has $25000 Bank of Commerce stock which pays a dividend of 8%. When money rworthr/ he sells out and invests in Bank of Toronto stock at 20^' which pays a dividend of ] 2%. What is tl e WerenccTn 111 income after allowing .% „rokeragc for ead Ss OENEnAL I'JtOBI.EMS. 281 losing 2r,7 out! otho w? ?' ?",'"!",«■-" % »" °"«. '^nd per cent. ^ '^ /°' "^ ^^^ ^'^^ 8"<^in or loss per hour, rc^, 4-th "v%v'\™ '.'.'""'■"''^ * '^"" « ""l^^ man, providoc l,e ft '..hv^ "• ^'l" ™"' "*' "'^ third two? ' '™'^'*' «-''°""■ IS raised on incomes below £m? '^•"^*', how much rotate'"': iT'samiTtxis''" wr™"'V'""''^ "^ ""> «='ock will the minute 1 nrt be mid «^/' ."'''' '"■'<"■ ^ o'clock, a A manuiacturir ^^ufol I ^fo"^, XfT' '^'' What was the I'ate of discount? ' ' '^ discount. betid'ir toi.!;: dXof'n^f^''"" ""'«" ^'°* -«st comes due, rea'l cLtotJi^fi.f^! fj,«,"'-' ^;^cf»''o " "e- 10. The whole time oppnn!o/i ? ? ^^P^^'^^^^uni? and travelling at 'the^al"^ ' 8 m^leVS^u'? f'^ '''"^' a bridge is 22A seconds Fin.! fi V ^^^' ^^ crossins- 11. A man %we7mo ^lm\^^ ^-"^"^^'^ ^^*^'« bridge. " due in 12 mon'hs l^Lm his^^Ll^ ? '"'T'''' ^^^^^ ^^OO months. How muoh Hn t S • ''?^'' ^^^' ^^^^0 due in 10 money beingllJrs^Vefrnrr '^ ^^^^ ^^^^^^-^ momhs^TolL'^1'^^^^^^^^^^^^ Tn ^^;;,J,^^0 capital ; after 3 more C also joins the firrf ZS^- '- ^^S^ ^^^^^^' ^ months at the end of theTea^wks^^^iTur'^K'''^^ ^''' ^^''^ divided? ^ ^^ ^^^^^' How should it be OEXERAI. rUoHLEMa 233 1.^. A conii.niiy takes a risk of .«?r,00() nt itv / (iominmy at 27 Wlnt wnnu .i •'.. "'''^ "^ ''notlier ca8e%hil.ou;cfis desttVc.Ty '" ''''' '^'^nnmy ^o.o in tl.e „u,„I,e.. of o^cl, l,,,„:,.l:'t """ '"" "^""'^ •■'« '^■■">"- t^'nl 6%'a„d Jhc .t^f,'ir, ■;;! ':f ;r' f«'- 1 >"--. i-t of it at in each case c ' s tl^^' '%:T, '•^'"■'™' "» "'^'-e^t 17. The Sinn , ?«• ,i ' • '' *'"' ""'" '«'»<■<' M C7. tributed SfSO more th.in A .,,„i i< ii ' ' ' ' ' • " ^^"1- did oiicli contribute ? ■ """"il man n. m..^ -^; f^' S '?;!f a^rJ^^f'^r^' of tM^cUfferent the inferior at V ccnt'T^omu] I, '""'' ^^ ^"^"^^^' ^"^^^ each did he buy ? ^ -^^"^'"^ "'^'">^ P0«"^« of 19. A speculator buys stock when it ik or,o/ ., and sells it when it is 19 7 below m, \n 4 f''^ P''''' of ^ain ? ° ^-^ /o o^^ow pai. 1\ hat js his rate itsllLu"" '^''■™'''"' °'''' ^O"^'-'' fi«M - «2 yard.. Fi„d pioceeds. At wl,at advance, per cent., on the c4t does J/j:tS:t:r *^^ - - ---^-^ wrr^e 234 ARITHMETIC. « ■■ ' . : ' tlf ^I' ^J'^P^^^ sold $4850 stock .It 871, and invested tbp mZm rati! "^""^^<="°"' ->d -"°««^ altogethe^r to paid bV 8^7 S'r '■"""'^'^'^ '"""""^^1- the premium. ^ A n amount of tlie risk $12b0 for itaonthsTisSfif f Pf to«>-«hiP. A contributes 6 mm,fh« w! , ' ,.f ^*^ '^"'' '^ months, and C $400 for fa months. How should a gain of $1960 be divided among a4nt^,?rt'';;^^"' f-^^'^ ^^^^ '^""l' »f wheat to his 5. What IS the averag-e time at which the folloivino. bills become due : Feb 19 i«Qn cj^aa o i^ne louowmg Ma..h 6, 1890, b7^''on lo ^fSft' 1^1 "itiJo' «m on 6 months' credit , Aug. I 1890, U,^Z 2 Lnths' ^voMh^T1;\^'""T^"*''f '^""'"'""'o^of^^'tae, one sort Se" wistf t^ t™ 'a mit:'of ■•75T,'; ''i ' ^=^"°" worth ti qfi2 o^ 11 '^^"^^^^"6 or "^ gallons from these fTom eali'sirt p"^""""- """^ '"'^"^ ^'■""'- ■™^« '>e take 7. A workman was hired for 40 days at 40 cents a dav Zl7y Z ^: wo^s^^f' ^' ^""'*'^ -""'«- «>at 101 every day lie was idle he was to forfeit 16 cLnt^ • work? ^'""^ ""' "^-^^ *^-««- How many jays dfd he a Find the compound interest on $324 for 81 years at 12% per annum, interest calculated half-yearly ^ E-aLd 2w" T^ 'WO horses for $200 each, on one he s^t pt^c:„\tti?etttL^i:: - ^-^' — nf Q n f "^^'* ^ pay for $3000 Stock which is selline- GENERAL PROBLEMS. 11 If th h ^^ on the same ceml^whf ' Tl^'^^ "«""« of a clock are «eco„d hand be mM;! f ^l^^w „1t „',f "'"'''''• ^'^^^^^ l^ve in paymenV;?,:,^f';^^°V'-''il^>'ay stock, at 40. and cash did J,. y, 16. 1A hat sum will be Ci « 1 '' 'f '' '' ^ ''«''««I ? .tased on a g-round rent of 7% 'o'" 'o' valued at?4200, ^^^1 the time and sum (use simple in t;;o z^J'uTz^:/>rr ^■""-'•' -'" h- takm. What was the rent o'fTe pt' ^''^ ™^ '-'•0''«^4 75 fS. per" cent. voulThlve' f '■"''"' "■''^' "^^^ 30 / less the . ■ gain per cent ^^^" '"'''' 'i "'"'es as ^tat ' Ru/ 1,;',' '^l' If it costs ^'^ what will if nn^f- fT ^^ ^^^''^^ ^'^ ^^^y wood inM q t 22. A ma„'?ts%'mit'°';" °'"°°'» ^^^^o i lent L"f ^"'' and back i„_ .13 mi.^r "p^nTIV: ^'.^ 'I? /" «"'»'■'«-, 2I T?'^!^ " ^''" »■■"«>■. ''" '■"'^ °' 'he stream, and '!•>• it 3(, beaxided to; n,.,.t ■ times the original nun",; r. ud'r'"' ";"''«™" « A™ 16 -^ "'^ tlie number. 236 ARITHMETIC. W ■;! ' I i 1. A person owns 300 acres of land, which brings him in an average rental of |3.37| an acre. Part of it he rents at $4 an acre, and the remainder at $2.50 an acre. Find the number of acres he rents at the former rate. 2. Find the compound interest on $187 for 4J years at 12% per annum, interest calculated every 4 months. 3. Which is the better investment, the 2| per cents at G5|, or the 4^^ per cents at 103| ? 4. A man "sells a house, losing 20% of the proceeds. What per cent, of the cost did he lose ? 5. The hour, minute and second hands of a clock are on the same centre. When Avill the hour hand be mid- way between the other two for the first time after 12 o'clock? G. Three men run a race around a circular course 100 yards in circumference. At the start A is 10 yards be- hind B, and B 15 yards behind C. If their rates are 10, 8, and 5 yards per second, respectively, when will they all be together for the first time ? 7. A workman was employed for GO days, on condi- tion that for every day he worked he should receive 15 pence, and for every day he was absent he should forfeit 5 pence. At the end of the time he had 20 shillings to receive ; how many days was he idle ? 8. A^ B, and G engage in trade. A puts into the busi- ness $3200 for G months, B puts in $4000 for 5 months, and C puts in $2500 for 7 months. C receives 10 % of the total profits for managing the business, and also his just share of the remainder of the profits. If C receives alto- gether $428.40, find A's and B's share of the profits. 9. A speculator had a certain amount of stock in the 3 per cents at 96, and an equal amount in the 4 per cents at 101. If he sells out and investo the proceeds of each in the other kind of stock, he finds that his annual income will be diminished by $12.15. Find tJie amounts of money originally invested. 10. The simple interest on a certain sum for 3 years is $600 ; the true discount off the same sum for the same time is 1500. Find the sum and the rate per cent. GENEKAL I'KOBLEMS. 13. A grocer bouo-iif onnn P?»nd. It loses ^"''o uf ° PT"ds of svgar at 6 cente a h« gain or loss by ^ell . ff . ""'' '''y"'^ out. F tt 6 cents. •> '^'""8^ « '« tlie rate of 15 ounces for 14. How many lionsiea o i av time, walking back at the me "f 3mn'.'° ^^'^ '•«'»™ i» ^0. imd the cost of o-r,irii- "^^ ''" hour? yard, a path 1 yard wMe^runn „^^' " '^ '='^"*« a square square garden, >hose stle'riry.:,';:;:"' "''' '"^"« °'- '' f^o^^^i:sri:irtnV?"^^'---'-fo^ nf-^fu o""^^'' "" ^^t^h transaoHnn fit,V/u ' ^' Repaid of the 3 per cents. "'^^^action, fmd the market value 238 ARITHMETIC. I iu m' 1. After A has received $10 from B he has as much f^n%^f? ^ ''''^ ^^ "'^''® 5 and between them they have ^40. What money had each at first ? 2. If 8 men and 5 boys can reap 71 acres of e-rass in 3 days, and 12 men and 14 boys can reap 50 acres in 12 days, how long will it take 1 man and 4 boys to reap 12 3. A merchant bought 777 yards of cloth at $1.50 per yard. In selling the first half he uses a 35 inch yard measure and in addition had tiie goods marked at an ad- yance of 2o % on the cost. At what advance per cent, on tlie cost should lie mark the remainder so as to gain only 25 /on the whole outlay, if he use a 37 inch yard measure m selling ]t? 4. A house worth $8937 is insured for $0000, so as to cover the value of the house, the premium of insurance and charges amounting to 5 % of the premium. What was the rate charged for insuring ? 5. Three men take an interest in a coal mine. B in- vests his capital for 4 months, and claims ,V of the profits ; OS capital IS in 8 months, and D invests $6000 for 6 months, and claims c of the profits. How much did B and each put in ? 6. A grocer mixes 30 pounds of tea with 40 pounds ^u "^{^^^^^ quality, and iinds that the mixture is worth 41 i cents a pound; the difference in the prices being 15 cents a pound, find the price of each kind per pound. ^ jJ^JtJ^Tl'''''^' P^'^^ •^^>-'53.60 duty on an invoice of goods, n 20/ ot the goods be exempt from duty, and 22 7 is charged on the remainder, find the invoice price of the goods. ^ 8. A man lends $1050, part of it at 4%, and the re- mainder at 7%. He finds that the income from both in- vestments is $1.86 a year more than it would have been mxa he lent the entire sum at the uniform rate of 51°/. Find the sums lent at the different rates. ° 9. Remitted $>200, including commission, to my agent to purchase goods on a commission of 5%. What amount did he invest in goods ? GENEnAL PROBLEMS. ^39 •i he bill h,.,l twice alC^rn,,, ? '"'™"°' """^^^ half af nZ;,;tT;'t'jrt,;:;:ter ■"^-'^ ' ^ -" -^^ - 13. A dealer receivorl . „ • • ' "'" 'I""'- Which was brol">'» 14. A mixture of hli.ck ,„^ 1 ""''"■''' 1"''«« ? pounds, cost S5.12J Tf h,„ :"'' ^''''™ tea. iveislnnff 14 Z:!:'^r r'^' '- -« I'T'J;';'-^ - i„tercha,^e1 tTe' Str '^ ^ ''°""<'- ^^-4 a';at,:f ^:?:,:;^'-n;;,';;.«; at ^"^^ ?"'« I-' -."are square field containinj. 2 acre " ^ "■°"™' *""= "'''-. a... how ,n„ch per ounces to the pound nnrt lul H * ?"«■''"'- s:»''"«- "nlv 15 to the pound ?' '"' ■'"'* *''<^ ""'cr half g-ivinj,- Tt ounces 18. Goods "Worth Jt^-'fifii 7k to cover the value of" t'heo-oodrL' "'"''"'? ''°'" *-™»' ^oas and $4„50 charges. Wlm w.i .?™'"'"'" "f^'^'^'ance charg-ed? * ""'" ""*■ the rate of premium 8 /at^tt^:,;?err srA ,^;^ t^'^^' '^ '« «'^ - ^ *« andBfofhis; at the c d 0^ H^e ''''''f i "^ >'is capital, $4000. How much 0? tl e J^u, ' ! Jlf *«"• ^^'>ole ^.tin i 20. A wine merclian mfxed s „ n'' '"''" <""'"ed to ? gallons of wine at 75 centTn^n"""' °<'^^ater, with 14 cents, and 16 gallons at m to ^ft"' "', f^""""^ at 90 the mixture worth v " - ' ' '^^ How much is a gallon ol 240 ARITHMETIC. [i; 1. A youn^ man^ receiving a legacy of $64788.50, in- vests 1- of it in 5% stock at y3|, and tlie remainder in 6% stuck at 108, paying brokerage at 1%. Find liis net an- nual income, after lie has paid an income tax of 18 mills on the dollar. 2. The duty on an importation of bottled Bay Rum, after deducting 2% for breakage, was ii^823.20, and the in- voice price of the rum was 25 cents a bottle. How many dozen sound bottk-s did the importer receive, duty at 20% r 3. If 170D metres are ecpial to 1 mile, and if a cubic foot of w^ater weighs 1000 ounces avoirdupois, and a cubic metre of water 1000 kilogrammes, lind the ratio of a kilo- gramme to a pound troy. 4. A man lends $843, part of it at 6% and the re- mainder at 8%. If his annual income from both invest- ments amounts to $()1.84i, find the sums lent at the differ- ent rates. 5. Received $4100 from my agent, Avho4iad deducted his commission of 5%, as the proceeds of a sale of goods. What were the goods sold at ? G. A sold goods to B, which cost him $10000 ; B sold them to C, who sold them for $11910.16. If they each gained the same rate per cent._, find the rate of gain. 7. A certain number of men and women earn $27 ; each man earns $1.25 and each woman earns 75 cents. The number of women is 3 more than twice the number of men. Find the number of each employed. 8. A grocer wishes to mix 47 pounds of tea worth 40 cents a pound, Avith a certain quantity worth 35 cents a pound, so that he can sell 16 pounds of the mixture for $6.07. How many pounds of 35 cent tea must he use ? 9. There are two places 126 miles apart, from which two persons start at the same time, with the design of meeting ; one travels at the rate of 3 miles in 2 hours, and the other at the rate of 5 miles in 4 hours. When will they meet ? 10. A rectangular field, Avhose sides are as 4 to 5, con- tains 1 acre, 20 sq. rods. What will it cost to gravel a pjitli 14- yav ds v.ide around the inside of it, at 10 cents a i^quare y iU'd? GENERAL PROBLEMS. 241 1 9 r^^ f . u?* ]^^^^ '^ ^^'^ v^l^® o*" the patent ? tr->q\hM , ^"^^' the passenger earnings to felS- ^67.29 the total expenses to $564798.20, and the companv was able to declare a dividend of U%. How much stock had the company issued? ^^° ^^ 13 A grocer has 280 lbs. of tea, of which he sell 80 lbs. at 30 cents a pound, and gains only 8% at this price ; he now raises the price so as to gain 10% on the whole out- lay. What does he now sell at a pound ''' 14. Find a merchant's gain, through dishonesty, if he sell goods, which cost him $84, by a false pound weight, I ounce too light. ^ ' nni?'ti"^f.n' ^'^' P^'^P^^'ty insured for $6500, for which he fn «qnnn ^^.F/'T'''"' ' ^'^ ^'^^^^ ^^ ^^^^'^^«^ the insurance to $9000. What extra premium will he be required to foTthe fonnef y ''' '''' '''''''' '' ^% ^'^^''^^ than that 16. A and B engage in trade ; A invests $6000, and at the end of 5 months withdraws a certain sum ; B puts into the business $4000, and at the end of 7 montlis, $6000 more • a^ the end of the year A's gain is $5800, and B's is $7800.' i^ md the amount A withdrew. 17. How much water is there in a mixture of 100 g-al- oZl fiZ?^ ^"n "^f ^'' ''''''^^' ^^'^^ '' g'^^^o"' if the wine cost $1.20 a gallon? 18. A liquor dealer receives an invoice of 150 dozen of bottled porter, worth $1.40 a dozen. If 10% of them are broken what duty does he pay on the remainder the rate being 24% ? 19. A man lends $375 at a certain rate of interest, and $412 at a rate 2% higher. If his interest for one year from both investments is $47.59, find the rates at which each was lent. 20. A boy engages to work for 30 da vs at 75 cents a da 7 and his board, on the understanding that for every day he is idle he will receive no wages, but will have to my 62^^ cents a day for his board. At the end of the time he receives altogether $12.87i . How many days was he idle? If ''I |..jM El 242 AniTHMETIC. 1. The difference between the true and bank discount on a sum of money for 8 months at 9% per annum is $4.95. Find the sum. 2. A bankrupt, whose liabilities are ^3000, can pay only 65 cents on the dollar, after the assignee receives $50 for his services ; what are the total assets ? 3. A farmer borrows $1500 at 7%, and a^-recs to pay both principal and interest in 2 ecjual annual payments. What is the amount of each payment? 4. Find the volume of a sphere whose surface measures 120 square inches. 5. The expense of collecting- taxes amounts to 2% of the total taxes. If a 19 mill rate gives the net taxes as $3G750, find the total amount of assessable property. 6. What must I pay for a draft purchased at 2% dis- count to cancel a debt amounting to $347.50? 7. What is the difference between the simple and compound interest on $400 for 2 years at 5°/ per an- num ? ^° 8. A stock company with a capital of $100000, 107 of which is paid up, can declare a dividend of 12 % on the paid up capital. They make a further call of 4 %, which is devoted to increasing their office accommodation ; what rate of dividend can they now declare ? 9. I send to my agent $34920, and instruct him to de- duct his commission at S}%, and invest the remainder in broadcloth at $2.95 per yard. When I receive the goods I have to pay in addition $1347.90 for freight, $479.40 for ins^urance, $169.83 for storage, and an ad valorem duty of 2|% on the invoice price of the goods. Kequired the num- ber of yards of cloth shipped to me, and what I gain or lose per cent, on the whole transaction, by selling- the iroods for $50000. fe b 10. If a merchant, in buying goods, uses a pound weight ^oz. too heavy, and in selling them one i ounce too lio-ht and gams $15 by his dishonesty, find what he paidfoinhe goods. 11. For what sum was a store insured, if the rate of in- surance was 7^- mills on the dollar, and the premium paid was $25.50 ? ^ ^ GENERAL PIU^BLEMS. 248 Qcasures 12. Three men form a pjirtncrsliip, each coiitrihutiiiff the same amount of capital ; one of them is appointed maiia^-er, and as such, is to receive 12i% of the total prolits. When «fi?wv?''"i?f '! ''^'''"''^, ''P' *^'^ manager receives altogether fbOOO. What were the total profits ? 13. A merchant, in New York gave $1950 for a hill of exchange of i^tOO to remit to Liverpool. What was the rate in favor of P]ngland ? 14. A sum of money together with the interest which ac- fo 4 days, or 12 men for 6 days. How many men in either case does the interest pay ? y i m r.vw ^P^^'.^Z l7 '''^''"«" ^- P^^"^^^^ ^f t^^^ f^i' ^ certain price gained 12 % ; afterwards he decided to give a less ?Q ro/^'' w?T'^' ^'^^^^^'^ '^"^'^ P^"^^^' '-^^^^^ ^^^ '^ ^'^^^It ff'-^ined 493 /o. What was the number of pounds he decided to >-,ive . «^r,nn-'^.^''''''''J ''^^ ^-'™ ^^"^ in 8 months; he paid Snc'e dueT""' "^' '''' "^ '^ ^^^^^^^^^^^ ^^^'^'^ ^^ ^^- \^k ? ^, ^y^^iels of wheat are worth 10 bushels of rye Whoi f ! "^^ ''^^ ''''' '"'^^'^^^ 14 bushels of oats, and 1 "wmiror;r' '' ^^^^^^ ^^^ "^-^^^^ ^-^^^-^ 18. At a 14 mill rate a man who has $400 of his salary exempt pays $7 taxes. What was his total salary ? 1 J. A man employed a number of men and G boys ; he paid the men $1.25 a day, and the boys 50 cents a day. The average price paid was $1.13 a day. How many men were employed ? ^ v^w ludiiy 20. A huckster bought a certain number of apples at the rate of 5 for 2 cents, and sold one-half of them at the rate of 3 for 1 cent, and the other half at the rate of 2 for didTe bf?''''^ altogether 4 cents. How many apples ^ 21. A person holds a certain amount of 67 accumulat- ing stock, and after the first dividend has been added to the original stock, he holds $1317,58 stock. If he secured tnc onginai stock when it was selling at 102, find the amount of money he invested. I I i W 244 ARITHMETIC. 1. A man borrows .f2500, and agrees to pay both prin- cipal and interest in 3 equal annual payments. What is the amount of each payment, interest being at 5 % ? 2. A commission merciiant bought a lot, 40 feet frontage, with the commission he realized from selling wheat at 5^% ; the net proceeds of the wheat, after deducting the commission, being ^3790, find the price paid per foot fof the lot. 3. A merchant buys a quantity of coal oil for $1500 ; by the use of a false measare, and by marking his goods at a prolit of 25%, he gains $750. Find the size of his gallon measure. 4. A house is insured for a certain sum, f of that sum at 1J% premium, and the remainder at 1|%. What is the average premium paid ? 5. Divide $500 among three persons in such a manner that the share of the second may be h greater than that of the first, and the share of the third | greater than that of the second. 6. If 50 barrels of flour in Chicago are worth 125 yards of cloth in New York, and 80 yards of the same cloth in New York are worth 13 bales of cotton in Charleston, and 13 bales of cotton in Charleston are worth 3| hogsheads of sugar in New Orleans ; how many hogsheads of sugar in New Orleans are worth 1500 barrels of flour in Chicago? 7. What rate of trade discount deducterl 4 times in suc- cession is equivalent to 59^;^,% off? 8. A sum of money, together with the interest which accumulates during the time, will pay the wages of 14 men for 9 days, or 5 men for 27 days. How many men's wages will it pay for 3 days ? 9. A bankrupt M^as able to pay 40 cents on the dollar, had not a debt of $500 proved worthless ; now he is able to pay only 24 cents on the dollar. Find the total amount of his liabilities. 10. Three men engage in trade with a joint capital of $13200. The first puts in $6 as often as the second puts in $10, and as often as tlie third puts in $14. Their annual gain is equal to th(^ third man's stock. What is each partner's gain ? fiENERAT, PROBLEMS. 245 if 1'^ Tho ocu.,. 1 1 •'^ iKiict, at ay per annum. *172.80 from a 3% commission V '" '=""'^'=*°'' '•<'««'^-«' timeofjfym'iit;" '°"""' *'**■ ^^'"'' ">« e.,uateci ciAtS,'ce "lo fc r'""" '"^"•'•"•^ "' ^ -■•«'«' -"«- divideitd o7(f'V''l"":?'."'""''^"y '^ '"''« '» declare a ordinary stLlchoSsSvI on V"-^'"'*T-T« stock of the company '•^'=''"''"'"') »t%. Imd the total ?5L'o,tri";Sd sreir::,"'' re:^",fir'" "'"°"' ^"■- tpii Ti.oc.iT ^ ''"^ piut^ceuh, less ins coniniission in 'nLiorb/ 1 5'.! '=™r;su;[rtt"' "" •'"^•"'*^ -"■ and the rate cha.'ged ^ '""^ '''""'' '" "''">'> "''^e. 18^ A merchant buys $7400 worth of goods Bv thp ,,«« a false measure and by ma-lcing hi! goals at^:'pro measi^rfuid.^ *^°°- *""* "^'^ ^<'"^* "^ 'he yard 19. Goods to the amount of .?20000 were insured in n Beparate companies, for $3000, «500 and $6m res pec a hJl^mu^o^tL'S^'elo'^rd " r •'"''"' ^ sustain? uamage &nould each company 20. A grocer lias a mixture of tea Avpio-hino. ca ^ ^^uld'' n^;™''"! f ^"'•'«' -^d '■'--"& srSfa ti>e number of po„7u,s of'ea:h kl^ fn the mixC' ''"' 240 ARITFIME'riC. \f i\ ' fl if t- m [^ lit*.. * it' f IP I I iir 1. A draft on Dublin for X':i«iO costs $il'.i{j. What waa the course; of cxchaii^'-c ? 2. A merchant wold a lot of goods at a gain of 107 • but liad they cost $50 more, and selling them at the same price as formerly, Jie would Jiavc lost 817. Find the cost price. 3. A person makes 7 %cleMr on his money, after pay- mgan income tax of 2 cents on the dollar, l)v investing in .) % stock. Find the market value of the stock, if a broker charged i % for transacting the business. ^ 4. An agent sold a consignment of wheat for $4120 and invested this sum, less his commission, in corn. Ills total commission on both transactions amounted to $240. What rate did ho charge, the rate in each case being the same? 5. A merchant buys $1050 worth of sugar. He uses a pound weight | ounce too light, and marks his goods at aju-ofitof 15%. Find his gnin. (5. A man pays $812.50 for insuring goods. AVhat is the total amount of insurance, if I of the amount is at a premium of 2 %, i at 2 J %, and tlie remainder nt 2^ % ? 7. Divide $;}G48 among three persons, so that the share of the first to that of the second shall be as 7 to 9, and of the first to the tl-ird as 3 to 4. 8. A boy hired with a mechanic for 20 weeks on con- dition that he should receive $20 and a coat ; at the end of 12 weeks the boy quit work, when it was found that he was entitled to $9 and the coat. What was the value of the coat ? 9. A baker's outlay for material is G0% of the receipts and his other expenses to 20% of the receipts. The price of fiour falls 30%, and his expenses increase 10%. What should he now charge for a loaf, Avhich formerly sold at 71 cents, in order that he may make the same per cent profit ? . 1^' "^^^ simple interest on a sum of money for 3 years IS $82.50, and for 5i years at 2% higher rate is $110. Find the sum and rates. 11. J. Brown owes $400 due in 6 months, $320 in 9 months, and $280 in 12 months. Required the average term of credit. OENEUAL IMIOHLEMH. 247 /"luit waa of 10% ; ;lio same Find tlie fter pay- ostiiig in a broker 120, and His total ). What e same ? He uses goods at What is It is at a 21 %? ho sliarc •, and of on con- e end of that he \^alue of 'eceipts, he price What Id at 7| jr cent. 3 years ). Find 20 in 9 iverage 12. A merchant ))ec'onios bankrupt and owes 8r)0(K). His assets amount to .13317.50. Wliat per cent, of his in- debtedness can h(^ pny, allowing tiie assignee 37, on the amount distributed, for his services ? 1 05 in his income. Find the market price of the 34 per cents., brokerage |% on each transaction. 18. A merchant shipped 2000 bushels of wheat to his agent, which brought 85 cents a bushel ; after deductin;>- his commission of 3% for selling, he was allowed 4% on the remainder for promptness ; after then paying firiglit and storage to the amount of $130.25, ho (tlie agent) re- mitted the net proceeds by a draft purchased at 117 premium. Find the amount of the draft. 19. In buying goods a merchant uses a pound wei<>ht one-quarter ounce too heavy, and in selling them a pound weight one-quarter ounce too light. Find his gain per cent, by his dishonesty. 20. What is the value of property, if it be insured tor ^, of Its value, at a premium of 2%, and 5% of prem- ium being charged for expenses, when the total co^t of insuring is 175.60? Ill 248 ARITHMETIC. ' I 1- 1 M ■'■f' ill; E --^ ^H ' 1 H ' w! li' at the oru? nf i^T f /^^^«' A VntB in $5000, and A. Ti ' ; ? ^""^ ^^® ^""^ ^^ ^ ^^«"^hs puts in $;3000 more What sum did A withdraw at the end of 4 months? vyl'ftmo%l1^ i^^i ^^'^'^^ "^ ^"™ '^^d 20 bushels fm. to7 ^ r? ' ^^ ^T^'^^' .^^ ^^^'^ ''^^^^^ 1^ bushels of rye foi 127. How much per bushel did he receive for each ^ days, but 12 days afterward 300 men were Ivilled. How bng will the provisions last the remainder of the ga^ 4. The simple interest on a sum of money loaned at a cei^iin mte for 2| years is |32.50, and the iXeston ?^nH H \^ f "^ ""' ^^^ ^'''^^^' ^'^^^ for 4 years is1l43 Jmd the hrst sum and rate. 5. A firm become insolvent, and owe j^SIOO Their assets amount to $4981.50. What per cent, of 'their in debtedness can they pay, allowing the assignees '?J} on the amount distributed, for their services *^ '^°' ins ntoS Too ^-^T '^''' f ^ ^"^'^^'' ^« P^^y-^ ^1200 m o months 5^600 in 5 months, and $500 in 8 inonths How long should the balance run before it becomes due? S5obooo'''''wif?'''l ^f assessable property valued at $500000. AVhat rate of taxation must they evv so as to secure in taxes $9775, after paying the coLctoV's fees It 8. A commission merchant had shipped to him 1000 bar- rels of flour, and 4000 bushels of wheat He paid I'U een s .r«^« 1 ' ^f ^^ ^'""'^^^ ^^'^ "^''^y- If J^« «oi^^ the flour 90 cents ; bfl""? "^ commission of 11%, and the wheat at .^u cents a bushel, on a commission of 2 cents a bushel what sum did he remit to his employer ? ' Mil!'- r^ 'J^fchant in Chicago, owes $4500 in New York Which of the following methods for paying the deTt is the more advantageous : To send to New York a draft on New New VTrk rrV " "^ n?^^^^ '' '°^^ P^'^"^^""^' '' '^ ««-! ^^ GENERAL PROBLEMS. 249 10. A man increases his annual income |72 bv trans- ferring $21000 stock from the 3%'s, at 78, lo the^im i incl the price paid for the latter stock. ^^° 11. A bankrupt can pay 60 cents on the dollar • if his J^md his liabilities and his assets. T.^^'n^ nierchant buys 2590 yards of cloth at $1 a yard Ind'o i7^-'\' ^^'^^^^^l*:^^ ^«^d a 35.inch yard meaCe and a 3 7 -inch one for the remainder. At what price per out'ia ? "'^''^ *^^ ^^""'^^ ^^ ""^ ^"^ ^""''^ ^^% ^^ ^i» ^ 13. An insurance company took a risk at 24-7, and re- insured j of the risk at 2%; the premium re^eiyed el the risk P'"^"^"^'^ P^^^ ^y '^42. Find the amount of ^^Ii'r^' ih^^ut'^ ^ft^'J''^'' partnership; A puts in 1347.80, B $359.2f>, and U $543.59. If the profits of the business amount to $500.26, how should it be divided ^^ 15. A, B, and C start from the same point at the same time, to trave around a lake 84 miles in circumference ttn^d r ?f '' -r^ ^f ' ".^"^^ '" ^^^ ^" *he sam-e direc: tion, and C 14 miles a day in the opposite direction. In how many days will they all be together ^^ 16. Gunpowder is composed of nitre, charcoal, and sul- ^f I''' ;.^.^^^,« P^'^P^f onof33,-7, and 5. How many pounds of each m 60 pounds of gunpowder ? f^^>-n^^ t^rtl'Jao/''^'^^^ "'^'''''^'^ T ^ '"'"^ of money fora certain time at 6/, per annum is $62.10, and the interest on 3 SM ""^ 'T ^"^^ ^^ r^"' '^^ ^% P^'' ^^"^^"1 i« ^^Iso $62.10. J^ind the first sum and the time. 18. A merchant in Toronto has $4800 due him in Hali- xax How much less will he realize by having a draft for that sum on Halifax and selling at 17 discoSnt h.r by ha™g a draft on Toronto remftted tllfm purch sed m Halifax for this sum at i% premium? 19 The expense of constructing a bridge was .«87S which was defrayed by a tax on the property of the Xh^^*"''^'""^'?^'"''"" ^^' ^^ mills on the dollar! nn .. tn. commission lor collecting, 3%. What was the as- sessed value of the town p '^ /° "^^ ''"*' ^^^ I 1 250 if j '■' i ¥ ' r ARITHMETIC. 1. B owes a certain sum, | payable in three mouths i m 5 months, A in 6 mouths, and tlio balan(;e in 10 months. Kequired the average term of credit 2. A farmer lias slieep AYorth $4, ^5, $6, ^8* per head. What number may he sell of each to realize an avenixre prjce of $5.50 per head ? civcid^e 3 A merchant had 400 barrels of ilour insured for 70% of Its cost, at 31% ; the premium was ^58.80. At what price per barrel must he sell the flour to gain 4. Find the present Avortli of an annuity of ,«r>00 having 4 years to run, money being worth 0% per annum. o. Innd the difference in area between a square field whose side is 10 chains, and a circular iield whose cir- ""''!! rl'TofJ'^ ?l"''^ ^'^ ^^^^ perimeter of the sciuare field. it d% bonds are at 80f, ^\■\n^t must be the price of 47 stock, so that the income may be unchanged, if the money be transtcrred from the first to the second stock, brokerae-c ^ on each transaction ? ^ 7. A commission merchant's terms are 6% on sales, with guaranteed payment of sales to his employer or ">i7 without any guarantee ; his employer accc^pts the latfcei- but ^90 of the sales proving worthless, how much does his employer lose by not selecting the former method the total sales being |2300. - 8. A and B run a race. A starts out at the rate of 450 yards a minute, and every successive minute he dimin- ishes his pace by 2 yards. B increases his ])aec ])y the same and overtakes A in 10 minutes. What was B's rate at starting ? y. J. Brown and L. Stuart enter into partnership and agree that Brown is to rec(Mve $400 of the profits for man- aging the business, and the balance of the profits is to be divided in i-roportion to their investments ; Brown imts into the business $1300 and KStuart |1500. How should a total profit of $1100 be divided ? 10. If 375 lbs. of sugar are bought at 6 cents a pound, and 10% of it Ix^ wasted, at Avhat price, per 11)., must the remainder be sold to make a gain of 11% on the whole transaction ? GENERAL PROBLEMS. 251 11. AiincnUiikeacoiiti-aetto^retlier, oiiAvhich for evorv whltt^^^^^^^^ I ^::^ '- ''^ ---- ^24 for his work, oi/Lw Oh'I "' ^^"^ ^^''^' purchased a draft for ^4680 on New Orleans, drawn at 30 days pa vine- «4fi97 VT What Wfis thA T'of^ ^4?^' . ^cijo, pciymg- -MoJ/.do. ^ndD was the rate of discount at which it was purchas- 13. A speculator sold a house at 507 profit hut thp by t- tra'Iirtfo'T'- '"" "^^ ^P'^'^"'^'-' ^-" -' '"- a.JJn^rr:^at1!y,T„l!^r>■--™p•-- tare^having: been allowed at 5%. iLw n.'ar^rbags we^e n,il I" 'f^'"^ ^""^^ * merchant uses a pound weiffhtl $1 50 ft'The''Lilevir«'l-->'°^-. 11 "^' P'""'™' ^''^ of the valnl !f ?? ^ ' ''^*'-' '■'"'' "'<' insurance is on S pertj ? ""^ P^P-'rty. >vhat is the value of the prc^ work thfohftf ™'" '""''' «"P'»yed to do a piece of worK, Which it was iiecessarvto finish in 25 davs Wh»n onlT^'of «'' "T ""'^ ^^"*«<» 20 day",ihe"; had Jone only of tlie work, and 3 more men had then to L Pm t^tlS'r^l::tr* '"'»-«-• H^^Jr^m^n Serof 41 L?> "■ '' r", .'1''' '■""'^'^ 30 f««' foni the of ground te tn '"'""'"' '"'""' *''"' '=■''" S-O' «"1 the area wi^ A " procure grass from ? semt t ?!: ;.f,^,.f:* '^;,";'''>' ™ake a profit of 20% by does he »Hrl il^^^ m """* '^ P""' How much water aoes he add to every gallon of the mixture sold ? 252 ARITHMETIC. I ai L III. 1. A man rows 3 miles down a stream in 40 minutes ; without the aid of the stream it would take him an hour. How long would it take him to return the 3 miles against the stream ? 2. A bankrupt's assets are 84800, out of which he pays 60 cents on the dollar on i of his debts, and 50 per cent, on the remainder. What is the amount of his liabilities ? 3. A merchant lays out £1000 in buying cloth in Eng- land at 3 shillings a yard. He takes the cloth to France at an expense of 3 pence a yard for carriage, &c., and in addition pays a duty of 42 centimes a metre. He sells half the cloth at 8 francs a metre, and the remainder at G francs a metre. What profit does he make ? (Express answer in dollars and cents, given £1=^4..86§, 1 franc = 19 cents, and 1 metre = 39| inches.) 4. A man invested ^15510 in the 4 per cents ; when the market price rose 3^ he sold out, and with the proceeds bought 3^% stock at 77, thereby increasing his income by $16.40|. Find the price he paid for the 4% stock. 5. A commission merchant's terms are a certain rate of commission with guaranteed payment of sales, or 3% without any guarantee. His employer accepts the former method (which is better than the latter by $19.50, owing to a bad debt of $96). If the total amount of the sales was $3400, what was the percentage demanded with guaranteed payment? 6. Reduce £350 5s. 7|d. to dollars and cents, a dollar being worth 4s. 2d. 7. A commenced business with a capital of $12000. Four months afterwards B entered into partnership with him and put in 37| acres of land. At the close of the year their profits were $4600, of which B was entitled to $1000. AVhat was the value of the land per acre ? 8. What per cent, advance on cost must a merchant mark his goods, so that, after allowing 5% of the sales for bad debts, an average credit of 6 months, and 7% of the prime cost of the goods for expenses, he may make a clear gain of 12J% ou the cost, money being worth 6% ? GENERAL PROBLEMS. o.^ 9 A b a ^«it ofZtZt^Z7i'tl!^TJ ' ";™"'« ^°'' «46 and of eJ^P'.^S^-fJ- f r;^/-"*''^-' Band A and C together to do it ^ ^ ""^ ^^"^ ^^^^d it take cost by rail v-^ tha^by Lr te f '"'V^ '"'"■ =^"^' "he 1^. Iind the lono-if„rip ^/n n ^* m »ute« to 7 a.m. amutth™ ^w "'"'"• '' '* '^ ^^ 13. A reduction of 207 ]n tL „ •"'*™/* Greenwich, enable a purchaser to oblin in P"™ "^ ^PP'^^ would What I the price before red ucti^.^"'" '. ""^ ^ ~%n. WhatwlVeru^,!^f -^^^^^^^^^ 15000 cost ,5075. t^dF^^^^'^^'<^'S''^: ti ^T - '''^ -« 81303.21. What was it -if fhl k • *™^ his capital is 14000 and C $6000. If the tot f^ ' i P"" '" *3*»0. B should they be divided ? " P™**'^ ''■■« «2600, how 17. A ship sails at the ra te nf i ■> •, ^he ,s a certain distance from «.'!"' '"' '«"• "*^hen leak which admits 20 tons o? ^'^ "^"'"^ '''« ^P""J?s a would suffice to sink her but t L c. P"'' '^°"''-- ^0 tons of water every 10 mi-iut^, p- / ';"' P"™?^ out 2J tons was from sho.^^ wh^^ she sprmlf th'"? '''^'™«« *ha! sh: I ounce too heaty" bu t''ZZ°t^ "-' apound weight i ounce too light, and glZ %^^.l ?■ "'''' »"« which is What he paid for the gofds * ''^' '"' dishonesty, find ly. What is the dufv n<- 10 valorem, on 4.^n k '.^A ^.^ ^^ cents a pound, anri 10°/ nd valued at 21 cents rpoundT' ^^^^ «^^^^^»"«^ HO pounds; 254 ARITHMETIC 1. A woman sold 90 apples, some at the rate of 6 for 5 cents, and the remainder at the rate of 10 for 13 cents. She found that those sold at the latter price broug-ht her 4 cents more than the others did. How many were sold at each rate ? 2. How many litres will a cistern hold that measures on the inside 5 feet in length, 4 feet in width, and 4 feet in depth ? 3. What must be the least number of soldiers in a regi- ment-to admit of it being drawn up 2, 3, 5, G, 8, 12 deep, and also of its being formed into a solid square ? 4. When, first after 4 o'clock, will the two hands of a watch be equidistant from the figure three ? 5. Two circular plates of solid gold, each one inch thick, the diameters of which are 9 and 12 inches respectively, are melted into a single circular plate one-half inch thick! Find its diameter. 6. The product of two numbers is 11154, their g.c.m. Is 13. Find their l.c.m. S • « 7. A speculator receives $103.50 income from a sum of money invested in the 3%'s. He sells out this stock at 75, and with the proceeds buys other stock at 115, by which he increases his annual income 30|§%. What per cent, dividend did the latter stock pay ? 8. A merchant ships $2700 worth of goods to his agent to sell. His agent's terms are 7|% commission and guaranteed payment of sales, or 4% without the guarantee. The merchant accepts the latter terms, but part of the sales being worthless, he loses $21.50 by not accepting the other plan. How much did he lose on bad debts? 9. A merchant sells goods at a profit of 60%, but his purchaser fails and pays only 60 cents on the dollar. How much per cent, does the merchant gain or lose by the transaction ? 10. Three men rent a pasture jot $165. The first puts in 4 horses for 8 weeks ; the second 5 horses for 11 weeks, and the third 9 horses foi 5 weeks. How much ought each to pay ? GENERAL PROBLEMS. £55 measure which is too ifn? w/L^ ^ """S^ ^ y^^^'^ the length of the yard n,e"fure ^ ""'^ *^*^- ^'™' and'i40oTem foH tT' HL,''''*' "" '■•"■" ^^^^ -'-v, to 11187.50, on account of thP^^n ^ "'' '"<=°""<' '^ reduced insurance at 2% onTe house 'f^^^ljlg' «^Penses .-$1000 ation on salary and on = o^e 1^'"' °" *"« dollar tax- on a mortgage^ forVth? va ue Z^^ "i'^^ ^'"^' '"««rest the value of the farm "^ **' '^^ «* 6%- Find andltoT^hlsoToper'tlf ^r T-lf^ ^'^ -"- fo"'- quantity of metal suSc ent to If a'mileVf'^''"^' "^ '^ Jnch bore and | of an inch thick' ^ "'^ ^'P'"^ "'^ » baikdSt ''"'■■'^ ""^ '"«'<'^^"<=« l^'ween true and Had the longitude of ^hlcMs 60" easf P'"' ''^ ^^ «' onenumbytrngl4:7„d"roThl;^ '''' ">« ^•<=-- l^,- SK-ei^efeL^Tpt.''^-' ''^^ -'- of .460 10s of 4i%^';rd Ue the'otjr «U 50"^ T ' ^»■»'"-'•- «ala What'did the house seulS','^ '°' ""' ^"are of the contain1nr25''yardsT*4T7/r" "!,'='"'"■ ^-^ piece months ; L in,^>edl;td/soid ^it^:*^ ^6^,°" "" ""'''">' ' 4—.^ What was his actual ^'^Z^^,^ yefLTS^/ra'C,;?' >"<'"> *^«-«0i interest in 4^ 21. The assets of a bankmnf ^.-m^i, are to be divided betwetn ^o e^^^^^^^ ^''''^' interests in the business are a^4 to i A . ""^ ^' "^^^^^ and for his M^ork receives 'U^/nf h ^ '''''*' '^' assig-nee, d^i. j. ^„ 1.. ,./,7^^6^ '^2/ ot the assets. a^hovAmoU "^i la to be djvided hntxir^,^.^, 4.1 . '"^lemain- What Share of u,:'i^t:i::^i:: ^.^^^ "^"«- :f i i 256 ARITHMETIC, ! il |i|' 1. A pound (Troy) of standard ^^okl (22 carats fine) is coined into 45 guineas. If tlie value of tlie alloy be \\. that of an equal weiglit of pure gold, find the value of the alloy per pound avoirdupois. 2. Two men starting from the same point at the same time and travelling in opposite directions are 39 miles apart in 4 hours ; but if they had gone iu the same direc- tion they would have been 5| miles apart in 7 hours. Find their rates per hour. 3. What is the area of a right-angled triangle that has a base 32 inches, and the sum of the hypotenuse and per- pendicular 50 inches ? 4. A merchant every year gains 40% on his capital, of which he spends annually llOOO in expenses connected with the Inisiness. At the end of 5 years he is worth 5 times as much as he was on commencing business. What was his original capital ? 5. A watch was 5 minutes fast on Tuesday noon, and on the following Saturday at midnight it was 3 minutes 40 seconds slow. What is the true time on the following Wednesday morning, when the watch indicates 15 minutes past 9 ? 6. If the interest on one dollar for one year at one per cent, per annum be the unit, what number will represent what a person owes for the use of $325 for 6| years at 6 per cent, per annum ? 7. The owner of 3i% stock sells out at 82 and realizes $10947, f of which he invests in 3% stock at 75, and the remainder in 4% stock at 89. Find the alteration in his income. 8. A man is owed $2500, of Avhich his lawyer collects 75% and charges 4% of the receipts for collecting. On sueing for the remainder he gets 40% of it, of which his lawyer's fees are 10%. What per cent, of the debt does the creditor ol)tain ? 9. A cask contains a mixture of wine and water, in the proportion of 5 to 1. How much of the mixture must be drawn off and water substituted ibr it, so that the mixture in the cask may contain wine and water in the ])roportion of 3 to 2 '■^ 11 K. GENERAL PROBLEMS. gS? bc'- of the firm .md pu i^";^?^''^;''' ^l*"^''^™" ^ "'«■"- been carried on one moMl, lo,?;. A''""- ^'' ''"''"«^^ '■'^s ■It tlie end of anotlior Znfh R*^ ^ withdrew *400, and be divided? ^ ^^^ WIOO. How should it a»c/;,ft^,™;::;,;trsl\T': '»■■"! "^ -">« *« Toronto, post- Ti,^ , '^^^ ^^ ^n Mvance of 9Pi°/ ^v, ^i S h,?r.l ?n"'';'riee""^, """' '^'' ^°''' «>-' ^o'r'^Uy i^: 'he seilin^^ pHee'o7Kltt'"''' '"' "" "^ '-'l- ^^ stead on-tlr;; T^: ^rs "■" ^.r-i'" ^ >-•• 'f. -■ at B% and buys one ^^io, h'! in. '""'?' ^^ ^°''°'''' «3000 H% annuallv^md Which i «,l f ^°}' '*^ ^"" ^aluc at -te Of taxation hi;'it„i^,ron"trc?on"r)'"'A::l"'' ("^^ 5)1. J5 a month w/jtpi- mfoc, „ ^'^'^^loiiaij. He also pays at the e^d of tho^^S r^^S^™ '2V"e,s1o;"^ t^ 'i"'''"^ 13. If 2 pounds of tea are vmrihV , '^" '"^ PaW- 14 pounds of coffee wort VZ d, ^''""''' °*' ""«■''«' ""d eocoa worth 24 pounds sLar and r T'"\ """ ^ P™™'^ 11 pounds raisins, how manvnonnr' ^.?""?' '"^^•" ™'-*h wortli 7 ponnds of tea 9^ ^ "* """'^"^^ would be ll"a,vt and cTI fcrcTta if S^" '" "^ "t^-^' « » -■•- - thiL^Xlir/:^-^^^^^^^^^^ «•« -'erestand est on tlie discount "^ ™oney is equal to the intor- buf-I; fl-lloV'^oirs'" t''l"r:S'- °''^ ""' «" «'• P«te- cha„j.ebeinff|,S.77for,5roufilip"'"'""'' "'" ^"'"^ «^- - Wl-S and';": reaUi:™tt"™t?'"'"""^>' ^^ ^ y-- same time and rate h 4" rn «" f ''""'' ™'" '''^■- "^e 'owed in each ^^^^^nCC,!:^ ^r^:^^;:^ 1 258 AKITHMETIC. 1. A mccli.'inic receives $2 a day for liis services, and pays $4 a wctilc for liis board. At tlie expiration of 10 weeks iu-. hn^ saved $72. How many days was he idle? 2. v\ f)r,v,r mercliant bought 120 barrels of flour for $6P.O payiiiM' ^o.75 a barrel for first (luality and $5 for second (luality. How many barrels were first quality ? ;}. A town whose property was assessed at $1500000 built a school house which cost $5092.50, the collector's commission boi.i^ .;^^. What was the rate of taxation? i. Divide $10170 into 3 sums, so that the amount of the first, by simple interest, in 1^ years at 8% per annum, may be equal to that of the second in S\ years at 5°/ and to that of the third in 2} years at 6%. ■ 5. A man invested $000 more than r of his money in a house, and $1200 more than !, of the reminder in a lot • he has^ $1800 left. How much had he at first ? T). Find how much a man will plough in 10 hours in a field 40 rods in length, if he walks at the rate of 3 miles an hour, cuts 8 inches wide, and takes 2 minutes turning each time. ^ 7. A grocer bought a quantity of tea at 40 cents a pound, and fixed a price on it to gain 23^;%, but in sell- ing It he used a pound weight f ounce^oo light, thus gaming ^32.50 more than he would have gained if the weight had been tru(\ How much did he buy ? 8. In running a l(»-mile race, on a quarter" mile track A overlaps B for the first time at the end of the 34th round. By what distance will A win at the same rate of running? i). Find th(5 difference between the true and standard times of the City of Hamilton, its longitude T>einff 79° 54' west. NoTK -The standard time of any place in Cana ^" > 10. A merchant ships to Iiis agent noo bushels of Avheat, whicli he (the agent,) sells fo,- 70 c(mts a bushel on a commission of 3% ; after deducting this from the pro- ceeds, and paying $380 for '.(ught and duty, he remits the net proceeds by a draft pure' sofl Mf. i°/ r,rPT>--"ni i'lnd fhv^ ani-mnt of the iiraft. r.EXEUAL I'llOBLEMS. 259 Lirs in a .s turned off. How long does it take-to'mUh ette"'™'' .'?31fi.5d Whnt™ ,,n nflT ' '";"'""* '"""""ted to • ""'''^'""•■'iiit of money did ho invest 'It Hi-=t9 13 A vessel containing iS-sle quart o^ wa^'r is emptied by a pitclicr whicl, liolds .04679 iriUoZ tt many times can tiie pitelier be (illed tnrt J, i « J^""^ a pint Will it contain wlien tlic la tmnnthv nf ™'l'"" °^ poured into it ? quantitj of water is 3 ho;se?rirrer B SinV'"" ^'f''^- ^ ?»'« '" C put. in 30 SI eep for /w .eks ThZ^ ^""'' ' ""'' pasture 2 hoi-ses as 3 cows and i cows TT ^ "'"','' '° nmcli ought each to pay ? "'^ ^ "''*'''?• ^^ 15. A machinist sold 56 binders -tt «i9n „„„i, ^ half of them he gained 30%':'and ^'th^i^Sdert ""rt actin Z""" """" '"^ ""^ ^'''" »■• '»- 0" *c whole^l^ans! 16. For what sum TTiusfc a vessel, valued at fp«(¥v^ r. insured, so that in case of loss tb . nwrf *^oOOO, be both the value of the vp..o1 , 'i ! ^""^''"^ "^^^^ ^'^^^^er 1 7 a\vn Vv • ^^^ ''"^ *^^^ premium of 1G7 ? .onJ: alrnr,"!' o'nlS^trfclcr:;^ '^"•^ /^^' ''^^^^ absf It, the same work would reouiro in ^V n ^^"^ the amount of work done b'/th^rn to ttfdoneT/:;;: .o;i^w^re;'t:Sot^n!'^ !-;:;t ^^.SJT the payment of the 1- nor if iv ,. i , '"'^^^^^ fo»* imm?' ' '^ "'""^^ '^^ ^^'"^li 1^>% per an- 2fi0 AHITIIMETXC. !• Find tho fliffcronoo hntwnni, fi. *. Orlcnmk^.fii/ • ' '^'"^<'^^'''«>n«-c direct on New tlu. custom's liouse. Wl a .s fh, ,^ "F ''"'""■'''' '" .-no,, in e,.e„ cask, llX^rof duty" ^^''ioy p"""' "' sir teSS"'^ -™---^ i^'HU) -uu] ^MiM\ ' ^i'^y";^^y' then- ^-ams are $960 nio!i(li.s lon-'cr tl-iii A's .vi./^ ? ^ ^^ ^ stock is in traded trade? " ' ''''"^ ^"''^ W'^« eac]i stock in standard it. loans for on New gain or n% pre- ns at 1% of wine OAved at niber of interest is equal JH3..']3J. of it at re. e same >f meat at the If the t'e they separ- ied by cistern cistern 10 are thaler ch 37 moo, I960, ade 3 ck in GENERAL PROBLEMS. 261 Of whieh^^m^;;;;;t''U!:[^;Jj!^ ''' '^ P'^^««. the longitude in longitude 178» west ? ""' ''^ "" ^^''^'' ^^^^^^ed w.ue;/:,?.eX\c^:;^^^^^ ^ p-^s of water substituted, for the x n ro In t '''^'''''^" °^'^^"'' wine and 4 parts of wate '? ^ ^^ ^''^^^" ^ P^»'t« ^^' 14. A speeulator holdliKr «i 1771 .jk no/ by selling- out nml i ,v s L h. ' ''4 "^'^^^ ""''^ »!">' 107J, he can i. c "oa e I is 1, -P™''""''' '" ^^ **'°«k '-i' value 2. Bv liovn nnh Tif- ""^^ '••''■« increased in increased r """ "'°""'' '"^ ■^""''1 income be now 15. A merchant shing 9500 i,~,.„„i, „^ aisrent, with instrnotioiis to ^o 1 ^ "! "'Pf'"*' "> '"'s mission of 47 frsemro-.,. V ti,'''?P[^'' '•'''•"«« '"« ™ni. to $300, and'^renUrtho"^;.o^;y'l^':'?,^t-hich amount was purchased at 117 preni i> ,„ Lh ", ""' •'™" What did the apph.sli-n'^";;;";;;;,;!:,''/'"""'"^'' '« ««ooo, and Ifln!u;L'«-I V:M;\?f'?f''" '"''"« <''' '-^ "^--''^in tin.e I shall ho 10 ninutes t ^VT "* "''* '"'"'^ "» l'""'' w I -all he 1 /tii-it ;-i-/- ;-j -:v;:j tw}Jcasm'uerea^™,',:r?''rI'.!''' ,"^° ''^'^'^ P°'^ i» first and second "^a 1 ?l,f ?,' "'« "'"-1 '« much as the three. How loul 1 a p o it or'48M";"' r "^ "." "'""' them? ' "^ ^^ ^■'^^ l^e divided among at't"; rltV;;?'^^^^^ ^"^^ '"l"'^^^ '""'''^''^ -^-^ year 4 years^Lfth'f is wtth I'oo t ^""."^^ ^^^^^•' ^- 19. A banker diseom?rn 7n ' ^'"^ ""^^^ '^^^ ^<= ^^^t? What rate of fnterestT^ '^''^ "^^^ ''^^^% P^r annum. 20. If sil ve isvo th ^T t^". "^ "'' ^"' ^""^>^ ^ ounee, what will Tthe wi>ht 7Tktf ^''^ I' l'' '^ tains 207 (hv weio-hn <,[!. on i i ^^^ ^^"^ ^^^i^h con- 21. Pi4;e-'tirt^tfi t<^^l^^.:T''''''' ^^^' or two numbers is on .? . ?u "" ^•^•''^- ''^"^ g.c.m bers. ' '^''^^^' ^'^ ^^^ V^'^^^^^t of the num. .rr«^' 262 M :j> In ARITHMETIC. and the cask flfled wfth 'wa f,-'""' I"""'""''* '^ '^'^^'^ oft, then drawn off, and heTask uV^'^'^J'!, *« ""^'"''^ i^ After this has been done 5 t L, f,f' " ?"'''* ^'"^ ^vater. of the original q^Jit^t wZT^^^T^t:'- '^^^' P""'"" 2 The ionfe^itude of Toldo I in^*' ^"^^ '" '^e cask? York 74- west. What is thl I- ° "'^''' '"'a' of New 5.10 p.„. on October 7tMn' New York Z""^"' "''<^" " '^ BJoinsMm^^^LV^-rf -;* in 15 days, and then eaves and A finishes the wS in tT ^ "^^^'^ '^^"'^ ^ ioj wonld it take B to do the wl,o1e Mo t'"? '""'• ^"^ net-pfSrr^:r JL-cI.^ 4^^V^.' S4000 as the for effecting the sale and >yt,r f '^■™'' ^^ commission was at iX premium pfni '?L"^;"f 'he ">•««, which sale? "^'"1 tne total proceeds of the and afterrrS"satd™vi:h 1 ''^ '"^^^^^ »" -^'. of the original quantity Whn'Vo^^ •"'"f >" ^e^h* to j really make, taking sa'nd IltZL^T'' P™«' ''°«« ^^ the m^Hnd^'s 'ry:^™;f «f -■ ^ f- 15 days' work of and for 6 days' work of the «f "'?,''"'' "'^^ --eceive $61 second they receive «?8 w ' ''?' '^ '''■'>'^' ^ork of the 7. Two brothe Ire^ain^ m S" '"''''' '^=''"' "'■•'■" «"" ? salaries which dilfer by s^-fo " ^''''';!-«"' J""'"^ have gross salary has to pay an income kx of"" ''*? ''^" *« '''"•ffer h'htr^s XTr^'ir "^ "^^ -^ *'•- ^^^^^^^^^^^^ re. How )00 as the )m mission aft which is of the e on cost, 3ig"ht to I k does he ' work of eive $61, rk of the m earn ? ve gross le larg-er e doilar, than his s on the 3' credit It of 20 money ^17000. on the ler ex- isiness. GENERAL PROBLEMS. 263 10. The numerator of a cerf-jin f,..,..f much again as the denominator n^i l] ""'^ '" ^ ^^'^^ as rator and denominator is 352 'ft !^?"' ^^'^^'^ "^»^^' 11. A merchant buy' loods a ilo ''"'"'"• and sells them at a gain of 'w/ ^^;f P^^' P^^nd troy, avoirdupois weight what must h^^' /^ ^^ '""' ^^^^"^ ^Y ^ 12. The longifude of cXu tti if ^^^^^ Rome 12° 30' east. What is ^L.- r""'*' '^"^ ^^'^'^ of 2.15 p.m. at Calcutta^ '^' *"^' ^' ^^"^^ ^^^^^en it is ^'^'^^C^'Z^^^^^r^ ^^ ^- trains 50 foinginthcsame^direS .ts's cS^^ '''''' ^^^^^ dn-ection in 3 seconds second*, and in opposite ^y^l^^^Sm^ ^% «^-k, finds that he can .^crease iis"lnSt!<^^: ^^^f 4" can effect the change the formerTnni; I ^ 1^ ' ^^^""''^ ^^ value and the latter has ryerrTf?h ^^' decreased 2 in fected, find by how much h^ fir«. • ^ ""^'^'^^^ ^^ ^^'^^ ^f- commission for sIungZnlldMfr'''''- ^^'"■''« ™'« "f V 20 men, aftei 7 I.tf me*!, .li"'" "f ^^ yards^. day and the woW goes on fi 1 v! , "^ '■"P'-iced by 15 boys, the whole lenjh uug is';^\'!S "'''- «"^"- '•' he money purchased ano^r whLh £ C?'/"l^'"" at a certain rate per cent is ?.iT'^I°'' '''''*''*''''■" *™e double the same princinal .It rhf/4/^f. "'« «™o»n' of one-haa the former time is $74r. Rnd tte p^rincS': '"' ' ! ,' 264 ARITHMETIC. 1. A merchant buys flanTiPl Qf qo what in-otit per ceu^r^ust Te sell If" ^ ^'"'^- ^' a Pounr^sa^^^^f ,« 2r . wort. 6 cents the ad valorem duty 26% ^ * "' ^"^'^ i"«"'d. and 16', 24' kst. ' '"" " '" - P-"'- '^ Vienna, longituL a. Vp'r^nnuin, t^rg::';;'^' ! ™ ir'Vi '^"^ "' '^ ™-«- count Kind the amount o?«,;1Sl' "'"" "^'^ '""^ '''^- »• A merchant sent his •io-o„f ericn ... to deduct his commiss at ■ ?v .^ ' '^'"' instructions in flour at |6 a baA'e ^f tlltst o^T'' fl' '•^™''"'^«'' ance amounts to $250 at wW ! '"'*'''" ^'"i "'s"''- barre, so as to mS^lJ^^^l^ """' "^^ ^"'^ '"' proceeds by a drafrpurclm °d tHV '''?""' *'' "'<' «>« do I receivi from the'^lt'c^ftht'pVon^eny i""' '^"^' ^"'" "• A has $8000 and n tiimA ! , ^. in as a pai tner C Id toT- "^ ^^'""^' ' *«y t«ko equal share in the bushfess u"^^^" "."'^' <^'^«'' "*« an increased, find what Cm v» fl 1 """iP"^' ''"Pitol i^ "ot share. *' ^ P^-^" '° "^ch of the others for his wlfiielj tekef l7but 9^/f;' '"'="• , ^ '^"^^ ^^ ^'^P^ B's. Who wins lie race a^,d h , T'""' '" '?"^"' *° ^0 °f 9 Finri +1. • t ' ^^^^ "7 liow much '-> annum "' "" '"'"■''^' "" «"«> ^r 67 days," at 5% per •nonVrpaJrt' Ce?,^' !*°^' "^^ ^"^ ^"^ ^^ 3 value at 18 mHls on ufe In """*■ "" ' '"' *« P^'chase spent «500 in improvempr,?. !,' "' T^^'"'' ^ ™™ths he he sold for S ZTXtJ' • ' "'" '"'' "^ ">« y<'ar 6% per annunt "^ *"' ^-ain, money being worth 11. Took a ri«;lr -if '"^0°/ net premium was $430.*i^hat wasTh. ^ "' " ^' ^i^" % ^ ^* ^^^^'^^^^as the amount Of the risk? yard. At r that the to his gain I'th 6 cents :)und, and )ec, longi- longitude 3 months ' true dis- tructions 3mainder id insur- be sold a >t for me me the iiat sum ey take I has an il is not > for his steps ;o 10 of Vo per i of 3 rchase ths he 3 year worth My risk ? GENERAL PROBLEMS. 265 12. What is the on^t np Qoor 4 cent, a pound, on^wLlT/'frst f,?^'"' """-"h' »* ^ cents a pound for dnn. ol ^ , }^^^-25 for freight and , .13- A cis\ern c/^'bt ;, ed ,n.'tw""''-"^ '2% fc^v^jj" ? minutes respectively and on n 'I " P'P"' '" 20 and 24 '" 30 minutes ; in wC h™» ^''.fmp'ied by another pip* ■•nnning-togeth;!.? " '""<' ^'" " be filled if all are value to Tl2"ounorTco?n.^^'"<.T ^"'" '^''^ '=^"^1 in weighing- 40 pounds troy !^nd /r'7.';',.^««=' soveVeiffns pound troy? ^ ''"'■' ^"^ shillings weighing 1 -h'le'^Cn'Sorlf/'riAof o^-'^S'''*'**^^ «^% "^ 'he How many fail to ,-,„ ' „ , ,/o ot the remainder im«c, the total umnber?' ''• '""' ^''"«' Percentage are thej of fe '' W^^PetorsJ^^Sr^.^^^^^^^^^ ^'h--*' o'clock ^ .17. A contracto, stds /nTten'i' "'^ 7']''='"'* 30° east? tarn work, with nothing to be odvf °^ *"000 for a cer- completed; a second sends in n?"'/" '"' *« work is stipulates to be paid t2mlZviT'^Z ^°' ^^OOOO, but ference between the tenderr,t\; """"'"• '"'I'l the dif- iiot be completed befo,"tc\ KT,? '"■•" '"e work can- be worth 2% per term of 4 months ^'"''''' ""** '"""'■>• '« 18. A man holds "Kfisnc i o, J • by selling out aiS in ves in?the T'' I" '"« ^ per cents ; m, ho can increase 1 i?am?*'^\r'='^'^fs i" 3|% stock a what the increase wo W be if h" "" r^' **"''-' A- Find charges 1% on each transac ion to SX \''™'^^'- "'"« IJ. A merchant shinned ■, n,,™; ^ "^ change, his agent who sold "'for 75 cents T,?^ ^T"^' ''^' «<•"* to sion of 2J%, he also rea ized 7Tj ^"™'' "" "" eommis- (calculated o„ the renni de. ./ « '" eommission of r/ Pa,yment, and a f .or pa yTn'fd.Uy /''?, P''°<'ecds) for pronT^t he remitted §27.3f, of!;^ ,,,^ ''"'^ *" *he amount of $170 10 of^salt were in the^'rigrnf^"'"''- «~y barrels fa25an7t,fJTh:r afwTo'^hor ^^'^P' «'<' <>"« at together $.54.7.5 more than inhcvtli , ^L'">' ''''''e'^ed al- form rate of $7.87i a head iSh"^ ''°''' ^"'<^ ■« the uni- J iiuw. 1. md the number sold by each » t 266 ARITHMETIC. M ! miles per Hour and fl,/^/i "'^"^'"S' =>■' Hie rate of 5 After LeZTn—.! Tf^, 'i ""'«^ P^-" hour- reaches Y ia^2 ho -u-s mm-e .tJ'^S'"^ ! '""'^ P'^'' "o"'' ''"d travefthe whole c i anee a soLrt f'^T ^ ""'I « «««" to 2. BouH-ht 5 ho..,hrd!' f , ° f "'' *''" distance, gallons, at 121 cen « f o-tn "'f'' each containing 84 Allcvii^g. lOrfor l" tl-t! T' '''"'* P"'? *-'5 for freight, bad debte, and 217 of ^he''L^''^''?' '4 "^ ""^ ^<'"<=« ^r liowmucl^agalldn nnL Isen'M""' 'f' '""^*""'?' *o'' on the whole cost ? *° "'''"'*' '"^ §'''in of 20% a^im''o^^f^eyrr'''r"o^V":?■"'""'^ "--"' ™ is 19.80. Find the snm. "*'" ''' **^ P*"' ""n"™. than 'b/lelTing' it'fr*"f. t ''\''"^ ^' '^^^ fo'' «210 price h/ve been » * •^°' "■''^' '>>««* 'he original atU!i:^;v1:4L^™ - «™^^^^ is it 4iel'risr'^]^5.dlhrz;;;-r '^ ^^ - «- 7. A retailer marks his ffoods it f? ,i,.i„„ a six months' credit price fnd ', tl '^ !?', '^ '='*'*'' P>''««. From his 12 months' -oWc; ]„/ , '""""'" '^''^dit price. 7*% for 6 raoMfe' cri t His r "."" Ir^ '"' <""«" '-^"d advance of the cost of thfgood "'S''.;?? '^ '°^^ \" STelltrTcrcarh"^ "" U^ — s't^ g^^ fell.' H^ii^trof^r d".!i&irr ^ -^'^-^ '- ^o- 2t(irfattonT^rl¥'^~-- proceeds in sugarlVeTer a S'^'R^rt:! """"' est ounce, the quantity of sugar I^'.^^ive"' "'''''■■ rj,^ days, how long ^U^'^ ^^^^t^^ 1 from X to 3 rate of 5 s per hour. t' hour and speed to 4 I B each to :k tahiin^ 84 'or freight. 3 sales for dieting-, for Lin of 20% scount on er annum, i for $210 3 original 3lock is it md their ish price, dit price. Jash, and is 40% in )w much it goods ; in Mon- n a coni- ission of s the net he near- d in SO end of ish the GEiTERAL PROBLEMS. . "• At ,vhat rate ,vm . ''"-^kecou and $6000 more at Ao)*^ ' ''ei-'^ured $4000 of it „t 90/ S.l£?s?iv-^|:'-,.,..oe.o. and B's time. '"^ '"»« « "WMhs. fioquJr '^^ '^^^ f f' !>' 15- Divide $1660 into 9 '^» g-.un on 0110 for 4 years -Q 10/ '"'"'*' '"^o tl'at the tn,f. .r hncj o , '^''^^ the area of a rfo-hf -/^^''^"^P^e interest) X\&'^- « ^'-4 an.rti;2ll^,-i:^:f^t^^^^^^ wopf-ii fio/ /•'^^J^ Tor f?57o. ( nnr? o)T • ^^^^"-'is Jie sold ^M^th|£Sf S^SJ^J;:; '--^ on a eorpo... '•fy, and one-fo„,TL n?n\"'"/ °'' '^''' "f a certain , 01 the latter ,.er',tu dTtl? ■f-'''"- «"" " e'^S He mixes them aiid^eii",!^ -^ "* ""'t of tlii f„,.„, ^ain pel cent. -' 'iWdUy. j, j^^^ j,.^ ^j^^. - 18 \f 'I tiWi M !ir;i 268 ARITHMETIC. 1. A vessel containing- 210 gallons of a mixture of coal- oil and water, of wliicli 75% is coal oil, leaks at the hot- torn, from which a certain amount of water alone escapes. On an analysis of it there is found to be 87|% of the re- maining- liquid coal-oil. Find the number of gallons of water which escaped. 2. A speculator holds $6840 stock, one-half in the 3 per cents at 72, and the rest in the 4 per cents at 05. If he sells out the first stock and with the proceeds buys more of the second stock, and also sells out the stock originally held in the second and invests in the first, find the change in his income. 3. A man removed from a town, where his income tax was 16 mills on the dollar, to another where the tax was only 7 mills on the dollar, and although in the latter place his salary was $500 more than in the former, his taxes were $6.40 less. Find his income tax in the latter place. 4. The ad valorem duty on 20% of the invoice price of a shipment of cigars was 40%, and on the remainder 35% • the whole duty being $810, what was the invoice price/ 5. A commission merchant sold 2 lots for $800, on one of theni he charged 2% commission for selling, and on the other 4% ; his total commission being $26, find the selling price of each of the lot. 6. A grocer sells one kind of tea at 24 cents a pound and loses 20%, and another kind at 42i cents a pound and gains 25%. He mixes them in equal p*roportions and sel's the mixture for 33^ cents a pound. What is now his gain or loss per cent? 7. A sets off from M to N, and B at the same time from N to M, and they travel uniformlv ; A reaches N in 16 hours, and B reaches M 25 hours after they met on the road. Find in what time each performed the journey. 8. If a grocer's pound weight is -03 ounces too heavy find his loss per cent, from its use. 9. Find the standard time of the City of Toronto, when it is 3 a.m. at Malta, if the longitude of th^ former place is 79° 25' west, and that of the latter 14° 30' east. brmer, his OENERAL PROBLEMS. .^n 10 A would buy 9 if paid ta eash '/ '"'"^ ^ l*»"ds for wh?a" cosUs his credit price?''- ^'"^ ">"«•> per cent above commi./oJ^„7V/" anrj"';'^ ^""^ '-^ house for me on « 19 A ^°°^'' ^'^11 fo'- ' *^ Pi-eniium. How much bou?ht1tXto7,raTd"''',!j'^ « 8% profit; if he had gete B to help-'hii,:' TW^Lr.* V" ^^ ^>'y^' '''"d then B leaves and A flnishes^tTn ,t°?f **"■ ''°'- ^ dkvs, when triple itseift';r;i:;7"' '"'^^^^'^ -«•-- ^m Of money tif ater„d?S'r-He i„.„rance of a house at ■"-111 receive the value of thfif ® ''""•'® '« destroved he ''nee and «125 besider Whaunhe'' f'""'™ "^'n-- 16- A bankrupt owes imn ,• "'*'"<' "^ 'he house' proportion of 3 to 5 1 'r»f'"''"'''' ^ «"■ Wh."t i-*. A merchant sends tiQw ■ ag-ent with instructions to seM . " T^ ''"'"^ ""tter to his proceeds, less his commiSons in t'/"'^? •''"'* '"^^^t the 5£o„the value of the S Zh^ Th« affent charges ston'='i:!Jje- °l'h« bufter'shfpV.f ff'?J„ 'o'"-:"'' ''^^^ ■"•>^"^"i'S CO 1x29.142 i-i— -J u aio tOtai comiiiis- I 11 iM.'l I- 'f • -il N 15 270 ARITHMETIC. 1. Pmd the Circumference of a circle whose area is equal to that of a sciuare, having a side 3 inches in length. 2. The rates at Avhich A and B walk are in the propor- tion of 2-4 to 3-7, and B walks 5-23 miles per hour. How !rtcYriIe mce ?"'^ '"'''^ ^^ ^''^'' '^ "'' ''' ^""^"^^^ ^''''^ ^'"^ "' 3 A speculator holds $10;);)2 stock, one-half in the 57's a Doi and the other half in the G%'s at 114. If he transfers ^'i)/:i '^""'u *^^''^ ^\'^ '' ""'"'^ ^"'^ o^iffinal 6% stock to the 5 / N hnd the alteration in his income, supposing he pays ^^ l)rokerage on each transaction. 4. Bought 3000 bushels of wheat at $1.15 per bushel payable at the end of a year. I immediately sell it at 1 nv ^^'' ^'^^^1^^^' ^'^sh, and put the money at interest at l^k.I^T.T''^ will I gain or lose by the transaction at the end of the year ? ^ 1 o' ?' /o?^^ ^ "^^®^* ^^^Pi^'^^ in the proportion of 7, 8, and 12 at 4^ per annum and realize $4050. If they in- vest It f.t 5|%, find the yearly income of each. 6. The Merchants' Bank of New York having declared a dividend of 8%, a stockliolder in Buffalo drew on the bauK for the sum due him, and sold the draft at a pre- mium of 11,//, thus realising $406 from the dividend. How many $100 shares did he own ? 7. I owe $719.02, and give my note for 60 days. What must be the face of the note to pay the exact debt, Avhen discounted at 8% per annum ? 8. What sum of money, with its semi-annual dividends of .)// invested with it, will amount to $12750 in 2 years ^ .i A merchant mixes 40 pounds of tea worth 371 cents a pound with 64 pounds worth 45 cents a pound ; he sells 21 pounds of the mixture at 50 cents a pound. At what price per pound must he sell the remainder so as to clear 25/^ on his whole outlay? .}u?\.rt- T'T^r^ ''"''''' ^''^'^ P^^^"^^ ^^^i^ht, which actu- ;o .fo7wr 1 ' "''"f '• ""^^ ^^^^^«'^ '^''^^ he cheat a cus- tomer who buys goods to the amount of .«!50'^ 11. A hatter buys 325 hats at $2.75 each ; he sells ■' of tZch '^^J^TT'" "^ f '/^ ."" ^^^^' ^^^ the remainder for ^^ each, i^ uia his rotal gain. 7913 miles, and at iCwO'i' I'k' "'"'''' ^'''""^'«'- '•« to one degree of lono^itX,;?, ^ '"'^ ^'1"''' "' '""K-tl, of tl,e „n,^ in inel^r J^v™ (f -'T''' """ ""^ '"'*^"" in^ed f ^n?:;;'^;""^;::;^ *-•; ^"'^-^ »' 2.%, and ... the prenfiuni paid bv«38 50 ' Tn"I" ''""'''"'""^ exceeded the risk? ' "J *^S-oU. ^\ Imt was tlie amount ol 15. A speculator inpr<.-ia,» 1.:. transferring $4850 stocTwml. 80 to" 5? /"?"'"" *" ">' 10. A grocei' 1 nv?„! .' i ''■^' "'" '°''""^'- stocl' «*'« "'nts of the book debts. He ,r vs r^ ! " '>ankruptcy are 5% the amount of his liiwiiui '' "'' ""^ ^°"'"-- ^'''"^ invcst„"em°t":.J,;^ r'rol'- dr'" ^'^ «^ ''"^"■«^' °" «>« bought to pay !.% interest" '^ '''''"''' " '""'« '« ''^ Of the stream is x niii; peVhour ? '*'''""'^''' '"^ *<> ™'^ an'dVysr ti;:i°^^.r,r "^cr ^^^^ - w. pays each man .^1.50 a d-iv .nui « f i ' -^ ^^ ^^^^- He How T»i«iiTr T ''^••^'-' '^ UcU aiKl each boy &>i ooMt^ <> rio^. now many ^yys were employed ? ^^ ^' II 272 AUirUMETIC. 11 r i II may ex- 1. A Iwmso -.nd I(,t tc^rethcr are worth 11400; one-iU'th o Mk' va.ne oi tiM, house, is (>(,ual to one half tho value of the hjt. Fmd Mio value of each. 2. A merchant sold a piece of cloth for $24.60 and thereby ost 18%. What would have been his ^ai n per cent, had he sold it for §3(J ? ^ ^ -tf)«rK/^'";p^'^'"'^'''^'' """"'^^^ ^^ ^•^''««« and cows for vl08(U There were 5 tinu^s as many cows as hor.es, and a horse cost 4 times as nmch as a cow. If each cow ^st .-JO, how many horses did he buy ? 4 Eifrht nion can do a pierce of work in 40 days ; after workiu^r .^;, cays they are joined by another man, and the whole work is completed in 31) days. What fraction of the ANwk IS performed by the last man in a day ^ o. Divide 8.) into two parts, so that the greater ceed the less by 41. ^ in ^V tf !''?r ^''"! '' '''"' ^^ '''^"^>^ ^* -^"/o ^""P^^' interest ; n 16 years the interest amounted to m less than the sum lent. \\ hat was the sum h iit? 7. A person invested |2804 in the 8 per cents at 95^ and after receiving- the half year's dividend, sold out at .'4A. Jnnd his g-ain or loss, brokerage being 1% 1 ^'iT-''^ f^ ^'""'l'"'® ''*' "" '''^'^'^ circular cone, whose height IS 4 feet, and the circumference of the base 9 feet .>. Aspeculator sold a Jot gaining 20% of the cost. What pel- cent, of the proceeds did he gain ? 10. A dishonest merchant adds 5 gallons of water to every 20 gallons of coal-oil. How much does he cheat a cu^stomer who buys 15 gallons at 15 cents a gallon ^ 11. Ihree men start together to walk around a circular coiirse 60 yards in circumference, their rates being Hi 1 !^, and 16 yards per second, respectively. Find when and where they will first be together again. 12. The hour, minute, and second hands of a clock are on the same centre. WHien, first after 4 o'clock, AviU the second hand l>e midAvay between the other two ? 13. A b.-inkrupt's liabilities were $3000, his assets $540- the assignee charged 3% of the assets for his work How many cents on the dollar is the bankrupt able to pay his creditors ? "^ CiENF^UAL PUOBLEMS. 273 miles. ()„e ton- .Me ^t,." f,I^^"" "" '•""' '^'"■"'"o is :W the rate of 4 mi ,'u,',!,^'"'V '""'''"" ""'' "-'vebat Toronto at the « , ,/ „ '" ,V\';,"« """"""' ^'""^ ''•">" miles an hour It ,v ,, , " ' '""''''>' '" "'« rate ofiii tliey Will met. """"" '" ''"""■ "-I'ere an.l whet Hftl-r .tr.iw;;;;.^'4;:";,t' *'''''i''l" •'''-^ ^^-i^ "' 9^?. "..d Find his ,.,in "r s on lufw f ,"?'"'"' '^'"1 °"' '^' «5 17 Fiml fP„. ,i!» , "''"'" transaction. (2) lialf-yearl/ ™' ^'^ '""''■««' Payable yearlj-, 18. If a number he incrfmpfl -xw j , be increased 25/, the result ti k^^ '"'"'' """ '""ount ber. ^'" ^ '''™" ^"11 be IC 1 uul the niim- 19. A g-rocer intended to »i'l- 14^., i-^..d Where the;''';;„"«^.:,t;n:',;r'' ""^^ '^^ ^^^ " '« dends of?f "™ rti'' '""=" H"'' P'^y^ half-vcarly divi he .nake o,t hfa moln- ;r""'- ^^''^'^' '■••'*« °f interest cl^es ^-ek a^ain in 51 ho.;." ''" ^^^j ;^ ^f^ \ '^^"^"^ ^'"^ qnarter of his mto in still wl;^ / .) '^'"^''"^ ^"^ ^'^^^ each way. ^^''^^'' ^"^^ the time in going "f votes polled, and wt ^^ ^.^trt'l-f "^ '"".'"'"l ""■"I'er What percentage of voters did not vote'^ "'''■""■"^^ "*' ^^"^^ ^. IMAGE EVALUATrON TEST TARGET (MT-3) & A :/. i/j 1.0 I.I 1.25 |Sb 13.2 M 2.2 M L8 14 IIIIII.6 Scmces Corporation 23 WEST MAIN STREET WEBSTER, N.Y. 14580 (716) 872-4503 i « i\ <> # '%' % ^. .4i 0^ #3 ^. "> i/.x '^ i 2r4 ARITHMETIC. IJ: 1. In a 100 yard race, A can beat B by 5 yards and C by 10 yards. If B gives C 5 yards start hi a 100 yard race, which wins and by how much ^ $100 tr"^ ^ ^T^^ '> ^''-'^^ ' ^ P^^^ ^"*« the business |i00 for 6 months, and B a certain sum for 7 months When they close up the business A receives $640 as Ms his''' w/'.' '^'"^ ""''f P^^«^^' ^^'^^"^ B receives i510 for his. Wliat amount of capital did B put into the firm ^ d. A man lias a net income of $745.80 after paving- an cept $400 of an exemption. Find his total income stock1t?.?n ^^^.^^T' f ^^^ ^^ ^% ^t^^k ^' 70 for 8% stock at 120. i md the alteration in his income stiU w.^of T '""'J '^^ '^' '^'' ^^ 12 ^^^'^« ^^ hour in still ^vater. How far may it go up a stream which is cupy 7 hrs. 24 mm. for the round trip ^ 6. A man has 50, 25, 10, and 5 cent pieces, the number tot'al v\l.Tif '%w\l r ^"^'"^^ 1 1' 2' '^ ^^-^ 4- If 'heh- total \ aline is $10.50, how many has he of each '? avoirdiinnl^^'^^n^'"'''' '? ^''"'^ *'^^ "^^^^'^^ «f ^'^ ounce avoirdupois. How much per cent, does he lose ^ 8. A and B work 5i and 7.^ hours a day, respectively for Tytork r'l Til 1 "" "^r ^'^'^ ' -^ theTeco'S ^n 08 ?i t^^ ^"f ^^}'''''? ^ '^^y' i-espectively; they receivl $13. 98 for their two days' work. How should it be divided ^ sold ti"'"'"^^''^ ^''' 1^% of certain perishable goods and ?n A T. , '^<=7 lat advance on cost he sold them. ill, 'v- P " ^/» '®''* '^»'' material and 13°/ more for ; dS' VuT''' 7r"^ ""™ '=°^' """ «^20.7^ less hS^ It did. AVhat was the contract price ? after dTiuctln^'ln' • '' '"^^'^^ '"'^^^^ ^^ ^% ^^««k, so that, arrei aeauctmg an income tax of 27, it inav viold fi°/ interest on the investment ? ^° ^ ^ ^ ^/- GENERAL PROBLEMS. 275 an ounce 13. A man can row 5 miles an hour in still water How tar may he go clown a stream which flows at the rate o l^^Z!:^^^ ''-'' '- -^y ^-^ -^- ^ !-«• ^^nin ^,'^IX =^ ^r^he^ir Ks!^^A^ the total loss sustained by the company ^ ' ^ business It is found that A is entitled to mTo "LZ^t and proHts, ,vhile B receives #456.50. Find the timTA left hjs money in the ' Irasiness. ^ PoiLMs"v«r«4l->'''-'f., """"^ ^'■'^'^" '"^ ^^'^-W^S 13 IJonnub IS \\oith H-'.>i ; if tlie proportions of each are in terchanged the mixture will be worth «4.44 m,e bkc -" ea IS wortli .30 cents a pound ; find the price of t e slct Te Sa..^ ''^ ""'"'^'=" "'-^"""^^ Of eacrkfndin in ^y aajs. If the work is worth 1120, and one ,r,.,n rZIa'tr '"^ *'"^ '"^ '''''^'•' ^''^ «>^ouMl,i:"nion;y 18. A man sold a lot for $512 g-ainina- 15°/ of ,h. iTonhe cl^t J ^^""'^^ '"^ ""™ -'^ *' ^°" '-• '^ ^aintd ehifedtt^i^Ti: a\Sf:'eetii\:^^btr';.::r.r d^-- dend, he sells at an advance. If l,c ^ains Im „ T Findtte'-a'ir''''""^ '"'''"'' "" *''° '■"^ 2 years is «50. wo^/' '■''^ ^'s^'^'ns of equal dimensions are filled with water and the taps for both are opened at the same thiie otnei 111 7 hours, find when one cistern will have 4 timB= as much water in it as the other. "^ 22. Four-fifth of the selling: price of ^oods i? ow io„ 276 ARITHMETIC. If' ' f'll: 1. A person set out to walk from A to B, at the rate of o miles an hour. • When he had travelled 1| miles he was overtaken by a coach from A, which was 10 minutes later in starting-. At a distance of ll^. miles from B he met the coach returning from B, where it had stopped 30 minutes. What is the distance from A to B ? 2. A man has 8 hours at his disposal. How far may he ride in a coach which- travels at the rate of 9 miles an hour, so as to return in time, walking at the rate of 4 miles an hour ? 3. Divide $140 among A, B, C and D, so that B may have three times as much as A, half as much again as A and B together, and D as much as A, B and C together 4. If I exchange 75 i-ailroad bonds of $500 each at 367 u "^ ^J!!' ^^'' ^''"^ ^^^^^ ^* ^% premium, how many shares of $100 each will I receive ? ^ 5. Find the compound interest on $40 for 9i years at 8/ per annum, interest payable half-yearly. 6. A cow is tied to the outside of a square enclosure (GO feet to the side) 20 feet from one end by a rope 100 feet long. Find how much ground she can procure grass from? ® 7. If the cost of an article had been 24% less, the gain per cent, (at the same selling price) would have been 3^ times as great. What was the actual gain per cent ^ ' 8. The length of a room is 25 feet, the height I'^i feet and the area of the floor is equal to ^ of the arealf the 4 avails. How much would it cost to carpet the floor at 80 cents per yard, if the carpet is 31- feet in width ? 9. Two trains start at the same time, the one from A to B and the other from B to A. If they arrive in A and B respectively 4 hours and 9 hours after they pass each other, compare their rates of travelling. <^^}.nr. ^J^'^'^^^'^^P^ o^^es $6000, his assets are $4500. If ^LOOO of the assets are secured to the creditors by a mortgage, And how many cents on the dollar he can pay the other creditors, the assignee's charges being 5°/ of the total assets. » /o 11. Stock wnich sells at 12% premium, paj^s 8°/ divi- dend. What interest is made on the investment ?'° GENERAL PROBLEMS. 277 them atata'^f 9y'"'B?^rf * "^ '°' "' ^^' ^"^ * "^ raise tl.at selling ^rieef„o"te"rh«rh''''' T" "l"^' "« iiiainder at the increas Jl ,-^ „ ^ ^ ^''"'"^ ""^ ™- whole transaction"^ ^ ^^ "^^ ""^"^^ *% «» 'he poit.^d'''''^ter',lIl,'?r" "-^^^ "^ "»«'^ as B already "r^'^/reH '-=-"«- «rp- -- ^^ - lii. It the cost of an article had been 10°/ 1p«« fi,^ • that one train has^rwe^?!? " .oj'"'",""'^ '"«"' " '« fo^nd fs T^r^-*""''^ " '^'^ ""-'"J^d among- them ? ^ ^ rate per cent is ^iT^ .L i o^ ^"""^ "* >^^^^*«' '^^ ^^ <^ertain investment "^41 lilli^LIT* nf'^ '°^ ■"'«■•-' ^ «>« 12% per annam, mterest payable everv 2 months' water aTd the?;" 1 f'?' ''^^"-""^ a-'e « led with i?the\:t.trVi i^ro^irrb^' ■•^' '"rr^ '"'-• than the large one, in going a mile ? '''" '"'""^^ I!'' iff JHIiil If a Is 'if 1 1 pi M. ,; is ^ 278 AlUTHMETIC. ni'., A "F;,'S'^» of A, Band C, by knowing that C's ag:eatAs birth was 5i times H's age, while now it is equal to the sum of A's and B's; also, if A were 1 year older his age would be | of B's. <- ^ yeai 2. If tlie cost of an article had been 25% less, the sain per cent, (at the same selling price) would^'hav; beefsi fines as great. What was the actual gain per cent! ? * i. The circumference of the fore-wheel of a carriage i, n*nce1wirtho"?'"'"," i^'""^/^""^' " f-t In Xt^di - lhni"d'-'w!:efr""'"'' '""^'^ ^50 revolutions more than attsO^orlx'st^^kSTs-/™"'^'"^ investment 5% stock per nS,tt:?:r7;ib;rotrr ' '"'• ^ ^^^--^ '•"i^" and tTcTnf nf \°''h''™' dimensions are filled with water, and the taps of both are opened at tlie same time. If othei in 6 hours, find when one cistern will have 3 times as much water in it as the other. . 7 A grocer bought sugar at 10 cents a pound • he sold us:d7;ot'/ '"^'/f ""' "^»"^'''' ->"' l^y " niiWake he ^n,rr!f * ', *?" '"' intended. Find the number of pounds of sugar he bought. and thl tl.?^'* '''•''"' -^ ^" '"■'"''''' ''«' diminished 10 cents, and the selling price increased 10 cents, the gain would be increased from 16|% to 60%. Find the cost y. A and B walk a race of 30 miles ; A gives B 50 mmatcs start , A walks at the rate of 5i mite an hou " bv who? d'%^ ^* T •'",•' °f -^ '""«^- ^i'll B'^ rate, and by what distance he is beaten. 10. A clock which gains 6 minutes in 24 hoars is 4 mmutes too slow at 7 a.m. Monday. What o'clock wi it ing Tuesday ?"'""' " '"""'"' '" ^ P"^' »° "^« f"""^" nfi rtlnm'^f '■""^'J "^ '"'?''•"■ ^^^ "^"''^'"^ i% brokerage, 136 (1100) shares of stock at 68f . Find what I realizld^ Fiid the':aTe'prcfj."'"'"' "" *'-" ""' ' ^^^^ '^ *««• i li GENERAL PROBLEMS 279 what iouW uTe forTrtlf .^'' ^'' 1 '"''^'^ '™« - «5, • 17, Bought ,^:;L"Vf'ru™zVonrarsio':v. , and the rest at «6 a barrel ; thrwhoeco^^wa-S How many barrels of each kiml did I buy? *™- B it 2 oVln";? ''■'"'" '■™"' ^ '^'^ « »'«'«<=k '^•■n- and reaches and reaehe: A'arVpT"'!'; '''r. ''''-'' f-m Bat" meet ? P'"- ^' ''''"'* *""« «f «he day do they 19. A bankrupt's assets are 1 of his liabilities h„t i „<• the assets prove to be worth only 4rcents on ,i,e li! How many cents on the dollar e.fn he pay ? " "°"'"■• tiofanl.r;,f;Lrndtt-r;T>.3^^^^^ sinalleft by $47 50 'o thaTthe' ?''''?"**•' ''^'=<'«'""^ '^e 2-rpntP«t ■,/«<>/' ^^ *"** simple mterest on the ^r^:th:s'i£rrrf^//err^r%Tr on the third at 5% for 3 years ^ '" """ JIh ^rT htti: t1LXV!r VKa« %^^ more for the house and 207 less for the h«^-n h.« A have cost «248 more. Wl4 did he pay fo "eaoh^ "°""^ 15 TtTiJlf,.^ ^■''"' "'' ^"* '" todays, Bt'l2 C in before completion. In what time is the 'work don7? ' 280 ARITHMETIC. ^ r 1. A man sold two horses for edii-ii mnnn,^fc< • • 10% on one and losing 10% ontkTotCThe^ZZ on the whole transaction, find the cost of eael, hm^e S" 90 bml/n""'^' "' '"* """ « ?"""''« of sugar are worth depre'cTtes .y" IT" '""^T"' ^5% in valuS, and the lea rer' poTnd of £h^ ^"""^ <=''^' «2.90i. Find the price Piets"^ *"« thlwtn.''^ Pt"^ '" twenty-five and ten cent pieces. It the whole number of coins was i') hr.™- „,„„„ coins were there of each kind ? ' ™*°y ;L .7- J , ' °y "'® sucond alone in 200 minutes and 1,^ be emptied if the three taps are opened together? land £7^0 '"hT ™P°'''' ^"^^'^ ^"ich cos^t him in Eng- S l^fm tfrouTan V'nt^f ll^r ^f fi°^' ^"' fn^ft^S '- -"^"^ '--"hlTin (LtngTb"e! pJ4>rbSyi^^^^^^^^ ttn^ti^n.""'^^^' "''° ""^"•^^'^ *% "rolcerrge^ o-lt ffo^sVorThl""" '"«i«h'i.nt sold a certain amount of m"^! of 5°/ «n^HT^'/ '",''"='• •'<' Inducted his com- WsemD'-Lr'^'f^^t""""*'*"''^ '■^■""'"'i the balance to uis emp'^yer ; for his promptness he reoeivprt IV ^„y.s.i! amounted to $6 414. Who/ ^i,i n.„ *"'^'^™i%i which transaction? *■ ^*' '*"' *« agent realize on the 9. How much water will dilute 9 o-nl i «t t ^ ^ alcohol 96% strong to 84% ? ^ ^ "l'- ^ P'- <" 10. A grain merchant "spent a certain snm nf m/^„„ m the purchase of wheat, s'times as much nlrie? aS $1500 m oats. He sold the wheat at a loss of 67 Z. barley at a gain of 9%, and tlie oats a a Vafn of 20y l» ceiving altogether $9390. Find the sum Lf out In wl'eat OENEUAr. rUOBLEMS. 281 12. Two men start together, and travel in the same direction ; one is going at the rate of 4 miles an hour Jind the other man's rate, if he is always equldistani from the starting point and the first man. ^ Id A and B can do a piece of work in 8 days • B and fheVtotf ^ "-1^ ^^"^ '' ^^y'- T^« ^-rk is done by the 3 together, with the exception of B not loininir the otjiej-s till after the first 3 days. In what t^merJUeloli' 14. A watch, which is set accurately at 7 o'clock a m mdic.tes 5 minutes past 8, at 8 p.m. of the same d^y' 15. Which is the better to buy flour at |5 a barrel on e^months' credit, or |4.87^ ca.h, mono/ being wortU T^'}fL^'''''^^ $100.75 between A and B, such that 217 of B s share may exceed 8% of A's by $3.39*. ^° 17. Property depreciates annually 8°/ for 4 years and at the end of that time is worth $4197.6U What w^rt. original value ? ^ ^^t.^i^. wnat was its ^ 18 A merchant buys sugar at 5 cents a pound • in sell ing It he wastes 7%, and 20% of the sales are bad deb " to I?n 2^7'o?.;' ^Tf "^^^^ '^ '''' ^^^ remainder so as to gain 24// on the whole tra^isaction ? 19. The capital stock of a railroad is $895750- thera.* sender earnings in one year were $74537 50 and fhe s'ofor.m^'T.,*^*^"-^"'- 'he disbursements were $107963 00, and $7397.00 was placed in a bank to tl p credit of the company. What rate of dividend can th^v .^^ mo.c uiciii ii dia. What was the cost of the house ?"" 282 ARITHMETIC. ';Ji' ! : i III': 1. A man holds $15600 stock worth 60, which if lie transfer to 4% stock at 78 he can increaire his annual in- come $12 ; before he could effect the transfer, each stock increased 2 in price ; find how his income is now altered. 2. A merchant buys goods for $304.50 on a credit of 3 months. At what price nmst he sell them^ on a credit of 8 months, to make a ready gain of 25% (money beinc worth 6%) ? /o V J b 3. How may a farmer mix 4 kinds of wheat wortli 75, 78, 93 and 100 cents a bushel, respectively, so as to forni a mixture worth 80 cents a bushel, and have ecjual quan- tities of the first two kinds, as well as of the hist two ? 4. A merchant marks his goods at 10% above cost, but throws off 2% of this price for cash. What per cent, above cost is his cash selling price? 5. A has 54 five cent pieces and 1 dollar; B has 54 dollars and 1 five cent piece. What sum will have to change hands so that B will have exactly 10 times as much as A ? 6. A merchant received an invoice of merchandise amounting to $618 on 4 months' credit, off which he was offered 4% discount for cash. If money is worth 9% per annum, how much cheaper can he get the goods by this offer ? 7. A can beat B 3 yards in a hundred yard race, B can beat C 9 yards in a 300 yard race. Find how much A can beat C in a 500 yard race. 8. A man in England invests a certain sum of money in Canada at 6%. All but $400 of his income is taxed a'^t 2%. If he pays $18.28 taxes, find the amount of English money invested, exchange at par. 9. A has $150, and 7% of his money exceeds 9% of B's by $6.09. Find B's money. 10. A, B, and C form a partnership, A contributino* $3500, B $4000 and C $2500. They agree that $800 of th? profits shall be placed to their account in the bank, and the remaining profits divided in proportion to the money invested. At the end of the year A's share of the divided profits is $2100. Find the percentage of profits realized on the entire capital. 9 m at th 11. What will [ GENERAL PROBLEMS. 283 .onths' credit; ^iJulr^iy^ET'^^i^F ^^^^^ invoice J i'm V II "ff thorn immod on iately for cash I'lt! invoice price mm^r.^r\ • """i^wmi 12. IfH'iy nf H ' "^^^'^y Ix^iii^- worth (>7 ? 13. A man Lorrowed a smn „/, ' "* ''''"'='1 '' ^™s "old. and invcted it in the 7 per ccm» "rf 4" 7' Perannun,, the year he receives the dTvM^d , °- ^' "'" ''"'I of out at li>5 ; lie find, h , f ^'V'"*^"''' «"xtuie of 144 pounds, worth 40 may'retver ttvXe of'trr" '"'' '" "-« "^ 'oss he insurance at 117 The [,0 1 ""T "'"^ *''" ?'■''""«« of of the elaim iltllowed Zl'", destroyed by flre and * ess than the value ^f the ho,t F^Vh '■«<=«i™s $756 house and the premium "^ "'® ^*'"c of the his'studteTTn o2Z Twlf!"' t? "»«" Progress with Fraser studies only ^ houS^';!; f ''""'''>'««," i" W, but On this basis, how should a toW TIV^ HamWy does, awarded to each ? °' ^^^ ''onor marks be hoJs'e wL"L"i"elr:r9 iL" "T' on a Journey on the same place and fdlows M n L'^?""'"' ^ '"'' «>« f™m rar^rX:r»£F'-"-^^^^^^ -'^^ =d it^Ta^to^lS^^- ^!l^.-/-^i deduct- ing yard v 19 expenses. For what w J^s the silk sold per 284 AlUTIIMF/nC. % ^ I 'I mi! 1. A, B, and C ran a 100 yard race ; A heats B by 1 yard, and C by 2 yards. By how many yards can B beat C in a 100 yard race ? 2. A man in England invests a certain sani of money in Canada at 6%, from which lie receives an annual income of $870.3G, after payiufjc an income tax of 15 mills on the dollar on all his income except an exemption of ^KX). Exchange being at par, find the amount of English money invested. 3. A note drawn on August 3rd for 4 months is dis- counted on September 12th at 9% per annum. What rate of interest is charged ? 4. Divide $338 between A and B, such that 3 times A's share may be $127 less than 4 times B's. 5. I owe $539.94 and give my note for 00 days. What must be the face of the note to pay the exact debt, when discounted at 8% per annum ? 6. If I own a vessel valued at $7791 and wish to insure it at a premium of 4;i% so as to recover, in case of tlie destruction of the vessel, both the premium paid and the value of the vessel, for what sum must I insure ? 7. A merchant buys 2590 yards of cloth at $1 a yard. He marks it to gain 25%, but in selling the first half he uses a 35-inch yard measure, and in selling the second half a 37-inch yard measure. Find the ditterence between his actual and his intended gain. •8. A and B form a partnership ; they carry on business in different places. A alone makes $350 total profits, 25% of which goes to pay his expenses ; B makes $375 total profits, 20% of which is used for his expenses. How will they settle, if they agree to divide the net profits equally ? 9. A grocer mixes teas worth 32, 41 and 54 cents a pound, respectively, forming a mixture worth 43| cents a pound, which contains 12 pounds of the third kind and equal quantities of the first two kinds. How many pounds of each of these grades of tea must he use ? 10. A speculator invests $9240 in the 3J per cents at 82|, and, on their rising to 87^ sells out and invests the proceeds in 5% stock at 112. Find the alteration in his income. GENERAL i'llOliJ.EMS. 286 3 times A's 11. An owner of a house worfh jRifSu^ di • %, so that in case of hJC^IfH il^'A^'^'t' "''"'^' '^ ^* ihe\ m. so that in case on«ri,o wm ***"'"• '"^ '"«"'«' " »' house, the premium nf I ' '^*''°'''''' '^« v»'oe of the What'was thVa Zto i "uraSe"','' ^V'^ ""^'-^^^ 12. A can do as null, wo, IT, a ,''''*''''''''" '*'«''°»««^ 11, but he does „ot " o, k the.!^.,. I '"""? "'' " '""' ^o in and B earns *30 00 el.mre 1 . ""'• J^ ^ '^'"•"'' ^-''''^ 13. If a company "S'a,t''r'.''''i'': '^"'•'' ^°''k«d. HZ, and reinsu,-es S ont Tn S™' "'^ "*' *=*"00 at what will the llrst Lllnya^"V fT'T""^ *»' -'1%, no accident occurs ? ^ * ''" transaction, If 24"enfs fp'^nd"'' Htil'"; ""T '■" r"'""<>' ''"<' -^^e at of cofifee than of te l,e 1,, h' ^ '"'" ^ Pounds more equal. What were he o,,.^? "' """ day ? 25%, and sold it at fts o^To"/ J^f,?.','"""^ '' ''"'y °f more, he would have cleared IV ^"^ ';" '""' '' '■»'• *26 did he receive for the waS? '^ " '"' '""•^'""- ^hat C 40 yards'tiSc;' "TJ '''"',' ^ ""'"^ ^^ y«"l^. and mile rice? ' ""'^ "'«''' would B beat C in a sow ■them°SeSlytrl3-/V" ^.H™"""'^' -«<*"• -<» credit as made my mmXfe o.-llo»/ '"tJ" '''.'"^'^"'^ "^ 18 give m \^. dmte ga.n 20^ How ong: credit -^^si^:^r!::^:z^^L. ., of andistr^Von ^ Xr'"'' Ah^'^'"'"^ ^0% «" «- gained |(on 4eVholet:nsactir "'' ''"' °'<''"^''' '"^ for'a'day Wk"'and 2^'' V '" ""^^ '•''«<"-« W''.40 or 10 boys what doe, 1?., "''"''' "' """^li «« 3 women tively, rL;.>e ftr'^daTs ^ork'.""'"''"' """^ "^^^ ^^P^- he solJ:?8Tcen7atrher'?H"^''<^'^ "^ "heat, which for selling it. S? investd 'th? f ''""'"'' 3% commission mission of 2% calculalefo^ It P™'?^^'' '«^^ a com- cs^cice. i^ina his total commission. ''''-^'^^^> i" real / 286 ARITHMETIC. 1^ m 1. If salaries above $1000 are assessed for the full amount, and those below have $700 exempt, what is the total salary of a man whose net salary is $2.78 less than that of a man whose total salary is $995 (the rate of assess- ment being 18 mills on the dollar) ? ?. i sold goods to B at 12% profit, B sold them to C at 8% loss, and C sold them at 5% profit, realizing $81.92 more than their first cost. Find the average rate of pro- fit on the goods and their first cost. 3. A grocer mixes sugars worth 6, 7|, 8, and 10 cents a pound, respectively, forming a mixti re, worth 9 cents a pound, which contains equal quantities of the first three kinds. In what quantities may he mix them ? 4. A merchant bought a quantity of goods for $340, on 3 months' credit ; how must he sell them to gain 25% ready money, after giving 9 months' credit (money bcin^ worth 8% per annum) ? 5. A man owns $29680 stock in the 3 per cents, which he sells at 82 ; he invests one-half of the proceeds in the 4 per cents at 106, and the other half in the 4| per cents at 112. Find the alteration in his income. 6. A banker discounts a 143 day note at 10% per an- num. What rate of interest is he charging for the money advanced ? V. Bought 600 barrels of flour at $7 per barrel ; | of which was to be paid in 6 months, and the balance in 9 months. If I paid cash in full on the day of purchase, find what I must pay, money being worth 6% per annum ? 8. The wages of A and B together for 14| days amount to the same sum as the wages of A alone for 25 days. For how many days v/ill this sum pay the wages of B alone ? 9. What must I ask for cloth which cost me $1.20 a yard, so chat I may fall 10%^ and still make 20%, after deducting 5% of the sales for bad debts ? 10. A grain dealer bought wheat at $1.40, barley at $0.50, and oats at $0.39 per bushel. In what proportion may he mix them so as to sell the remainder at 76| cents per bushel on a credit of 6 months, and make a cash gain of 20%, money being worth 4% ? n : GENERAL PROBLEMS. 287 or the full vhat is the '8 less than be of assess- em to C at ing 181.92 ate of pro- l 10 cents a h 9 cents a first three Dr $340, on I 25% ready >ein^ worth ents, which Beds in the [^ per cents 10% per an- r the money larrel ; | of alance in 9 )f purchase, )er annum ? ays amount "or 25 days, wages of B me $1.20 a 20%, after ), barley at t proportion at 76| cents a cash gain lavoiame turn, lie increases tlik rii-i<.,. av j s " '"uie the latter price $22 Wtnf w-Vf ff 4: ''"'' «l«a''s at 13. A huckster bought baananr^l "' ''" ^""'^ ' cents, and retailed tlitS ,u I e ?ato ot 20'''"'%°^ ''/"' * for the fli-st 12 dozen on tvhlh IT *",^0 .c«nts a dozen lay, the renuunder^SlcTold af ^tZZt ^S^'- T gain per cent, on the whole transaction^' "' '^"^ '"^ 14. 1 sold an article so as to a-ain 8V ' tc t 1, ,1 .t for 8% less and sold it for 48^cente ie,. t\. ''•i'T'^'" gained 16%. Find the cost price ™"^ ^^^'^ of Wstof'A^^'stl'nomC?"''^'?"'^ ^"■* -* -P''»' oTZ/^ c. 5;inr"Ts ';ir;^% 2"'r^/r„d^ bu'siL'f °- "^"^ "" ^'O'^'' -"-" -eh P.U inti The incote''rSro;i8';nills":n the'dLr'' *'"',t'''^'- P^>-^ -^ cpnf- ^ic\(\ -xv^ \. .} ^^^ dollar on al h s salarv pv eept |>400. \\hat was his salary ''' -t^diy, ex- 17. By buying 3% stock at a certain nnW T « ^ .u after paying an income tax of 183 n?in^ \^"^ ^^^*' net income is ??83 1 •>J T^in^ f,^ ^ ^" ^^^ ^^llar, my nmke 3i% interf on^my m" """^"^ ^^^^^^^ ^^ I capital 30%. How should a gait of 81858^^^^^ between them at the end of th^ year? ^' ^'""'^^^ 19. A commission merchant chare-pd ocl debts $17, and There times as of each for $25, 50 a day onth for and $2 a sold his le. v^eight of et wide, • Aveig-hs s, and 7 dll keep r all the em to C, %, B 8%, !S in suc- , costs G d. Sup- that the which is 11. In a shooting match, a bull's eye counts 4, a centre 3, and an outer 2. If a company of marksmen, consisting of 20, fire one round and score 40, 6 misses and .: bull's eyes being made, find the number of centres and outers. 12. Divide $2567.50 among A, E, C and D, so that A's share may be to B's as 4 to 5, B's to C's as 6 to 7, and C's to D's as 8 to 9. 13. Two men and 5 boys can do a piece of work in 20 days, 1 man and 8 boys can do it in 18 days. In what time can a man or a boy do it ? 14. What is the difference in the expense of fencing 2 fields of 25 acres each^ one square and the other in a form of a rectangle, whose length is twice its breadth, the fence costing 62| cents a rod ? 15. A commission merchant's charges, together with the money he invests, amount to $1700. If he charges a commission of 2% for investing the money, and his ex- penses amount to 17|% of his total commission, what clear profit does he make ? 16. If hay costs $12 a ton, and oats 40 cents a bushel, what will it cost to keep a horse from March 10th to Oct. 25th of the same year, supposing that the horse eats 24 lbs. of hay and 3 gallons of oats daily ? 17. A merchant in Hamilton buys 25 tons of a cheap grade of coal for 75 cents a ton. He pays $1.75 a ton for freight ; it contains 25% of bituminous coal, on which he pays a specific duty of 60 cents a ton, and he also pays an ad valorem duty of 20% on the whole quantity, reckoned on the prime cost. Find his gain by selling it at $3.50 a ton. 18. A farmer has 14 bushels of wheat worth $1.50 a bushel, 19 bushels of barley worth $.48 a bushel. How much oats worth 34 cents a bushel must he mix with the wheat and barley, to make a mixture worth 65^ cents a bushel? 19. A Toronto firm h owed $4020 by a merchant in Vancouver. A draft t.. che amount on Vancouver is at J% discount in Toronto, and a draft on Toronto bought in Vancouver is at |% premium. Which is the better way to cancel the debt, and by how much ? 292 ARITHMETIC. •^^i ( ' I [ i \V { i I ill 1. A person has a sum of money to invest which will buy $1200 more 4% than 4|% stock, v.hen the former is selling at 87 and the latter at 99. Find the difference in the incomes from investing the sum in these securities. 2. A can do a piece of work in 8 days, B in 10 days, C in 12 days. The work is done by the 3 together, with the exception of B, who quits work 2 days before it is com- pleted. In what time is the work done ? 3. What rate of trade discount deducted twice is equiva^ lent to 33«| % off. 4. Water expands 10% in freezing. Find the weight of the ice stored in a building 40 feet long, 32 feet wide, and 17 feet high, supposing there is no ice above the plate^ and I of the space is filled with sawdust. 5. A grocer sold a quantity of sugar at 12% advance on cost, and gained $40. Had he sold it at 15% advance he would have gained $50 ; find the cost. 6. Divide 635.75 into two parts, such that the simple interest on one part for 6 years at 5% may be $29.37 more than the simple interest on the other part for 4 years at 8%. 7. In a constituency, 26^% of the voters refused to vote°; one candidate polled 70% of those promised him, the other polled 75% of those promised him, and was elected by 150 votes. Had they each polled all promised, the suc- cessful candidate would have been defeated by 200 votes ; how many voters were there in the constituency ? 8. A can earn $10 in the same time that it takes B to earn $8, but at a piece of work on which they are both engaged A works only | as long as B. How should $55 which was paid to both be divided? 9. A building society borrow $30000, and with this sum build 12 houses at an equal cost. They sell 4 of them at once for $2800 each, and rent 6 of them for $25 a month, and the remaining 2 are idle. If the houses which they hold are assessed for their full value at 15 mills on the dollar, and they pay 5% per annum for their borrowed capital, find their gain or loss at the end of the year. 10. Find the amount of money a person invests in the 3 per cents at 84, so that after paying an income tax of 2 cents on the dollar, his net income is $102.90. GENERAL PROBLEMS. 293 ich will irmer is fence in rities. days, C with the is com- equiva- weight Bt wide, le plate, ance on ance he simple 57 more ■sat 8%. to vote ; le other ited by the suc- ► votes ; 3S B to re both lid 155 th this ;11 4 of r ^25 a 5 which tills on rrowed ir. 1 in the ax of 2 11. 14% of a shipment of goods were admitted free of duty on account of damage received, and on the invoice price of the remainder a duty of 26% was charged ; the duty amounted to $111.80 What was the invoice price of the shipment ? 12. A quantity of goods invoiced at $1278, cost me in store $1452.38, after paying the duty and $14.63 for freight. What was the rate of duty ? 13. A wine merchant has 35 gallons of wine worth $3.25 a gallon. What quantity of water must he add to lower the price to $2,60 a gallon? 14. For what price should a man sell a lot, which costs $550, so as to gain 12% of the proceeds on the sale ? 15. A farmer has oxen worth $45 each, and sheep worth $6.25 each. The number of oxen and sheep being 35, and their value $645, find the number he had of each. 16. The compound interest on $5000 for 3 years is $1298.56. Find the rate charged. 17. One-fifth part of some goods was destroyed by fire, one-third of the remainder Avas sold at a loss of 5%. At what increase per cent, on cost must the balance be sold so that no loss may be sustained ? 18. A person sold a house for $700, gaining 12% of the proceeds. What would he have sold it for, had he gained 12% of the cost? 19. Max Pierce rents a farm for $300 a year. If he does not pay the rent for 5 years, what will be the amount due, interest at 7% ? 20. A tailor bought 50 yards of broadcloth, 1| yards wide, but on sponging, it shrunk 5% in width and 5% in length. He bought fiannel If yards wide to line it, which shrunk 1 yard for every 16 yards in length, and 2| inches in width ; how many yards of fiannel are re- quired? 21. What is the diff'erence between 60% discount, and 20% taken off 3 times ? 22. A, B and C engaged in trade with a joint capital of $2128 : A's capital was in 5 months, B's 8 months, and C's 12 moiiths. A's share of the profit was $228, B's $226=40, and C's $330. What was the capital of each ? 294 ARITHMETIC. 1. The Commercical Company issued a policy of insur- ance on a vessel for ^ of the value of the vessel and cargo ho5^°'n "^^"lediately re-insured | the risk in the Man- hattan Company at 5%. During theVoyage the ship was IZmu * r ^^^^^^^^^ial Company lost |120 more than the Manhattan Company. What did the owners lose ? -• A man m vests his whole capital in 4 successive spec- 2\f7r.%i. V-'^. ^'f ^^ ^^^"' ^^%' ^^ ^h« ««cond he loses Tnv w^.^"''l?'^ ^'''''' ^^%' '^^^ ^" th« f^^^'th he gains wMi .V/''? ^ o ""i' '' hundred yard race ; A takes 7 steps 13 ot R . W, 'h^'^'-n' «f ^'^ «^eP« are equal in length^ 13 ot Bs. Winch will win the race, and by how much ? fl ^* f^P^^f on having a certain sum of money to invest hnds that by investing in the 3| per cents at 78, his an' nual income will be $7.50 greater than if he invested in the 5 per cents at 112|. Find the sum he had to invest. 5. I wo persons travelling together agree to pay ex- penses in the ratio of 8 to 5. The first (who contributes second $24.35. What must one pay the other to settle the Dill according to agreement ? 6 Sold goods at $3.15 and gained | of the cost price. What part would I gain if I sold for $3:33 ? 7. A vessel had two taps running into and one runnine- oat of It ; the taps running into it can fill it in 5 and 7 hours respectively, and tlie third can empty it in 4 hours It the vessel be empty and the three taps started simul^ taneously, how long before the vessel will be filled 8 The compound interest on $15 for 3 years is $4,961 1^ md the rate per cent, charged. ' ^ 9. What per cent, does a fruiterer gain, who buys lemons at the rate of 21 for 50 cents, and sells them at the rate of 40 cents a dozen ? 10. A and B start at the same time to walk in the same direction. A is 8 miles behind B and travels at the rate ot 5 miles an hour, while B's rate is 4 miles. Where will have to start from, and what will be his rate nro^ided he wants to remain equi-distant from them while travelling? GENERAL PROBLEMS. 295 «;i the 11. A merchant marked his goods so as to gain 207, but sold them for 8% less than his asking price. He gained altogether $130.65, what was the cost price of the goods ? 12. What is the final value of an annuity of $750 for 9 years at 7% per annum ? 13. A merchant gains 20% of his capital in each year for 3 successive years, and at the end of that time he is worth $10800. Find his original capital. 14. The law requires that a teacher's salary shall be paid quarterly. Find the value of assessable property in a school section which pays the teacher $500, and is com- pelled to give 3 notes for 9, 6 and 3 months, respectively, at 8% per annum, to pay the salary as it comes due, if a 4^ mill rate leaves it a surplus of $124.81 after paying the teacher. 15. A grocer mixes teas worth 36, 48 and 60 cents a pounds respectively, forming a mixture worth 49| cents a pound, having equal quantities of the iirst two kinds and 10 pounds of the third kind. How many pounds of each of the first two kinds does he use ? 16. A man buys 4% stock at 86^, and after receiving one dividend sells out at [)^, and clears altogether $108.75. How much did he invest at first ? 17. A person bought a house for $3000. He insures it at H% for I of its value, pays $1.25 a month water rates, and the assessor assesses it at | of its value (the rate of taxation being 19 mills on the dollar). If the property depreciates in value 5% during the year, what rate of in- terest does the owner make on his money, supposing that he receives $35 a month rental for the property ? 18. A wine merchant buys 1240 gallons of wine at $3 a gallon. After adding 1 gallon of water to every 5 of wine, he bottles it in pint-and-half bottles, each of Vhich costs 5 cents. At what price per bottle must he sell it to clear 25% on his whole outlay ? 19. A mixture of black and green tea, weighing 16 pounds, costs $6.15|. If the proportions are interchanged the mixture is worth $5.85 ; the black tea is worth 35 cents a pound. Find the number of pounds of each kind in the first mixture. 296 ARITHMETIC. i 1 I i I i 1. After paying an income tax of 18 mills on the dollar, a man has $245.50 left. What had he at lirst ? 2. Find the compound interest on $17 for 8^ years, at 6% per annum. 3. A merchant buys 1440 yards of cloth. He sells J of it at a gain of 8%, | at a gain of 12%, | at a gain of 14%, and the remainder at 9% loss. Had he sold the whole at a gain of 5% he would have received $23.50 more than he did. What was the cost price per yard ? 4. A merchant sold ^ of a lot of goods at a profit of 12%, J at a profit of 21%, | at a profit of 25%, and the remainder at a loss of 45%. How much was his average gain per cent. ? 5. A man rented a house for $25 a month for 2| years. What sum would pay the entire rent in advance, interest being calculated monthly at |% a month? 6. Twelve men engage to do a piece of work in 9 days. How long may 5 men remain away, and the work be fin- ished in the same time by their bringing 10 men more with them? 7. A man, whose net income is $1156.10, secures it from the following sources, a fixed salary, and the rent of a house. What is his fixed salary, if on the house which rents for $25 a month, there is a mortgage of $1000 at 6% per annum," a $2000 insurance at 1|%, taxes (calcu- lated at the rate of 19 mills on the dollar) on an assess- ment of $2500, and on his salary (with $400 exempt) the same rate of taxation ? 8. A, B, and C engage in business ; A puts in $400 at first, and $400 more at the end of 6 months ; B puts in $900 at first, and withdraws | of his capital at the end of 6 months ; C puts in $200 at the end of every 6 months ; at the end of 2 years they gained $6700. What share of the profits should C receive in addition to 25% of the total profits for managing the business ? 9. A man borrows a sum of money at 6% per annum, and invests it in 5% stock at 105 ; he receives the dividend at the end of the year, which is subject to an income tax of 2%, and immediately sells his stock at 112|. If his net gain is $48.80, what sum did he borrow ? GENERAL PROBLEMS. 297 3 dollar, ears, at lells J of of 14%, vhole at than he of 12% nainder :ain per I years, interest 9 days. : be fin- jn more 5 it from ent of a e which ^000 at i (calcu- i assess- npt) the $400 at puts in 5 end of nonths ; share of he total annum, ividend )me tax '. Ills 116 b •!?*i?^^!^'^'^^'l'"®^^^^-l''^ ^ e^^'lo^ must be mixed with 16 gallons at $1.80 a gallon, and 46 gallons at 90 cents a gallon, to make a mixture worth 11.14 ^ per gallon? - T7 F^* Q.^'^aoiir'^f ,^??^l"'^''^^'''^^^^® ^« follows :--July 3rd $35.26; July 4th $48.65, on 30 days; August 17th, $6.48 ; September 12th, $50. What is due on account on October 12th (interest at 9%) ? 12. A merchant mixes 9°pounds of one quality of tea with 3 pounds of another quality, and the mixture is worth 45 cents a pound, but if the quantities were inter- changed in a mixture it would be worth 35 cents a pound. Imd the price of each kind in the mixture. ' 13. A man has 7 hrs. 48 min. 45 sec. at his disposal. How far may he walk at the rate of 4 miles an hourfso as to return in time riding at the rate of 11 miles an hour ? 14. l^ind the compound interest on $23 for 10 years, at 6% per annum. "^ 15. A person bought cloth at the rate of 48 yards for fbO, and sold it at the rate of 50 yards for $48. Find his loss per cent. 16. Received an invoice of crockery, 12% of which was broken. At what advance per cent, on cost must the re- mainder be sold to clear 25% on the invoice? 17. A man spends $25 a year in tobacco for 30 years What would this amount to if placed in a bank, at 47 per annum ? '/or 18. A commission merchant sold flour for his principal at a loss of 10%, but if the flour had cost $1 a barrel less he would have gained 5% on the sale. What was the cost 01 the flour per barrel ? 19. A and B start from different places to walk around a circular track ; their rates are 6 and 4 miles per hour respectively. Where M^ould C have to start from and what must his rate of walking be, so as to be alongside B every time that A is ? ^ . ^?k \ ^^^. ^^ ^^ "^^^^ ^^^k ^n 9 days as B can do T? fv? '• \ 1 ^^^^^ ^^^? ^^ ^^^^® ^^^ ®^®^'y 12 B works. If their total earnings on a job be $109, how should the money be divided? 298 AIIITIIMETIC. I 1. A man worth $10000 was offered a salary of ^2000 per annam-, and a chance; to invest his $10000 "on a tirst- class mortgage at (>% per annum. He, however, decided to go into business, and invested his .^10000 in goods, on which he paid a duty of 24%, except on 12% which were damaged. He sold the undamaged goods at 65% advance on prime cost, and those wliicli were damaged at 20% of tiio cost. He lost 5% of the sales in hud debts, and paid 1% on the remainder for collecting. Did he gain or lose by not accepting the first offer, and how much ? (Inter- est reckoned on mortgage only.) 2. A man increases liis capital yearly 20% of what it was at the beginning of each year What per cent, does lie increase it in 4 years ? 3. Three persons were to share $10000 in the proportion of 3, 4, and 5, but the first dying it is required to divide the whole sura equitably between the otlier tAvo. 4. A man's income from the 3% consols is $720. It he sells out I of it at 84, and invests the proceeds in otlier stock at 120, he will increase his income by $40. Wliat per cent, does the latter stock pay ? 5. Ten per cent, of an army was slain on the field of battle, 8% of the remainder was mortally wounded ; the difference between the killed and the mortally wounded was 504. How many men went into battle ? 6. A vessel A contains 2 gallons of wine and 3 of water ; another vessel B contains 3 gallons of wine and 1 of water. How many gallons must be drawn from each cask so as to produce by their mixture 1 gallon of winr and 1 of water ? 7. A square box, whose depth is 14 inches, ^ • l; cal content of 4 cubic feet 113 cubic inches. Find the length of a side. 8. A man buys a farm for $8000, which he agrees to pay for in 12 years by equal annual instalments, without interest ; -,ht owner being pressed for money offers to take $6000 cafh. Wh.-ch is the better for the purchaser, money being worth o% per annum? 9. Find tlie compound interest on $27 for 4| years, at 8% per annum. Interest payable quarterly CiENEllAL HtOBLEMS. 299 of ^2000 I a firat- dccidcd •ockIs, on leh were advance t20% of uid paid II or lose (Inter- r what it 3nt. does 'oportion 3 divide . It he in other . Wliat iield of led ; the tvounded >f water ; 3f water. 3k so as ind 1 of ■ ':ii'?'ical e length grees to without 's to take [", money y^ears, at 10. A merchant in London (Eng.), owed another in Pa^ wV^ f T''^''^ ^^'*"^*' ^« ^•^"^i"^^ through Pans, when he exchange was 25..35 francs for £1, and between Pans and Petersburg S-ai) francs for 1 rouble f^'^^^iVJ ' *. ^.^^hange l>etween London and Paris wa« 25 625 francs for £1, and between Paris and Petersburg 3-37 francs for 1 rouble. How much would he have gainc-1 by the delay ? or^l'if- ni^^';«hant, who puts 5 gallons of water into every 25 gallons of coal o I, sells it for 2 cents a gallon less than \rT21 ^''T'^ "' elsewhere. If good coal oil is wort wLTuys^4f)^!^^^^^^^^^ "^^^^^ '^^^ '^ ^^-^ ^ — -• dpif 'iJil'^vS'^''^'*''''^ "'^'^ ''^'^'''''y ^« SI 750000, and its d^bt 18 $(>750()0; Its gross earnings for a year are ^5(;5- 000, and expenses $384500. After paying the interesf on the debt at 6%, and $52500*of the d J,t?Vhat rate of dx^ vidend are they able to declare ? flvin ^^ ^'''^^'' intended to gain 10% on a stock of tea, and fixed his price accordingly. When he had sold f of the loUie was compelled to lower his price 8 cents a pound and so gained only two-thirds as much as he had intended' What was the original cost per pound of the tea ? ^ 14. If stock bought at 5% premium will pay 6°/ on the SuntT' "'^'^ ''"' ^'''' '' ^"^y ifbo^htat 15% 15. I sold a consignment of goods through a factor who charged me 1^% ; I was allowed 2|% commission and Uy V^JX' ' ^""'^ .'"'^^"^ payment, and I cleared $51. i^ind the sum remitted to my employer 16. A farmer rents a farm of 450 acres on the followine- terms : he pays a fixed rent of $1.50 an acre, and a corn rent of 100 bushels of wheat, 40 bushels of barley; and ?5 ^^75 4«1. The price of the wheat, barley,'^nd oats eZ% reit paM. '" ' ' '"'''' respectively. Find the 17. How many railway shares ($100 each) at 407 dis- count must be sold in order that the proceeds, inve^^d -p bauK siocK, Which is 4% below par, and pays a dividend of 7%, may yield an income of $1680 ? 20 ( I n !? u [I- i 300 ARITHMETIC, 1. Find the decimal of a foot, which differs from an inch by less than the millionth part of a yard. 2. A certain article of consumption is subject to a duty of 12 cents per pound ; in consequence of a reduction in the duty, the consumption is doubled, and the revenue is increased one-quarter. Find the duty per pound after reduction. 3. Find the compound interest on $25 for 8| years at 47 per annum, interest payable half-yearly. 4. A merchant in New York wishes to transmit 4500 marcs banco to Hamburg, and the exchange between New York and Hamburg is 35 cents for 1 marc banco. He finds, however, that the exchange between New York and Lisbon is $1.08 for 1 milree, that between Lisbon and Paris is 6 milrees for 38 francs, and that between Paris and Hamburg is 19 francs for 10 marcs banco. How much will he gain or lose by the circuitous exchange ? 5. What is the value of a perpetuity of $450 a year to be given at the end of 15 years, money being worth 8% per annum ? 6. Three men invest capital in business, in the propor- tion of 4, 5 and 6, on the understanding that the last is to receive 10% of the total profits for managing the busi- ness, and the remainder is to be divided in the proportion of the capital invested by each. The manager receives in all $4600, find the total profits. 7. 14 oxen eat 2 acres of grass in 3 weeks, and 16 oxen eat 6 acres of grass in 9 weeks. How many oxen will eat 24 acres of grass in 6 weeks, the grass on each acre being equal at first and growing uniformly ? 8. A merchant buys 12 dozen of port at $18 a dozen, and 48 dozen at $10 a dozen ; he mixes them, and sells the mix(;ure at $14.50 a dozen. What profit per cent. does he make? 9. A bankrupt whose total assets amount to $6000, owes A $5000 and B $4600 ; the assignee charges a cer- tain rate per cent, of the assets for winding up the busi- ness, and other expenses amount to $132. A receives for his share of the estate $2931.25. Find the rate charged by the assignee. 3 from an to a duty duction in revenue is )und after i| years at ismit 4500 ween New lanco. He York and jisbon and kveen Paris How much 3? ) a year to worth 8% the propor- the last is g the basi- proportion 3r receives nd 16 oxen :en will eat acre being L8 a dozen, L, and sells t per cent. ; to S6000, trg-es a cer- p the busi- receives for i,te charged GENERAL PROBLEMS. 301 chi^Vy^t'th^ti^ ''"'!,''• P^^^^' '' ^"^--^ With 3, andVe mixture Is soM^^A '" '^'^ P^'oportion of 7 to .ain per ^^^1^:^^^:^^^ ^^^^d. .ind the broL^wLXfes V/on e? h T'"'''^ ^"^^ '' ^ '' ^^^ find the amountTstoYk hLdled '""''"" "^^^^ ^''' fo/U'ertf t^fo?iTboV'rr?^" ^^^^^ '^'''' - --«h material and 77 more fo. 'l.. u T^^ ^% ^"^^^ ^^^ cost $10144. A7ZIZ ^^' ^'^ '^^^^ --^^ h-« anfac^blTot^fwit^S^^^^ ^^^^^ -«^-. weightofaquarrorwater^o^^^^^^ '^^"^ ^^^ mals. toriect to three places of deci- he gavt Tolo^ll'tl f ^"^ f f ^^^ 3 «ons ; to the first third I as much ;s to bo r^f ^ ^?^'^" ^^ole, and to the did thi farm contain P '^'^ ^th^^^- How many acres the' firsm r^ns'arf rr ^ Vr''''^ ^^ "^"^^ ^ ^-"^ Ions ; the quTnti y remt VS fn S.f ^ T^"? ^^ ^^^- that remaining in the second tt ^'^^ ^^!^ '^ double of originally contain P ^^"^ '''^^^ ^^^ ^^ch cask inside Ind 'o:t"it lllenT"* ^ "'^^^^" ^^^^^^^^ ^ —, is 30 feet longSf fLV;re,Yn^d^^^^^^^^ ^^-- FiiLh^e'^rcCg^^^ - $90. er^ool.^ ^^Jl^'S^-^t l^^y ^^000 in Liv- francs 25 centimes for ^1 ^/i?^ xx ^^""^ ^" ^^^^S' ^ guilder. The Txchfni bWwl ^^^""^^"^ ^^ ^^^^^ *^ a the same time L 25 lnnp« T V^'^u^^ '^"^ ^^^1^"^ at England l^TguHd'eVSr^^ii. ^h ;^;^^^^^^^^^^ Ha-bu.g on t^age^ous, to transmit direct, throu^^ ^19. Find the present value of «n annu^'^v r^ <*— ■ "egm at the end of 7 years and foTL p n ^ ^^'''' ^^ being worth 5/ ^ *^ *^" ^^^' ^ years, money 302 ARITHMETIC. 1. Find the measure of the altitude and the area of a triang-Ie, the measures of whose sides are a b and c, re- spectively. 2. A farmer rents a piece of land for $120 a year. He lays out $625 on 75 sheep. At the end of a year he sells them, having expended $12.50 in labor. How much per head must he gain on them, in order to realise his rent and expenses and 20% gain on his original outlay for the sheep ? 3. A merchant imported a quantity of goods, paying 15% for freight and insurance and 10% for duty (reckoned on the prime cost). He sold them at a loss of 10%, but ' had he sold them for $600 more than he actually did he would have made a profit of 2%. Find the invoice price of the goods. 4. A person invested $8001 in the 4 per cents at 95^, and when they rise to 98 sells out and invests in the 3 per cents at 84. What amount of the latter stock does he obtain ? 5. What per cent, of the first loss is the difference be- tween 8% loss on the cost and 8% logs on that selling price ? 6. A speculator bought 368 acres of liind at $57.50 an acre, borrowing the money at 4%. At the end of the year he sells f of it at $63 an acre, and the remainder at $50 an acre. How much does he lose by the transaction ? 7. A merchant in London owes another in Petersburg a debt of 460 roubles, which must be remitted through Paris. He pays the requisite sum to his broker, at a time when exchange between London and Paris is 23 francs for £1, and between Paris and Petersburg 2 francs for 1 rouble. The i;emittance is delayed until the rates of exchange are 24 francs for £1, and 3 francs for 2 roubles. What does the broker gain or lose by the transaction ? 8. A grocer by selling 10 pounds of tea for a certain price, gained 15% ; afterwards he increased the price, giv- ing only 8 pounds for the same money. What per cent, did he make at the increased price ? ^ 9. Having received a stock dividend of 6%, 1 find that i own 291 J ($100 shares). How many had I at first ? GENERAL PROBLEMS. 303 irea of a nd c, re- ear. He ' he sells nuch per his rent y for the ;, paying reckoned 10%, but lydid he oice price bs at 95J, , in the 3 5k does he irence be- ing" price ? $57.50 an )f the year ler at $50 LCtion ? Petersburg d through oker, at a 'aris is 23 •g 2 francs I the rates iincs for 2 >se by the »r a certain I price, giv- it per cent. [ find tbat I first? 10. A person leaves $12670 to be divided amonL'- 5 children and 3 brothers, so that after the legacy Xtv has been paid, each child's share shall be twice as 7vLtll each brother's. The legacy duty on a child's 'hafe be ng one per cent., and on a brother's share three per cent find what amount they respectively receive ^ 11. A horse dealer sold two horses for $160 each gain- ing the same per cent, on one as he lost on the other 'ind on t^ie whole he lost $13^. Find the per cent ' 12. An English mile is -2136 of a German mile. What time will a tram, which travels 20 English miles an hour take to travel 3| German miles ? ' T^iih?^^ coinpound interest on $435 for 6 years is $205. r ma the rate per cent. toHanfbrJ'^HifT'f"' ^l '"^"'"^^^ ^^^ marcs banco to Hamburg. He finds exchange between Montreal and Hamburg to be 36 cents for 1 marc. The exchange be tween Montreal and London (Eng.) is $4.83 for £1 ; that be ween London and Paris is 26 francs for £1 ; and tha between Pans and Hamburg is 47 francs for 25 marcs By which way should the merchant transmit P mil irr^i^9n ^"^ a ^''^^.^'' '"^ manufacturing shoes. A puts m ^1920 for 6 months ; B a sum not specified for 12 ZT^l ' T^ ^ ^^^^^ ^^^' ^ *"^^ ««^ «P««ifl«d. A received O ^2080. Required B's stock and C's time 16. A speculator bought 35 ($100) shares of stock at foriS'^^h'f ^'?J^ P"^^^"' ^ ^''''' «- N«- York for $4000. What was the rate of premium of the draft ? <^J1' ^"^""^ ^'''''^"^ ^y ^ corporation were $38600 3iTfor oTT''^' rate Of 2^, ,^^.^^^ ^^ ^ commission of ¥« A ^^^/f ^*^"% F^^d the value of assessable property. at a iofs of 6il '/n^ n \^^'' "^'^°^' ^ ^^"« *hem to C fn^r «i°/ wH\T^ ^-^^^^^ ^^^"^ *^ ^ fo»' ^^130.20, gain- mg 8^/. Find the prime cost of the goods 19. A baker's outlay is 70% of his g-ross receintq nnd soy"" a„1%.^'^r"^ "'■*' '*^^- ''^^ I^-ice oTflTTet Whi.^T' expenses are thereby increased 25°/. \\ &at advance mast be made in the price ofa 71 cent Imf that he may still realize the same amount of profit from U ? 304 ARITHMETIC 11:,. 1 SOME PROPERTIES OF NUMBERS. 1. A number is divisible by 2 when its last digit is divisible by 2, or is zero. 2. A number is divisible by 3 when the sum of its digits is divisible by 3. 3. A number i^ divisible by 4 when its last two digits taken in order form a number divisible by 4, or are zeros. 4. A number is divisible by 5 when its last digit is 5 or zero. 5. A number is divisible by 6 when its last digit is divisible by 2 or is zero, and the sum of its digits is divi- sible by 3. 6. A number is divisible by 7 when the sum of once the units^ or iirst digit, 3 times the second, 2 times the third, 6 times the fourth, 4 times the fifth, 5 times the sixth, once the seventh, 3 times the eighth, etc., is divisi- ble by 7. 7. A number is divisible by 8 when its last three digits taken in order form a number divisible by 8, or are zeros. 8. A number is divisible by 9 when the sum of its digits is divisible by 9. 9. A number is divisible by 10 when its last digit is zero. 10. A number is divisible by 11 when the difference between the sum of the digits in tlie odd places and the sum of those in the even places, is divisible by 11 or is zero. 1 1. A number is divisible by 12 when its last two digits taken in order, form a number divisible by 4, or are zeros, and the sum of the digits is divisible by 3. 12. A number of three digits is divisible by 13, when 4 times the hundreds digits, increased by 3 times the tens digit, and then diminished by the units digit, gives a re- sult divisible by 13. 13. A number is divided into periods of three figures each, beginning at the right. If the sum of the odd periods differs from the sum of the even periods by a multiple of 13, the number is divisible by 13. SOME PROPERTIES OF NUMBERS. 305 lig-it is 3 digits 3 digits B zeros, ifit is 5 iigit is is divi- of once lies the nes the 5 divisi- e digits or are ts digits digit is fference the sum is zero. digits :"e zeros, 3, when the tens '^es a re- '. figures periods Itiple of 14. The theorem of example 13 is true of divisors 7 and 15. An even number is divisible by 14 when the test of example 6 is satisfied. 16. A number of three digits is divisible by 14, when twice the hundreds digit, increased by the units digit, and diminished by four times the tens digit, ffives a re- sult divisible by 14. • 17. A number is divisible by 15, when its last digit is 5 or zero, and the sum of its digits is divisible by 3. 18. A number of four digits is divisible by 16, when the result of eight times the thousands digit, less 4 times the hundreds digit, increased by 6 times the tens digit, less the units digit, gives a result divisible by 16. 19. A number of 4 digits is divisible by 17 when the sum of 3 times the thousands digit, twice the hundreds digit, and 7 times the tens digit, diminished by the units digit, gives a result divisible by 17. 20. An even number is divisible by 18, when the sum of its digits is divisible by 9. 21. A number of four digits is divisible by 19, when the result of 7 times the thousands digit, less 5 times the hundreds digit, increased by 9 times the tens digit, dimin- ished by the units digit, gives a result divisible by 19. 22. A number is divisible by 25 when its last two digits, taken in order, form a number divisible by 25, or are zeros. 23. A number is divisible by 125 when its last three digits, taken in order, form a number divisible by 125, or are zeros. 24. Any number formed by writing down an odd num- ber of digits, and then repeating them in order, is divisible by 11. 25. A number, formed by writing down three digits and then repeating them in order, is divisible by 7 and 13. 26. A number, formed by writing down four digits and repeating them in order, is divisible by 73 and 137. 27. Show that the difference between two numbers con- sisting of the same digits, {u-ranged in different order, is divisible by 9. ■II 306 ARITHMETIC. 1 li u 28. If the sum of the digits of a number be subtrkcted from the number, the remainder is divisible by 9. 29. A number is divisible by 8 when 4 times the hun- dreds digit, twice the tens digit, and the units digit form a sum divisible by 8. 30. A number of three digits is divisible by 19, when 5 times the hundreds digit, added to the number formed by the other two digits, gives a sum divisible by 19. 31. A number of three digits is divisible by 9, if eight times the units digit equals the number formed by the other two digits. 32. A number of three digits is divisible by 13, if 9 times the units digit equals the number formed by the other two digits. 33. A number formed by writing any units digit and placing before it double the units digit is always divisible by 3 and 7. 34. A number of 2 or 3 digits is divisible by 7, when the units digit is one-ninth of the part on the left. 35. A number of 3 digits is divisible by 7 when twice the hundreds digit, added to the number formed by the tens and units digits is divisible by 7. 36. If a number be divisible by 11, the number formed by writing its digits in reverse order is also divisible by 11. 37. A reversible number, consisting of 2n — 1 digits is divisible by 11, if the remainder on dividing the first n diQ;-its by U is onc-luilf of the nth digit. 3S. The difference between the square of a number of two digits and that of the number formed by reversing thcw dibits' is divisible by 99. 3.). To multiply a^ number by 25 affix two zeros at the right, and divide by 4. 40. To multiply a number by 125, affix three zeros at the right, and divide by 8. 41. To multiply a number by 125, affix two zeros at the right, and increase this result by | of itself. 42. To multiply a number by 328, multiply first by 8; multiply this product 'lo, and take the sum of the two products. SOME PROPEIITIES OF NUMBERS. mbtrkcted 9. the hun- digit form ^9, when 5 formed by 9. d, if eight ed by the ;, if 9 times 3 other two digit and s divisible • , when the hen twice led by the )er formed } divisible -1 digits is the first n number of reversing 3ros at the !e zeros at zeros at first by 8 ; of the two 43. Show how to m ultiply 307 '28144, number by 1' Ai Ti:r "T"* "' ^^"^^^ products. resulting- number anri thill ? ' '^"'' *'"' P''«l««' of the and afflf 25 To this pl-^^cT '"'''""''"' *"°'« "«-"b'^'-, fratio^al%^:r/l?L"f<''' "™''«'-, of wl,ich the product of the whoirnLL^ "^ ^^fo^ows :_Find the number, and aihx the ?ractio' i?nH*-' "'f «0"^ecutive 49 Show th.,t 1 ^t'"" i to this product. two diguTerlS.* '""^'"'''' '' ■•'''='"-"°? decimal with a thre°ciguVer^!*' ^''^'""'' ^ ■•«<="ri''S decimal with a a flvc.d?gT period*^ P'*^"""' * '•'^'=""'"S decimal with a flveil^7p'ertP^'"''^"-^«^ ' -««""? decimal with With a thrle dfgit^irTod"^ Produces a recurring decimal flvet^igH^ertad. "^ '"°^''''' " ■'''-"-^ decimal with a sevenl'gT^e^U'- '"''^""'^ " P"'''' ^^P^'^^d with a an^elgScUyt'^periS''" Produces a pure repetend with nine^dfgu'^erM'''**" ^"^'''"" "" P"''^ ''^Potend with a 58. Show that ^4iii-'6 78 9 r^-nrii-.^^ with a thirteen-dillt pJdS. "" ^'ecurring decimal 59. Show that the number of r^lar.^., ;« ! '!f I' f 1} i08 ARITHMETIC. . ■■ 60. A number, whose digits are all 9's, is either a mul- tiple of 11, or is 2 less than a multiple. 61. Show that a fraction whose denominator is of the form 10" + 1, and whose numerator is unity, produces a repetend with a 2n-digit period. 62. The number of figures in a repetend cannot exceed the number of units in the denominator of the common fraction producing it, less 1. 63. Reduce | to a repeating decimal, and hence show that a number 'consisting entirely of nines must contain at least six digits before it is exactly divisible by 7. Also, show that a number consisting of six one's is divisible by 7. 64. How many digits must there be at least in a number consisting entirely of 4's, so that it may be exactly divisi- ble by 7? 65. What is the least number of digits greater than six, which must be in a number consisting entirely of 4's, so that it may be exactly divisible by 7? 66. Prove that a number consisting entirely of must contain 16 digits to be exactly divisible by 17. 67. Prove that a number consisting entirely of must contain 28 digits to be exactly divisible by 29. 68. Why does a proper fraction Avhose denominator is 37, a prime number, repeat after three decimal places ? 69. In reducing -^V ^^ ^ repetend, when we obtain the remainder 16 we have found one-half of the repetend, and the remaining half may be found by subtracting the terms of the first half respectively from 9. 70. When 10^ •' is divided by 17 the remainder is 1. 71. AVithout dividing, show that 37 is a factor 718249()321 i. 72. Given ^jV = -0588235294117647, show how to find by a short method. '73. Show that ..ivi, produces a recurring decimal, with not less than a 67-digit ])criod. (It gives a 268-digit period). 74. Show that ,"'. ^^'i^l produce a recurring decimal of not less than 23 digits. 75. Why docs the squni'c root of 87 produce a non- terminating, non-repeating decimal ? 9's I's of 6 TT iv a mul- ' the form repetcnd )t exceed common nee show iontain at 7. Also, divisible a number ily .divisi- than six, of 4's, so ly of 9's rl7. iy of I's r29. [linator is places ? )btain the Jtend, and the terms iv is 1. factor of to find ^ mal, with :it period), iecimal of ce a non- SOME PROPERTIES OF NUMBERS. gOO process of division a vemZLvilVt°'" '^' "''"«" '" "'« to tlie difference betwon ih„ Pi-oduccd wliich is ecmal of tl,e fraction, 'ne!^^uf of he T:^ZT ?'"* "«»on>inl?o the remaining l.alf of the neW^ "^'l'^'"^" ^"""l ! an-l tracting i„ sfccession'earh '^XfoZl U^^r^l''' "^ -''■ ^o;««»„._Let ^represent the fraction n the number Then4 = b n + a n + 1 _ a lO lO-- n + b-a n + 1- 10'- n : + 10' (10 -l)_n + ^ b 77 Oh ^^"^ lO'^'' one digir bf dfeiding\rir'"«^ "'''"'"'' »'"-'•"'»=- »"ly onrdigrt'bfdlliirg'byTT"'"^ ""'""''''• "'"'^'"'"^ °nly 2 dfgi[;\7^iifr4" bTiT''""^ '"""'^''' "'^'---^ -'y onfy lXt'VSg;;^f '"- V 17, obtaining ve^s^eiy'^Torof^^^re^eTnSrr^V^^-' »<>■ - power, is even. numoer, which is a complete is formed"Ty^;rs1ng the'Xll^"^ ^n^""''^'''- "«■»"- H;f¥v.,.^^„. t_". ^^, "s cne dibits. One dio-it of ih^ ..ff..=nee .x.ween the two numbers being 4? find th^ 310 ARITHMETIC. 4- ' I \ I I I : 84. Any power of an odd number is odd, and, con- versely, the root of an odd number, which is a complete power, is odd. 85. In finding the product of two numbers, why are the partial products placed in a diagonal column ? 86. In multiplying one number by anotiier we can ob- tain the partial products in any order we please, and arrive at the true result. ' 87. Multiply 46087 by 4967, beginning with the 4 of the multiplier ; then using the i), and so on. 88. Explain clearly the reasons for the different steps in a problem in long division. Sd. Prove that any number containing 7 digits is greater than the square of any number containing 3 digits. Hence show that the square of the number represented by the 3 right-hand digits of any number containing 7 digits is less than the number of units represented by the remaining 4 digits. 90. Every time a remainder is obtained in the process of extracting the square root of a number, what kind of a number has been subtracted from the original number? 91. Between what numbers do the successive complete remainders indicate the differences, in the process of finding the square root of a number? 92. Between what numbers do the successive complete remainders indicate the differences, in the process of find- ing the cube root of a number ? 93. In long division, if we neglect to introduce into the work at the proper time the successive digits of the divi- dend, we can, by neglecting in order the successive digits of the divisor, beginning at the right, obtain a certain number of digits in the quotient. Show that this principle will hold in a case in square root, after a number of digits in the square root h«LS been obtained by the ordinary method. Example. — Extract the square root of 11 to 10 decimal places : first find 7 digits in the answer by the ordinary method, and then use contracted division. 94. A prime number cannot divide the ])roduct of two factors without dividing at least one of the factors. u :■ , and, con- a complete vhy are the we can ob- please, and the 4 of the erent steps s is greater igits. 'epresented >ntaining 7 [ited by the e process of -t kind of a 1 number? e complete process of /e complete sess of find- iice into the of the divi- issive digits 1 a certain 30 in square )ot bus been • 10 decimal he ordinary duct of two 3tors. SOME PROPERTIES OF NUMBERS. 311 95. A number cannot be resolved into i)rim ' If ■• K I il ■ '} 102. How many units arc expresBcd by 10101 in tlie scale of radix 5 ? 103. Express in ordinary scale 12H4r> which is in the scale of G. 104. Divide 125 by 4 ; divide the qaotient by 4 ; divide the second quotent by 4. Show that 125 = 1 x 4 ' + 3 x 4-' + - 3x4 + 1. 105. Express 125 in the scale of radix 4. 106. Express 12345 in the scale of radix 5. 107. Express 52301 in the scale of radix twelve. 108. Express, in the ordinary scale, the number 4321, which is in the scale of 7. 109. Add 4078, 5236, 7134, 8216, which are in the scale of 9, and express the result in the scale of ten. 110. Subtract 4136 from 7014, each number being in the scale of 9 ; express the remainder in the scale of ten. 111. Multiply 1687432 by 6, each number being in the scale of 9 ; give the result in the scale of ten. 112. Multiply 41625 by 254, each number being in the scale of 7 ; give the result in the scale of 7. 113. Square the number 425, in the scale of 6 ; give the result in the scale of 6. 114. Divide 1738 by 144, in the scale of 9; give the quotient in the scale of 9. 115. Divide 1000002 by 110, in the scale of 4 ; give the quotient in the scale of 4. 116. Form the multiplication table up to six times in the scale of radix 7. 117. Write down the first ten numbers in the scale of 2. 118. Any number in the scale of 7, when divided by 6 gives the same remainder as the sum of its digits divided by 6. 119. How can a grocer in one weighing with a balance obtain 3| pounds of sugar, if 'the only weights he has at his disposal are a ^, 1^ and 4 pound weight respectively? 120. How can a grocer, by using not more than one of each of the weights, 1, 2, 4, 8, 16, 32 pounds respectively, weigh at one time 51 pounds of sugar? 121. Bj?- using not more than one of each of the following weights : 1, 3, 3^, 3'^, etc., pounds, how can a grocer weigh 427 pounds at one weighing ? n the scale SOME PIIOI'EIITIES OP NlTMBEItS. 313 122. Every power off, necessarily ends with fi. U.i. I he product of2 nuinLers, dilteriii.. by 2 is one les, than the square of the intermediate nun.hen*^ ' ""' be d"iv>,fi;nLf •.""""■'' ""'"'""■ ''''''"h '" " Perfect square 125 Slmw^h;,",^"" ""^"'' '""^« 2 for a ren.ainder. less tha!^ r """ "^« ^""»'-« ••«<" of 3 differs from IJ by .,(■ 'i!q". ,''''"<'.^2'>'"'« "f 12345 is 15231»i)025. Find tlie souaro wwfh",''^"'.'''"" "'" f»"»«'ing' method of determinins in h,ri aJ'the t,"""^"". ""' '"<' <'^<'" "«»"'«■• of coin Tnow" ro^'r„"m,rin""h::trer'^''" ""'"'"^'' '" »"« '■-<» -^ Innd^brom!"!. '""■"'" «o "mltiply tlie number in the right and by any even number whatever, and that in tlie left P.Uuc^'tsf.nfh''' ";™ •''I '»■» -dd together thl'two pioaucts and if the whole sum be odd, the even nnm ter will be in the right hand, if the sum be even? in the 128. If we take any two numbers, then cither nn^ nf 129 V wttT' •"■ """'• ''"''«'•<'"««• '^ d'vlsib by 3. 129. If we talce any term of the series U 2' S» 1' etc., in which the whole numbers and the numerators are :«:crvTrdrnrsTeSr;;r3^"T tr^Vr"p~tt:t' v^~'^^^^^^ whirh the h vn^fh "'''' "^ '^ right-angled triangle, of Cr greater ZltbT' '"'"''' ''I'-epresented by the num' mu *"® numerator by 1 Thus 4j = .V>, gives the sides 9, 40, 41. Explain riiVi.r' ^^^^^^^g"? te the reason for the following- rule for dividing a number by 365 :— ^ fh5wf *^5 decimal point in the number two places to 11 thl''';^-^^^' ^^ ^' *^^" ^^^i^l« this quotfenTby 20in th. «' ^^'' ^^^'^'^' ^^20; then again by 11 and :,, '"^ *h?, ^?™e way, and so on by 11 and 20 continn ally until the required degree of accuraey is reached then the Slim nf qII fi,oo^ ^,,7,..._„.- .,, , / ^" leacnea ; quired. "'' ^ quouunm wih oe the quotient re- !f ■ !■ . ^ i 1 r % '1 i; 1 ? . .■, ' ! - i .i :i 314 ARITHMETIC. 131. In dividincT bv 73000 it is advantftg-eous to do so by the following'" method : — Having written down tlie number to be divided we write under it one-third of itself, then one-tenth of this second number, neglecting remainders, and lastly one- tenth of this third number. The sum of these four num- bers with the last five figures, reckoned as decimals, will be the quotient required. Establish the correctness of the method. To what extent can its accuracy be depended upon ? Indicate a slight extension of the method which will enable any required degree of accuracy to be attained. 132. Prove the correctness of the following approximate method of dividing a number by 73 : Multiply the number by 137 ; subtract ji^^oy of the pro- duct, and point off four decimal places. 133. To divide a number by 137 ; Multiply the number bjs 73 ; subtract jy^xnr ^f the pro- duct, and point off four decimal places. 134. If a is a little less than the cube root of a number N, then ^ -^ .^ a, is a closer approximation. 135. By successive applications of the principle in ex- ample 134, find the cube root of 5, true to 3 decimal places. 136. The square of the sum of a series of the natural numbers, beginning with 1, is equal to the sum of the cubes of the same numbers. SELECTED UNIVEUSJTY PROKLKMS. 315 n Ls to do so ivided we ith of this astly one- four num- mals, will . upon? diich will ,ttained. proximate of the pro- of the pro- a number iple in ex- 3 decimal le natural lim of the PROBLEMS SELECTED FROM TORONTO UNI VERSITY MATRICULATION PAPERS. 1. The value of the old Spanish dollar (which was the unit of exchange between England and America) was 4s. Gd. sterling, but gold became the standard of the US weigh 258 grains, being nine-tenths line. The u ^h CO nage ,s of metal 22 carats iine, 40 pounds be ingcciii^i nto 18b«J sovereigns. AVith these data explain why he b^pa. exchange between New York Ld L^l^t 2. Prove the following rule for computing interest -it ' SIX per cent, per annum for a period of months ndd-vt the substance of which was given in the Leade^^ofUar^^ Multiply the number of months by 5, and add one-sixth the number of days ; multiply this sum by the prhS expressed in dollars ; the result will be the intire t cx pressed in mills. iiucicbc cx- 3. A tradesman who gives six months' credit abates 5 per cent, for cash. Find the rate of interest in order hat this may be the true discount. tt'*-;'^^J'^ ^''"^ advised by telegraph that, on Fridav Uni ed States Five-Twenties were sold at the Londmi S Exchange for G8-i gold. The same class of securHcs brought here 110| currency. Taking the average o-oh IH-emium at fifty per cent., and adding the par of'ex change, which is about eight per cent., the reader c-in calculate for himself that G8i gold in Londc^ s prJt: y m^r ^1 cm-rency m New York, or, more exactly ^p^l^ mgllOf. New ioi'k Times, TuvMhiv Au^. 21 18GG trive a full exi)laiiation of the preceding extract , ^Tf ^'''•^'' ^''^ P'^''"'^' ^^' ^«ff^« '^t 14 cents per pound, and oi) pounds of chicory at G cents per pound -he pays an import duty of 12i i>er cent, ad valorem, and mixes and sells them at 25 cents per pound, but by the usee a false balance gains I ounce on every apparent pound sold. Find the profit per cent, made on his outlay IIP*' 316 ARITHMETIC. i 1 6. By the Canadian Statute it is provided that the silver coins of the Canadian currency shall bear the same rela- tion to the pound currency that the sterlinj^ silver coins bear to the pound sterling-, being also of the same standard of lineness. Sterling silver is 1)2.5 per cent, line, and from one pound troy of this metal are coined C6 shillings. The pound sterling is said to be equal to £1 4s. 4d. cur- rency, or S't.SGf, the pound currency being $4. In Martin & Trubner's "Currency," the Canadian ten cent piece is said to weigh 38-42 grains and to be /^ fine, but an an- alysis by Prof. Croft shows that the fineness is that of sterling. Tiie American mint asserts the value of this piece to be about 9-^- cents, their dollar containing 345-6 grains pure silver. Exa,niine the consistency of these statements. 7. Two persons, A and B, borrow 8300 on joint mort- gage from a building society, A taking 200 and BlOO, the amount being repayable, principal and interest, by equal monthly instalments, to wliich A and B contribute propor- tionally. After a few payments have been made, they desire to borrow each $200 additional and propose to merge the old debt and tlie new into a single mortgage for $700, whereupon the account stands thus ; Amount required to pay off old mortgage .. $245 " of new loan ". 400 Surplus 55 $700 Discuss the interest which each has in this surplus of 855, and the ]n'oj)ortion in which tliey should contribute to the instalments payal)le in future. 8. A (luantity of pulp, filling a trough 3 feet deep, 10 feet 7 inclies long, and 11 inches wide, is made into paper of the same width and of such a thickness that 12 sheets would measure a quarter-inch. Find the length of the paper made, the pulp losing | of its bulk in manufacture. 9. Explain what is meant by inierest and discount. Find the time for which the discount on a certain sum of money will be ecpial to the interest on the same sum for a year ; the rate of interest in both cases being 5 per cent. at the silver same rela,- silver coins lie standard ;. line, and »6 shillings. 4s. 4d. cur- In Martin & nt piece is but an an- ; is that of blue of this Ining 345-6 joint mort- 1 B 100, the it, by equal lUte propor- inade, they propose to e mortgage . . $2-15 .. 400 .. 55 $700 surplus of . conti'ibute set deep, 10 into paper 12 sheets igth of the Anufacture. I discount, lin sum of e sum for a 5 per cent. SELECTED UNIVERSITY PKOBLEMS. 317 A giving IJ 31,2 yanls'tt.n'^"'" opposite directions, B's by o„e-tcm|,; a, ," ^e j ;, 1'?'"*, 'I ^'•«"'"' ">» uonicrbv'lKa v.ir-ff.5 J, , ' re'ichod the onmsite area of tl,e (Jld Va , ,„M ™ ' """""' «• I'™^« «'«« e diagoj,al less tlt:rilo\™.cr ' '^'" ^O^Oy'"^^. ""■■ il^ 11. The Sovereign wciL'-hs'i 9 Q. 9 7K carats, and the Napoleon Mul^t I ^''''''"^' ^^^"^ ^^ 22 being nine^enthsK ^^tisS^^^^^^ '''"^^^'^'-''^^^^^^ twenty-hve francs, and to na^oTh'*^ ^o coin a piece of interchangeable. Sui nosh.i h . l^'' ''"^ ^^'^ sovereign in the standards on^^^T^'^:"'^ '^\^ ''^^^'^^^ alteration can this be done with t^ielo-^^.^ ?''';' ^^ ^^ ^^^ «^^"^«. how and What will be et'ii'^lfir^"^''^"«^«fthe coinages, The Chancellor o? he pt?/ ^'^.''^ ^^^^ ^^ ^^^^^^ ? June 7), to hav each sovereign (tl is cHiUr T""^^- ^"" ^'"''^"^ ^^^ of grain) by w^y of o . ^^^^^^^^^^^^ ^^iglit by one the 25-franc piece 1^ i f ' V ?J^ ^^ identical with Chancellor probably me-,,n 9 '"^ "^^ ""^^^ ^h^<= did the En!;ilsl!'a;ron t^ why the France. "''' ^" ^^ «^ ^he Pri„ce Imperial of A person holds stock in the Enxrb'.h -^^i are at 98, to the amount of ff^nn .^ ^f."* ''^^ transfers to Canadian Governmlffi "'^'^'''^' ™' ^'^ at 105. Find the alterS^fn^ f« • P""'' ''^'^^'•' ^^^^^ are pound sterling is wonh "4 87 '"'"''"'^ ^" ^""^^'^' ^^ ^'^e What rate percent nn.. , "''^ei est and discount, note for one v'i^ar^u 7 per S"" ""'"'' '^ ^"««»"" »" ^ ff "■ Pounil sterling to £2?S\.ndC£"M Xho^nH''''"rf ' ^ ««"tributes .«50, l^ '"Hlone year after Bdr^^'l.ro'^r'^f ''''•'"^ •^'«»- is wound up at the end offnm "" *''" P"rtnci-'g tl,o original'Un^,';:^' V mn:"'" " *"*^ '•"^"^ <"• ofir a'l"'Sc,ffi- "" ^'^^^ ^ "'ere -d add together r^JS;\trfS^^^^^ the number wliich is their ^m ft. i, ' ^''^'^ subtract with the decimal Pdnt reLw. i 1 "'^ '''"^" ^^^^^^>«^' Prove the above Tie ^^''^' ^^ "^« "g-^^^- 24. A grocer has two kinds of too «.i • i seventy and eighty-five ceiKn .? ? ^'^^'^''^^ ^^^^ ^i"" mixes them in th^propor^^^^^^^^^^^ I'^spectively, and to one of the h,,tt^' A v h ^ of two pon.id. of the formc-r ture in order to^aliz^^Mr v Z^r" '"?"" ''' '''' '^''' "^i^" " '^'" cniity per cent. i)rofit? m li^ 320 25. at?ttttmt;tic. Find tlu! square root of .3 to twelve places of de«;i. luals and deduce the values of —,, , ^' , 4/0-33 s'3 1 - ^'3 ^ 2{\. The value of a diamond (brilliant cut) varies as the s(iuare of its weight, and a dininond of one carat is worth £!(); find the value of the Koh-i-nor, which weighs 1021 carats. 27. The Yorkshire coal field extends over 940 square miles, with an average depth of seam 70 feet ; coal is about 1| times as heavy as water, of which a cubic foot weighs 1000 ounces. The annual consumption of coal in Great Britain being 80 millions of tons, how many years will this field furnish the supply at thn.t rate ? 28. Find the present worth* of a promissorv note of $728, due 2 years hence (a) at sinijjle interest, (b) at com- pound interest, money being at G per cent. 29. A person holds $4000 of Bank of Toronto stock, and directs a broker to dispose of it, and invest the proceeds in Ontario Bank stock, after deducting his commission. The broker sells at 179 and buys at 107, and charges one- third per cent, commission for each transaction. How is the income of the owner affected, supposing the Bank of Toronto to pay twelve and the Ontario Bank eight per cent, per annum ? 30. A dealer sells at a profit of 25 per cent. His pur- chaser fails, paying 75 cents on the dollar. How much per cent, does the dealer gain or lose ? 31. A person bought an amount of tea at the rate of 8 pounds for 5 dollars, and sold half of it at the rate of 5 pounds for 3 dollars, and found he was losing money. He then sold the remainder at 3 pounds for 2 dollars, and on the Avhole transaction gained |1. What was the loss per cent, on the first sale, and the gain per cent, on the last? 32. A person has three notes, one for a certain amount, and each of the others for half as much, and all three due in twelve months ; he gets the note for the largest sum discounted at 10 per cent., and the two others at 8 per cent, and 12 per cent., respectively, and finds j^^hafc the sum of these two discounts is less th.'vn the discount on the other note by $3/^'V- Find the nniounts of the notes. SELECTED UNIVERSITY PROBLEMS. 321 n JLf ^f '^''' ^?r ^""^ ^""^^' ^^'^'"^^^ ^" ''^' ^^'-ti^'-'^l position agnmst ^i tower. Ihnv miicli will the top of the la Irler bo lowered by drawing out its foot 10 feet^ in a hol^cmtal r^<;?•nf'''''''^'^''■^^ ^^"^ "^^^'^^^ «f extracting the square ItoLtnJ number giving reasons for the successive steps in the process. ()l,tain the square roots of fnf 1 . '^^' '''>'->«10482 ; 894651027; to four decimal places. ' qqoLnn^"^ is purchased in New York to the amount of TorLtfX; >^ ^''''^^^'^ brought to 1 01 onto, where it is all expended in the purchase of United States currency notes at the quotation 88^-. These notes are then expeiKled in the purchase of gold at The quota- and wl ir..,^r. "'""'^ was gained by the transaction, blid out ? P^'^^"'"''^'^ '^^' '''^^'^ «n the amount originally cubeVSnf\:tmlref '""^'^^" ^" '''' '""'^'^''^ ^' ^'- ll'^I'T-^"''"'''^''! ''^'^''''''^ ^^^ «f ^6 feet surface and ni[ 1 wwr' f '' '^^'"^ ^^^^^ ^^^^^ of another kin metal, whose sur aces are as the numbers 1, 4 and 9 F?^ r. 'f ^?^'^^^^«^^ ^^^^i^l^fc equals that of the box Find the lengths of the edges of the cubes, the weight of 37. $500 is offered by a buiidiiiff society to be remirl d"ebtTs nTf IT'T'' °'^-''' dollars el:.h so thSe sent F nd fh! '" !''P '"" "^ '™ y^'-"-^ f™^ *h« Pre- sent. J'md the society's rate of interest. 38. A bank wishes to realize 4 per cent inte-f>st on its discounting operations. Form for it a .table of tie rates dljsl'respeSy.""""' ""'" '"^''''' "^ ^^- <"^ -^-^ «« 39 A person who holds stock in the bank of Montreal vesf in Vl'r" '''.' "-^'"^ "' ^20,000, sells out and f^ th„ M . R Cnnadinn bank stock, which is at fli). If RoiiT '■*" r^*™" ^P'' '"^ ""'■'^'""1 «'■ 10 per cent., and the Royal Canadian a • l-md the imniber of dnys ex?LS^t,X4":;^'of T'"''";''^ ««•"- - '^e may obtain as many more ei onri'""'' .'""^ ""^^ "'<^ of division. ' ^ °"®' by a simple process unf;':?;'fcS.tl pir ^«<"o*-^«^-«2 to Within a Bto\Torm14"aH^VS'ti?,-r "'. ""' --^ '•™"' person travel^, opfto^^^n ^'*^^ ^'^^'^^ ^''^'^ ^ to C. A tl^n crossrbv't l^e ^/e^^^^^^^^^^ '' ^«--'ds A, 'and C to A and finds li^Zf1l^:i ^.l^ ',r' ^'^"^ ^''^^"^ from C. Arriving- at A he finds Til ^'"^ ^ """^ «^^'^'" one-fourth of the^istlce tt B to%T' 'r'^'^^^' ^^>^ have done had he not lof^ fi. J .•!.-! ^ , ^^'"' 'le would 4| dista nee of B from A and C eft tlie (lir ect road. Required the 324 ARirnMETIO. 5a. On .'I r;iil\v;iy nro two p/inulcl tracks; on one of tliosc, tniJiis paHs n certain jujint every 1',).', minutes ; on tlie other, the same point, every 52Jniinat(rs. A train on the former track lias just passed this i)()int, and in 27^ minutes one on the lattiu- will do so. Will trains on these tracks ever ])a,s.s tiiis point at the same instant ? If so, in wliat time from* the ])a,ssaffc just mentioned? 54. Prove the following rule for lindin^- approximately the interest on $1 for any numher of days, at six i)er cent".: Divide the number of days l)y six and call the (piotient mills. The interest on a certain sum of money for 73 days obtained by this method, differed from the' true interest by 5 cents. What was tlic sum of money ? 55. A and I^ a,rc to race from U to N aiid back ; A moves at the rate of 10 miles an hour, and gets a start of 20 minutes. On A's returning from N he nieets B moving towards it, and one mile from it; but A is overtaken by B when one mile from M. Find the distance from M to N. 5(). A mortgage dated 1st January, 1872, payable in three equal annual payments of ^200 each, with interest on the M'hole, jxiyable half-yea I'ly at six per cent., is sold on the 1st July, 1872. What sum must the purchaser pay so that the investment may be worth 8 per cent.? 57. The cajitain of a privateer seeing a trading vessel 10 mik^s a head, sailed 15 miles in direct pursuit of her, and then observing that the trader steered at right angles to her former course, changed his own course so as to overtake her Avithout changing the direction of his ship's motion, the privateer running 10 and the trader 7 miles an hour. Find the whole distance travelled by the priva- teer before overtaking the other, and the time occupied. 58. A person has a certain amount of bank stock which he^sells at UOJ, and invests the proceeds in 5 per cents at 71)^. When this has risen 5^ per cent., he purchases the same amount of th(^ original stock as he held at first at 10!)^, which noAv jiays 8 per cent., and finds that while $110 remain in cash his income has fallen $8. ^Ind the percentage originally paid by the bank stock, allowing l pcr cent, brokerage on each transaction. m '<• ! SELECTED UXIVEnSITY rnOHr.EMS, 326 Stops, al,d c^ch of'z o :,^ w^;;";;';;;." ",""'•, """" <""' the .Mul of tl,c s,x.,„a Ka. "mfl r, "'T'V"' "' the w„rk be com,,loJ'.d ratio, m wbat ti,n„ will !.n.™ expanded u„iform^::Sil?<.,7.m!fu;: *;';,."'''' whole distance and the disCeo Iv f'^ l"^" '"^'''■'^™ "'« the distance l.etwl the wo «oint fi, ""l °"'', '""' "*« forms tlie same o, o.' ,t on winf? ', *""'"'• "" l*''- the distance betTe i ;"t -o' ,, ir^r^rr e'' I ^ '"""" """^ ho one inch less tha,n in the?, ' Vh t s h'c'disr '" from A to B ? ^i« w juii is tiic clist.'ince of^ok^''''""" *' •''"'"' ^y^'^"" of expressing the c,„alitv carat SJgrs.? ^ ' "'^ ^^^''''^ ™Ported worse 1 fi2. An oarsman finds that durin? the flr«,t h.,if „<• „ which flows at the ..fi^fffr^ ','"'"' ''"'™ "**''•«"" finds that bVloil down 1, ";'t" "'""?. '"' ''""•' '""' '"^ wards it takes hinf hon, Ton '^." "''' '"'"''."P "*■''''■- than by eoinff flr, , ^ .. h "7 *^'" *"'<"' "'« «o"'-so in^lJl^Art^^:;;-:?! --- r ;^;^;;„j';r'^-''^r taxes. On both income an.l n,™ oVtv lo I ',"'"' '"' rate of li). „,„„ on the do r 'a Vl ..^^:: ■* '■" property taxed altogether? *'' "* '''^ 326 AniTirvTr/rrc i'A. Find the amallost f^nm that c/m ho oxactlv i.'ii.l w.th hoth shiIli„,.H .t.rlin,^ and currency, one ^0, sterling- hein«- worth $ LHl. ^ ^ «5. Bank of Commerce stock is wortli 120 and Davs •. dividend of 8 per cent, per annum. Find the in!;^! "i^n„; 00 sliares and tlie amount ohtained by the sale of them ailowinff the l)roker a commission of i per cent 40^.ft* %^ .^'•^^^V"^''''/.'^ ^^^^"""' ''^' ^"''''^'^''>' ''^^ 75 cents, 40 at W.r>0 and a certain nuinhcr of p,ll(,„s at one doHar «-;illon, he would liave gained twenty per cent, profit and SIX per cent. Interest on his capitnl ; hut owin^ to a ^ kl n^^e he ff.nis us interest and l(;i per cent. proHt. Fud the number of gallons thnt leaked out. 07. A niMii is to row over a certain course in a cortnin time By rowing at the rate of four miles an hour 1 e would arrive 5 niinutes too late; and by rowbig t tl e late of five miles an hour he would arrive 10 minutes too row^g. ""' "'' ''"^^' '' '''' '^'''''^^ ^^'"^ ^'-^ ^^^^ (W. A room is 20 feet long, IG ft. wide and 12 ft. high with openings of area 94 S(,. feet. It takes as much plaster ing as another room which is as long as ])ron I fl high and whose openings have an area of 70; sciuare ft Find the length of the second room. nuH^n*^ >'?? '^ *"" ^^- distributed among A, B and C. From part o it they receive equal amounts, and of the rest B^ share is 10 per cent, more than A's share, and C's is 10 per cent, more than B's. Altogether B's share is 8,1-' per c^nt. mx)re t^ian A's, and 7- per cent, less than C's.' ' FM the part of the $1200 that was divided equally O.J ,f'/T^'-"' '?^^«^"^^^>f'«ix per cent, stock at 95 is sold mic'e Vi f'"'^/r'''r^ "^ '^^" ^^ P^^' ««^t« ^^t '<^ certain piice, 1 is found that the resulting income, after deduct- ing an income tax of one per cent.'is two per cenrmore per con? P^'T .'"'^ ''''r ^'^^^^^^^^"^ '^^ ^"«««^« tax of 2 per cent^ Find at what the second stock is quoted. nrJir "^f^'^^'^ ^«^^ C'*^""«n balls of diameters 3, 4 5 and 6 inches, respectively. Show that the weight of the largest is ec,ual to tlie combined weight of all fhe otherL SEi.ECTEl) UNlVEUsriT I'Uujjlems. 327 How ,„;u.,„ i. iiis'^u;,:?-:.".;;!;;,"'"^'' " ^'''""*' '^' ■"'.■ V(i. A jioi-son lijis Ji2U0(X) in vested in -,expressaya.d metre. ' '^^^ ^^" ^'^^Ii in terms of the centi- ^'^^^^r.^^JZ^^'^'^ -•«, -i-'iveiv kilometres. ^ <'J tannics, i^^xju-ess these in square |oid;4^Se"S t'^dSy'art^' ",' ^'^''-^-^ SP-->' Spanish e-old havins- been tak ..l .f ' ' '•°''«''J«ence of only worth DiJ eonts on the dol , -'""i' "'."''■'"' " '-"'' a^'wanis claimed. Calcnhfte e ^nu'tf "'' ^'"'^ -- •M. How much Avill «!innn '"^ '""ouiit. pound imerest t 4 pcr^nt'T'"' '» '" ^ years, con,- "•''• ^^Pei'son pays |2'J2.50 for «!!nn 1 • hence. Wliat rate ,>er cent / . ° ''"" '" ""'•^e niontlis '••fi. What is e.-,m h i, e""'™''' ""''^ "^ ■■eceive" change i.J- p.e. i.ren.u'n'ri.. "'« "^^pn'ssion, "Sterling, ex- A person pays ,?l81..r,o fo, £07 ,„ , cent, prenuum is sterling excI..ufJe-i*- ^''^- ^^'"'-^ l«n- in >' i'ou.rard''rdTTh;?r';f,f '!"- « -^ -^-f^ ^gam stream in tlie same timeJr/mU ■" ""'"^ ^vith the "'- each way and thrra'S of" ^strc?;'™"' "• ^""'^ "- «ie length of a give,mr, ^ ' te 'C''"'"-'^' '" "''^'^"'«' together at any instant -itTl ! '''"PPo^nig tlicui to be move in it from cJt^^nt^JZmT "',""' '"'"' ""^"" -It 94. Fmd the income from each^ ^ ''''"^- "^^^^^^^^ If 3' M '] J'ii IHII : i 330 ARITHMETIC. 100. Two trains, 92 and 84 feet long-, respectively, mov- ing with uniform velocity in opposite directions on paral- lel rails^ pass each other in 1^ seconds ; when moving in the same direction the faster passes the other in 6 seconds. Find the rate at which each train moves. 101. A cistern is 2 metres long, 5 decimetres broad and 8 centimetres deep. What is the quantity and weight of water it will contain ? 102. A crystal weighs 1-53 ounces in water and 1-73 in nai)litha of specific gravity -85. Find its actual weight, bulk and specific gravity. 103. Given one pound sterling equals $4.8o§. Obtain short methods for the conversion of sterling into currency and currency into sterling, and illustrate by examples. ' 104. A man bought $500 three per cent, stock at 93^, and after receiving one dividend sold at 96f . What did he gain ? 105. Explain the metric system of weights and mea- sures, and give the equivalent of each metric unit. 106. The issue price of certain railway shares was $50, to be paid in 5 instalments of $10 each, the first on appli- cation. After a "call" or second payment of $10, the shares stood at one dollar a share premium. A person then invested $750, and after paying a further call of $10, a dividend was declared of 8| per cent, per annum on the paid-up capital. What is the amount of his dividend, and what interest has he got for his money? 107. Three boats started at the same moment at inter- vals of 100 yards apart ; in 6 minutes the third overtook the second and in two minutes more it overtook the first. How soon will the second overtake the first ? 108. The interest on a sum of money for two years is $319.58, and the discount on the same sum for the same time is $310.74; simple interest in both cases. Find the rate per cent, and the amount. 109. A person shooting at a target, at a distance of 545 yards, hears the bullet strike the target 4 seconds after he fired. A spectator, equally distant from the target and the shooting ])oint, hears the shot sti'ike 2^ seconds after he heard the report. Find the velocity of sound. SELECTKD UNIVERSITY PROBLEMS. 331 Of lxch^i"o'!lnTf r'" ^ ? ^"'''^ 1000 francs by a bill if« .roi,,^- ^'^^liing. j^jnd the amoant of the bill and for which tKu sellf '"'^ *''" '"'^''"°' ^"^ ^^a"<== the"l!;u?inir'?"'','"'°'l"^/ '''"'^'^ ^ ^ '"«hes long.nci o4 V.!; r'h^i^^^^^^^^^^^^ i^ra- r-- ^ '- ^ co^kwefehs '"la.l'T'' '™<^« - heavy as water; out of B and put In toV ^Tr " / 1','' "'^^'"'''^ '^ '^k''" 114. What must be the erosd nmrin^^ ^4? ~iS :::r;aauF 4?----rd'a" remain $11312? ^""'" °" '^^ ''^^''l"''. *ere may $8 thin he eld of coffee for «10 '^W! '. "'"•\°f «»^«r ^r pound of each ? * ^'"" '''''' *« P™e of a ofTi^^:X:^t °;^«-«ng to any root wiiVL fe~dTn t^rnsiretroVrcir Ar ^'r"^ external dimensions are to to be 3, f andl foe'?' ^'"" Plankne es , ; fr?,^";onstrS"„ oT' I' "^^--J-'-'-^'f .•nte™al dimensU nCrrcTa^^ ^^3^??' -"''"^^ and"-in, tVl ,:rof \Xr'' .*rV"'r'^' -^"^ *« -our an angle of 60 de^.r^l'^' F^d the '^ime!' '" '''^'' "''"'' ^' 22 332 ARITHMETIC. t ui 3 + ^2 5 + v'C X -■■' — • 120. Multiply 359,999,999,999 by 799,999 and divide the result by 599,999. 121. Simplify 2 + V3 2-JZ 3-^/2 5-v6 122. The crew of a boat row G miles down a river and half way back again in two hours. Supposing the stream to have a current of 2| miles an hour, find at what rate they would row in still water. 123. Show how a piece of paper 8 inches square may be cut so as to form a parallelogram, whose sides are very nearly 13 and 5, and whose angles are very nearly right angles. 124. A vessel 5 ft. 10 inches long, 4 ft. 2 in. wide, and 3 ft. 4 in. deep is filled with water. Find the weight in pounds of the contents of the vessel ; a cubic foot of water weighing 1000 ounces. 125. A Glasgow merchant ships to his Montreal agents for sale goods for which he pays £116 sterling in Glas- gow. He pays an ad valorem duty of 12 per cent, upon the goods and a commission of 7 per cent, to his agents for their services. The goods realize in Montreal $780. Find the merchant's net gain, a pound sterling being equal to $4.86. 126. A person ordered $150 to be distributed among some poor people, but before the distribution had taken place two more unexpectedly appeared, in consequence of which the former received $2.50 each less than they other- wise would have done. What was their number at first ? 127. A grocer can sell coffee at 30 cents a pound, and realize a profit of 25 per cent. He, however, mixes the coffee with chicory, which costs him 6 cents a pound, and selling the mixture at 25 cents a pound realizes a profit of 40 per cent. How many per cent, of coffee does the adulterated mixture contain? 128. The area of the base of a cylinder is 2 square ft. and its height 30 inches. Find the height of a cylinder the solid content of which is 3 times as great, but whose dia- meter is only twp-thirds that of the given pne. a,nd divide SELECTED UNIVERSITY PROBLEMS. 333 I, river and the stream what rate ijuare may 3s are very early right . wide, and weig-ht in )ot of water 'eal agents ig in Glas- cent. upon s agents for ^780. Find g equal to ted among had taken sequence of they othcr- er at first ? Doand, and mixes the pound, and zes a profit e does the iiare ft. and rlinder the whose dia- laccs of decimals 129. Find accurately to three cube root of 78836421 perform 1 ILZf ZiVlZZ'T'"' ' "if"™ ^ '"^y be required for 8 men nnrt^ , ^ ' '"''"■"y''"y'^™W day to perform the s:"rto.k";?T,r'''''"f " """'•■' ^ rate of 8 pe^cm Dn v^iS ^ *°^f '""" the privilele of mW ? t "l"'-"'"^'' "" '«'-ro«'er having of prineipa? annuX^if £ T'' '"'^ *"'"'" *»" °" '^««0'>"t id4. A goldsmith mixes gold of 15 iq 9^ onri o* tTtv Of J' V"""P°r ""-^^ ^^ aom^Le' " What mvlT tity of each must he use ? *viicicqua,n- Any consistent answer will be sufficient, twopenct '^elk tr/T."! ^''^'' "^^« ^' '^^ rate of 5 for Point out the conditions Of possibility. ■I -. .1 ^ hi l!;i i:^ I 334 ARITHMETIC. 137. A room 24 ft. long-, l^ ft. wide, and 11 ft. high lias two doors and two windows, each 8 ft. high and 4 ft. wide. How many sciuare yards of plastering will it re- quire, including the ceiling, and how many yards of car- peting, 27 inches wide for the floor, allowing no waste ? 138. Find the sixth root of 2,505,726,409. 139. A square number cannot be of the form 12n + 5. 140. The product of three consecutive numbers cannot be a perfect square. 141. Define the terms circula,ting decimal and perfect repetend. If 2wTl ^® ^ perfect repetend, and n figures of the re- petend are known, show how the remaining n figures can be found by simple inspection. 142. Given 1^ = 0-0588-^4, write down four more figures by simple multiplication, and the remaining eight by inspection. 143. The scale of a map is j.^oVoo ^^^^ the distance on the map between the positions of two places is 2| inches. Find their actual distance in miles. 144. Find the radius of a sphere whose volume is equal to the sum of the volumes of three spheres whose radii are 7, 8 and 9 feet, respectively. 145. The present income of a railway company would justify a dividend of six per cent, if there were no prefer- ence shares ; but as there is £400000 of the stock consist- ing of such shares which are guaranteed 7| per cent, per annum, the ordinary shareholders receive only 6 per cent. Find the amount of ordinary stock and the com- pany's income. 146. A man has a piece of land 109 rods one yard TO f^"- lo»8'» ^ind 66 rods 1 yard 1] » ft. wide, which he wishes to lay out in the largest possible square lots of equal size. How many lots will there be ? 147. A person annually ir-reases his capital 20 per cent., less a yearly expenditure of |500. At the end of four years his capital amounts to .«18,052. Find his original capital. 11 ft. hig-h I and 4 ft. will it rc- rds of car- ) waste ? 12n + 5. 3rs cannot ad perfect of the re- igures can 3re fi^'urcs eight hy istancc on 2| inches, le is equal hose radii my would no prefer- 3k consist- ' cent, per nly 5 per . the corn- one yard which he ire lots of al 20 per the end of is original SELECTED UNIVERSITY PROBLEMS. 148. A i)erson lei't |6000 to he divided 335 uioiitlis U yciirs and 2 months, re.si.ecdvely in sticli ? m'i ve at h '""°T> f" "'^^ '"""' ^""' "'''•^'' ""^^ ''■'""W sluu-e? *''" -' ^'^'"''- '*^''"" ^^"'« «'«h one's wii'f'^^ merchant bought a certain <|uantitv of corn for wl icli he Paul a certain smn of money, but 6n nw.Ju-'Z linin""'tr ii;ff 1 "'" f """'>' '" '^^P-'^'' ' '•« -l' if the rate otV'll ,' '"k' ??'™" *-!*''''' ^'"^h ^"« '« m w 1 ti , " *^"''/'' ''"''^"' ■"•»•« *•'«" l'« ■'vould have paid liad ho received tlie quantity he expected How many^busheis did he supposi he Ma bought'and'at wh"u :: oi ^? s :n^ ^, ^^^: i ii^r s^- „t-: miinor ho?u^'"^^ '^" ""-• ^^'-' ^^ "- --^e '" 151 The recent 3J- per cent. Canadian loan of £5000000 was placed m London at the rate of £91 2s. Od for each fn T An.! !f P''^'^^^^' "^^"^ P^^«^^ i^ Canada, bv drafts exchange), what was the total amount in Canndi^n currency realized from the loan ^ Canadian ruf ^V ■xwL'''"\'^r^^ ^' ^''''^ ^"^' ^^ ^^100 debenture to run foi 20 years at 4 per cent, per annum, in order thu the^ investor may realize 5 per cent, per knnum on Ms 153. A contractor engaged to complete 1000 vards of radway m 50 days, and employed 100 men workin.^ < hours a day, but at the end of 30 days he found only 450 ynrds hnished. How many additional men must he hire m order that all working 10 hours a day may finish the work in the given time ? ^ ^ ^^^ 154. Standard gold is |18.94.^^ an ounce. Find the least number of ounces that can be coined (1) into an exoot number of $5 pieces, (2) into an exact numbn.. ;T4^^r eigns, and find the number of coins in each case. 380 ARITHMETIC. 155. Tho valiu; of diamonds varies as the square of their wciglits, and the 8(iuare of the value of rubies varies as tlic cube of their weights ; a diamond of a carats is wortli m times a ruby of h cara,ts, and l)oth together arc worth £c. Find the value of a diamond and ruby, each weighing x carats. 156. A wateli wliich is 10 miimtes too fast at 12 o'clock noon on Monday loses 4 minutes and 12 seconds per day. Wliat will be the true time on the following Saturday morning, when the Avatch sliows 8 o'clock ? 157. Tlie L.O.M. of two numbers is 100793; the G.C.M. is 1 7 ; the difference of the numbers is 1224 Find the numbers. 158. At what advance on cost must a merchant mark his goods so that after allowing 10 per cent, of his sales for bad debts, 8 per cent, of tlie costs for expenses, and an average credit of 9 months (money being worth 4 per cent.), he may make a clear gain of 20 per cent, on the first cost of the goods ? 150. What will l)c the true interest on $1000 for 6 months, it being supposed that if this interest is invested for the next six montlis that the whole interest for the year shall be exactly six per cent. ? 100. A merchant in London remits to Amsterdam £1000, at the rate of 18d. per guilder, directing his Am- sterdam agent to remit the same to Paris at 2 francs 10 centimes per guilder, less -| per cent, for commission, but the exchange between Amsterdam and Paris happened to be at the time the order was received at 2 francs 20 centimes per guilder. The merchant at London, not ap- prised of this, drew upon Paris at 25 francs per pound sterling. Did he gain or lose, and how much per cent ? 101. CalcuK'ite to within yJ^y,,- of one per cent, the rate per cent, that paid quarterly is equivalent to 8 per cent, paid annually. 102. Two boats start to row a race at 3 o'clock. The race is over at O;:; ]ninutes past 3, the losing boat being 40 yards behind at the finish. At 4 minutes past 3 this boat was 700 ya:'h from the Avinning post. Find the speed of each boat ':\i miles per ^lour. s(]Ur,ti th^l ,i ^ . "* ■"!£•".' rr" *.'■• ■>•"■ '~»* .Si ""■ "■'" an^rthP l^r i r^'^P* ^'' expenditure of $2000 a vear first^and'^kfm'''^ ^ ^""f^f ^ ^" business. A puts in $400 at Som . « ^ "^^^^ '^<= the end of six months. B puts in SVsixl;;ts"^^|^'T^.T;^^-^ ^^'h- iapL^aUh^ mon^h. ^,T ? P'l^' '" ^^^ ""^ ^^^ «"d of every six iTym wl . ? ^"'*. ^^ ^"^^ ^^^'-^^^ they have gained •^(;700. What share of the profit should C receive in nH dition to 25 per cent, of the total profits for m''a-r~ th^ business Y ^ i-i"najjing the Ulr 838 ARITHMETIC. li^ I : t' ft- '' ( I It si i 'I II '1- III |5 ifi I i :ii i- 170. Lumber worth $20 per tliousand feet is exported from Canada to the United States, to be manuftictured at a cost of $25 per thousand feet for re-iniportation into Canada. If the export duty on lumber be 10 per cent., and the import duty on the maimfactured article be 30 per cent., find tlie cost of manufticture in Canada that the Canadian manufacturer, after making a profit of 20 i)er cent., may sell his goods at a price which would leave the American manuftictiirer no profit. 171. A ship carrying- a certain number of passengers m becalmed with provisions for six weeks, but it is found that one person dies at the end of each week and the pro- visions last for 8 weeks. How many were on board at first ? 172. Find the time between m and m + 1 o'clock, when the hands of a watch are exactly 7i minute spaces apart. 173. A ladder when placed upright is 4 feet higher than a wall ; when the foot of the ladder is removed 16 ft. from the wall it just reaches the top. Find the heig-ht of the wall. ^ 174. A rope 512 ft. long has one end fastened to a corner of a house 25 ft. square, around w^hich it is then com- pletely wrapped. How f;ir will a man holding the other end of the rope, and keeping it stretched, walk before the rope is re-wrapped round the house in the contrary direc- tion ? -^ 175. Find the effect of adding the same quantity to both terms of a fraction. Employ your result to compare the values of the frac- tions 176. A cubic mch of gold wcjghs -\^f\p^oz. (avoir.), one of silver -VfW- oz., and of copper f ?§§5z. A coin whose volume is -04 cubic inch is composed of gold, silver and copper in the ratio (by volume) of 12:2:1. How many grains of each kind of metal does the coin contain ? 177. A mortgage for $1000, ])earing interest at the rate of8 per cent, per annum payable half yearly, has two years to run. Find its present value if inone'y is worth six per cent, per annum. I exported icturcd at ;ation into per cent., icle he 30 hi that the of 20 per I leave the isengcrs m t is found d tlic pro- . board at ock, when ces apart, ig-her tlian v^ed 16 ft. height of ;o a corner then com- tlie other before the ary direc- ity to both the frac- 09 (IT* voir.), one Din wliose ilver and ow many in? Lt the rate , has two is worth SELECTED UNIVERSITY PUOBLEMS. 339 n^\?V'''''^' '•' ^^^^ ^^^^^^ investment : doHar ? ' •^ '" "" ""'"'""^ '"'^ oi' 1^ mills on the 179. A person borrows «1000 for two vcars onrf Hi« diaries tlic debt bv vavhw «(U«i .., *. ^ , ;. ° "'*" and mm at tl,o e,fd^o/two vcnrs Wh'?;' i "/ """ ^"'"" per annum (co,n„„„nd) intc°relt did 1 'IJ'-V'"'^ '"' """'■ per innltm ''"hi"T-''? '"'^ '"■" ^°'' *"• '"^ ^ Premium of « j:meo-zx;. t^iXof i:;?,.s.rc.t 't- rr reckomng compound interest at the raT ;^?rp"er cent.'U and A and B 'tigethe d'v "irt^ ^TT' '"'* * ""■^^' take by himself: ^ ' ''*"'' '""^ '''^"h would 183. DcHiie exe'iange and par of exohTiiD-r. wi„,i .i par of exchange betwicn England anrthcUnuoJs,,,'" •58 gS *^;;i^"''\"r United States-TlO golcK -.>«gis an( IS ,»^hne, and that 40 lbs trov of I.'n.Hkh standard gold, ^ fine, are coine,l into 18«9 s^ovorc^gS"" 184 If a person invests in the tliree per cents so s to have three per cent, clear on his investment after mvi,,," d en^re^cS^e" witn'^th'"^ ''™""' ^"^^ P- -nHl^^ the pound V '"" "^'^ '"""■"'^ t'-^^ >•« reduced to 6d. in a<.-ot;ri()(T(rn"„J'<'°,''''!'*''''' '''■»"«•' «"'l '«t sometime ot"n, ni • T ^u,'"' """" "' '"« ■*™«1' *47()0, the value off per ^nr*" ^v^T-u'"" ""''* "'' ""^ "°"^« "'^« ^ "" uii lu pel cent. What will it cost hw^ fn ,>o,,^p ^.u^ i.„„-., at f per cent, on 4 of its value ? ' " '''''^^ 840 AlllTUMETlC. m m ': . 180. A note for 11050, dated Aiif,ni8t 30, for three iiioutliH, and bearin^^ interest at the rat(5 of (5 pereeiit. per aiiuuin, is diHcounted at a hank on October 5 at 8 per cent. Wliat are the pi'oceeds of the note ? 187. Which is the better investment, 3 per cent, consols at i)()[ff, the dividend being payable 5 months hence; or, U. S. 4 per cents at 130|, dividend payable in 2 months, money being worth tlireo per cent, per annum interest? 188. Two men, A and B, enter into business ; A is to re- ceive $25 a month and 10 per cent, of the sales for manag- ing the business, the remaining profits to be divided e(iually. The goods are sold at an average profit of 30 per cent. Find the total sales, in order that B may re- ceive 25 per cent, of the profits. 181). A gallon of fresh water weighs 10 lbs., and has a volume of 277-271: cubic inclies. How many cubic feet of salt water (which is three per cent, hejivier than fresh water), will a vessel weigliing 500 tons displace, the weight of the water displaced being equal to the weight of the vessel ? 190. A and B are travelling on the same road towards Toronto, A at the rate of a miles and B at the rate of b miles per hour. At noon A was m miles, and at 6 p.m. B was n miles from Toronto. Find how many hours from noon A passed B, a being greater than b. Interpret the result when m = 40, a = 5, b = S and w = 26 ; also, when 11 = 18. 191. A man divides $1300 into two sums, and lends them at dilferent rates of interest. He finds the incomes from them to be equal. If he had loaned the first at the rate of the second he would have received $36, and the second at the rate of the first he would have obtained $49. Find the rates of interest. 192. A circular plate of lead 2 inches thick and 8 inches in diameter, is converted without loss into spherical shot of the same density and each -05 inch radius. Find the number of shot. 193. If 10 square cliains make an acre, find the length of a link in inches. Find tlie distance in links between the opposite corners of a square field containing 10 acres. E 16 . .: SELECTED UN[VERSITY PnOIII.EM.S. 841 194. A iimn'H income consists of a siiiarv ( f S.MO riJr nnnun., of < ivioney_^,s worth « per cent, per annum, iryalle half 1117. State clearly the difference between simple and comiK,und interest. Show that at 10 per cent. p. r mmm SZr'e'lhr:: '' '""■ "'■"'""'-'y ""' loubie i^»e;?in a iiLcio moie tnan seven years. 11)8. A in Toronto owes B in Liverpool £2500; in wliat different ways may payment be made ? Explain the method usually adopted. If exchange is at mr wh f sum in Canadian money will disehar,.f the debt ?' "' era! IZ •''™'' !''"^' ^""^^^ies, Consols, Brok- erag-e A person mvests $3840 in the 3 per cents it 84 and when they have risen to 8(> transfers^ J of Ms cfpitai nf ho.^ P''i T'f ^' ^^- ^' ^^« i'^^-"^^ ineieased or S n inished, and by how much? 200. Water is dischargea at the rate of 500 e-allons -^ Tonsrs^r"/"^'^'' 2700 square feet 7urS into H kt fn H ^T'^. ^^^^ '^''^'^^^«- ^^ind how long- it will take to dimmish the difference between their levels bv 6 ^ ^x^' '' ''^''- -^^^"^^^ '^ — :?n5 1 em1?iov(Tl%or'Sn '1 """"'^f^ ^^"^^ ^^ ^^^^^ ^^ masons were painters for 1(, days. If a mason receive 5 per cent less and the total cost of the woi-k is ^21)r,5.r>0 find how mm-^' each workman receives per day. " ' "" ^ ^''"' "^^^" 342 ARITHMETIC. 202. A workman in the city of A finds thcat lie has to pay 50 per cent, of his wag-cs for food, 15 per cent, for clotlies, and 12 i)er cent, for rent, and saves tlie reniM.inder. On moving to J^ lie linds that food costs him four-fifths as mueli as at A, clothes 3 times, and rent two-and-a-half times as miicli, but wages ai-e 50 per cent, liiglier. llovtr long will his savings for a year support him in each of the two places, respectively. 208. An importer entcu's for duty (at 30 per cent, ad valorem) an article which he values at $200. The value of the article is however found to be $330, and the cus- toms regulation in such cases is that there shall be levied on such goods (in addition to the duty payable when pro- perly entered) a sum equal to the same percentnge of the proper duty as the percentage of undervaluation of the original entry. Find the full amount payable in this case. I : 1; EXAMINATION PAPER& 343 EXAMINATION PAPERS. UNIVERSITIES OP c^tJEEN's, TRINITY AND VICTORIA. MATRICULATION, 1888. 2. Findihe Value of '' ^'""^ °*' ""^ P'''""' ^ ofl's Lif ct • ''^'''' ^"^ ''^ ^'^' ""1 B'« «hare the sum and'/ft'deen'in'^f/ *''"'"' f ^0 yds. wide nvmv Ln ?ww , f ', '""''"'"'S' ^^ hoars a day: how 5. A n,an left ,f of hi. elirto o .^^ n i 'Z,': a.'^'so per cent, of the remainder to another and he rest of the was ,k8.. How much did his widow receive 9 VcL-l',en,.c ."t ' x'*" '"'"'""' °" ^ ""'« «f *-*59.05, due 4 years! ence at i| per cent, per annum. «qm i «'"'■?""« on a certain sum due 2 years Jience is FM';];;'smu ^r;?' "V" """''^""' '-■ ^^y<'-S42 «. J he area of a i-liom bus is ll'JO sq. \ds. and the leno-th a 4 w^s«7r m'■"',"!*'^'' ',''■'■"''"■ 1^'"' °<>i-o""d a.t4!1.25 a yd was $(15,00, l-'md tjie length of a straic-ht iv,th I'unmnf, from si.le to side through 'the centre. ^ '"* , ,10. The pressure of wind on a plane surface varies win •^'v:i!^ur"»° *;/■'" ""■''"^•^ ^'"^ "'«"™- «f "-■ umih \Liocity. If the pressure on a soimre foot is i ih r , , " ±'.' ""^ ^"™* '^" 'novin^ With i:ij| f 3city es an liour. If :i '[ i M i V I ! „!i 344 ARITHMETIC. Q., T., V. MATRICULATION, 1889. 1. How do we know by inspection wiien a number is divisible by 2 ? by 3 ? by 4 ? by 5 ? by 6 ? by 8 ? by 9 ? by 11? 2. Work out to a single linal result each of the follow- ing : sl2 + 2 V2 + 1 to 4 decimals. 0-12 0-196 X - -- expressed in vulgar fractions. 0.3 3. A and B ran a mile race which B can accomplish in fij minutes. A gets 10 seconds the start and comes in 16 rods ahead. How would the race finish if B got 9 seconds the start ? 4. If I buy f of 4 J yards of cloth for $0.65 and sell ] of 1| yards for 24 cents, at what rate per yard must I sell the remainder so as to gain 20 per cent, on the whole transaction ? 5. A, B and C form a partnership for a year. On Jan. 1st, A puts in $300, B $500, and C $600. B draws out $3tK) on May 1st, and C draws out $300 on Sept. 1st. For the first six months the gain is 10 per cent, on the mean stock, and for the second six months it is $112. Find the total profits, and show how they are to be divided. 6. A person sells $6250 of 3 per cent, consols, when they are worth 96, and invests the proceeds in railway stock at 75 and paying 2| per cent. How does this transfer afi'ect his income ? 7. A fioor 16 by 20 is to })e completely covered with carpet 27 inches wide, and having a pattern which matches every 3 feet. If the carpet is to be matched, in which direction must the strips run so as to give the least waste, and how much is the waste ? 8. Of how many gallons capacity is a box 3 feet long, 2 feot wide and one foot deep ? >! number is 8 ? by 9 ? he foUow- Tactions. )mplish in omes in 16 1 9 seconds nd sell ^r of must I sell the whole On Jan. draws out ;. 1st. For the mean Find the led. when they ay stock at Qsfer alfect ,^ered Avith jh matches , in which east waste, 1 feet long, EXAMINATION PAPERS. 345 9. A silver plate of uniform thickness in tliP fn^rr. ^ ■(■'. !: Q-> T., V. MATRICULATION, 1S90. 1 1. Siraplifv r— 'J? — and "'^'Th , ,. ., •' 1 -^_» a™ f:f-r--r' and divide their (lif- fcrcnee by their prodact inerelled'by nnity to yaSs*!""' ^-'^^ of 0-518 of 0-081 of 0-00875 of 405 miles 6. There are 50 coins consisting- of 25-cenf ind 9n nnr., pieces worth $11.75 If enoh 9^ nn^l t.- 20-cent tr^^^wrdtiiH"^-^^^^^^ each kind P ^ ' ^^"^ ^'^''^^y ^^^^^ '-^^'^ there of 8. What two numbers, each of four fig-uros h'.vo im for their g-reatest common measure and IwVfnr.i least common multiple? ' ^^^ ^^^ *^^^^^' _ 9 The discount on a certain sum for 21 years i^ 4o ^f the interest for the same time, simple intent be n' rock oned m each case. Find the rate per cent nn^ ?f^i • ' terest and discount t-r^o^he- ar^ '^-r^ m r" i ^- '^^^''' '"" pal sum ^-fc,-^ne. arc s7oGo.40; find the princi- t 1 346 ARITHMETIC. 10. The external length, breadth, and hcig-htof a closed rectang-iilar wooden box are 18, 10 and G inches, resjiec- tively, and the thickness of the wood is half an int-b. When the box is empty it weighs 15 pounds, and jften filled with sand 100 pounds. Compare the wei>ats ot equal bulks of wood and sand. EDUCATION DEPARTMENT. THIRD CLASS, 1 . Add together of £13, 1883. 1 7id of 9j:of ;^ of £2 12s., and ^ f9d. Reduce 1 3s. 4,id. to the decimal of 19s. 6d. 2. Find by Practice the value of -8596 lbs. at £10 18s. eacu. 3. A person borrows ??500 on April 10th, and on June 22nd pays his debt Avith -^510.20. At what rate per cent, per annum was he charged interest ? 4. A man liaAing a ccrtMin sum of money to invest has an opportunity of purchasing 7 per cent, stock at 95, but dehiys until it has risen to 110. What per cent, is his income less than if he had purchased at the first price? 5. At an international exhibition one country was awarded 5 gold, 9 silver and 11 bronze medals ; and an- other 4 gold, 15 silver and 10 bronze. Find a ratio of values for such medals that these countries may l)c re- garded as equally foi'tunate. 0. In a l)ox there is a certain number of sovereigns, three times ns many guineas^ and twice as many marks (13s. 4d.) as guineas. The entire amount in the box is £815. How many coins of each kind are there? 7. Find, when iirst after 2 o'clock, the hour and minute hands of a clock make an angle of 60 degrees with each other. 8. For each of three succeeding months the population of a North-AVest town rose 50 per cent. ; and at the end of the third month was 2700. W^liat was the population at the beginning of the time ? of a closed es, resi>cc- f an iii'--'i. and ^en wei>ata ot EXAMINATION PAPERS. 12s., and i It £10 18s. d on June e per cent. invest has at '.15, hut ient. is liis 5t price? Lintry was i ; and an- a ratio of nay be re- 50vereig"ns, ^iny marlvs tlie box is ind minute with each population the end of julatiion at t W'S:^.;^ -^::^^^^^^^ century, except tJie average length of a yZ^ " '^'''''^^^ ^>' ^' What is l*^- A cube is fbnnPfr^^- ' l>oun iV , pound costs -07031-5 o( Fts wj "''• "^'"'S^''' where one ?«o: They ^Jr: r,'e'S;t '° "° '' P'-« «^ -rk for to work, which was 4 to^Vnd^Ke 7 '" *^"- '''^"^ which was 3 to 4. How much d d e C "'" ':''""' ^"''kod: 4- A quantity of silk vvi? ij ^'''^ receive ? had it been sold ?or is "w '"^ *" ** '"^^ »f 1 Per cent • been a gain of l fei- ctt'^'ra leV'T ^■'"'"""ve 5. A person rides to town at til f"' "'""'^ P'*o- hoa.^ and after resting- 35 nSnutes waS'^°^«i '""''^ «" o' -'^ miles an hour. The ,Xni„ V ^''"'^ ""* 'he rate ''o«'-s 2o,», mi„„j3^ Find the S'""' '"^"■Pied Was 7 «• Instead of a vai-d ,„„ distance. which is 36.35 in^hsong™;:^/'-?'^'- °^«« a stick by so doing? ^ '"'* does lie lose per cent SI money) is said to be at -^nor " <;^«''a"S-e (ie. Kng- tins calculate tl,e par of excl,;r "'"'• 1"-^™™"- Fro^;. »• If silver is worfh «i in , ^''• ounce, find the weight of. ten'd^n"'"'' "•"'' ^^"^^ ^" Per parts in 40 of gold,1i..d1i;: ^^^f ,^'^^^_''«'" eontainiiig-^'sr 23 If-i ]> 1 1 1^ ^^^^^^^H ' ii < 1 ,; i ! 1 ..} i ■i 1 i 1 f i i !f i f ! i ' h 548 ARITHMETIC. 9. Equal volumes of iron and copper are found to weigh 77 oz. and 81) oz. respectively. Find the weif^ht of 10| feet of circular copper rod, when 9 inches of iron rod of equal diameter weigh 31-,»^ ounces. 10. The expense of carpeting a room 15 feet wide was $52.80; hut if the length had been a yard less, the ex- pense would have been $46.20. Find length of the room. 11. A rectangular solid 41 feet long, 3^ feet broad, and 11 feet thick, is increased 11 inches in thickness. By how much must the breadth be diminished, so that the solid may retain the same bulk as before ? i THIRD CLASS, 1885. 1. Define prime number, factor, common multiple, dis- count, exchange. Draw a diagram showing that there must be 30^ sq. yds. in a sq. rod, if the linear rod contains 51 yds. 2. A merchant bought 124 vds. of cloth at $3,621 per yd., and 871 yds. at 14.121 per yd. At what price per yd. must he sell the whole to realize a ])rotit of 20 per cent. ? 3. Simplify the following and give the result in Jj s. and d. : — 3 1-125 -J of 1| ^.,^ -^ (3-3 4- 1-25) of £1 + 1 of -^T^fTTT- of 9s. 2-lCi + — r-d. 2-09 4. A farmer sold two loads of wheat, in all 110 bushels, for $94.95. One load was sold at 97 cts. per bushel and the other at 72 cts. per bushel. How many bushels were there in each load ? 5. A and B engage in trade ; A invcFt ■* $6000, and at the end of 5 nionths withdraws a certjiiu sum, B invests $4000 and at (.ho end of 7 months $6^(1: > more. At the end of the year A'a gain is $5800 and B ;. 's $7800. Find the amount A withdrew. ''* id to wei^h frht of 10| iron rod of ; wide was ss, the ex- »f the room, broad, and IS. By how it the solid Lultiple, dis- be 30^ sq. yds. 13. 62 1 per It price per Lt of 20 per suit in £ s. : 9s. 110 bushels, bushel and mshels were >000, and at ai, B invests At the end 3. Find the EXAMINATION PAPERS. who] .'•.Ar^.:^^^.'^„^."*^.«5^htelofl, 349 -It $2 at a profit of ,«?120 • h d hV Tf ^^^ «"^ «old the lesshe would have lost g% IT '""^'^ '^ ^t 20 per eent ^ 7. What will be tlctLft'"^'"'^^ f 47580, at the rate o '^^^^^^ ^^ P^'^perty won. loss the owner may recover both Z ' f" '^'''' '"^ ^^«« ot and the premium paid ? ^ '^''^''^ ''^' <=he property ^r^^^'r:7S!^r^^^^ '^ ^^- ^ --ths' in erest on B's share at it per oPn.' ^'^^'i^""«»^' ^ months' est on C's share at U per tent m?> h^k '^ "^^"^^«' ^^^^r- 9. I owe a m-m d-.vs • coniDlction nin •). w I .V'^ , " ''-^ ^^•'.'^^ b^^orc the comi)iLtion ii in Ji days bclorc the coinpJution In wl...f timo was tliG work done ? ^^i-ion. m what 4. Prove tlie rule for division of decimals. Divide to <; decimal places, -00785^31) by '\n)2Ui\i ^ ^sm ^\^^:'''''^\ ^^/'^l ^ ^>''»k s-ives me |8i5 for a note of 18 ,0. Wlien is the note due, interest 87 v ^'' ""^ 6. iind the cost, in sterling, of 181 tons 17 cwt •} ,irs Zr7i.o/ ^ "'*' °"' '" WJ, !Uid found th.-it I l,M, i). I mvested |10,000, but sold out at 207 discount TTow much must I borrow at 4% so tliat by in4thiriU a?87 I may just retrieve my loss ? ^voscin^ ail at 8/ 10. A square field containing- 27X acros h^^^ n riin^. i pajh across it What is the Icn.nh V the^th in yS' -3 ~nt Wn/fS realize? 9. A begins business with a capital of $3200; after " months B is admitted as partner with $2400 ; after 3 months more C is admitted with $1600. What fraction of the yi'-ir's gain should each have? 10. What is the cost of polishing a, cyb' drical marble ])illa.r, 2 ft. ' in. in diameter and 12 ft. long, "t $1.25 a Sduare foot? EXAMINATION PAPERS. 353 re(iuire«, ki'.H. All would it lot which 5 })cr ctiiit. 360 days id Mt $80 10 rest at 1)3 a day, 50. Ih'ul ;lie whole lie as the r of each. i. of Hour 1 tiiids oil 's, and C's hole lot at )n. What times .IS ei^hs 7-44 1 bulks of in 765 lbs. , if sold at iv cent, of per cent. r cent, bo ir did his ); after '* V 3 months ion of the il marble 't $1.25 a loni'nfv'.M? ^''V-'' *?' '''*'f"''" ''-^-^^room 25 ft. 3 in. ioii« 1.) ft. I> m. wid(,, and 12 ft. hiu-h, when the mnor h iyd. WHle, find cost of the paper per lin^r var d^ n allowance for doors and windows). ^ ^^" 12. A square field, containing- 16 ac. 401 so vds h'm -i walk around it outside 12 ft. in width. Find 'the area of the walk in yards. '^ ^' THIRD CLASS, 1889. 1. fa) Simplify ^;;^^--.4.i«tii_><(.t)^ ,, m fractional form.) *^ i o x b«5 ^^^ ll';^ ^{^ "^'^^ ^^7!^'^:^ I'l^^ees of decimals, of 1^3 5, 21, 7|, -034, 3-125, 0, 24-58 and 12..V Note.— No marks will be allowed for oitlmr /« i /j. f answer be perfectly correct. ' "^ """^ ^^^ ^""^^V^ the 2. In What, time will $30441 «ain $221010 if -it tho ""?5%'^2f ^h^r T''-'"' ^^^ ' year ar^ 15 day day; tf a year)?'' " ''' ^"'^ ^^^ ^^"^^ ^^ — ^^ (^65 is Lure'dTr tto^l^'^^^^^^^^ ,t f^f ' "^T^ -,P on an assessment of $1^45^, Td 1]^' 5 s'^^e'^:. i"^^^^ on^repairs. What rate of interest does thTLv^i^^;!; 4 A rectangular field, av hose width is ^ of its Ipno-fh than to -fTu .iround tlic two sides ? ai''« ^vork, unci wo •] r ll^H I '" ?^'"'' ' ^''"^ ^" "''''^'^" ^^> complete the niLii loi the 8ith (lay. How mucii less or mom tliuii -m nvcu.^0 day's work on the part of these 17 mel^" m^ Mr^^c^l-M^l'' ''t^iiTr/' " >'^^^- ^^'^^^ ^^"^^ »'^'«^^» 7 Tl.o .;. ;. ' V"^ ''^^ '>() quare feet ; the area of the floor is HHi sc.uarc feet A lowing. ^V of area, of walls fo? doors and\vndovvs how cylinder fitted with a piston be 20 lbs. on the s u re inclf md when the piston is forced in 2 inches tl cm mssim'. Srcyi^r i'^- ^^^ ''"^ ''-'- '^-' ^ -^- is r iS;:^ vests in Stocks at 108, which pa^^ 31% ha f!yeTdy Fiml the change in his half-yearly income ^ ^' 10. Bought s-oods at $5.70 on 4 months' credit and .old them immediately at $(112 on such a term of Jre it .^ made my immediate gain 6f %. Reckoning interest U 47 per annum, how long- credit did I give ? '^'"'^^ ''^ ^/^ ^^* //"^^ ,^^?\'^ "'"^'^"^ ^'y ^iveraging accounts? (b) hnul the equated time for the payment of the following account "^ ^ ^^- John Smith. q^. "1888 June 10 July 15 Aug. 20 Sept. 1 To mdse. @ 30da,ys, IS050 " " "45 '' 800 " " " CO " " " 30 '' 1888 July 10 Aug. 15 2501 Sept. 5 150 ^i If n 356 ARITHMETIC. 12. A merchant in Montreal drew on Hamburg" for 10000 guilders at $-415 ; how mu(;h more would he have received if he had ordered remittance tliroug-h London to Montreal, excliange at Hamburg on London being 11^ guilders for £1, and at London on Montreal i)|%, broker- age being 1|% for remittance from London? SECOND CLASS, 1883. 1. Prove that \ of y = jV Simplify (2f of 3A-) + i - (1^ of 1 A) - (If of ^ of ,\). 2. The pendulum of one clock makes 24 beats in 26" ; that of another 36 beats in 40". If they start at the same time, when lirst will the beats occur together ? 3. A can do as much work in 4 hours as B in G ; and B in 3^ as C in 5. A does half a certain piece of work in 12 hours ; in wliat time can it be finished by B and C, work- ing separately eqiuil times, and C succeeding B ? 4. A note for $500, made March 9th at three months, is discounted April 11th, at 8 per cent. What is received for the note? (True discount.) 5. The unclaimed dividends on a certain amount of stock which pays 6 per cent, per annum amounted in 3 years to 11152. The stock was sold at a discount of 12|^ })er cent, on its par value. What sum was realized ? 6. Teas at 3s. 6d., 4s. and 6s. a pound are mixed to produce a tea worth 5s. a pound. What is the least integ- ral number of pounds that the mixture can contain ? 7. A man buys 150 lbs. of sugar, and after selling 100 lbs. finds he has been jmrting with it at a loss of 5 per cent. At what rate per cent, advance on the cost must he sell the remaining 50 lbs. that he may gain 10 per cent, on the entire transaction ? 8. The hour, minute and second hands of a watch are on concentric axes. When first, after 12 o'clock, will the direction of the second hand }n'oduced backwards bisect the angle botvv'ccn the hour and the minute haiido ? jurg- for he have oiidoii to iin^^ Hi broker- EXAMINATION PAPERa 357 9. Each member of a pedestrian club walks as m.iiv or tiic tup lb foi each member as many pence per m le 'is here are members in the club. The t^otal expl3n e ' £50 l^b. lid. How many members are there ? in 26" ; he same ; and B )rk in 12 d, work- onths, is I'cceived lount of :ed in 3 b of 12% [3d? lixed to St integ- in? selling oss of 5 the cost gain 10 atch are will the [s bisect 5? ■ SECOND CLASS, 1881. 1. Simplify — (1|-1 -002)^( 1 - -OOfi) . . •002 - -Ofi ^ -•>•> X 3-fi 2. A mnn mixers 28 lbs. black tea with 36 lbs of nn in tenor quality, which costs 20 cents a pound les's am? bv" selling the mixture at 58.^ cents a po^ li^ied '^^^^ cent. Find the cost of each kind of tea. ^ ^^' frcnn ^^^F to'> 1 ^T''''^^ , "' ^ "^^^^ ^^ ^"^^^ '^ ^^^ cen FinH X •' '''''''' f^""«nsion is increased -3 per CO t. 1^ md the percentage of increase in the bulk 4. On a quantity of tea a grocer fixed a price to make a h ve tin /nnl ""^ '^^' \ '' '''' ^^^^^^^''^^ was found to fhil ^^.\ '^"'"'.''"'^^'^ had to reduce the price on cent ioss t. '' h"""^' ^'".^ ^^ ^^^« ^^^^1« 8-«in was^l8^per ^m^:^^^^:r -P--^-.ai"- What .md^was beaten by 176 yanls. Find the\x"X;te of fjyl.y ^?^ "'''^'''^'' ^^ ^^1^^'^' t<^ "' "^iles, and if a cubic ^"^olT^;^;^:'^' ^^^ ^^- -^^^ ^^ ^ kilogramme 7^ A tradesman marks his goods at two prices one for W^tisThTLwS' other at a credit Jf six'Zntfl com? ^^'^ ^^^^^'^^ l'^'^^^«' if "^oney is worth 10 per 14^ i^^t TT" ^^ ^'"7^^^' " ^^«^'''-"<^y is e<,ual to £500 14S. bfl. It gold IS quotiid at 151 and the course of o^ change is 489 ? -' course or ex- f r] I 1 ! i ii 358 ARITHMETIC. I ; f1 9. The external diineiisions of a rectaiififular iron chest are 2 ft. 3 in.^ 1 ft. 8 in., 1 ft. 2J in., and the sides, lid and bottom are one inch tliick. Of liow many cubic inches of iron is it formed ? 10. A dealer has three i>rices for his g-oods — a year's credit price, a six montlis' credit price, and a cash price. The year's credit price is 35 per cent, in advance of cost^ his six months' price is I) per cent, off his year's credit price, and his cash price 10 pi!r cent, off his year's credit price. At what advance on cost must he mark a six months' credit price on an article whose cjish price is $12.15? SECOND CLASS 1885. 1. A man bou^»-lit a house, which cost him 4 per cent, on the outlay to put it in repair ; it remained empty for a yeai; during which time he reckoned he was losing- 5 per cent, on his total outlay. He tlien sold it for $1192, which paid fur repairs and loss and also gave a profit of 10 per cent, on the cost price of the house. Find the cost price. 2. A railway train moving with uniform speed is met and passed in 5 seconds by an engine and tender SOj feet long and running 30 miles an hour ; the engine and tender return shortly afterwards and pass the train in 25 seconds after overtaking it. Find the length of the train. 3. A person invested $5420 in 8 per cent, stock on the 7th day of January, at lOOi, and on the 12th day of February^ of the same year, sold it out at 117|, paying \ per cent. ])rokerage on each transaction. Find his gain per cent, on what the stock cost him — money being worth 8 j)er cent, per annum (o()0 days). 4. I bought French goods for 7490 francs, and paid an import ad valorem duty of 15 per cent. I sold the goods for £420. Find my gain or loss in dollars and cents, if the ^:== 25.22 fr. =$4.87. 5. A triangle, altitude 00 feet, is l)isected by a line drawn paralhd to the 1ms(>,. Find the ])erpendicular dis- ••ance between the base and the dividing lino. n I II EXAMINATION PAPERS. 859 I'«lf of it 1,0 gave oil}sr.uTZ TvJ Vf-'"^,^-'" *"''' .■emain,lcr l,o g-ave 37 in" fo V vrt H "'.^^l'"'? '!'« whole transaction iJSSDT, AV " t ,li, th. f m""''^ "" ""* per yd.? '"" '"^^ <=lo"i cost liiui 7. I invested in 7 per cent sfnni- „* -oi , , received a l»ilt-y<>ar-s\livS;iTl 'd ouJ f 7^^ """^l"^ i per cent. l,rol tian lie did ;,rw.,,dt,4''' Jainedt^ *'" """■" on tl,e wlioie eost. Find that cost ^ ^•' •''"" ''""'• m;iied'T™:;'*a^,:.:;fLi;::;'.'::'''"-''''^ '" '"•••'""■*-■. - --.esin,a,i,,,t:''i^^;;;t-;;--:;^l;:;y-^^^^^ 360 ARITHMETIC. I ' ; 3. The men employed in a certain factory numbGred three less than twice the number of women employed in it. The men received $1.55 per day, the women 85 cts. per day, and the total weekly wages amounted to $469.80. How many men were employed in the factory ? 4. A and B agree to share the profits of a certain trans- action in the proportion of $11 to A for every $7 to B. In connection with the transaction, A has received $960 and paid out $470, and B has received $1370 and paid out $330. How much must B pay to A to settle the accounts of the transaction ? 5. M and N starting at the same moment from the same place, and in the same direction, w-vlk around a circular track, M at the rate of 8^ yds. to every 5| yds. by N. At what point of the track will M first overtake N, and how many rounds will each have then made ? C). At an election the successful candidate received | of the total number of votes cast, and had a majority of 832 over his rival. Of the total number of electors in the constituency fg- did not vote. How many electors were there in the constituency ? 7. Between 1871 and 1881 the county of A lost 24-73 per cent, of its population by deaths and removals^ but during the same time it gained 42-41 per cent, by births, etc., the percentages being reckoned on the population in 1871, In 1881 the population was found to be 26478 What wa^ it in 1871 ? . " 8. Find the d'fference between the discount on lOth Sept., at 8 per cent., on a bill for $128 drawn on 3rd Sept. at 3 months, and the interest at 8 per cent, for the same time on the proceeds. (In reckoning the discount include the 3 days of grace, but no other charges.) 9. The length of the sides taken in order of a quadrila- teral field are 20 rd., 21 rd., 21rd., and 22 rd., and the angle between the first and second of these sides is a right angle. Find the area of the field to the nearest square rod. EXAMINATION PAPERS. SECOND CLASS, 1887. 361 1; (a) In reducing- a vulgar fniction to a deciiinl ov (/>) Express as a decimal -, i of fi-SOTGO-^ x 1 -^^Rr.ji sold t'^Trof^^V, "'-'hs- c?eait, ,u,Ja,t; ^ ^l ' the ^ilv^er is 13^ ^nd Tn'tL Jw irs'^'tS is thT '" centage of dross in the mixture P ' ''*' '" "'" t"^''" of simple '^Sue.Lt?'' ^^ ''"'■^'"«" ''"■"P"™'' "-'^'''d of excT^go :. """"" '^ *" ^"'^ "' '^^«'-"'»^« ? The course New Yori" fouoZ'^t"^ """"""^'^ '"^'^'^<'" London and ... ' ,"- '^qnoted at 2 per cent, premium what -ill ho obuunedm New York money for a bill TmO 12s "cf? 6. A 60-day note was discounted at the bank it I .ii- :hat:d'"° Al^ot"*" *?'' "r *^" Tr'e' DSseount'." charged. Allowing days of grace, find the face of the of li.e'lTntwM P^,-''"'^ '^'^/'''^"P *■<"• **•«■ *e animals oi inc same knid costmg a uniform price. Sold foi Wm "^o Sp'^ Fi^theT'-rf *", <=°"^' '"cl the cost of each a head. ccni Sir" Lr'. '"^''^' "^«^*^^8-« f^r -$2500 at 5 per tercst is oK^^^^^^^^ ^"^'^^'^ ^« ^^"- What rate of in- terest is oota ned if the mortgage is satisfied at maturity ^ Ssl' ^naZ "elHnlhf ''"' ^ ^"'^ "^ ^^^^ ^ pe" cen^sVt oo«f, ana tnc rest in the 5 per cents at 102i brokeriP-o I K'L^?:!'-..™ "•■^'^^ ease. His total income beLg jSn] uiic oUm luvusced m each stock ^' J ■, m^ ARITPIMETIC. 10 A fTfivden whose width is 9 rods and leiifftli 15 rods Is to have a wall 8|- ft. thick around it outside. What will be the cost of digging a trench for it, 21 feet deep, at Ifc. per cul)ic foot ? ^' 11 A circular race course 22 yds. wide covers 12 acres, tind the diameter of the inner circle. (- = 3 ' ) ,Jr \^'«"f '^^ ^('"t wlK.se slMnt height is 12'ft. requires l.>.sq. ft. Of can vns to make it ; how much ground does the lloor ol the tent cover ? ^ » Hi SECOND CLASS AND JUNIOR MATRICULATION, 1888. 1. A person sells out 3 j^er cent, consols at 04* and in- vests the proceeds in h.-ink stock which sc^ls at '225 and pays yearly dividends of 8i per cent, If his income is changed to the extent of $57, how much money had he invested ? j c vx n^ 2. The profits of a loan company for a year were suffi- cient to enable the directors to add |20.o6o to a reserve und, to pay $51)05 for cost of management, to pay two lialf yearly dividends of 31 per cent, on a paid-up cnnifil stock of $301)050, and to have still on hand $423 F n the profits for the year. ''^- Aand B enter into partnership for 3 years. A puts m $20000 and B $5000 ; B is to manage the business, ^^l theprohts/M'etobe ecjually divided ; but at the end of the (irst year A increases his stock to $30000. How slrill they divide a gain of $28500 at the end of the three years? 4. Find, most readily, to six places of decimals, ./., v5 V5 ^ + 2s'5, where \^" = 2-230 + 5. A tram lea.ves Toronto for Hamilton at 5 55 d m running at the rate of 20 miles an hour. Another leaves Hamilton for Toronto at 0.35 p.m., running 40 n.iles an hour. Before they meet the first loses 5 minutes and the second 10 minutes by stoppages. At what time will the trains meet. Toronto and "^ '■ ito and Hamilton being 39 mil es apart ? EXAMINATION PAPElta 3^3 circular order. Explain thif " "* Produced in What IS the shortest method of flnding ^% when it i. Icnown that ^ = .941 1 764705882352 ? " J. A merchant buvs a nnnntu.r ^^^ it at an advance of ifper cem "L iff"^' T"^ '""^ i °*' of 20 ,,er cent. He nowTliTcov;;! th^ . o '" ^'" '""^""'^^ pods are qUte unsalable Wi^t per eenr'Z«; °""^ on^^,:!etSt?r--^^"-i^er t'St pure gold. The soferefeTi; m if T- 'f ^' '''"'' '^ g^in., and is worth .4.866!^ 'U^^^^^t,^^ the"mixt?;e?T44"'ibf ^a?"?"" """^f '>*' '^ '^"'^ -« cost, receiving fbrit'sfio ,n i/TI' °^ "" ?<"' «''"'• on tea kt the sanfepriie per pound I^h"/ Ti^"'"^'' '^""' «f t.^ in the n..ture:t:."S ^"^^-^1;-,- yei^Js's :;.T^^p:,'.?r„\'«' :;-' -vings hank ''"■-*,' the 1884. In 188.1 tL depo'il we ^ 'w ^ ^T'"''' '" than in 1885, while the'mi 0,^72? ' ./ .I'."' ««'«. greater the average of the three T», •'"''1'"'""'' '" 1«87 exceeded The aggregate ^ :^Z^;:^J!rZ^^r T'^ the an,onntdeposit«l in oaeh year *l*"'^^7.50. imd 24 364 ARITHMETIC. 13. If 76 mon and 50 boys can do as much work in 299 days as 40 men and 33 boys can do in 557 days ; how many men will do as much work in a day as 15 boys? li SECOND CLASS AND JUNIOR M ATUKIULATKW, 1889, 1. A note for $876, dated May 17, for 90 dn,ys, and bear- ing interest at the rate of 8 per cent, per annum, is discounted at a bank on July 3, at 6 per cent. What are the proceeds of the note ? 2. Explain the terms Exchange, Bill of Exchange, Par of Exchange. What is meant by saying ^' the rate of sterling ex- change is $4-87 for 60-day bills"? How is the par of exchange between two countries arrived at ? 3. What (capital should be invested in 6 per cent, stock at 104 to produce an income one-third greater than that deriv(Ml from $1500 invested in 7 per cent, stock at 115 ? What rate of interest is received en the money in- vested in each case ? 4. Prove that a vulgar fraction may always be reduced to a terminated or to a repeating decimal. Explain any short method of reducing f?- to a re- peating decimal. 5. Explain the method of contracted multiplication of decimals. Employ this method to find the number of cubic yds. in a cubic metre correct to 4 decimal places, a metre being equal to 1-093(13 yds., Vnnytir measure. 6. A rectangular solid is hammered until its length is increased 10 per cent., and its width 3 5 per cent.; by how much per cent., has its thickness been diminished ? 7. In 1837 the U.S. half-dolh'ir was changed in weight from 208 grains to 206| grs., and in fineness from -8924 to -900 ; find the least Avhole numbers which will show the relative values of the coins before and after the change. EXAMINATION PAPERS. work in 299 days ; how 15 boys? s, 1889, i&, .'uid bcar- ' nil 1 mm, is , What are change, I'ar sterling ex- ro countries cent, stock 3r than that ock at 115? e money in- j be reduced ; ff to a rc- ])lication of )f' cable yds. 3S, a metre its length is mt.; by how ihed ? . in weight from -8924 ill show the tiie change. 365 vxniy :: r l"'";-'l;:rr.''!;;fa'\;:;;T"' ■■"•«<"« "openc^ '■'"V material. Waffos re 2^J'\"'\ "'" <=<«' "»' "'-= of one-sixth in the cost o •m■Uer^^. o'"!V '"" " '•«'•"=''"•■ turer to produce 10 of th^nrMV .■""*" "'^ "'.■"miac before tlie elum-e How n„ ^ "' *"'' ^'"" l^ «ost liin, cent., ami the .'einai,, 1 ., " «! '"""*''"*''>' '<■'■ ^ por Wha,t nmst be the a™Le wt. l ''' *'''• '"='" "' «!'•■' '«»• the sharehohlers 4 pJr cent Hu? ^ "'"''''"' ^° '''^ «<> l'"y rental of one per cent, on I,e c„Vt If T'" ''' " '"""""y hiKls tluit the lot will rise 5 «.?w '"^ P'-<'l'erty. He l.rceiate4perce„, in vu'uA.'ve .'v" V' t /'" """^^ "- ■I of tlie value of the Drom.rf,^^-* ',.'"" ""*'"'»n'^^^i! (on Po'-cent, every three ve,^^^> ??•"") "'" «ost hin, U "MU on the dollar an/ ti't h' " ""^'^ *'" l« 1« I'fe property at 10 pe. ee "t tss tl.Tr^^"''^ "'^-« valued ™t« per cent, will l,e receive on thi ^^'^"^ *»'• ''• What vested? 'i-ceive on the money he has in- ^^Ofi^lnZnZcIto-^^, merchandise, amounting, to is received at Niwrork a^'ri' nT;'?'^"^^^ P-'rc^nt? iit tlie rate of if4-8.U h, m,,. """vc'-tcd into U.S. nu.ne; «-8GC,5, thetrue value how ^^'^ ''"'-""g i..stead of- tlicdirterenceandbywho,n7 '" Seined or lost by 12. A leaves P for n «o , m t tl.at B loaves Q for /;' fhe"" "'; ir^"*' »'."'« «a'»e tin.e siJCed till they meet B then ,? '" """■'"■■" rates of eighth and reaches Pin 5 XsS-r ''' """"' ""''- P. I"ina the rate .t^tidlir^ir ~r l'^'," '•-'- r h. ■ » 1 . 1 ^' I 3()() ARITHMETIC. SECONIi CLASS AND JUNIOU M.VTUK^ULATION, 1800. 1. Define a fraction and establisli a series of i)roposi- tions based upon your definition leading up to the rule fur the division of fractions. Examine how far oacli of the following statements is true : (a) I of ? :^ « X f . fc) A traction represents the quotient of the numerator by tlie denoniiimtor. 2. A l)ou«-ht ^\jods to the value of ^5101.53, and g-ave in payment his note due in 3 months. Wlmt nmst be the face of the note so that when discounted at 7 per cent, it will realize the amount rcipiircd? 3. A person luivin^ a quantity of 4,^old bullion may either dispose of it at once .-it the rate of £3 I7s lid per oz., or take it to the mint and ha,ve it coined for him at i'3 17s lO^d per oz., waiting in the latter case 10 days for his money. Which plan had he better adopt, if money is worth G per cent. })cr annum to him ? 4. Find the distance (in inches correct to 3 decimal places) between the opposite corners of a cube whose vol- ume is 2 cubic yards. 5. In a certain municipality 33| per cent, of the taxes (n,t 18 mills on the dollar) go to pay interest on its indebt- edness, the remainder being apportioned in the ratio 3:5 to school and to city purposes. In a subsecjuent year when 10 per cent, of the del)t has been paid off and the interest on the remainder reduced to § of the former rate, it was found that for city purposes there would be needcri 13| per cent, more than before, and for school purposes 22 ;| per cent. more. The value of the taxable property having increased by 3 J percent., what rate of assessment will now be recjuired ? (). A train 110 yds. long overtakes A who is going at the rate of 4 miles an hour, and passes him in 9 seconds. Ten minutes after leaving A the train meets Band passes him in 71 seconds. In what time after meeting the train will B meet A ? atcmciits iy I numerator EXAMINATION APK/?S. J^(;7 ,1 J; ^ f ^^^^""^^"^ ^vhich rloriv(.H a revenue of 20 million lollars from the duty on imported fro(xls tinds itnecM^ls -1 k> ohtam an additional two millions IVom this not .(f Assum.ng that if the rate of duty l.e increased by ^nv fraction, say one-lifth of itself, the y.-.Iuo of the L^oods f ported will be diminisluMl by one-tenth and so mi f,' l' .ppn)xiniately by what ,,er e^nt. of itseJf i,:^r ^e^'d^' quired? '""'''"''^^ "^ "'■'^"' '' I^^'"'^^^^^' ^''^' ^'^'^'^"«- ^l 8. A cubic inch of water weighs 252-458 xrriin P..hi ..19-3 ti„,c« and silv,,,. lO-r, S.ncs ,s .oa^Vy «\,1^^' I'lml the w.iiffl.t of ,1 (,„bie ind, „f ,, mixturo c m.?; ,,; ffold an.1 silver i„ the rati,, (by w.^i^H,!') ol "l-1 *• m,!l pll"ee; "''"'''"''' '" ''" """''''' '"*-''' '" '"" ''"'"•'■I' 'I"'!- (a) the value of 2fl_,_l 1 1 \ ^5 3x5-»"^5x5^^"^7T5^>'' r/>J the number of cubic centimetres in a cubic inch (1 metre - l-O'J^iCa yds.). .v.^-^^ f '\''''''''* ^''^^^ '^ quantity of ]i(juor at ;! of its vnlue which he keeps for two years and then selll. The V- ,: ncreases 10 per cent, per annum by a^e, 1 pei ceni is lost each year by evaporation, and thc.^ 'is a^wnste ';<• ' m- cent, in handling- while it is bein^- sold. Wl .Tt r t^ or cent, pcr annum (com,. ound) in terl^t does he n.e m Ins money if he sells at the enhanced value? 11. A person invests money (a) In bank stock at 128 payin.^- half-yearly dividends (b) In city pro]ierty yielding a rental of 10 per cent costing hiin one-fifth of the rent for insurance and rem rV c:^i)Ss.x!L"^""^"^'""^ "'■'"^ ^'' ^^^ -nt;^";,;::; If the Mdiole amount inv(^st(Ml is M*)8^ hoAv q1,..ii i,^ % IMAGE EVALUATION TEST TARGET (MT-3) 1.0 I.I ■^ IIIIIM 1.25 UUI- 1.4 2.5 2.2 1.8 Photographic Sciences Corporation :/, ^ 40' fV v <-s> ^9) V ■\ .V^o'^'o^ '" 5. Four men start together from the same point, and lun in the same direction round a ring at ditferent uni- form speeds. The first runs at the ra?e of 10 mUes the second at the rate of lOf miles, tlie third at the rate of 111 niiles, and the fourtli at the rate of 121 miles each ner hour. At what part of the ring will thiy be firS aVto gether again after starting? i-^t du lo G. Tlie discount off a note drawn at 4 months is $10.50 : ^rulTr' "'' '^^' P^'""""^^^ ^'^^^^^^^ f«^' th^ «^^nie time and at the same per cent, as the discount, would be $10 '>0 1^ md the amount for which the note is drawn, and the per- centage taken off as discount. ^ 7. I'oint off any number into periods of three figures the sum of the odd perils, numbering them from the m^h ; lo'l '^''? "^^^^'^ ^^^^ P^^^od^' t)e divisible by 7 01 by 13, the number itself will be divisible by 7 or bv 13 as the case may be. Prove tUs. ■ ^ ' 8. The perimeter of a semi-circle is three yards find Its area in square feet. .y^us, jiiia EXAMINATION PAPERS. slant height 145 inches ""'""*• '">'* 'he ix^i^z onrs!^ ^"'r" ^ ?''"r ^"■™"' ""■"- scrited about it •ir,. .^^ . ' " *"''"''=> "'^' "'l'"'' " " "'*^'^'""- and B and C i-i « « of th:.'^™ a T '' '''"•' " ''^'oo'd ta ke, the time "ich pe'I-son s „ J fl" m "' " "■""'" ''"^o- I'''"< 7. Find the ^qu^.^ „"ft' JfTw, (*/P ^ "" ""'•'^• millionth, and the value of Ju-l '"T «\\'T "'™.''' less than a ten-thousandth ^ ^ '^ ' *° "''""" 372 ARITHMETIC. 8. Two tniins, whose lengths arc 420 feet and 4()0 feet respectively, pass each other in 30 seconds when moving in the same direction, and in 7^ seconds when moving in o])posite directions. Find the rate of each train in miles per hour. 1). The circumference of one circle is 27|^ feet longer than that of another, and 11 times the diameter of the first is equal to 5 times the circumference of the second. Find the diameter of each, ~ being assumed — 3i. 10. The length of an iron cylindrical vessel with closed ends is four feet, and its outside circumference is 40 inches, and the thickness of the metal one inch. Find the entire weight when the cylinder is filled with water, iron being 11 times heavier than water, and water weigh- ing G2| lbs. per cubic foot. 11. I hold some three per cent, stock; on receiving my first half year's dividend I invest it in the same stock at 1>3|, and my next half year's dividend is $1905. AVhat amount of stock had I at first? 12. The area of each of the longer walls of a room is 3()0 ft., that of each of the other walls is 1*J2 ft., and that of the floor is 480 ft. How many yards (linear) of paper 18 inches wide, will be needed for the walls, deducting one twenty-fifth of the whole area for doors, etc. 13. Find the depth of a ditch, the transverse section of which is a trapezoid, of which the longer Jde is 20 ft., the sloi)Cs of the sides 2 in 1 and 3 in 1, respectively, and the area 140-25 square feet. -^:^s-WEK.s. PAOK. (1) ^, V,JV^, Y, ¥• (2) M^ =V^i-, w. i,n^^ ^:^^ ^uoQ 17. 1-,? X5 ( T!) > -"to (3) 1 r> r J 20.-(l) ,1 *.. (22) f 21— (1) i. • (15). i (21) \m- (7) 22. -d) i^,S 6,3., 00:-^ai^, (2)^5^0, TAI 807 6 ir'SiL^has-^^"' ,1?%;,. ,.© iw„ i?!i, 'ii^, 21 2 Q 26.-.(l) I 1 fi 2(; 7 5- f(V\ 255 00 7 ""4ts^'V'-»"' J 0000- V^V ^7:i, 2.S' i;(7> xf- .„, -. (2) H. (3) ]■§. (4) fl. (5) 13 (8)f. (!>)§?. (10) ^ (n)f (i^^r (i3),Vlljlf. (ir^)i3|. (i«no,-. n^)4i)Ji Kahx^^- (19)22f3|i. ■* • ' >..- •-•.'-'i87- vio) l:>i^j;ji^, (373) "14' ,3 5_ I « 1 » 2?' (7) ;{5;j' Hi 374 PAGE. 28- 29.- 30.- ANSVVEHS. (20) 37l]l (23) ISoV,. f24) 1 lil (1) aU. (2) (8Kf^«n (14) / (20) I 1 '?47rtii/f''^---<"'--- <"'"• ^'^ ft 2!r («) 38 1 T2 (^) ?U»V(T (3) t!.i (7) rk. 31. 33. 34. 36. 37. 38. 39. 44.- (2«.) h V (15) II (21) T%- (12) 2h- (io)2;3i5i. (11) ,1,^ (12 4 (rs)iil ("25) 15-0i>4o -v29) If Jl iV- If 1 . (30> (2) _ (9) 10 (3) U- (10) 2 (4) Ti^^- 12S 141 • (5)^ (11) 2. (12) ar)),,;],. (if>) ic. (17; j^. Ti^H _^) . -w,. ^^"-..(31) 13a. (32) («) (14) (20) r (21) If; -;i) ^ (2) I. (S) 2: (4) 34. (5) 1,/ 14) 99. (lo) S.ViS. (16) {'^. (17; - (20) 9. (21) 1^9. ^ ^ (33) ,,T&Lo. (6) M^ 2S a 2' (18) I («M ?• (12) iM- (18) 1. 5d- (7)f. (13) ir4. (19) 15. (29) ;||?,V ^7) 42T V'' 4 Its (13) ,V (19) 1 1 Ift'Ai- '?o'?r!v (?S1#'-/.?^>^3S-„(?»'^V,. (2!>)4AV; 21 (7)1^ ^S„05 7 1 St I i> 2 « • ■ 022; |. 46. 49. 60. -(1)144- (2)5,V (3) Si" (4) ^,."75)1. 76) Wti^- (9)nk aO)7,V (13)3^. -(12) ^,. (13) 1J,\. (14) 8. (15) .^i!^4. (Ifi) ijlf l^;t ^^^^^^- (Wir'(2S)4^L -(1) Seven, and six-tenths. (2) Thirty-nine, .ind three-tenths (cJ) J^our, and eighty-nine hundredths. (4) Seven hundred and sixty-two units. (5) Seven hundred and sixty-two thousandths. (6) Seven hundred and sixtv-two, and seven hundred and sixty-two thousandths. (7) (3ne thousand two hundred and thirty-four, and five thousand six hundred and seventy-eight ten-thousandths. (8 ) One hundred and twenty- three, and forty-five thousand six hundred and seveniv-eight hundred-thousandths. (9) Two thousand four hundred, and thirty-six ten-thousandths. (10) Two thousand four hundred ?i ov S^'^^^y-f} ' Jen-thousandths. (11) Six ten-thousandths. (l^)Sixminionths. (13) Six billionths. (14)7689. (15)14-003 16) 1000003. (17) -0103. (18) 30070-001083. ^ 9aI ?!!! 5.^^- ^2> 61-23737. (3) 2503-61876243. (4) 14654-5429118. (5) 10011101. 91 t^tTiit (2> 5-477082. (3) 44-395031413. (4) 57-8687277. (5) 77123875248. (6) 26484624 ■00000072. (9) -0000001. (8) (7) 12025-21200096. (10) 1 -'1) 1-2, -12, -012. (2) •9, 2-7, 2 18-7, •0(X)3. ■625. (5) or, -00049, •4, -04, -16, 12-8, 64. (3) 8-1, 81, (4) -015(}25, 78-125, 625, 000025. 00343, 2401. PAQB. 61.- PAGES 51-71. 375 62. 53. 64.- V^^ »a» Til. TTw-. 7f.',r^ 875. (2) -1875, -3125, 59. 82. 63. (54. m.- ea- 70.- 71.- ( ^!L^^' ■^' "75, 125, -375 (525 •4375, -5025, -6875, -8125, 9375 ' •38(>38() (h\ ■rvvvi'i^ ./'i-iuuu'' ?^5b75, -531531, -438438, •727272,* -&7 ' ''''''"' ■^^^'^^^''•' •«^^333, -777777; -(1) -947308421052631578, •631578947308421052 •5882352941176470, -3529411704705882, ' -616384 •3913043478260869565217, -380952 (2) •6086956521739130434782 ^^.r'^^fJ^f'''''^'"''^''^^^' •^«7741935483876, 972, -02439, -3584905660377 *'^'S ■'*''''■ '''"'''' '^"^- •"«• ••w^^i. V^/ Tr> IT' 1 to J -d) 2J) ior> j_ 1 UOsi TT (2) ^i\ i4t- . (3) Vk. (4) i\ii.''(5) I'm _43 JiJlO 5 7J 1 .'. .111 T> T> ik» A- Iciwt^Q^?!! (Vl«' 376 ; i wi I- n^ ANSWERS. PAGES 87-100. 377 PAGE. (4) 27 cu. ft. (0) J8 cu. ft. 87.-(imr,ft. (17) 11 ^{M; in. (lH)204?8.,.in. (l<.))2(»J)m)28.i in (20) 128-l!)2«,i.ft. (21) 117-5 Hq.V ' ®®'~i!f.Uor "'^- ^- (^^'^ (i7(5.0(i3 8i7 in. 22 01oni., 0-281 in. 89.— (20) 00 cu. ft., ()5 cu. yds., 29^ cu. ft., H5a-;J12 cu ft 37-300 cu. ft. (21) 144 8,,. in. (22) 12 in. (23) 8 ft. 90.-(24)4-773ya8. (25) 04-058y-^*»<'»-»t. -(23) I08cu. ft. (2) 108 cu. ft. (3) (50cu. ft. (7) 40 cu. ft. (8) 54 cu. ft. -(ll)84cu.ft. (12)352cu.in. (13) 124,« cu. ft. (14) lOsc.. ft. 15 2-256 ft. (1(5) Oi« in. (17) 1-048 in\ (18) 4-0(;3 ft. ^ -(17) 60 cu. ft. (18) 32 cu. ft. (10) 18 s,,. in. (20) 4'535 in. (21) 46)1 cu. ft. (22) 18 cu. ft. (23) 137 -(J73 cu. ft (10) 3-349yas. -(17) 1437^ cu. in. (18) 268-lJK) cu. in. (10) 21 in r20)-1792cu. ft. (21) 296-508 sq.ft. (1) 10m., 100 dm., IDm. (2) IKui., 100 Dn.., lOOOOdnu (3) 1200 Dm. , 15000 m. , 21000000 mm. (4) 12345()7 89 dm 123456-789 m., 123-456789 Km. (5) 856(KX)cm., 5632 nmi."; 96.-(6) 1635639-87 m. (7) 554151-1 cm. (8) 1152-48768 Km m 91-441 cm. (10) 18-3 m. (11) 50 m. (12) 17907 cm ri3) 749-301 mm. (14) 4 m. 6 m. (15) 13 hr. 32 m. 30 sec" (16)6250. (17) 900mm. (18)952, 400mm. (19) 25-441sci m (20)123-4m. (21) 12 Ha., 120000 ca., 120 Da. (22) 12 -345a ' 5 -678 Da. , 10000 ca . (23) 1017 "32 a. 97.— (24) 9-9 Da. (25) 20424-15 Ha. (26) 10-8 Ha. (27) 172 m (28) -1 ma. (29) 54 ca. (30) 1500000 sq. en.. (31) 13 ca" (32)4-5a. (33)6000cu.m. (34)154750. (35)la. (36)721-lm' (37) 2-471 acres. (38) JK) ca. (39) -66 a. (40) 39 6 in' > (41) 27 s. (42) 1000 cm. v y . 98.— (43) 1000000 c. cm. (44) 1 s. (45) $1080. (4()) $105 (47) 100 cu. dm. (48) 35-31 cu. ft. (49) -2759 cord. (50)248* (51) 3 m., 12 m., 15 m. (52)45000. (53)(i0ma. (54) 2500 Ku 91.- 92. 93.- 94.- 96.- W (57) 125 ;/. (58) -4 m. (55) 909 J^ Kg. (56) 1006s. (59) 1 min. (60) 200 m. 99.— (61) 36 m. (62)1000,1000. (63) 1 Kg. (64) 2 m. (65) Equal. (66) 4-004004 s. (67) 1000. (68) 1 cu. dm. (69) -03531 cu. ft (70) 4-543. (71) 135-9 ml. (72) Equal. (73) 36 min. (74) 10 m. (75) -1576258. (76) 10 m. ICX).— (77) 28-31. (78) 1-3 Kg. (79) 1298 Kg. (80) 1000 c. cm. (81) 1000000 mg. (82) 1000000 g. (83) 10000 Kg, :*, . r ■1 i i •3i' i if t 11 i I i j 378 I'AOE. 100. 101. (87) 37.3-24 «. ,12.1 •45(578!) T. (U4) 11551. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111.- 112.- ANSWERS. - (84) 1.3'5 Kg. (85) 4(5:25. (8(5) 5M>7-2 T. (88) 4{M)Kg. (8!)) 4 -ST. (!H)) 12;{45(5-78J) Kg. (!)l) 2-204 11.8. (U2) -84 ca. (1)3) -375 in. (95) 3(54 -8 Kg. -(!M5) 1311-5(5(5 Kg. (f)7) 5-7(5 niin. (98) 138-8 cm. (99)l-12m. 1(K)) 20-11(5 m. (101) 101-17 lla. (102) 3(58-529 Km 103 1(509-3 m. (104) -984. (105) 14-21. (1(M5) 39(5 in. 107) 4h. 8«l. (1()8)1(HMK)01. (109)1()(K)1. (110) -83(5011. '40411^ 111) 509(HM)9()9,\ Hh. (112) 1539 8. (1) •0202, -2025, 202-522. (2) 22(M57. (3)5-439. (4) -033 (5) 13828-059. ((J) -83, 83, 83(M)00. (7) •03125 5^1mfP?5??f- (?> T*"Wn, i'H^lH^, ItHM- (9) (5-5504777! (U) ^-J?^ ' W'^s^^^ ■'^'^''' ^''^ '''''"-' ^''^ '^' ol! ^oiii'^^nJ^f^ti^W^- (1^) •«^^'^«' •«^^7. (19) -318,3098. (20) -808. (21)1. (22) £134 28. 2d. (23) JTO- (24) A^V 16-8476196. (26) 110/^. •000156^5. (28) ^,",^, 1|. (29) -2. (30) |h[, i^?. i¥t. (32)x"iYi. (33) 3n;.J, L\j|. (34) -4047 or -4648. V'' L^:l;^^lt*^' ^^^i; F> ^^' m.M3S) -30375. , . A. (40) 7925-591 m. (41)1. (42) -{^fJi^f. (43)4741.. 48,^2- (44)1, 4,^ (45) lf5. ^'^^^^h? ^ >''i5>., (4(5) £2 08. (5a. (48) £4 5s. 4(1. (49) 38,V;M. (GO) ,VA. (Q^Umi (61) 3^141592. (65) lOf. (66) jm- (70) 66. H, I (73) -3183. (74) (77) -131870. (78) (81) •41()()86. (82)615 0703204. (8(J) 2^7182818285. (87) ij m. (90) 3^14285, 3-14159. (91) j^, (25) (27) (31) (35) (39) A' (62) 7. •43429. •00003. -(58) '62.- (59) ^i^^. (63) 6. (64) -0000002. (67) ff. (69) I 18. -(71) mh- (72) H, (75) 7^. (76) -^0 (79) 3-1415926. (80) -()042. -(84) 1-3999... (85) 7. (88) -9984097. (89) -1. 5.\, 4?!. -(92)409. (93) 84866 •(56.-i, $48(56-56. (94) -1715. (95) -00004. (97) 1 •(5961. (99)3-14159. (100)26-2856. (102)7-745966. (103) The metre would be nearly 39-41 in. (104) -00001 (105)2006. (106)5,^m. (107) 704 •037 lb. (108)909 (109) 2^1 oz. (110) 60. (Ill) I7015f. (112) (113) $2-20. B \ J i}}i} Li^^^) ^^- (^1^^) 286750 ft. (117) 70. (118) 1440. (119) f23^25. (120) Merchant 50c. (121) 10-52 pm (122) £13 9f. Ic. 3.VJm. ^ ^ ^ -(123) 6d. (124) •00278 .. . (125) f 36300. (126) 65 ft. (127)30. (128) 3819-:i lb. (129) 11 121 ox. n.30^ 81JLl,r (J31) '■' '-—-. ...w,..- ..7/ui .1 ,„v--/ --.I--'- (132) 21. (133) 84. (1.34) 13,^^/',. (135) 13 PAGES 113-127. 379 PAOB, 113.-(i;m) ir'''*'^- (A'J7)10?,«. (138)14. (l.'M))224(>. (14(1)7.80 (141)!?l,m. (142) 3^, 51(}2(;47, 14,1. 71.. 11m iVf 2(i74) Every 66^\min. (187) 50c. (188) 7-4 (180) sec. (100) A. (191) $240. (102) 3027. (103) 7. (104) $5-40, $7-20, $3-(50. (105) $3-57. (100) 10;i?J> hr [202^ 2'* ^'' ^^'*'^ '''" ^^'^'^^ ^^' ^^ -^^^ ^*- ^-^^^"^' »'''^' (203) Dec. 3, 3 p.m. (204) £432 lis. lid. (205) 3 5 7 9 13. (200)7. (207) 1,1, hr. (208) 14(i007. (200) 140000 d.l 21 hr. 10 mill. 40 sec. (210) 35202;;. (211) $3000 etc. 116. 117. 118. 119.- 120.- 121.- -(212) 12. (213) -8 gr. (214) 1()8^J,. (215) -7433950. (210) lim. perhr. (217) T(jtal $14-3lT (218)307;' -(219) $38-25, $38, $37-8 ). (220) $4070-103, m. (221) 73 (227) 2(LV (229) 91:81. (230) 3 ft. 2 yd. 7 in. 122. =(231) 2600. (232) ,it.|cr„. (333) siui (2' 235 lU- (230) 1yd. '237) 1 (it. ■'7238^ 97 T« sq. in. ' !) 8 i) • (235) m. (230) '1 yd:'"" (237)Tqt."^'7238) ^^ i^% (239) 2i-, 10, 1. (240) 1 red. (241) 1 link. (242 1 yd , 1 chain, 100000, 0272040. (243) $4-03, 23-0384 ft (244) 121b. av. -(245) 10000. (240) 15 gal. (247) 00480. (248) ^1 (249) 12-78. (250) 2 yd. (251) 33-84. (252) (253) 883346-17. ^254) 10-83. (255) 32 days. -(250) 2100. (257) U. (258) 30. (269) |a ft., l^^ sq. ft (260) 1530. (201) Between 237620-5 and 25r7033. (202) 55 (263) Gold •48in. , and 9-31 . (204) 59 92 lb. i2(*6) 103,'' lb. -(266) 510 lb. (2(J7) 27(KK). (268) -00021 in. (209) 4th . (270) 87293-9. (271) 4290-8 lb. (272) '•> . (273) llA. io« Ei^ ^'''^^- (2^^) '^1323, $1260, $1200. (270) 28571-^ ft. 126.— (277) 46-5 gal. (278) 2-75. (279) 5. (280) U. (281) ISr, .^^ •^'4%-17r. (282)44802. (283) 880ft., 54:^* (284) 3:2. 127.— (285)^ £45-55. (286) 415:408. (287) 8672 yd. (288) 20:7, 5s. l^d. (289) Men, $1251-70. (290) 3:7. (291) 433|. 25 123. 124. 125. I- 380 FAOB. ANSWERS. 128. -(292) 6-66. (293) 2 hr. 40'. (294) 2 m (29R\ iaa q Jf'J?;??''- (61)600. ,52)10291* (4") W6000. WmO.^(2),m. (3)W. (4,%.28|. (5)*16.66S. 1 S^O. (32) «4V *(!«) fe.^^ *^- (3») 12i%. 145.— (34) 60c. (36) 22-6+% ■" {33)|6%. (34)35j,.<^^55f4.<»i. Tl)ilt -(39) $1129i«. (40) 48c. (41) Lost 73 o/ ..^. „,, (43) 33U. (44) $71?. (45rk-^)m^- (4^^-)' (47) 46-6 + c. (48) $200. (49) loT" ^50)1511^90 '^' -(1 $240. (2)^1-62^. (3)W85o'^V4)662f f^T^IO-iQl (% fe. ^^^ '^- ^'' '^- ''> ^^^ (10) $94^ Til) si; ^^•~n^8^lit^^-./a\^^l/?2- (1^>» (16;$1715M. (17)1870 22 Ss^.l ^^aloh¥- ^^^^ 25263 A lbs. (21) $9720 152. — (34) 40058,^^^, Ihs r.qK\ ®Fio'7n /o/>\ -..j^v ,„^. (38) 1|%. (40) 1^^, 1%. ' ^ loo+V ('^7) lU. 148. 149. PAGE. 163. 154. 156. 156. 157. 168 PAGES 153-165. 381 -^^1.^^''"' ^''^ ^^'*^- (<^)2%. (45)li%. (1) aa .(2) ^105. (3) $56-25. -4) $100. (5) 1%. (6) % (7) (10) $2133J. (11) I ^ ?f2) i/ (10; $4()()0; ^ ^ '^ *^^* "ii!i.^S?\(l^)^^000. (19)1-^ (13) $40. a»ij%. (15) $8000. 2,S 111 &S«) #^>»«^^«A;,^Si^1b4tAi^- 159. 160. 161. 162.- 163.- 164.- 165.- (36) $184d6^i:.L ^„,^ ^__ iol f}\^' .J^^ ^^^■^^- ' (3) $240. (4) $30 r5) S^W9. (12)lSMuills? (3-$5^lL*'''' ~/^!!?i^ ""^^«' IS) $507500. (IfJ) $4 nf\ tQo 122113-% (i«>^19«<^ (20) ^2-i7f • (ilYVS: (1) $3-70. ~m Iq.^-' ^'"^'^f- ^^^ ?^^- (^) *'^00. (5) $3-78. (6) 7^c -04) $20. (15)10%. (lfi)20,loz a?^27i„ /im n ■ —(2) $9375. (3) $2100. (4) $4050 m^«9q99''7Pi /r'\ >i:.^^.»k (12^ %V7.Wf^- (^'^^^ wu!^^ nfi lf?«^^ ^.it\r?:^^^- ,(^^> ^'^1-50. (16) $87-50. W $118-25. (17) 5^%. (18)5,«^%. (19)8%.^ (20) $4000, (5) $45, $5-40, $146-88. g.1111. (6) $30, $84-58.^, $18-612. (7) (a) $144 /M«ii«^i.o« re; $129-71, (/) $256-864, f ) $j4-487 \7)'^«108 chu' Csr; $404-416 (h) £216 la 6d /sT «106 /oT «i f/f' (10) M $1488 (i) $2100 (c;$l(^?9^4f6 6/) ^216^8 759 '11^ ft-fi'O-ddff (JO) ^/iilf5^-' (^s)^- ■1i^^ Wobdivys, (c) $106'32 (,ft «ri.";'u-^'i S H ¥'"■• **• <') W a/ys, (0) I31-86 ?> Ii?^.u 170. !?: M:;/3*a,*'/./SX/r» ?i)*i?|:l?- '"•11?} ^k '|%1^. '(^^^ «**««•"• a«) I397-60. 174.-(19) $296 07. fif.„tr- ^Jt^ ,s^», a«)/«/r' "ML' !l?i .'2)«104 (3,«i08-lB.'(4)$108-I6 (K>Sl,o..«64 W*Uz-4S.J4. ,7) «1-1248«4. (8) |121.55/-(9y«r8i'K; 175.- 176 PAGES 177-191. 88^ PAGE. 177. 178. 179 180 181. 182. —(10) $32-,%. (11) $109-29. (12) $197-03. (13) 1-157625 (14)1-169858. (15) $689-83. (16) $1201-3L ^ "^ ^'''''^''■ —{17) $m2-4b, (18) $9-72. (19) $4448-99 + . (20) $1025-76 (21) $135J-61. (22) $111-49. (23) $1689^7. (24) 11 35|c (25) $4328-25. (26) 43923. ^ ^ ii ^o^c. ~S2?H^^"2^ -^(l-04)-3-l|.. (34) $(l'06)«x(l-03) I'll ff T42)^6'o^9r'^- ^''^ ^''-''- ^''^ ^^^-- —(43) 2-95 + %. (44) 17^,^%. (45) 5%, $200. (46) 47, $1250 47 4%, $1500. (48)6%. 49 $I0(>00. (50)Wk57-15 -(7) $769-23. (8)'96-15c. (9) 90-71C V) .$7^-45 (11) $23-54. (12) $48-80. (13) (a) PA-U (h)iu7-)7 (.)J93-14, (./) $2852-98, (\) 78-35c. (14) \il089-30 (15) $1200-62. (16) $384-82. (17) $97-60. —(18) $3013-86, $3322-78, $3663-36. (19) $(V^)3c. (23)^. (24) J i.^y^^'^T- (27) $860-48. 183. 184, 185. 186. 187. 188. 189. 190. 191.- (20) $a250. (5) $10000 (7) $5137-04. (8, $4274-02. (9) $1421-36. (10)$I727^* —(12) 3-377 times the annuity. (13) $295 -92. (14) $3290-8 1 ' a9) ilo^6'" ^''^ ^'^ ^^^ ^''■'"- (1W31-57: —(22) $7122-92. (23) $ J 876 "52. (24) $9803 92. (25) $5386 -08 ^m Im67-20 ('^>^'^''-'''^- (28)^1118-39: (Vj)l5965-5?: -(31) $2675-49 (32) $4178-91. (33) $404-21. (34) (1 •04V' (1-04)2, (1-04), 1. (35) ^3401 .39, (36) $220-67. (37) $77-88 (38) $5.385 12. (39). $105-24. (40) $350+ ^'-'^''''^^' -(41) $33-43. (42) $46,3 -93. (43) $28-38. (44) .$'>'> •2'> (45) Al).. lit 10 years (46) Never. (47) $18,395-44 ' (1) $1250 each (2) Jones $1120, Smith $1680. -(3) $4000 (4).$2000. (5),$20000. (6)$]200. (7)A$1001-74 B $459-13, C.f 939-13. (8) $.5422 ^ (9) $2618-18* (10) 8 10 and 12 mos. (11) $2000 each ^ ^ ' -(12) $37(54 25 (13) Lock $1000, Smith $1012, Kniaht $960 (14) B $6315-79, C $5684 21 (15) B $13331, C $1?566^' (16) Smith $8000, Jones $9000, Cook $3000. (17) $14472 + ' (18) 8 mos. -(19) Terry $273?., Tucker $1.33.^ Taylor $731. (20) .4^722-06 (1) $4010. (2)$2.509-37A. (3) $79(;. (4) $12090. (5) .5764 50 (t;; ^<4ol .;9. {i) £281 OS. (8) $25*0-69 + . (9) ;»i '/ (10) $14433^. ^^ 2/0. t! i ; r I r . Ill: 384 PAGE. ANSWERS. 192.-(11) $7275. (12)£50C. (13) 9J. (U) Direct $4010 • /9n( 1 M ^?-..o(^^) P744-186. (24) $342-857 ^Q^ ^nV ^^•^"We = ^-012fr. (26) ldollar=.2-487fl. ^^^•-is^^^^f >/) l?/t (2) 56ft. (3) 48in. (4) lOOfsq. ft. hi oil ;• ^ ^ ^^^ y'^'- ^> ^"2 ft. (8) 2-699 ft. (9) 28-6 sq. ft. (10) -0000918. 196.— (11) 30-805 ft. a2) $266 CIS^ 302 ff nA\ ir.K • ^^"^"S o"/.'V *^*' (26) 364-641 sq.ft. (27)672. (28) 22-27 ft. 3? 152tfi3'"- (30) 190 1149,,. yds. (31)21. (32) 2005-3ao. , ^ ^#1 'mi ';l^" (.?^ 2i,it' i-'s ''• ^''^ "^^ ^•^> «^« »"• " (42^ ^""■Ij','^';;''-^ Mjf^S ft., 12-64!) ft. (41) 30-688. (4J) 18 »,8ft. (43)8ft. (44) 2 ac. 46 reds. (4B)Squai-e,8«r '"■ W «eh.™s.^ (W434-;q. ft,. 199.— ^51) 1-118 in. (52) 76-157 ft. (53)243-721 (64^ 28-281 ff 69 I"i87";';i- ^|!S?*A'1-« ,W)-577in. S I'L^W.' (0.3) I16OM '"«in257T.' "'• ^"'- <•'!> '"^- <«2> 134-114. 200. -(6.6) 10 ft. p„r hr. (66) 444444-4 (67) 12A (68^ 647-764 ??2)''48-olf 'T?rn^-y<'^- (|«) 1795 «q'.T„.T7i) u,!a '^^'^(l\\tf^"';J1V, ="• "<^9> 8 to 27. (80) 104 ft I5 ll\u '''- (S4)lW-168yI 202.-(»2, $203-20. (!.3, 12. (1)4) 82-083 ft! (96) 12381 so ft nom™'?*- 'S?2; (»«)S)*466-005' (100) 6 ch. (101) 22-488 ft (102) 10-2. (10.3) 4708 97 203. -(104) 1.326-481 s,|. yds. (105) •■ /12 aLuXr 7' (107) 100 (108)' iim-d lliifh, i \ll0< »2476 alsj ^512-, *"'> *•**• <"3) 30*d;.ys:-(l}l)%cl*B0: Jioi; iTq'^, ■■ 9.^"' ■*** ^'1- «'■- '121) 8400 sq. yds 9. ft ??lA"-,',",- .(123)2.36?. (124)20rods (125) 32%^ 12ift (126)31-Jm. (127) 113! in. (128) 172?in . PAGES 205-216. 385 PAGE. 205.— (129) 1 -445 sq. ft. (130) 368 sq. in. , 90 in. (131) 5028i so. ft. (132) 6. (133) $98 (134) 65-948ft. (135) 8-808sq: ft. (136) 92fsq.ft,37.fft. (137)126in.,CV48q.in. (138)84in., 339-477 sq. in (139) 189 -52 sq. ft. 206.— (140) 676 -869 sq. rods (141) 80000 miles. (142) 1131 cu. in. (143) 29icu.in. (144)thrs. (146)$79-66. (146) 398 '363 sq.ft. (147) 8-64 in (148) 212f sq. in. (149) 148i sq. in. (150^ 24-589 in. (151) 14 -661 in. (152) 40-737 sq. in. 207.— (153) 10879f sq. yds (154) 68 938 mi. (155) 34-469 mi. (156) 48-849 mi. (157)5 128. (158)2836. (159)1173333331. (160) 212f (]61)366§sq.in. (162) 229i sq. in. (163)12ft. (164) 96 sq.ft. (166) 2-659 ft , 3159ft. 208.— (166) 7^4 in. (167) ^/l/Vi^ in. (168) 1396-825. (169)488. (170) $25-099. (171), 16-165 ft,, 11316 sq.ft. (172)63ft., 45 ft. (173)22|. (174) 28f. (175)553}. (176) JUi f 60in. (177) f #^1540 in. 209.— (178) 3748 075. (179) 3466145-382. (180) 792-261 cu. in. (181) 12-728 lbs. (182) 11407-407 oz. (183)25. (184) 49 in. (185) 2799-107. (186) 44-9. (187) 39434 square fest. (188) 758 556^u. in. 210.— (189) ^ii- if 18. (190) 28704-761 sq. ft. (191) 1390027. (192) 43-216. (193) 49. (194) $59-15. (195) 48 sq. ft., 105sq in. (196) 278,-^. (197) 75 ft. (198)973 631. 211.— (199) 18962.50. (200) 59-37 in., etc. (201) 15 ft., 36-496 ft. (202) 6-928 in. (203) 5-55. (204) VTMx ft- (206) 25. 212.-(1) A $30, B $29-70. (2) $1648. (3)^325. (4) $144, $180, $202-50. (5)48|c. (6)196. (7) $133-87*. (8) $4383. (9) $3-92. (10) A $75, B $126. (11) ^istimce from starting point (12) 69ic. (13) 20|§' past 4. (14) $54-26 $71-75. (15) 26 lbs. (16) $180, $240, $300. (17) $270. (18) $600. (19)25. (20)20fa'past5. (21)$9-18|. (22) A $69, B $249. (23)1^. (24) 6.^? days. (25) 6^0. loss. (26) 5A. (27) i^. (28, A $900, B $1800. (29) $78. (30)4^. 214 —(1) 49ic. (2) 5^ sq. yds. (3) A $12, B $10. (4) 18' past 3. (5) B pay A $16. (6) 35. (7) 21 yrs. (8) $2576-25. (9) $1015, $1421, $2233. (10> $22-50. (11) $14-28f. C $1750. a3) $40. (14) $891. (16) 32. (17) A $8-40, B $7, C $6. $3486. (20) 27 tV' past 5. (21) $1260. (23) 5-26. (24) 41ifc. (25) $4. (26) <• '., $192, $180. (27) A $650, B $371-25. (28) 12c. 216.-(1) 5Hays. (2) $4-85^. (3) $180. (4) A $240, B $210. (6) $140 -4_1J. (6) $2617 -50. (7) $348. (8) $65 40. (12) A $6, B „ , (16) $2826, $2355. (18) $217-50. (19) (22) $S2-25 loss. (9 \ A #0 B ^x'A, v> ^-;u. (xu; A ^^v, n ^OZV. {XI) ^t>-44 (12) 720. (13) B 2f yards. (14) $29 16. (15) 84 386 I'AOE. ANSWERS. (18) $10000. (19) 75, 00. 8. (23) 21, 28, 42. (24) Oj-. 216.— (10) $588-24. (17) 1000000 (20) $(;43-50. (21) 25. '22) (25) 40. ^ ' -* ^20-(l)^ (2)100rod8. (3) $674. (4) $152. (5)4*°/ (a\ 2^- (8) 111087-52; (9)' 1st by lie 14? 1%J,'P .^'"T^ i}^] ^1"^^-^ (13) |4m m * ^"j^) *i^ (19) |17a-86. ^2^ 236-16 sq. in. (4) $28938-24. (9) o^^o i\ V%; Jl'*> ^^^79. (15) 5^ w. (17, (6) 1 ^li. ^3^ lbs. ' (871 mii: !^:? fu ^*1" ^^'^^' I'l sq. yds., 6 •33. (8). 81. •6«.;f. (12) $10-50. , , $1856-72^. (18)8 $3587-50 @ 8, &c. (5) $10000* sq. ft., 132 5 $720. (16; $33333.33i.\lf)rodh ^''^""'^'^ sq. in. 260.— I (19)4 (9)^19', 36". (10) $3b()0 ^" 43|. L^c. (18) 3 to 2 15 «r'(f6^) \n'iL?-a^^^^^^^^^ pflm, (19)8rt,V (20)ii|0or (^''* *^^' *«• (18) $88 11-51(1. $8-60. $888 -88':^. ^?^.*?«-_ W^S, $2. u)$1120, .tc. (8)$360. (9) A. $3612-50. 264.- (W' ^( 6)'6iV3'?'7A"\(^^>^^'^«- ^^'>^«-l^i" (19 ^25. ^ ^ *^^- ^^'-^ ^- ^i »»"- (18) $3200. 46. 34. 266. /trV . 18.^- <16 2 p.m. (17) $340 nearlv VlS\«i-«lr (19;4(XX). (20)273. V t. (10) $1804-275. (Il%l51^8n^^^ ^fAfh days. (14 4AV past 11 am. (15) 1st (16; *A ^M ^^ (17) $5859-37|: (18) 8Ac. (19) 6 (2()V Jsm ^ ^• 282.-(l) Inc. $15-65. (l) ^390. (3)^33 33, 7 7 (4) 7^ (5, ..... (11) 4|. (12).55A. (13) $1320. (14) 731 (it >9 )« S, (16) $3950, $50. (17) 1(55, 162. ^ (18 12 mi ^ n^^ l^ -ft 284.-(l) 1 1- yds. (2) £3005. 4s'. 3-6d. (3) gl?!? (i^ i t^i^^ (5) $547-60. (6) $81662 (7^ ao-Kn /S^tV' ^ \ 1?^L^- OQQ /1\ (:»i/tn (6) A»2.34;'-^7M^ S ^- it'i&..„(S) 44i day, (8)'«sa2-68:^ (974I?: 390 PAOR. 288. (14) 6. 200. ANSWERS. ab)$Um. (l«)A44yrs. (17) 2f:. (18) 20 nou 1 /« /^ „z. (7) 8 (8) $1000. (!)) 25. (10 20a (11) bcontres. (12) A $480. (Hi) man 00 davs 4p a4)|9m US) $27-50. (10) ^$6^32" y)Tl7-5() (18j 27 bush, f 19) Draft on Tomnto, lOo 296 292. -])*20a (2) $10-89. (3) 8.V,c. (4)2!!. «) SU'H-72 6 « days. (7)11000. (8) $c. (3) $10. (4) $117. (6) Sfe]77^-9'i qt. 298.— ( 302.— 40. 308.- 311— 312.- 315. 316- 317. 318. )}i< i^' '^^"'^ • ^^^> ^'^^^- W i'^tter. (15) $3200 15 mo (16) 5. 17) $2000000. (18) $133 -331 (19) S^c -((J4) 6. (65) 12. ■' ^ ' ■'''' -(100) 19. (101) 27. -(102) 651. (103) 1865. (105) 1331. (106) 343^40 (107)26375. (108) 1534. (109) 18537. ^ (110) |)86 (115) 3030: rem. 102, (117) 1, 10 11 100 101 Tin iiV 1000, 1001, 1010. (120) 32 + 16^2 + 1 = 51 ' ' ' ^^^' (121) 3«-3s~3*+33 -32 + 3 + 1 = 427 * (3) 10}« (5) Nearly 83. (7) A $36.2, B $18§, 4 : 3. (8) 73151 ft. (9) 1 i (12) $189-93. (13)7-52,7-38. (14) $973-78 '" (16) $20-88. (17) 50c. Gold basis, 30c. currency (18) 22 -23 lb. Av. ^' yr. PAGES 319-a%. 391 PAOB, 0. (14) 6. (5) ICJ5G. (10) 20nf. fliiys, etc. 7) ^17-50. ^333 -n.*]?,. 10) $204(). ') 11 oxen. 20)42,^J«. (5) $15-05. idwjiy 1)0- 13) $()250. ) lCJM-72. 0) 58 g!vl. 4) $18-18. (19) Mid- (5) 18000. )) $10 80. (14) 7/r. $1773-23 -) $22400 ?h1., 1 (it. )-38. J) $18-40. . (11) 20. >0, 15 mo. 343340. 0) 2080. (114) 12. 110, 111. rr. 319. -(19) 36-2. (20) A £12(53|. (22) £20. (24) 97Jlc 320. -(25) l-7320508075 ^"' ^«' ^^- (79) £5 inc. '(80) 9 m: (81) JO loss. 328.-(82) $13077-07. (83) £7 10s. (84) £7 10s. 7 W. (85) 7H c loS To} ./f > ^ 40' ^ «0 d. (88) 339-7 grs. (89) 12-61. (.»U) bj%, 3fg. 329.-(91) 5-17%. (92) 314050, 9064800. (93) $948-015. (94) $1104-08. (95)12-37. (90)8-9. (97) .^, hr. , ,^ hr. ; i rPd (c2 - d2) m. per. hr. (98) 2079 min. (99) $317-86. ^^•■~M^O^-'^0'"-P^^^"'- (101)801it.,80kil. (102) 2-8630Z., /^."S^^^-'"-' ^'^^^^^ (104) $31.25. (106) $91-80, 8X % etc,, 1?I^. t^a'?o ^10?)«^^*^^^^-ly' 'S2796-82. (109) 1090 f t. pe?sec. 331.-(110) £39 12s. Id., $19802, 99921 fr. (Ill) 4-5A 4-38^a. i}}^2 Lead 36-35, cork 43-04 lbs. (113) 2-004353504072 gall 1H0 f^tf^^' ^^^^^ 2^^-' 12*c. (117) lOlf. (118) m 332. -(120) 480000199999. (121) 4-005. ... (122) 5m per hr atf)m'ft'- ^'''^^^'^^^- (120) 10. ^ aV 05m: ^^•~f}?^i f.f ^f; ^^^?Uf^- (131)$212-71. (132) 4 in. nearly. (133) 8ft., 7 ft. (134) 4:3:1:5 by wt. (135)720. 334. -(137) 113*, 48. (138) 47. (143) ISIjl (144) 11-032 ft (145) £500000, £54000. (140)470. (147) $10000 335. -(148) $2250, -li^iOOO, $1750. a49) 24(X), 80c. (150) 35?S (151) $22047187-50. (152, .$87 -53. (153)05. (154yi26oO o'zt 336.-(150) 8 hr. 10' 10-4". (157) 2057, 833. (158) 46-5 per cent. ^'r^-.J^'^.^'''^^^"''- (1^1) 1-942. (162) 9m. 660 yd., 9 m. 1035 yd. i 392 l'A«B. AlfSWEKS, 337. (!♦;;{) 4)|. (1,54) $r,:]:m, o? m) «•;;, pmX niHr^i;;:;- f'*'^) ^^'i^- (i««) in, 24 ft. r>r><-k \ - ■-• / " III l» "^24 ,..„,. («) f247-«5. (S) 49U.49; 351. J3 If;, '?ix»l»2f «». W7.B0. (J 6.,1.d (4) -007916. (5) 80d. («) £i«2gi y. ^ ^^'i^i^i^- „f^^l (4) 29 ^,. PAOR. PAGES ;« 1^172. ms 1 l» 366.-(ia) »iia(w- "• ^*- 3e0.-(a) ;i9. C4) Ws. (5) r. f^ . ,. (7)22500. (8r4«c. (f5)4j() '^"*^'"*'' I'^' ^«) ^(m (II) «14(X), WKH). "^^ *'^' *''*"^i- WSi3%. qao f,?'!;^'^. wrlrU; „!'■'*'■ W -wax;., .w^K)!.!, ^•~(Tl)1l634!l2r-=«' ''-• <«) -4055. lr..38«.. (,o) 20}. (1) -998. W ^,0'iV (9)1,^4 11,. (10)30^3 ,,, (?) Shm'fmi'^ '^ ^''- <«> ^ « '-. B 10, etc.