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Lorsque le document est trop grand pour dtre reproduit en un seul clich6, il est film6 d partir de Tangle sup^rieur gauche, de gauche d droite, et de hui.t en h9«<. en prenant le nombre d'images ndcessaire. mb Ji?«^r<*mmes suivants illustrent la mdthode. -» >^ 1 2 3 1 2 3 4 5 6 ] Ci Su A NBW Practical Arithmetic, CONTAINING NEW AND COMPLETE INSTRUCTIONS FOR ALL THOSE WHO DESIRE TO BE "QUICK AT FIGURES," DESIGNED ESPECIALLY FOR Commercial Schools and Business Men. By the Rev. J. L. H. Roy, Superior of the Seminary of Sherbrooke and Professor of the Business Class in the said Seminary, Ittontveal: PRINTED BY JOHN LOVELL & SON. 1892. K.it.r6d according to Acl of H,e P»rli,„„e„i „f c»-,a.la |,„ ,.,.„■ 1892, b, the Rev. J. H. R„v, „i ,he Depart,,,,,,, „f A..ric„li„r,.. i DEDICATION. TO THE BUSINESS MEN OK THK EASTERN TOWNSHIPS, AND TO JIV PUPILS, PAST AM) PltESKNT, ■iiiis PRACTICAL ARITHMETIC IS .aflrrrttoiioirltj PrDtmtcd. 1 ! ■•I I ■i- 4t PRKFACE. The principal features embodied in this work are simplicity and rapidity. The ability to make business calculations with ease, accuracy and rapidity, is an all-important acquisition to the business class of the community. In actual business, the smart, practical man rarely uses the methods ot calculating taught in school books ; the value of time compels him to substitute easy and rapid methods for the clumsy rules found in many a text-book. la order to meet the wants of the community, the author has put aside all the old methods and introduced in his book all the short practical and labor-saving rules actually used by business men Pupils m commercial schools will find in this book all the requisites to make the study of arithmetic an agreeable pastime. The author has been connected for the space of twenty-seven years with commercial colleges, 17 years of which with the commercial department of the Seminary of Sherbrooke. It was for the pupils of that department that his arithmetic was primarily prepared. He takes this opportunity to acknowledge his gratitude to all those Who have assisted him in the preparation of this work, and he would deem it a special favor if business men, teachers and others would communicate to him any suggestion, new or improved methods of cal- culation that may add to the accuracy and completeness of this arith- metic. :« /'.H ^ < I ... ill I 1: I ■ 1 PRELIMINAKTES. A collection of several similar objocts, or the repetition of the same event gives the idea of a number : v.g., in ascending a stair we have the idea of the number of its steps. The idea of number is found when we compute magnitudes. We call magnitude or quantity anything admitting i'^creaae or decrease, if not in reality at least in the mind: u.g., a length, a weight or the area of a field are quantities. A unit is a single one: v.g., a house, a town, a man, a foot, an acre, a dollar, a cent. Kach quantity has its unit. Unit of extent is a foot. " " duration (( 1 hour. *« " weight K 1 pound. " " value <( 1 dollar. '* " area (( 1 square foot. To measure a quantity is to compare it to its unit. Number is the expression showing how many things are to be con- sidered. An integral unit is a whole one. A fractional unit is any quantity less than a unit. Thus !;• is a fractional unit, f represents three fractional units. A number is concrete when it is followed by the name of the quan- tity or object represented, as a horse, a buggy, a harness. A number is abstract when it is not followed by the name of the quantity or object represented, as any number taken alone, 6, 7, 8, 9, etc. In grouping numbers and comparing them we are led to make cer- tain operations which constitute " Calculation." Arithmetic deals only with this class of numbers. NUMERATION. SPOKEN NUMERATION. When the descendants of Adam and Eve began to associate together, they felt the necessity of inventing names to designate the objects which fell under their sight, or those that imagination could conceive. For instance, the name tree was given to those big plants shooting up from the earth ; and, then, dividing trees into different species, par- ticular names were given to each of them, such as the poplar, pine, apple tree, etc., etc. i ' t i > 1 i 1 : 8 Numeration, Tlu! Hiimo thinj; took place for aninmlH. Kciually, the word number WttH given to a C(!rtaiu nuuibcr of units; but a;? these numbers were unlimilt'd, it became indispensable to create names for all possible numbers. The knowl(!djj;e of these numbers took the name of *' spoken numer- ation" or simple Jiumeration. Hence: Spoken numeration consists in the knowledge of the names adopted to express all possible numbers, v.g., one, two, three, four, five, six, (itc, etc. Written numeration or notation consists in representing all num- bers witli a small number of characters. There are two methods of notation in common use, the " Arabic " and the " Roman." Arabic notation comprises ten numbers, viz. : — 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. one, two, three, four, five, six, seven, eight, nine, cipher. The cipher is called zero or naught. It represents no value itself 5 it is used only in connection with other figures to indicate their proper relative order and value. ARABIC NOTATION.* The Arabic Notation is divided into Periods and Orders. The Periods are : Units, thousands, millions, billions, trillions, etc. The Orders are : Units, tens, hundreds ; thousands, ten-thousands, hundred-thousands; millions, ten-millions, hundred-millions; billions, ten billions, hundred-billions ; trillions, ten-trillions, hundred-trillions. etc. Hence the following : — 5th Period. 4th Period. 3rd Period. 2nd Period. 1st Period. Trillions. Billions. Millions. Thousands. Units. of of of of n3 CO CQ n3 ■T3 •^ v s> f- tm u «H Ui n3 CD n3 M na _ CO ■^ m rs Hun Tens Unit Hun Tens Unit Hun Tens Unit Hun Tens Unit 1 of en -S 13'S in :o (M t-05cc th »o -^ the 6 lown under t from pre- 4, to which 3m preced- which you lone and 1 ind of the thus prov- down the ^1 ^1 SEVENTH METHOD. IN8TANTANKOU8 ADDITION «V C()MHINATIO^f . Write two, three, four or mon; rows of miscellaneous figures, then write such tiguros as will make an ecjual number of nini's in each column; under tliuHo again, write another row of miscellaneous tigures. 4S()7:{4 4r)(i.S75 2(;H74t)/ \«;4r)4:{5 5i:i2()5 8975H7 4897503 The first four rows are miscellaneous; the next four arc the com- plement of the first four, forming 9's all the time ; the last row is mis- cellaneous. From the unlfs of tlit; lowest row subtract 4, which is the number of rows of 9's, and putting it at the beginning (jf that same row you have the answer. Or then : — g/ 84". 072542 ^ 54:52 lO.'UG 8 10 This last row contains the answer. ^,3/4r)2i94(;:{7 \ 058910478 fj /725(]:U54:J ' ^ 2748084.50 /4052 14802 V428i)74520 / 502405878 ^ 548045787 / 548207840 V 450782053 / 278450534 vol 5482854 / 750784899 \ 354870211 [-787567079 miscellaneous 10 87S7567071 24 Addition. As there arc 8 rows of nines subtract 8 from the unit 9, and put it at the beginning of the row. Explanation. — A row of 8's and a row of lO's are equal to two rows of nines. In all we have 8 rows of nines forming a total amount of 7,999,999,992 or 8,000,000,000 minus 8. Now add a row of mis- cellaneous figures to your addition. Deduct from the unit number of that odd row the figure representing the number of rows, and place it at the beginning of your answer. The above method is a very entertaining, amusing and profitable exercise for developing the calculating faculty of a young man. The row containing the answer can occupy any place in the combination, provided it is not coupled with any row. EXERCISES AS A STUDY HALL TASK. No. 1. The sum of the numbers in each row of the following table, whether taken vertically or horizontally, or from corner to corner, is 24156. Let the student be required to make these 24 distinct additions . TABLE (borrowed from Sangster's Arithmetic.) 2016 252 2448 684 2880 1116 3312 1548 3744 1980 417G 4212 2052 288 2484 720 2916 1152 3348 1584 3780 1620 165G 4248 2088 324 2520 756 2952 1188 3384 1224 3816 3852 1G92 4284 2124 3G0 2556 792 1296 3888 1728 4320 2160 396 2592 3492 1332 3924 17G4 4356 219G 36 2628 468 3060 900 936 3528 1368 3960 1800 3996 2232 72 3132 972 3564 1404 3600 1836 4032 576 3168 1008 3204 1440 3636 1872 2772 612 2808 1044 3240 1476 3672 1908 4104 2340 180 216 2412 648 2844 1080 3276 1512 3708 1944 4140 2376 2988 828 3420 432 3021 864 22G8 4068 230 i 144 2736 26G4 504 3096 108 2700 540 1260 3456 and put it ■m ual to two al amount ow of mis- aumber of id place it profitable lan. The abination, . V 3, whether is 24156. )ns. 216 412 j48 ^44 )80 276 H2 08 f44 76 Addition. No. 2. 25 In the following columns, add all the figures enclosed in each brac- ket as one number, and name only results in the same manner as when the figures are taken separately. Thus, in adding column No. 4, say 7, 11, 16, etc. When a figure is repeated several times, count the number of times it occurs, and multiply by the figure. Thus, if the figure 8 occurs seven times in a column, multiply 7 by 8 for the result, instead of adding seven 8's together. When three figures occur in regular order, - -as 4, 5, G or 6, 7, 8,- three times the middk figure will be their sum : when five figures occur, take five times the middle figure. When there are four figures in regular order, take twice the sum of the extremes ; when there are six, take three times the sum of the extremes. (1) 324 235 143 421 312 234 343 423 225 123 334 212 324 123 431 212 4419 (2) 678 789 97('' 899 989 988 878 673 789 968 887 987 798 976 687 997 13959 7212 No. 3. ADDING HORIZONTALLY. In some branches of business the ability to add numbers which are written horizontally instead of being placed under each other is often required. Thus : — 824 + 865 + 526 = 2215 i i SSS BSSSBS 26 Addition. *l! il-: 'ill; '' til ft IH ;: :i :fii t All the units are first added, then the tens, and then the hundreds. A little practice will soon overcome any difficulty which may be expe- rienced at firsti Add 437 + 461 + 579 + 385 624 + 395 + 671 + 452 " 347 + 621 + 374 + 987 Add the following numbers as they stand : — 325 + 116 + 365 431 + 275 + 218 Ans. 1730 463 - 657 + 146 234 + 532 + 534 — — _ Ans. 2566 No. 4. (1^ 246875 (2) 689745 (3) 278956 394654 372164 376543 279531 531763 487654 104645 297534 598765 461589 4567ti9 609876 275341 G54321 718987 158905 789247 346579 275654 657325 275398 387567 904137 754369 244658 563425 123456 492372 639562 234567 690145 4H6756 345678 549023 234(554 456789 167346 789346 5i;7890 385432 356543 678901 563109 123456 789012 754354 789012 890123 643715 345678 901234 573490 901234 012345 234567 567890 123456 890123 123456 789012 456789 789123 352734 124356 456789 639576 789134 343736 384653 257986 527282 289104 734675 175635 563795 389146 294516 789534 254324 763465 376379 i 741045 234605 476532 Students should not be allowed to 1 ;ave Addition until they can add up without the least hesitation. a h se til 8£ $4 Addition. 27 PROBLEMS ON ADDITION FOR MENTAL SOLUTION. [iey can add 1. How many dollars in four boxes, if the first coutains S12, the second $8, the third $10, the fourth 85 ? Am. $35 . 2. If you pay 85 for a fur cap, $25 for a coat and 85 for shoes, how much would it make? Af.s. 835. 3. Jolin has 13 apples, 15 oranges, 12 lemons, and 16 plums; how many has he in all ? Ans. 56. 4. How many eggs in three baskets, the first containing 2 dozen, the second ^ dozen, and the third 11 eggs? Ans. 41. 5. How many bills has a man who has a 85 bill, a 82 bill, a $10 bill, a 84 bill, and a 820 bill ? Ans. 5 bills. How many dollars ? Ans. 841. 6. A family owes the baker, 827 ; the butcher, 846 ; the shoe-maker, 8t le multiple 075 mt^tlioJc onliiiairo 075 dc (i 240 L'46 ■1 m Wi Ml! 4050 produit par 6. 10200 100050 4050 2700 1350 100050 Exi'LlCATiON. — Multiplioz par la plu8 petite partie du multiplica- teui, 6; ensuitc, multiplitjz le produit de (} par le iiombrc de tbis que 6 est coutunu duns I'autre partie du uiultiplicateur. Ainsi 24 -^ = 4- Maiiitonaat, multiplici! 4050 -; 4 et vou.s aurez le produit do 24 qui est 4 lois le produit de 0, Multipllez 720 < 248 079 x 240 892 X 549 054 x 549 " 507 X 508 490 x 497 Multiply 994 X 872. 994 U«ual way 994 872 872 7952 71508 H00708 1988 0958 7952 800708 In the above multiplication, the multiple i.s 72, begin with the 8 and get the 1st product or sum. Then 72 -^ 8 = 9. Multiply 7952 by 9. Put 8 under the units and add up. This rule is well adapted for mental drill, and should be used frequently. 070630 EXERCISES ON ABOVE RULE. 1. Multiply 4765 by 142 Answei 2. (( 7654 (( 357 i( 3. {( 8546 u 426 u 4. <( 4898 u 369 . « 5. It 3725 « 424 (( M u It i plication. 87 ur est un (5. Multiply i'tW.i \>y S-'in .1 8245 0'> 10. 11. 12. 13. 7749 49G1 9754 7(i54 4107 »27 972 24* i 872 (;72 182 217 iiH)rrr It u No. 2. How to ujultiply by 10, To multiply any number by 10, 100, 1000 : add to the ri.i^ht of any number if you multiply by 10; 00, if you multiply by 100; 000 if you multiply by lOoO, otc. NoTE.- abovo rule iiori! is no necessity in <;ivin,u' osamplos applicable to the No. •]. To multiply by numbers, the half, third, fourth or fifth, etc., is a convcnicut multiplier. To multiply by \. Usual way 24 2 = 48 18 ^ 2 = 9 m <■ ' 192 24 432 432 Multiply the multiplicand by 2 and divide the multiplier by 2. By so doing you simply diminish your work without altering in any way the value of your numbers. To multiply by J. 25 X 8 = 75 21 -f- 3 = 7 26 50 626 i: 'M .^\h 1 ! H w ^ t'l' 38 M Multiplication. To multiply by J, 48 X 4 = 192 36 -T- 4 = 9 &c. illi^:-' 288 1728 144 1728 To m ultiply by ^. 55 X 5 = 275 45 — 5 = 9 275 2475 220 2475 To multiply by ^. 42 X 6 = 252 24 -h 6 = 4 168 1008 84 - 1008 &c. & dot le almost < ifinitu m. ES. CERCIS Byi , 1. Multiply 48 by 32 Answer 1536 2. K 18 (( 14 (( 252 3. l( 24 By* 18 (( 432 4. « 51 by 24 t( 1224 5. (( 36 (( 21 i( 756 6. (( 42 Byi 27 <( 1134 7. (( 56 by 36 II 2016 8. « 92 (( 48 K 441 G 0. « 96 « 24 <( 2304 Multiplication. 10. Multiply 55 by 15 11. 12. 13. 14. 15. (I 1944 2304 1728 &o. 3250 Subtract the multiplicand 65 95 " 45 75 " 35 • Byi 54 by 36 96 " 24 48 " 36 &c. No. 4. To multiply a sum by any number ending with 9, Rule.— Multiply by the next higher number and subtract the multiplicand. Multiply 65 by 49. Add 49 + 1 = 50 ; now multiply 65 by 50 65 50 Answer [ n85 ime problem by ordinary method • ^^_^ 65 49 585 260 3185 1. Mutiply 75 by 29 Answer 2175 2. 165 " 39 (( 6435 8. « 89 " 49 u 4361 4 " 198 " 50 (( 11682 6. " 275 " 69 <( 18975 6. *' 385 " :y <; 3G415 f. « 72^- " &ry ,j 64792 a " 836 - 129 (( 107844 ■ ■ •% '^ " 7829 '' veu <( 6020501 ill l\ 40 Multiplication. h % No. 5. To multiply whou the units added together equal 10, and the tejis are alike. Rule. — Multiply the units and write down the result, then add one to the uppv'r number in tens and multiply by the lower tens. Multiply 86 x 84. 6 and 4 representing the units form 10, 86 and the tens are alike. Now say, 4 times 84 6 are 24. Write down 24. Add 1 to the upper number 8 + 1 = 9; now say, 8 times 7224 9 = 72, prefix 72 to 24 and you have your answer. Usual way i\f a proof: — 86 84 344 688 7224 hii th fo be ad EXERCISES ON ABOVE RULE. 1. Multiply 86 by 84 Answer 7224 2. « 38 32 (I 1216 3. <( 69 61 (( 4209 4. (( 22 28 616 5. it 78 72 (( 5616 6. (( 68 62 (( 4216 7. (( 88 82 u 7216 8. (( 28 22 n 616 9. *( 56 54 (( 3024 10. (< 98 92 (( 9016 No. 6. To multiply any number of two figures by 11. KuLE. — Write the sum of the figures between them Multiply 35 by 11. Say 3 + 5=8, write 8 between 3 and 5 a tot CO] your answer will be 385. i Multiplication. 41 ,nd the ca add Eemark. — When the sum of the two figures is more than 9, in- crease tlic left liand figure by the one to carry, thus : — G9 x 11 = 759. Say [,6 +9 = 15. Put 5 between the two figures and add 1 to 6. EXERCISES ON ABOVE RULE. 1. Multiply 58 by 11 Ansiver 638 737 902 429 495 792 374 319 1023 825 With regard to mental drill on multiplication, the above methods or hints are more than suflBcient, if well practised As in the above methods there are questions involving fractions, if the rules which govern these contractions are considered too difiicult for young pupils to understand at this stage of their progress, they may be omitted for the present, and attended to when they are further advanced. 2. ' 67 3. 82 4. ' 39 6. • 45 6. 72 7. ' 34 8. « 29 9. ' ' 93 10. 75 (< ^ s? 385. DIVISION. DEFINITIONS. Division is the process of finding how many times a number can be contained in another. The number to be divided is styled : Dividend. 'il.e number by which we divide is called : Divisor. The answer to be obtained is styled : Quotient. EXAMPLE. Dividend Divisor 36 87516 72 155 144 111 108 2431 Quotidicnt 36 36 ,'1 m '■\ .!li \u mm !|jii 42 Division. Explanation. — We first find how many times 36 is contained in 87, and being contained 2 times, we write down 2 as the first figure in the answer, then say twice 36 = 72. Write 72 under 87 and subtract. To the remainder 15 we annex 5 of the dividend ; then say 36 in 155, 4 times, we write 4 as second figure of the answer. Now say 4 times 36 = 144, set down 144 under 155, and subtract, and so on till you are through with your division. If anything remains, make a fraction out of it. For instance : — Divisor Dividend Quotient 12j 1867 I 1 12 I 155i V2 66 60 67 60 Explanation. — Proced as above for the division. Now, as we have the figure 7 remaining, put it down in your answer IS a fraction of 12. SHORT DIVISION. When the divisor is not greater tlian 12, the process of dividing may be greatly abridged. We have to divide 87567 by 7. The operation is set down in the following form : I 13 64 S75672 125096 Say 7 in 8 is contained once with 1 as a remainder, we set 1 under 8 and place the remainder right above the following figure, the result will be 17. Then, 7 in 17 two times; write down 2 and carry 3, which we put over 5, and so on till the end. li ■ Division. 4H EXERCISES ON DIVISION. &t 1. Divide 7652 by 2 Answer. 2. a 6543 K 5 (( 3. (( 9436 it 8 i< 4. ic 6456 (( 9 >( 5. (( 34567 « 12 il 6. u 834675 (( 18 (( 7. (( 964534 (( 24 (( 8. u 1559376 U 36 ii 9. (I 2487567 u 47 K 10. i( 8965439 li 59 <( 11. (I 6753464 <( 68 « 12. (< 5647653 n 75 (( 13. u 6789012 l( 89 '( U. u 7890123 (( 315 K 15. (( 8901234 (( 781 a 16. (( 9012345 <( 891 (( 17. u 7040506 il 987 (( 18. 1.1 1234567 ii 1236 (( 19. u 2345678 (f 2463 (( 20. (( 3456789 (( 3764 a 21. (( 4567890 u 4653 li 22. li 5678901 i: 3769 11 23. (( 6789012 ti 7098 li 2i. (( 8965407 11 8140 (. 25. u 6754679 ii 9989 11 26. u 1234567 a 6010 il 27. (( 2345678 ii 7146 11 28. u 1040065 il 8104 <( 29. (( 3100040 (( 5019 il 30. a 4006054 (( 6146 i< PROBLEMS ON DIVISION. 1. A train on the C.P.R. runs 40 miles an hour, how long will it take to go around the world, the distance being 25000 miles ? Ans. 26 days, 1 hr. 2. The large wheels of an engine are 15 feet in circumference ; how many turns will they make in a distance of 14019255 feet ? Ans. 934617 turns. i Hi: i 'ii; - ; ■ Vi I 1 M i mm 48 Cancellation. (I il" 4. 10 '^i^l^*^'^"c«by5- ^ns. .4 liO 60 45 J " p. 24714 147(136 , , (^ ft 1 2 4 r, 7 3 A 1 '• 4 U 7 15 8 9- ^^^'IH Q 22 {» 12 5 . ,(. ., 3"Tf c'"4' ■^H"'^' i 5 units. 9 25J__li_36 .^ 117 ^* 4 10 21 &4' 18 >7 21 15 18 54 7 . 'J 9 10. 28 27 21 T 54' GREATEST COMMON DIVISOll. The greatest common divisor of two cr more numbers is the greatest number of them that will divide each of them exactly : — 168—210—252 84—105—126 28— ■15- 42 4— 5— 6 Explanation. — Find a number that will divide all the numbers exactly. 2 divides them all, then 3, then 7. Now the common divisor will be 2 X 3 X 7 - 42. Divide 168 -f- 42 = 4, which is just the remaining number at bottom. Divide 210-f-42 = 5, the remaining number. Divide 252 -f 42 ^ 6, " <( Rule. — Divide as long as you can by numbers that will divide ex- actly. The product of all those numbers will be the Greatest Common Divisor. Find the greatest common divisor of the following: — 1. 16. 20. 24. 40. Answer 4 2. 8. 24. 36. 48. " 4 3. 15. 20. 25. 40. " 5 4. 36. 84. 108. 144. " 12 III' Common Denominator. 49 COMMON DENOMINATOR. Fractious liavc a common Denomiuator wlieu they have the same number for a tlonominator, if 7 2 Multiply 5 X 7 X 2 = 70. 70 is the common denominator. Thou divide 70 by each denominator and multiply tho nsult by the corres- ponding numerator. 70-^5-14. 14x3 = 4^ 70 70-f-7 = lO. I(>x4 = '*« 70 70-^2 = 35. 35x1=35 70 Remark. — The Common Denominator is used when numbers are odd numbers or not divisible. In all other cases, use the following me- thod. LEAST COMMON MULTIPLE OR DENOMINATOR. The Least Common Denominator is the least denominator to which all fractions or numbers can be reduced. 2 3 •1 3 8 5 G 2 4 3 6 1 2 3 3 2 1 1 Explanation. — Take the smallest number that can ul/iuo two of those denominators exactly. Now multiply together all the divisors, and the figure remaining at the bottom — 2 x 2 x 3 x 2 = 24. Hence 24 is the least common multipb or denominator. Note. — In finding the least common multiple or denomiator, it is necessary to divide by the smallest possible number, dividing two or more figures exactly. The following example will illustrate the case. ) J \V > m , I :i !h 1 i Mi i 1 Wtt 60 Least C(ymmoii Multi'ple. What, is the loasf comriiori ruultipU) of 1 EX. 1 12 1 IH 1 30" 18 2nd. EX. 12 3 1 12 1 1 18 :w 2 (J 9 1 18 3 3 3 9 9 ^o „ i 3 1 3 3 Q ^ 4i 1 1 21 G is certainly not the least com. diviaor. 2x2 >c3x3 = 36. 36 is the least com. div. The 2nd Ex. is wrong, because 12 has a number that can divide it in the last line, 6. Find the least common multiple of 2-2 ¥-^^ Ans.48. ^- iH ^ 15 ^»«- 30- A '111a o/' 4. 3" U IT 12 ^"''- '^^• 5-rff-2^4^A««-42. li- REDUCTION OF FRACTIONS. deduction of Fractious is the process of changing their form with- out altering their value. FiusT Case. — To reduce a fraction to its lowest terms. Eeduce jg to its lowest terms. Operation : — Ans. I. 2)- = - 3)§ i i5°'l We divide the terms by 2, a factor common to both, and obtain f . We divide again both terms by 3 and obtain \, and the frac- tion is reduced to its lowest terms, means the same thing. The form alone having been changed and not the value. RcJuct J) 4 \i J 38 00 m M7 Reduction of Fractions, Ans. Ans. 61 ; " to Its lowest terms. « a (( (< « <( (( It .'( (I (( 1 r>' I 4* Ans. r,. Aus. |. An,. =. All8. -, 7 Second Case. — To reduce a mixed number to an impropor fraction. In 9jr Iiow many fifths ? Ans. "^^ Operation : — 9 46 8 48 fifths = *^ 5 Since there are 5 fifths in I unit, there will be 5 times as many in 9 units, and tlie 3 fifths bcinj^ added make 48 fifths, wl^ieh wc express in the following way -,. In 8j how many fourths ? Aus. ^^ " sevenths? Ans. ^ i( (( « 62 " thirds? Ans. " sixths ? Ans. -1 16 77 Third Case. — To reduce improper fractions to a whole or mixed number. 37 How many units in j^ ? Ans. 2'^ Operation : — 16 37 32 "5 5 2,-6 Since 16 sixteenths make one unit, there will be as many units in 37 sixteenths of a unit as 37 contains times 16, or 2^. 1. Reduce j to a whole number. Ans. 19|. 2. Reduce f " 3. Reduce ^f " 5 4. Reduce ^ " Ans. 9. Ans. 29. t I I . .( 1 i I ■ 1 ; i' Ill 52 Reduction of Fractions. Fourth Case. — To reduce a compound fraction to a simple fraction. Reduce ~ of p to a simple fraction. Ans. J. Operation : — * of 1 = ^ As we can cancel 4 by 12 and 7 by 7, we obtain J. 3 If cancellation is not practicable, we then multiply the numerators together and the denominators together as : — 4 2_ 8 5 ^ 3~i5 Eequired the value of § of f of f Ans. i|. Required the value of | of y\ of ^ Ans. ^''jj. Required the value of ^^ of li of -^j Ans. ■^. Required thfe value of ^ of ^ of f Ans. j^^. ADDITION OP FRACTIONS. Fractions may be added by means of a common denominator. We can add halves to halves, thirds to thirds, fourths to fourths, etc>; but we cannot add directly halves and thirds, etc., any more than we can add other things of different kinds, as dollars and days. Thus ^+i = l unit. J + ^ = §. Add up the following f + |^ + | + ^'^ Operation : — The least common denominator is 30. 30-^3 = 10, and 10 + 2 = 20 30^5 = 6, and 6 x4 = 24 30-f6 = 5, and5 X 5 = 25 30-;- 10= 3, and3 X 7 = 21 Amount 90 Com. Denom. 30 Cancel by 10, ?? =f or 3 units, and by 3. f fm: Addition of Fractiona. 53 EXERCISES. 1 Lli Ans. 2A 2 3 24 2. 4 7 13 »„„ oil 5 10 15 ^^^- "^30 3. 7 9 11 1122 3 6 Ans. 1^ 4. 13 5 7 15 12 20 Acs. 1» 5. 2 1 3 17 Ans. 2!? 6. 3 6 4 18 5 7 3 36 Ans. 2: 19 8 10 5 16 "80 Add the following mixed fractions : — Put all the units together, then add up the fractions as usual. 3 5 6 4 Units = 18 4 1 4 £ 5 3 "*" 5 "*■ 10 "^ G The common denomitiator is 30. 304-3 = 10, and 10 xl = 10 30-f-5= 6, and 6 x 4 = 24 30^-10 = 3, and3 X 3 = 9 30-1-6 = 5, and 5 X 5 = 25 2^B Ans. Cancel by 2| = ^* Amount 68 Com. Denom. 30 15 The sum of your fraction is ^g ; but ^j. is an improper fraction. Div- ide 34-f 15 = 2j Now carry that amount under 18 units to be added together and it will give you 20jg for your answer . EXERCISES. Add 1.3^ + 4,^ + 8^3 + 9^ Ans. 26* 2. 4 + 2^3| + 5^ Ans. is;* 3-64^-55 + 4-^7^ Ans.23g M l^il •1, % 64 Addition of Fractions. To add compound fractions : — Add 2 . 2 12 + It Ans. 12 Cancel as many numbers as possible, then multiply what remains. In the above fractions, we have ^ and ^ representing the two compound fractions. Adding up by the usual way, we obtain :^^ for the answer. EXERCISES. Add 1 iof^of^ + |oflof| + ^of,^ 6 ' 8 7 ' 5 6 1. A merchant has sold from a piece of cloth 12, 4| 8^ and 2^ yards ; how many yards in all were sold ? Ans. 27^J yds. 2. Bought a handkerchief for ^ of a dollar, a vest for 2^ dollars, a pair of gloves for ^ of a dollar, and a hat for 4^ dollars ; how much did the whole cost ? Ans. $7 11 16 3. Bought three lots of coal, weighing respectively ll X ^^^ ^1" tons ; how much is there in all ? Ans. lli tons 4. A farmer sold sheep for $62g, cattle for $102,, and a horse for $125^ ; how much did he receive for all ? 5. In two different occasions | and ^ of a certain work were made. What part of the work was done ? Subtraction of Fractions. 55 SUBTRACTION. FIRST CASE. Subtraction is the process of finding the difference between two fractions. Rule. — Reduce the fractions to the same denominator, then sub- tract the numerator of the smaller fraction from the numerator of the greater, and under the result put the common denominator. Example — 1.5—8 from 4^V ~~ 20 "t ■ i With two small fractions as above, we multiply the numerator of one by the denominator of the other , Say 4 times 2 are 8; write 8 above the 2 ; then 5 times 3 are 15, which we place above the 3. Now subtract 8 from 15 = 7. Place 7 at the end of your problem. Multiply the two denominators together 4x5 = 20, set 20 under 7 and your answer will be 5^^ EXERCISES. 1. From^-J 6 5 2. 3. 4. i_ 3 2 8 7_1 8 9 11 _ 3 12 4 Ans, Ans. Ans. Ans. 5. From ^""7 6. 7. 8. SECOND CASE. 9 4 10 "7 6 _1 11 3 15_£ 22 11 Ans. Ans. Ans. Ans. SUBTRACTION OP MIXED NUMBERS. From 5?- 12—5 Take ^ ixi == 7 2i-(} Proceed as in first case. Set down ^ under the line as a part of your answer. Then say, 3 from 5 leaves 2, which you set down under IT the numbers subtracted. Your answer will be 2,^^ Note. — In the case of mixed numbers, it happens that the fractional part of the under number is greater than the above number. When this happens, instead of reducing both quantities to improper fractions, i\^ ■ 'I '. ' . 'i T W I' I 56 Subtraction of Fractions. we borrow cue from the above uaits and we consider it equal to the common denominator; then we add it to the deficient number. Thus :— 1 k i \ i 1 - 1- 1' i From 6| Take 3* 5 4^^5 20 21 20 Proceed as in 1'* case. Now we have from ;,^ take ~ , a thing which we can't do, unless we borrow 1 unit from 6 - which will be equal to 20 the common denominator. 20 and 5 will be 25. Now say 16 from 25 leaves 9. Sot 9 as the numerator with 20 as" denominator. But, by the borrowing, G has been diminished by 1. So we now subtract 3 from 5 leaves 2. The answer is then 2^. EXERCISES. 1. From 4- 4 Ans. i| 2. i- >>3 "4 Ans. o9 ^20 3. 8j- i Ans. 4 4. 3J- 4 Ans. >f. 5. 9|- ^l Ans. 3- ^4 6. 24- r4 Ans. o32 ^63 7. 34^ ^ ^^i Am. .53 ■*60 8. 43 - 9'^ Alls. 34 9. 8 - ll^ 20 Ans. «". 10. 315g-102. Ans. 213S 11. 3 8 of 1 1 ,. 2 -4 -2"^ IT Ans 1 352 12. 2 '3 of G-n«^8 Ans 137 • 300 13. 7 8 of 3 5 " 20 Ans 29 60 PROBLEMS. 1. If from a barrel containing 2^ gallons of wine, 7f gallons '>re lost by leakage ; how many remains in it ? Ans. 17^J gallons. m Subtraction of Fractions. 57 2. Bought a lot of coal for $31| and sold it for $37 ^r> ', how much was gained ? Ans. $6^^,- 3. Paid $113|- for some goods, and sold them at $71 under cost ; for how much were they sold ? Ans. $106^. MULTIPLICATION OF FRACTIONS. Rule — Multiply the numerators together and the denominators to- gether. Example — Multiply | by | 3 7 X -^ 2i 4 2 1 X 1 9 z= 3 9 !) To multiply mixed fractions — 5Jx3l Reduce into improper fractions ^^V being an improper fraction, we divide 399 -f- 20 = 19^§ the answer. The following method might be ibund pi-eferable by some : — 8Jx4^ Multiply the whole nuu-bers together 8 x 4 = 32 Multiply 8 by the opposite fraction 8 x J- = 2§ or ^ Multiply 4 by the opposite fraction 4x^ = 2 Multiply the two fractions together i x J = ^ Ans. 36 Usual way as a proof of the correctness of the above. 1. 2. 3. 4. Multiply :- 17 V 13—^21 U ^ IT "" 6 221-^6 = 36# Ans. EXERCISES. lOfby 6J Ans. 68-rV 181 by 7\ Ans. 133^21. 201 by le^ Ans. 338/^. 19J by 10| Ans. 19(>^. TO MULTIPY COMPOUND FRACTIONS OR FRACTIONS OP FRACTIONS. Set them all in one row, cancel as much as you can, and then mul- tiply all the numerators together and all the denominators together. ill Mm ;'■« :u 58 Multiplication of Fractions. Multiply 1 of 3 of 4 Of 2 Of 2 Of 5 of 3 of 1 1 5 7 8 12 11 i 112' If you have mixed fractions, reduce into improper fractions, and cancel in the usual way. Multiply I of i of 31 of 55 11 98 or 7 4 1 55 _ I or 1 unit. 11 5 2 98 1 14 2 ai EXERCISES. 1. Multiply : — T^"" n>^ *Xt'2x56J^ J Ans. V2V 2. ^x 3fx 4^x ^ X 6^x51 Ans. 1561H 4 3. 4f X 6f X 71 X 8^ Ans. 1841y»f 4. ^x 9J X 11 2 X 84 Ans. 4478-^|i 5. 12|xllf X 7f x5i Ans. 613717 PROBLEMS FOR PRACTICE. 1. What cost § of a bushel of corn at S of a dollar per bushel ? Ans. ^ of a dollar. 2. How much will cost 2^ yards of cloth at f of a dollar per yard ? Ans. $2.27^-^ 3. What will be the price 7;f cords of wood at $5f per cord ? Ans. $41fi. 4. When cloth is | of a dollar a yard, what will j of 12 yards cost ? Ans. $6. 5. If a man travels -^^ of a mile in an hour, how far would he tra- vel in -^^ of an hour ? Ans. J of a mile. 6. If a bushel of buckwheat will buy x ^^ of a bushel of salt, how much salt might be bought for f of a bushel of buckwheat ? Ans. ■^i of a bushel. 4 7. If f off of a dollar buy one bushel of corn, what will ^ of it of a bushel cost ? Ans. ^^ of a dollar. 8. What cost 6f gallons of molasses @ 23| cents per gallon ? Ans. $1.52|f a 6 7" Multiplication of Fractions. 59 9. What cost 361^1 acres of land at $25§ per acre ? Ans. $91 6t U3. 3 2 0* 10. What is the price of 10^ yards of calico at 15J cents per yard ? Ans. 81.64fo- DIVISION OF FRACTIONS. Rule. — Invert the divisor and multiply the numerators together and the denominators together. 1st Cask. — To divide a fraction by a fraction. Divide Operation. — 4 x^ Invert the divisor, cancel 3 and 9, and 4 and 2. The answer will be rf . 2ni) Case. — To divide a whole number by a fraction. Operation. — a V « = •'L4 1 •^ r. r. 54 -^ 5 = 104 Ans. Divide 9 -^ 5^ Ans. 104. Place a 1 under 9 as a denominator, invert the divisor and multiply. You obtain 53 an improper fraction. Divide 54 -f 5 = 101- the answer. 1 1 3 28 3rd Case. — To divide a fraction by a whole number. Divide |^ -f- 8. Ans. -^g-. Operation. — Place a 1 under 8 as a denominator, invert ^ the divisor, and multiply as usual after having cancelled 6 and 8 by 2. 4th Case. — To divide a mixed number by a whole number. Divide 15g by 8. Ans. 1^^- Operation. — Reduce the mixed number into an improper fraction. Place a 1 under 8. Cancel 76 and 8 by 4 after having inverted your divisor. 151 ■ 1 1 J§. X i = 19 10 19-^10 = 1-3 10 The answer will be ,-. or Ir^- 10 10 f#l' f'i- EXERCISES. 1. Divide g- -^ 2. Divide | -^ 3 10 3 6 Answer 2- w 60 ^r! Biviaion of Fractlo7is. 3. Divide 27 - n 12 --^ 12 11 • ^"^ -- 9 43 • ^ -.- 75 4. Divide 23 ~ 1 4 5. Divide 6. Divide 7. Divide 8. Divide 9. Divide 10. Divide 17? 4- 7 o 11. Divide 18^ ^ 6 5 12. Divide 78t -~ 12 450 533 6 G 6 7 1 3 1_ 4 29 92 7 i3ii 3^ 43 6_ 633 11 3^ ^7 85 i_ 30 17 30 13. Divide 1 of I by | of ^^ Aas. 1- Operation _7 ryxf X 6 = U Ans. ^t Invert the compound divisor, and as you have now a multiplication to make, cancel as much as you can, and mul- tiply numerators together and denom- inators together ; ^ being an im- proper fraction, divide 7->6 = 1^ the answer. 14. Divide i of |. by i of |. Answer 1 J. 15. Divide f of X of § by x of ^ of §. " J. 16. Divide | of § by f of J. 17. Divide 4 of I byf of f. 18. Divide 130J by 30^. 19. Divide 75a by 19f. 2J_ 26* 4IO7 3303 PIIOBLEMS ON DIVISION OP FRACTIONS. 1. If 3 cakes cost $4^, how many can be bought for $38^ ? Ans. 28. 2. If $14 will buy U. of a ton of copperas, what quantity will $1 buy? " Ans. aVo- 3. A young man having $10 gave f of the sum for paper at $38^ per ream, how many reams did he buy ? Ans. 2 reams. 4. When 4 bushels of peaches cost $4.^, what is the cost of 1 bushel ? Ans. $1^. I !|;||ili^ :; li Division of Fractions. 61 5. How many pounds of cotl'ee at ^ of a dollar a pound can be bought lor $7i? Ans. 30 lbs. 6. When for I of a dollar 3^ bushels of apples can be bouL^ht, what is the co^^t of 1 bushel ? Ans, J of a dollar. 7. At ^ of ii dollar a gallon, how many gallons of molasses can be bought for $5.i ? Ans. 8|. 8. When 13 tons of eoal are sold for $71 i, what is the price ptM- ton ? Aus. 85^. 9. Paid $10^ for tea, at the rate of ^^ of a dollar a pound ; how many poumls did I buy? Ans. 15 pounds. 10. When broadcloth is $4| a yard, how many yards cun be bought for 3 cords of wood at §7 a cord ? Ans. 4a yards. REVIEW OF VULGAR FRACTIONS. 1. A farm is divided into 5 field.s, containing respectively, 20^ 56j9p 36^, y,!' and 23^] acres, how many in all? Ans. 146f^-^. 2. A man has 4 lots, the 1st containing 320J-i, the 2nd 136f, the 3rd GO 2, and the 4th 78^^ acres. How many in all ? Aus. 597/3. 3. By adding ^3^ to a certain number we obtain S1G§; what is that number ? Ans. 13§. 4. When $60 are paid for 12 yards of cloth, how much must be paid for 9^ yards ? Ans. S49.37J. 5. If ih pounds of butter are worth $1.44, what are lly'^g pounds worth ? Ans. $3.66. 6. A ship and her cargo are valued at $60,000, and | of the value of the ship is equal to J of the value of the cargo ; find the value of each. Ans. 7. From a hogshead iiiolasses of 100 gallons, y\ of it leaked out; 2 of the remainder I kept for my family ; what quantity remained for sale ? Ans. 24^» . 3 3" 8. If 5 barrels of flour cost $48f , how many barrels can be bouglit for $263| ? Ans. 27 barrels. 9. If f of a barrel of flour cost $5, how much will 2 bags of flour cost, one containing | of a barrel and the other -f of a barrel ? Ans. $12. Hi w 62 Review of Vulga7' Fractions, 10. Bi.u^ht ^ of f of V yards of cloth, at the rate of $3^ per yard Required the cost of it? Ans. $8^'^. 11. I purchased 7 loads of coal, each eoutaininj; 15| bushels («J I24. cents per bushel. Required the cost ? Ans. $13. 7^^^. 12. Arthur has G* acres in one lot ai)d 7f in another. Bernard has 5| times as much as Arthur. How many acres has he ? Ans. 83;J^ 13. What will be the cost of 654 feet of boards at $15^ per 1000 feet; 1344 feet of siding at $1.62') per 100 ; and 2216 bricks at $4^ per 1000? "^ Ans. 41.94;,,. 14. At |- of a dollar per bushel, how uuiny bushels of wheat may be bought for si 8.90? Ans. 21 «. 15. John has 6^ times more has John ? )|, James has 2^ times Y uiuius ^o'^ ; how much Ans. $442 5. 16. Richard owns f of a mill worth $48,000. He sells 3 of his part to Anthony and ^ the remainder to Harbeau. How much does ho re- ceive from each, and what part has he remaining ? Ans. Richard still owns $8400. Ans. 1st $25,200, 2iid $8400. 17. A farmer has loaned money to 4 persons : to the 1st | of tlie whole amount ; to the 2nd 1 ; to the 3rd ^ ; and to the 4th $30. How much has he loaned, and how much to each man ? Ans. lie has loaned $350. ■ 1st $100, 2nd $70, 3rd $150, 4lh $30. 18. J of a population can read ; -^ of the remainder can read and write, /. of the remainder can read, write and cipher ; while the rest, 243600, can neither read, write nor cipher. What is the population ? Ans. 19. Bonght 18f yards of silk at $2| a yard, and 27^ pounds of cheese at $.2'o per pound ; how much money did I spend ? Ans. $49^. 20. A person worth $40,000 dies, leaving ^ of his property to his wife, h to his son, and the rest to his daughter. The wife at her death leaves ^ of her legacy to the son, and the rest to her daughter ; but the son adds his fortune to his sister's and gives her ^ of the whole. How much will the sister gain by this ? DcciDiah. 08 DECIMALS AND DECIMAL FRACTIONS. DEFINITIONS. If a unit bo divided into tea equal parts, each of those parts will be one- tenth. If each tenth be divided into ton equal parts, each part will be 10 ^^ 10' ^^ ^""^ hundredth. If each hundredth be divided into ten e(|ual parts, each part will be ^ of — , or one thousandth, and so on. NUMERATION OP DECIMALS. The relation of decimals to whole numbers and to each other maj be learned from the following ' TABLE. es O o a w o m a c 7 n3 •a 4> (4-1 U o n-{ a a w o n. , n-i r/3 o a a • "a H H-l H P a 'o '3 •y ft 6 5 4 3 2 1 i> 0! o 13 CJ « o « -2 C8 "El 'p. o «4 o S >-• o ^4 V (ri ^ 0) a; OJ a; -o -o n-i TJ M wi ^ »H o o o o M ^ ^ rC •*3 -1^ •^ -M t- «o la -i< eL, pL, o « o CO o o a 1) o o o ■73 a o a o a p< CO I "5 fC . a TS ^ uz fc- a "a 'O c a « o a a ^ 5 ft H H a rr 53 S § o a 23456789 a M 3 c3 c3 'p^ P- Ph Ph Oh Ph o ;h y o t-i o a -y O o o o « n3 o o o « CJ 0) o « o rt ci CS 0< Ph "Ph li £.4 o o O is >-' o o o •7S 'S ^ ^ ^ J ,a .j= CO in b- CO C5 Whole Numbers. Decimals. < 1 We proceed in this numeration as in the numeration of units. The only difference is, that counting from the decimal point to the left the figures increase tenfold, while counting from the decimal point to the right the figures decrease tenfold. 4 lii 64 Decimals, The periods to the left of tlie deciruiil point may be called the ascend- ing werieH, and those to the ri^ht the deHcending series. The above table consists of a whole number and decimal. Read the whole number from left to decimal point: Seven millions, six hundred fifty-four thousand, three hundred and twenly-one. Head the decimal number from the decimal point: to the right : Two hundred tliirty-four millions, five hundred sixty -seven thousand, eight hundred and ninety-three billionths. THE METHOD BY ZEROS. % ii i 1 i i i ■f \ l ,;'■■ flilii't i yj I A very plain method, consisting of zeros representing the different periods of decimal numeration, will greatly facilitate the work of writing and reading decimals. CO a o ace a O j3 « t^ CD •s o a '^ '^ us a a 3 O r/j a tH t/j S z a fflH a o H in fl en +- 3 g-p m •5 • 4) a a 2 a • ca O i) _!- U +3 r^ ca -£3 =: if rz: a> o a •- 3 S=^H^ I II I I .34 .736 .000064 The above should be loff written on the blackboard, and students told to write under them as long as they shall not be familiar .vitli <*«. numeration. Now let the pupils understand well, that the zer .osenting th' units is the starting point for both numerations. Having studied the numeration of whole numbers, a pupil will have no trouble to understand the decimals. Suppose you are asked to write 34 hundredths. Write 4 under the zero representing the hundredths, and 3 bclure — 736 thousandths. Write 6 under the z^iro representing the thousandths and the other two figures before. You have seven hundred, thirty-six thousands, 64 mil- Decimals. G5 liorith.s. Write 4 under the zero ropresentln;^' the millionths, and G before, then fill up the Hpace from the deeimnl point to with zeros. NoTA Benk. — Teacliers shouhl noteomoience addition before their pupils understand thoroughly the writing and reading of decimals. Write and read the following decimals: 1. .0 5. .475 9. .506 2. .15 6. .374 10. .7153 3. .18 7. .126 11. .0174 4. .25 8. .2i)l 12. .00<;501 13. .300141 17. .82007451 14. .(J700356 18. .72859005 15. .00034589 19. .121000386 16. .00013604 20. .23(1582174 Teachers would do well to give numerous exercises on the blackboard. Decimals, sinoo they increase from right to left and decrease from left to right, by the scale often, as do simple whole numbers, may be added, subtracted, multiplied and divided, in like manner. ADDITION OF DECIMALS. Add together 5.875, 6.087, 18.810, 25.004. Operation. — 5.875 6.087 18.810 25.004 55.776 55 units, 776 thousandths. We write the numbers so that the figures of the same decimal place shall stand in the same column, and then beginning at the right hand add them as whole numbers, and place the decimal point in the result directly under those above. EXAMPLES FOR PRACTICE. 1. Add together 171.61111; 16.7101, .00007, 71.0006; 1.167897. Ans. 260.489777. 2. Add .16711; 1.766; 76111.1; 167.1 ; .000007 ; 1476.1. Ans. 77756.233117. 3. Read and add .01012305 ; .000027 ; .500006 ; 45.678045. Ans. lilli ! ; imm IP' i ''' ' ' 6^; '1|^-: III 1 III 1; ,i„ ^1' .■ IHi< 66 Addition of Decimals. 4. 207.0084; 7080.00607008; .006; 3456.823401. Ans. 5. .002005505; 600.06; 1000.001; 1045.2765456. Ans. 6. 25000000.000250; 206.00000206; 2387.014567. Ans. 7. .030056; 7051.013005; 7400.0056; 5780.45092513. Ans. 8. .7534; 16.64; 5002.875; 2345.S93456; 3456.03. Ans. 0. 400302005006.04003020001; 400.00011; 378.905703. Ans. 10. .16041; 1900.0909009; 40004.50002; 3765.675349. Ans. 11. 1623598474; 40056.019068; .001409; 27.4560; 2.6543750. Ans. 12. Six-hundred and forty-four ten-thousandths. Seven thousand and eighty-two ten-thousandths. 13. Fifty-seven hundred-thousandths. Seven hundred and eight hundred-thousandths. 14. Nine thousand and forty-eight hundred-thousandths. Seven thousand six hundred and forty-three millionths. Forty thousand and sixty-three millionths. SUBTRACTION OF DECIMALS. Exercise.— From 74.806 take 49.054. Ans. 25.752. Operation — 74.806 Having written the less number under the greater 49.054 so that figures of the same decimal place stand in the same column, we subtract as in common num- 25.752 bers and place the decimal point in the answer, as in the addition of decimals. EXERCISES. From 36.890450 " 123.100453 " 75.040056 «' 36.406579 " 439.909109 " 364.548094 take 25.8045675 " 93.614364 " 29.456014 *' 16.579809 " 238.V65798 " 139.875671 Multiplication of Decimals. 67 MULTIPLICATION OF DECIMALS. Multiply 67.831 by 8.34. Operation — We multiply as in whole numbers, 6t.F31 8.34 271324 203493 542648 565.71054 and point off on the right of the answer as many figures in the mul- tiplicand and multiplier. In this example, the mulplicand has 3 de- cimals and the multiplier 2, making 5 figures to be cut off in the answer. Multiply (( (t <( EXERCISES. 61.045 4.604 18.4056 75.1049 by. 3.971 .375 5.834 1.1045 189.6746 " 23.7890 reater H ud in H num- H er, as 1 Divide IS Operat 5.34 18.156 1 1 ario !.156 ion — 3.4 2136 2136 DIVISION OF DECIMALS. by 5.34. Ans. 3.4 We divide as in whole numbers, and the quotient or answer must have as many decimal figures as the number of deci- mals iu the dividend exceeds that in the divisor. In the above, the divisor has d three, then two decimal s, the divi le in the answer. EXERCISES. Divide 183.375 by 4.89 (( 67.8632 " 32.85 (< 56.7891 '' 8.123 tt 139.1045 « 6.42 (1 42.1902 " 3.015 tt 254.2045 <' 16.275 tt 1841046 " 25.183 ill 68 Decimal Fractions. DECIMAL FRACTIONS. A decimal fraction has unity with one or more zeros to the right of it as denominator. Thus ?_ to a decimal = .5 10 ^ to a decimal = .09 100 A to a decimal = .009 1000 _L_ to a decimal = .0008 10000 To reduce a vulgar fraction to a decimal fraction. Rule. — Divide the numerator by the denominator. Point oflF in the answer as many decimals as there have been ciphers annexed. Reduce — i 10 100 100 .33+ .25 51 10 .2 I 64 8[ 7000 .875 MONEY. Money is a piece of metal, usually gold, silver or copper, stamped by public authority, and used as the medium of commerce. The word money is derived from the Latin monerc — to warn — as it informs the holder oF its value. The latin word^jectntif/, money, is supposed to be derived from jjccms — a " sheep"^ — because in ancient times sheep or stamped skins were used instead of money. In different countries and in different conditions of society various articles served as money. The Indians of America and, in general, uncivilized nations still use articles instead of money, such as bra- celets, beads, cattle, etc., etc. Among modern nations, gold and silver have been adopted for the purposes of mouey, as they possess great value in small compass and can be easily transported from one place to another. Paper money is a legal substitute for metallic currency. Money. MONETARY UNION, 09 We call Monetary Union an international treaty establishing between several countries uniformity of weight, title, dimension and value of currencies and the free circulation of them in the countries belonging to the Union. By this agreement, exchange has been suppressed, and private individuals have realized economies without any prejudice to governments. France, Belgium, Italy and Switzerland belong to the Union since the 1st of August, 1866, and Greece since 1868. Spain, Hungary, Serbia, Roumania, Findland, Philippine Lslandsj Columbia, Venezuela, Peru, Chili and Uruguay having adopted the same standard shall soon , we hope, join the Union. We can now consider as the international monetary standard the silver 5 franc piece, weighing 25 grammes (metric system), ^9„ fine- ness. The gold coins are of same fineness and are 15^ times the worth of silver coins of equal weight. The silver coin of 5 francs corresponds exactly to two 50c pieces of our currency. France and the United States of North America are making laudabK' efforts in the way of bringing all the civilized nations to ioio a general monetary union. DOMINION OF CANADA. The weight and fineness of Canadian silver are regulated by the " Currency Act," chap. 30, Revised Statutes. Under that act the coins are of the fineness fixed by the laws of the United Kingdom and of weights bearing the same proportion to the value which the weights of the silver coins of the United Kingdom bear to their nominal value. According to the above mentioned act, the weights and fineness are as under : — Weight Fineness oz. lOOOths. 25c 0.1875 925 50c .0375 925 By the same Act, authority has been given for the coinage of gold coins for Canada. No coins however have been stau)ped, as the British and American gold have proved sufficient for our needs and are made legal by the Act. Approximative value, in Canadian Currency, of different foreign gold and silver coins. !'v!S i , 70 Money. ,ii; : ili II iii ENGLAND. Gold Coins, Silver Coins. Guinea is equal to Half Guinea " " Sovereign or Pound Sterling. Half Sovereign Gold Coins, Silver Coins. Gold Coins Silver Coins. Crown is equal to Half Crown Shilling Half Shilling FEANCE. A piece of 100 francs is equal to. u 20 " " A piece of 5 francs is equal to. (( 2 " " " 1 franc '< " 50 centimes " « 20 " " GERMANY. A Double Frederick is equal to. A Double Augustus " A piece of 10 Thalers " A Ducat " A piece of 5 Florins " A piece of 2 Thalers is equal to. Crown " 1 Thaler " I Florin " 1 .Mark " $ Cts. 5 05 2 52 4 86 2 43 1 12 56 22 11 19 7 3 1 28 71 85 92 96 96 38 19 09 04 7 97 7 94 7 89 2 27 2 04 1 44 1 11 72 41 24 Gold Coins. Silver Coins. Money. UNITED STATES OF NORTH-AMERICA Double Eagle is equal to Eagle " Half Eagle '* Quarter Eagle " Dollar '* Dollar is equal to HalfDoUar " Quarter Dollar Dime Half Dime 71 20 10 5 2 1 50 50 25 10 05 PROPORTION. Proportion consists in the equality of two Ratios. A Ratio is the result obtained by dividing two numbers. Thus the ratio of 7 to 21 is 3, because 7 is contained 3 times in 21 ; or, if you prefer, 21 is 3 times 7. The same result is obtained if you divide 7 by 21, for we then have :jj or ^, which means that 7 is 4 of 21. A Proportion is either direct or inverse. A direct proportion is when two proportions are so connected tha when one is increased by 3, 4, 5, etc., times, the other is also increasea by 3, 4, 5, etc., times. Ex. — 12 men can make 48 yards of cloth ; how many yards can 30 men make ? It is evident that if 12 men can make 48 yds. of cloth in a certain time, 30 men will do more in the s ime time. Hence the proportion is directly proportional to the number of men. If 6 men can do a piece of work in 5 days, in what time can 9 men do the same work ? Here the time is inversely proportioned to the number of men ; for if 6 men do a piece of work in 5 days, it will take 9 men a shorter period that 5 days to do the work. Up to this present time, arithmeticians have solved problems of propor- tions by the Unity System or by the old Proportion Method, v. g.^ 4 : 8 :: 9 : 18. 4 are to 8 as 9 are to 18. The two terms 4 and 18 are called the Extremes. 8 and 9 are called the Means. 72 Proportion. 1 \ '■ 1 i ! i 1 N III ; l!; ; Now, these four numbers being in proportion, the product of the Extremes=the product of the means. Example. — 4 x 18 = 72 and 8 X 9 = 72 From this, it is evident that if tliree out of the four numbers that form a proportion are given, we can fiiwl the fourth. The Unity System consists, as its name indicates, in finding the price of one object, or the sum a man can gain in one day, etc., etc. Ex. — 12 men can make 48 yards of cloth ; how many yds. can 30 men make ? By the Unity System, wo say : If 12 men can make 48 yds. in a certain time, 1 m;in will make 4 yds. in the same time. and 30 men " 30 x 4 yds. = 120 yds. in the same time. A long experience has convinced me that those methods are too long and (when the problems are complicated) too diflScult and out of the grasp of an ordinary intelligcace. I have thought of substituting, in their stead, another method, wliich I have used with the best result in teaching Proportion. The following chapter will explain by itself. * SIMPLE PEOPORTION. Simple Proportion is an equality between two simple ratios. Example. — If 76 barrels of flour cost $456, what will 12 barrels cost? Ans. $72. barrels $ Explanation. — Say 76 = 456, the price am^ 12= 76^ ^456 X, representing the unknown price. Always put similar objects together ; for instance, barrels with barrels, and dollars with dollars. Kow draw two lines crossing each other at the sign of equality, thus connecting 12 with 456, and 76 with theX. Then draw a line below the problem, and write over it all the figures not connected with the X, 12 and 456, and below the line all the figures con- nected with the X, 76 and more if you had any. 12^^X 3 12 X 456 76 19 Simple Proportion. Now cancel by 4. Then multiply 456 x 3 = 1368, 7a 456 3 which sum we divide by 19, and the result will be 19| 1368 $72, the price of 12 barrels. ' 133 72 38 38 NoTA Bene. — This practical and short way of solving problems in proportions will be of great help to business men in general. EXAMPLES POH THE CLASS ROOM. 1. 7 gallons of oil were paid $5.88, how much would 27 gallons cost ? . Ans. $22.68. 2. If 12 tons of coal can be bought for $96, what would 32 tons cost ? Ans. $256. 3. If 13|^ bushels of wheat make 3 bai rels of flour, how many bush- els of wheat will be required to make 40 barrels of flour ? Ans. 180 bush. 4. If 12 boarders eat in a certain time $25 worth of bread, what would be in the same time the bread expenses of 55 boarders ? Ans. $114.58J. 5. Edward, unable to meet all his liabiliiies, promises 64 cents on the dollar ; how much shall I get for a debt of $2563.50 ? Atis. $1640.64. 6. If I of a bushel of prunes cost 'r of a dollar, what part of a. bushel can be had for 7 of a dollar ? 20 '25 Ans. 7 of a bushel. 7. A certain piece of labor was to have been done by 144 men in 36 days,ibut a number of them having been sent away, the work was per- formed in 48 days ; how many men were discharged ? Ans. 36 men. Explanation. — men days 144 X 36 48>5^1 labor 1 3 6 3 144 X 36 48 4 = 108. 144—103 = 36 We always put the amount representing the work done or the money earned next to the sign of equality. Now,as in the above case,the propor- tion is inverse; the work has to be represented by the unit 1, the con- m w 74 Simple Proportion. struction of the equation as above is the only good one to be used in such a case. 8. If 3 men can earn in one day $24, how much will 11 men earn in the same time ? Ans. $88. 9. If $153.60 were paid for 32 yards of cloth, how much should be paid for 17 yards? Ans. $81.60. 10. In the annual movement of the French population, the statistics give 84 deaths to 100 births ; what would be the proportion of deaths to 350 births ? Ans. 294 deaths. 11. If 25 barrels of flour cost $165, what will 35 barrels cost? Ans. $231. 12. If 7 yards of velvet cost $24.50, how much will 12 yards cost ? Ans. $42. 13. If I of an acre of land cost $320.25, what will f of an acre cost ? Ans. $288.25. 14. If 7 sheep cost $21, what will 9 cost at the same rate ? Ans. $27. 15. If 1 of a yard of velvet cost $2|, what will be the cost of 7 yards? Ans. $30. 16. If 150 cows cost $1800, bow, many cows can be bought for $132 ? Ans. 11 cows. 17. For 30 days' work a man received $45, how much would he have received for 25 days' work. ? Ans. $37.50. 1 8. Maurice, failing, can pay but 70 cents on the dollar. Now he owes Arthur $1690, Bernard $2000, and Charles $1100 ; how much will each receive ? Ans. Arthur, $1183 ; Bernard, $1400 ; Charles, $770. 19. If $3000 in gold be worth $3363.75 in currency, what amount of gold can be bought for $3500 in currency ? Ans. $3121.51.+ 20. If 96 bushels of wheat cost $128, what will 15 bushels cost? Ans. $20, 21. If 17 days' work pay for 2 barrels of sugar, how many barrels will 279 days' work pay ? Ans. 32ii barrels. Simple Proportion. 75 22. If 190 bushels of sour apples make 16 barrels of cider, how many barrels of cider will 36 bushels make ? Ans. 3? barrels. 96 23. If 55 yards of cloth cost $28.42, what will 1 of a yard cost ? 5 7 •' , Ans. $2.80. 24. A mud turtle does not walk more than 300 yards in a day. How long will it take to crawl from Sherbrooke to Montreal, a distance of 103 miles, reckoning 1760 yards to a mile? Ans. 604 days. 25. If 221 cords of wood last as long of 15 L tons of coal, how many cords of wood will last as long as 11^ tons of coal ? 26 Ans. 16 1 cords of wood. 18 2b'. Two pieces o cloth are each 36 and 48 yards long. The longer piece co.'^ts $60 more than the other. What is the price of each piece ? Ans. 1st, $240 ; 2nd, $180. COMPOUND PROPORTION. A Compound Proportion is one which involves two or more simple equations. The mcthou already explained in Simple Proportion will be used for compound proportions with as much facility. Example. — If 12 men in 30 days earn $270, how many dollars will Ans. $486. 18 men earn in 36 days ? Explanation. — Always put the numbers representing the individuals doing the work first, then the days, then the hours, etc., and all that will equal the gain or work done. Now draw two lines crossing each other where the sign equal is, thus connecting 12, 30 with the X, and 18, 36 with 270. men days 12 X 30^ S270 18 X 36>*^X 18—36- 9 -270 ■$486. 12—30 Then draw a horizontal line below the problem, and write over it all the figures not connected with the X, viz., 18, 36, 270 ; and below the line all the figures connected with the X, viz., 12, 30. Now can- cel 270 and 30 by 10, and 3 and 27 by 3, 36 and 12 by 12. Multiply 18 X 3 X 9 = 486, the answer. m M lilil 76 Compound Proportion, ! I i ',■ I ,' 1 1 '■ i > li 'WW' PROBLEMS. 1. 3 men in 14 days working 10 hours a day have made a wall 75 yiirds lon<^ ; how long would it take 7 men, by working 9 hours per day, to make a wall 108 yards long? Ans. 9^ days. 2. 3 men working during 8 hours have gained $12; how much could 5 men gain by working 9 hours ? Ans. $22.50. 3. 7 men in 12 days of 11 hours each have gained $G00.GO ; wiiat sum could 12 men gain by working during 15 days of 10 hours a day ? Ans. $1170. 4. 8 men working 10 hours per day and during 18 days have done a certain work. IIow many men would be required to do the same work in 15 days of 12 hours a day ? Ans. 8 men. 5. A regiment of soldiers, consisting of 939 men can eat 351 bush- els of wheat in 3 weeks ; how many soldiers will it require to eat 1404 bushels in 2 weeks ? Ans. 5G34 men. 6. If 8 men spend $64 in 13 weeks, what will 12 men spend in 52 weeks ? Ans. $384. 7. If 8 horses consume 42 bushels of grain in 24 days, how many bushels will suffice 32 horses in 48 days ? Ans. 33() bush. 8. If 24 men can saw 90 cords of wood in (5 days, when the work. ing days are 9 hours long, how many cords can 8 men saw in 36 days of 1 2 hours each ? Ans. 240 cords. 9. Received $21 for 15 days' work of 7 horses, each drawing at an average power, 1350 lbs. ; how much shall I receive for 12 horses, working during 20 days, each drawing at an average power 1200 lbs ? Ans. $4i.66. 10. If 144 men, in 7 days of 11 hours each, build a wall 200 feet long, 3 feet high and 2 feet thick, in how many days of 7 hours each will 30 men build another wall 350 feet long, 6 feet high, and 3 feet thick ? Ans. 259^ days. 11. If it require 45 tailors to make 300 coats in 36 days how many will be required to make 200 in 27 days ? Ans. 40 men. 12. If 18 men in 24 days, 12 hours a day, can make 2880 yards of a certain work; how many men, in 9 days, by working 10 hours a day, can make 450 yards? Ans. 9 men. 13. If 6 horses eat 70 bushels of oats in 9 days, how many can be fed with 280 bushels in 27 days ? Ans. 8 horses. iit; Compound Proportion. 1 1 14. Iff) men in IG days of 9 hours oacli huild a wall 20 feet loiiif, 6 feet high and 4 feet thiek, in how many days of Hi hours each will 24 men build a wall 200 feet long, IG feet high, and G feet thiek ? Ans. 90 days. 15. If a man travel 117 miles in 15 days, employing only 9 hoursu day, liow far would he go in 20 days, travelling 12 hours a day ? Ans. 208 miles. 16. If 12 men in 15 days can build a wall 30 feet long, 6 feet high and ',i feet thick, when the days are 12 liours long, in what lime will 30 men build a wall 300 feet long, 8 feet hiyh, and G feet thiek, wlicn they work 8 hours a day ? Ans. 240 days. 17. if a regiment of G79 soldiers consume 702 bu 'lels nf wheat in 336 days, how many bushels will an army of 22407 soldiers consume in 112 days? Ans. 7722 bushels. 18. If 2 men can build 12.^ rods of wall in GJ days, how lung will it take 18 men to build 247 j^j rods? Ans. 14 days. 19. If 248 men, in 5^ days of 11 hours each, dig a 'trench of 7 degrees of hardness, 232^ feet long, 35 feet wide, 2^ feet deep, in how many days of 9 hours each will 24 men dig a trench of 4 degrees of hardness, and 337^ feet long, 5| feet wide and 3j4 f^^^t deep ? Ans. 132 days. 20. Knowing that $500 give ^10 interest in 3 months, what princi. pal should 1 place at interest to give me 0200 in 1 year ? Ans. $2500. 21. During how many days of 8 hours each must 49 men work to do as much work as 7 men did in 28 days of 10 hours each ? Ans. 5 days. 22. A piece of cloth 30 yds. long, | of a yard wide, was woven with 26 pounds of thread ; what will be the length of a piece J of a yard wide, and which requires 32 pounds of thread ? Ans. 39^*.j yards. 23. If 5 men, by working 10 hours a day, can mow a field of 30 acres in 10 days, how long will it require 8 men working 9 hours a day to mow 54 acres? Ans. 12^ days. 24. If the carriage of 575 pounds at a distance of 150 miles cost $24.58, what must be paid for the carriage of 765 pounds at a dis- tance of 32 miles ? Ans. $6,97,+ ( ; 78 m Conjoined Proportion. CONJOINED PKOPOIITION. Ooiijoiucd I'roportioii in a kind of C()iu])oun J proportion. It roliitos princii)ally to excliunj^i'H bctwc(!n difl'orcut countries, in rcwpect to spe- cie, wc'i;i;lit8 and lueaHures, but is applicable to common business trans- actions. When it relates to spoeio, it is called Arbitration of Kxchaas^t!. The followint^ problems will explain the above definition. PiioiUiEM. — 12 yards of cloth in Slierbrooke cost as much us 15 yards in iMontreal, and "JU yards in Montreal cost as niucii as 28 in Quebec ; how many yards in kSherbrooke will cost as much as 54 yards in Quebec ? Ans. 30^ yds. Slierbrooke. Operation : — 12 yds. Slierbrooke = 15 yds. Montreal 20 " Montreal -28 " Quebec 54 " Quebec = X " Slierbrooke Write the corresponding terms, as they como, right and left of the sign of equality, taking care that terms of the same name shall always be on opposite sides. Draw a line, and write over ^ * it all the figures not connected with the X, as 12, 20, 54 ; and under it those connected with the X, or that are on the side where the X is. Now cancel by the usual wfiy 15 and 20 218-^7 =30f by 5 ; 12 and 28 by 4 ; 3 and 3 by 3. The result will be 216. Now, as we have an improper fraction in -1'', we divide 216 by 7, and We get 30^, the answer. The X is not always at the same place. All depends on the article or name beginning the proportion. You must always end your pro- blem with the name beginning it. Thus in the above, the problem begins with Sherbrooke, it must also end with Sherbrooke. Example : — If 25 sheep eat as much hay as 19 goats, and 33 goats as much as 10 cows, and 3?! cows as much as 22 horses, how many horses will eat as much as 60 sheep ? 12—20—54 216 15 3 28 7 Conijolned Prui)ortio)i, POSITION OP THE PROni.KM. 79 25 Shcop = 19 Guats 33 (ioats = 10 Cows 38 Cows = 22 Horses X IIorscH = 00 Sheup » a ilO 19—10—22 -60 25 - 33 - B .1 ■i Writo on the line 19, 10, 22, GO ; and under it 25, 33, 38. Cancel 33 and 22 by 11 ; 19 and 38 by 19 ; 3 and (50 by 3 ; 25 and 20 by ;') ; 5 and 10 by 5 ; 2 and 2 by 2. Now we have 2 and 4 remaining, which being multiplied together give 8 for the answer. Remark that the X is on the left hand side this time, because the above equations finishing with the figure representing the horses, the following term must begin with the same merchandise. NoTA Bene. — The sign = in such questions merely means equal in value. PROBLEMS. 1. If 17 cords of wood are worth 116 pounds of sugar, and 87 pounds of sugar 12 barrels of flour, and 19 barrels of flour to 34 days' work, and 92 days' work to 57 baskets of peaches, and 31 baskets of peaches to 24 dollars, and 12 dollars to 2 tons of coal, how many cords of wood may be purchased for 35 tons of coal ? Ans. 259?? cords. 9li 2. 32 cords of wood pay for 100 turkeys, and 155 turkeys for 200 chickens, and 96 chickens for 65 pounds of tea^ and 54 pounds of tea for 85 pounds of coffee, and 36 pounds of coff'ee for 12 barrels of flour, and 16 barrels of flour for 1 cow, and 10 cows for 2 horses, how many cords of wood would buy 6 horses ? Ans. 335^- 3. If 32 yards of lawn pay for 24 neckties, and 30 neckties for 24 handkerchiefs, and 28 handkerchiefs for 35 yards of ribbon, and 40 yards of ribbon for 10 yards of cloth, and 25 yards of cloth for 150 yards of cambric, how many yards of cambric will pay for 15 yards of lawn ? Ans. 16^ yds. cambric. w|i T^ 80 Conjoined Proportion. 4. If 11 barrels of apples pay for 80 bushels of potatoes, and 21 bushels of potatoes for 11 bushels of barley, and 19 bushels of barley for 29 busliels of oats, how many bushels of oats will pay 20 barrels ofappk"-? Ans. 116 + bushels of oats. PERCENTAGE. Percentage and Per Cent, are terms derived from the Latin ^er centum, meaning by the hundred. The term per cent, is usually employed to indicate the price paid for the use of money, buLis also used to express so much the hundred units of anything. Business men regulate their affairs and compute their profits and losses by taking 100 as the basis. Thus, the term 5 per cent, means 85 on SlOO, or 5 gallons on every 100 gallons, or 5 barrels on every 100 barrels, etc., etc. Lending money at 5 per cent, means that at the end of the year you will get $105. Whea we speak of a clerk getting 3 per cent, as a commission on sales, we mean that from every SlOO he makes he has a right to $3, etc. When we say that Sherbrooke has increased 20 per cent, since 1880, we mean that if the population then, had been divided into groups of 100 and the present population into groups of 120, the number of groups would be the same in both cases. Be-.r in mind that the expression " Per Cent. " does not mean $10 or 10 cents of Dominion money, but i2 ; 10 per cent, of $50 is $5 only. 10 per cent, of $95 is $9|only. The rate per cent, is always expressed in decimal fractions. Thus : — 1 per cent, is equal to L or .01 of a unit. ^ 100 2 (( ii i( il 2 100 (( .02 (( a (( a n u << 3 ioo (( .03 <( (( (( 5 a (( (( « 5 100 u .05 (( (( <( 6 (( a n << 6 ioo (( .06 (( a (( 10 i( a « u 10 100 (( .10 a (( (( 125 li (( (( « 125 Too (( 1.25 (1 (( (( 340 u (( (< li 340 100 (( 3.40 n (( u etc., etc., etc. Percentage. 81 All problems in Percentage refer 'x) two or more of the following five terms : — ^ < 1. The Base is the sum on which the Percentage is computed. 2. The Per Cent, is the rate to be levied. 3. The Percentage is the sum levied from the Base. 4. The Amount is the Basj and Percentage added together. 5. The Difierence is the Percentage subtracted from the Amount. The sign % is often used in business, instead of the words " percent.," ijuch as 05%, 06%, etc., etc. Percentage is the soul of all commercial computations, such as : — Interest, Commission, Brokerage, Discount, Insurance, Profit and Loss, Partnership, etc. Percentage being^^then, the foundation of all business calculations, a student should not proceed any further before he is perfectly familiar with the four following Gases. CASE I To find the Percentage of any number or quantity, the rate per cent, being given. Rule. — Multiply the given number or base by the per cent, or rate. What is the -02^% of ;i;975.50. Operation. — Multiply the Bisc by the rate. ^ of a hundred u. equal to .005. Ans. $24.38+ 975.50 .025 487750 195100 ff 5 decimals. $24.38750 EXEllCISES. 1. What is .03% of $640? Ans. $19.20 2. .03^% " 475? 16.625 3, .04% " 750? 30. 4. .04^% " 175.70? 7.9065 6. .05% - 890.65? 44.53^ 6. .05^ " 675.30 ? 6 37.14+ w^ ^ 82 Pen lentage. 7. .06% of 964.40 ? Ans. 57.864H 8. .06i% - 750.80 ? a 48.80+ 9. .0G|% " 660.75? (( 44.60 10. .06^% " 737.90 ? (( 46.709 11. .06^% " 360.25 ? (( 22.51 12. .0620, u 290.65 ? u 19.35 13. .071% " 779.89? '( 57.71 U. .081% " 376.74? u 33.15 CASE II. To find the Rate per cent. Rule. — Annex two ciphor.-> to the Percentage, and divide by the Base. What per cent, of $450 is §81 ? • Ans. 18% Operation.— 450| 8100 .18% Annex two ciphers to the Percentage ^450 and divide by the Base. The result will be 18%, 3600 3600 1. Wiiat per cent, of S340 2. 3. 4. 6. 6. 7. 8. 9. 10. 11. EXERCISES. S340. is 630? Ans. .088% 475. i( 25? u .052% 480. i< 19? u .039% 490. u 28? (( .057% 675. (< 36? ii .053% 739. li 40? (1 .054% 796. a 55? u .069% 2890.25 u 185? u .064% 3750.80 n 275? ll .073+% 4560.90 (I 390? « .085% 8375.75 a 496? i( .059+% A student, at an examination in which 80 questions were asked, answered 60 correctly. What per cent, did ho answer ? Ans. 75%. Bought a horse for $840 and sold it at a gain or percentage of $560. What was the per cent. ? Ans. 66f %. Percentage. 83 THIRD CASE. To fiud the Base when the Percentage and Rate are given. Rule. — Annex two ciphers to the Percentage, and divide by the Eate per cent. 8354 is the .OGZ of what number ? Ans. 85900. Operation. — $35-1 is the per- coutage obtained @ .0G% from the Capital or J]ase which is in this case tlie unknown number. ■0(j|:j54oa |59oo 30 54 54 Annex two ciphers to the percentage and divide by the rate per cent. The result will be the Base. EXERCISES. 1. $26 is the .03% of what number ? Ans. 8866.66. 2. 30 .04% Ans. 750. 3. 36 •05% Ans. 720. 4. 475 .05i% Ans. 8636.36. 5. 380 .06% Ans. 6333.33+, (i. 565.10 .06^% Ans. 8693.84+. 7. 390.75 .06f% Ans. 5788.88. 8. 975.18 .07% Ans. 13931.14+. 9. 1750.30 .08% Ans. 21878.75. A man sold a house, and gained $2850, which was .12.^ per cent, of what he received. How much did ho sell it for ? Ans. 822800. A farmer saved annuall} 8145.50, which was .33J of liis annual income. Required his income ? Ans. 8436.50. FOURTH CASE. To find what numbci' is a certain per cent, more or less than a given number. Rule. — Wlien the given number is viore than the required number, add up the rate per cent, to 1,00, and divide. When the number is less, subtract the rate per cent, from 1.00, and divide. A 84 'i::A Percentage. I have received $720 which was 25% more than what I expectei What real amount was expected ? Opeuation. — You have received more than you expected, tlien add up .25% to 1.00 = 1.25. Now div- ide 720 -f- 1.25 ; but as you have two decimals in the divisor, you must annex two ciphers to the dividend in order to equalize both sides, and then divide as above. The result will be the expected amount. 1.25 720.C0 625 Ans. 657G. 576 950 875 750 750 Isold a house for S3750, and lost by the sale 15%. What was the real price of it ? Operation. — In the above pro- blem there is a loss of 15%^ there- fore subtract .15 from 1.00 and the result will be .85. But as $3750 is not the real sum, being 15% less,3750 is then equal to 85% only. Annex two ciphers to $3750 and diTiae as in operation. The result will be $4411.76y\, the real price of the house. Ans. $4411.76+. .85 3750.00 4411.76+ 340 350 340 by 5 40 = 8 85 = 17 100 85 150 85 650 595 550 510 40 PROBLEMS. 1. I sold merchandise for $1680, and lost by the sale 20% ; what did the merchandise cost me? Ans. $2100. 2. Aristotle wishing to sell a horse asked 25% more than cost, but, unable to get his price, he sold it for 15% less than his asking price, and gained only $7.50. How much did tLf^ horse cost him, and what was his asking price y Cost $l*.iO A oVi^^^^ price $150. Percentage. 85 3. A gentleman sold two horses at $420 each ; tor one he received 25% more, and for the other 25% less. What is his loss ? Ans. Net loss S56. 4. Dick received $75 for some fruit, gaining 15% ; what price did li^' ptiy ? Ans. $05.21. 5. Francis sold three cows for $138, or $46 each, and gained 15% ; what was the price of the cows ? Ans. $120, or $40 each. 'j. Bought rice which Isold $375, thereby losing 25%; what was the price of the rice ? Ans. $500. 7. The population of Sherhrooke is 9,840 inhabitants, if it has augmented by 20% per cent, in 10 years, what was the population 10 years ago? Ans. 8,200 m)u1s. 8. After paying 42^% of my debt, I find that $2,050 would settle the ace(junt. How much did I owe ? Ans. $4,008.69+. 9. Freddie, after having lost 15% in an unfortunate transaction, has still $8,500 ; bow much had he before ? Ans. $10,000. 10. A cargo of wheat having been damaged was sold for $3,750, which was at a loss of ,32%, what was the price of the cargo ? Ans. $5,514.70+. 11. Thomas spends 45% of bis salary, and still saves $858; what is hi.s salary? Ans. $1,500. REVIEW ON THE FOUR CASES. 1. If I purchase 729 gallons of molasses and lose 11% by leakage, how nianv gallons remain ? Ans, 048 ^-^^ gallons, 2. The average number of deaths in one year in the city <»f Montreal is 5250 ; if it is .01^% of the population, what is the popula- tion ? Ans. 350.00(^ ^ouls. 3. Joseph, having received a iegacy $8,750, deposited | in La Baiif|ue Nationale, and gave 10% o^ the remainder to the city hospital ; how much remained ? Ans. $3150_ 4. An army composed of 100,000 men was twice decimated, how many soldiers remained after the second battle ? Ans. 81,000 soldiers. 5. The number of students in a certain college has increased in one year from 125 to 240; what is the increase per cent. ? Ans. 92%. TT" 86 Review on Percentage. 6. I had $15,000 in a bank; I drew out at first 22%, then 34% of the remainder, and, finally, finding that I had drawn too much, I returned 1?% of what I had drawn to the bank ; how much have I now in bank ? Any. S8595.3<). 7. A merchant imports 2740 boxes of oranges, and finds, on receiv- ing them, that 20% of the whole amount are decayed, to how niaiiy boxes was liis loss equivalent ? Ans. 548 boxes. 8. A gentleman purchases a farm for $7490, agreeing to pay 10 por cent, down, 17 per cent, at the end of the second year, and 40% at the end of the third year ; what is the amount of each payment ? Ans. 1st } car. S1273.30 ; 2nd }car. §2022.30; 3rd year, S3445.40, and $749 down. 9. What is the difference between .04i% of $740, and .02^% of $1680 ? " Ans. $8.70, 10. What is the .04*% of 587 yards ? Ans. 26.415 yards. 11. I lost .10% of $975, how much have I remaining? Ans. $877.50. 12. Sent to Liverpool 5000 bushels of wheat, which cost me $1.25 per bushel, but 25% of the wheat Avas thrown overboard in a storm. and the remainder was sold at $2 per bushel, whiit was gained on the wheat? Ans. $1250. 13. T. Page received a legacy of $8000 ; he gave 19% of it to \\\> wife, 37% of the remainder to his sous, and $2000 to his daughters, what sum had he remaining ? An;*. $2082.40. 14. If a muii travels 48 miles one day, and this was 20% more tliau he travell'jd the previous day, how far did he travel the previous day ? Ans. 40 miie?. 15. A mercliant failing was able to pay his creditors but 55/'. He owes Robert. $3500 ; Bayle, 81200; Cayley, 81134 ; Day, $1600; what will each receive ? Ans. E. $1925; B. $660 ; C. $623.70; D. $880. 16. A merchant sells 40% of his stock of goods for $3500, what is the value of his entire stock ? Ans. $8750. 17. 15% is what per cent, of 60 per cent. ? Ans. 25%. 18. Anthcny, at his death, leaves an estate worth $15,000 ; 10% of whicli lie received from his fatlier, 20% from speculation, 30% from augmentation of property, 25% from the estate of an old aunt, and Review on Percentage. 87 the remainder from liis wife, how much did he receive from each source ? Aqs. Father, S1500; Speculation, $3000 ; Augmentation, 84500 ; Aunt, $3750 ; Wife, S2250. 19. A person whose annual income is $450 pays $125 for board, $140 for clothing, $25 for books, and $30 for sundries, what pi r cent, of his income is each item, and what per cent, remains ? Ans. B(iard .277% ; Clothing, .31-i% ; Books, .05f% ; Sundries, .06p ; Remainder, .28^/. 20. A. mtrcliant failing owes $3500 ; liis property is valued at $2100 ; what per cent, of his indebtedness can lie pay ? Ans. 60%. 21. A man having $10,000 lost 15% of it in speculation ; what mm had lie remaining ? Ans. $8500. 22. The sales of a hardware firm of Sherbrooke amount to $90,000 a year ; | of the sales were made at a profit of .25% ; "^ at a profit of .35% ; and the remainder iit a profit of .20% ; what was the cost of the goods? Ans. $71,300. 23. A merchant expended the same sum in buying wine, gin and coffee. In selling theui, he gained .08% on the wine, .05% on the gin, and lost .14% on the cottee ; having received from his sales $2522, how much did he pay for each article ? Ans. $843.46+. 24. Which is the better investment, a house that will rent ior $1200 a year, which sum is only .08% of its value, or the same $1200 per- centage from a real estate worth $20,000? How much better in one year? Ans. House Kent, .08%; Real Estate, .06% ; House renting is prefernble by .02%. 25. Kdmund and Edward have, respectively, .06% and .04% more than Maurice, and the three have togLilier §22,320 ; how much has Maurice ? Au^. $7200. 26. An estate was divided among fonr children, the 1st receiving $7580; the 2nd, $8650; the 3rd $1290 Ir^s than the 1st; and the 4th $283 more than the 2nd, What was the value of the estate, and what was the rate of percentage received by each child of the whole amount? Ans. 1st, 24+%; 2nd, 27+%; 3rd, 190%; 4th, 28+%. Value of the Estate $31,453. 27. Austin has an income of $1100 a year; he pays 10% of it for; board; ^v for washing; 2 ' for incidentals; 15% for clothing; 9% for otlier expenses. What does each item cost, and how much has he left? Ans. Board, $110 ; Washintr, $5.50 ; Incidentals, $22; Clothing, $165; Expenses, $99. Remaining $698.50. 88 Commission, Brokerage and Stocks. COMMISSION, BllOKEllAGE AND STOCKS'. Commission is a certain porccntagc paid to af^onts. An Afiont is an authorized person to do some act or series of aetn in the name and place of another, called a ])riucipal. Agency is based on this principle of law, tliat whatever a man can do in his own right he may appoint another to do it for him. Agents are known under other names according to the business tliey do, such as : Brokers, Factors, Commission Merchants, Collectors, etc., etc. A liroker is a person who simply negotiates sales witliout tlic property being put into his hands. The percentage charged by a broker is called brokerage, and is com- puted at a certain rate per cent, on the amount of money received or expended. A Factor is an agent for the sale of property, and his commission is always charged on the amount of money received from sales. A Commission ^lerehant is an agent receiving a certain quantity of goods from a person named a Consignor, to be sold at that Consignor's risks and perils. A Collector is a person who settles accounts between parties. He may be an officer of the government, as a Collector of Customs, Collector of Inland Ilevenue, or an officer of a municipality or corpo- ration, as a Collector of taxes. Stock is a term to denote the capitalof financial or industrial institu- tions, such p- Banks, Eailway Companies, Gas and Electric Companies, Navigation Companies, Sub-Marine Cable Companies, Manufactures of all sorts, etc., etc., etc. Stock is divided into parts called shares. A share is generally of SIOOO, $500, $100, and $50 each. Individuals holding these shares are called Shareholders or Stock- holders. C ;rtificates of Stock are issued to each stockliolder, stating the num- ber of shares to which he is entitled. Those shares, once bought, can be sold to other persons, called sub- sequent holder.'', but cannot be returned to the company. The Association of shareholders is called a Company or Corporation. These companies or corporations arc formed under the authority of an act of Parliament called a Charter, or under a certain act called : •' The Joint Stock Companies Act." Commission, Brokerage and Stocks. 89 acts A Charter is a parchment specifyiiif^ the amount of subscribed capi- tal, powers, riglits and privileges of the incorporate company. A company formed under " Thd Joint Stock Companic^s Act" is strictly bound, in the Province of Quebec, to deliver to the Prothono- tary of the district and to the Registrar of the county in which the company desires to carry on business, a declaration in writing, ^^tat- ing the object of the company, the names of its members and the time from which it dates. The subscribed capital is paid up at certain fixed times and by por- tions called Instalments. Stocks are " At Par " when they sell for their original value. '■'Below Par," when sold at a discount or below the original value. " Above Par," when sold at a premium or above the original value- The Gross Earnings consist of the entire gain. The Net Earnings arc what remains of the gain after the expenses are deducted. A Dividend is the net gain divided among the shareholders. PROBLEMS. 1. A broker purchased for me 50 shares of railroad stock of the par value of SIOO. His charge was h%, what is his brokerage ? Au'^. $25.00. 2. What is the commission on the sale of a quantity of dry goods worth 8675G.50 @ .03% commission ? Ans. 8202.G9- 8. An agent collects debts to the amount of S878.80 ; what is his commission at 2h% ? Ans. $21.97. 4. A broker bought rice forme to the amount of $7193.16, what have I to pay him for his services or brokerage at ii^X ? Ans. $224.78+ 5. Richard bought for me in Quebec 6000 bushels of barley at $1.37^ each and shipped the same to my agent in Montreal, who sold it $1.62^ per bushel. How much have I made after paying S543 for expenses and a commission of 2-^% ? Ans. $723. 6. A factor received $1600 for selling a building lot on Wellington street, Sherbrooke. The lot being sold for $25,600, what rate of com- mission did he charge me ? Ans. ^\. 7. My collector charges me $25 for collecting $800. What is his rate of commission? Ans. .0'S\ 00 Commission, Jirokerafje and Stocks. 8. !My agent in Ottawa informs me that he lias dispoHod of 500 barrol.s of flour at 8'-^/ V Ana. $151.95. 9. A country merchant buys for uw. an invoice of goods for $2550. If his eouimission is 2/', how much shall I remit to pay for the goods and commission together? Ans. $2601. 10. A merchant in Three-Rivers sends $1500 to a commi.ssion merchant in 3Iontreal,witli instruetion to deduct his commission at 2^%> and buy goods with the balance. What amount shall remain for buy ing goods, and what will be his eonimissiou? Operation. — $1500 -^ 1.025 =$14(»l>.414, amount for buying goods i 1500—1463.414 = $36.58i; commission. Since the agent is entitled to 2^% of the amount he lays out, it is evi- dent ho requires $1.02^ to purchase goods to the amount of $1. Hence, he can expend for goods as many dollars as $1.02 J is contained times in $1500, or $14(53.414 ; which l)eing subtracted from $1500, tho amount sent him, leaves as his commi.'^sion $30,586. 11. A wool merchant of Sherbrookc remits to his agent in Chicoutimi $4*740 for the purchase of wool, after deducting his commission at 2% ; how much will be spent for the wool and what will be his commission ? Ans. $4647.06 for wool. $92.94 + commission. 12. A fiictor sold a farmer on 4% commission and remitted $10095.- 3G as the net proceeds; for what price did he sell the property, and what was his commission ? Ans. Full price $10516. Commission $420.64. OPEitATioN.— 10095.36-^96 = $10516, full price of farm; 10516 —10095.36 = $420.64 commission. He kept .04% commission, and remitted $10095.36, which is equal to 96/0 of tlie whole amount. Divide 10095. 36 -^-. 96 =10516 full price (4th case of percentage). From $10516—10095.36 = $420.64. 13. Miller and Griffith have sold for me, at auction, a lot of goods to the amount of $14500, Their charges are : commission on sales .02% ; guarantee .02 on sales ; advertising $22.50 ; cartage, labor and storage $32.50 . How much is due me ? Ans. $13865. Commusion, Brokerage and Stocks. 91 14. Paid a broker $24 for investing woncy in the Jacf|ues-Cartior liank stock lor mo selling at fiur, at fi commission of ^/i. JIow nuicli did lie invest? Ans. «Hi;O0. 15. If I pay a commission mercliant 8125 for commission on a sale of 82500, what rate does he receive ? Aus. .05/^. Hi. An agent from Coaticook receives 84,920 to expend in buying cows at 832 a head ; after deducting iiis commission at 2.^%, how many cows did lie buy V Ans. 150 cows. 17. A Kichmond agent has 82,000 to invest in bank stock ; after deductinir his commission of 1A%, what sum will be invested and what will be his commission ? Ans. Sum invested 8l,!>70.44:{. Co)iimission 82'.'. 557. 18. T hav(! sent to my agent in Quebec 810,000 to be expended in flour ; after deducting his commission of 3J;/, what will be his com- mission and the value of the flour bought ? Ans. Commission 8314.76. Value of flour 80,085. 23-k 19. If I remit to my agent 813,000, instructing him to purchase apples at 83.00 per barrel, deducting his commission at 3^%, how many barrels of apples will he buy ? 20. I collected 65/^ of a note of 837.50, and cliarged 5% commis- sion ; what is my commission and the sum paid over? Ans. Commission 82.84. Paid over 854.03. 21. A broker receives 8110(iO with instruction to invest it in Bank Stock, deducting his brokerage at ^% ; what sum did he invest ? Ans. 810,904.58+ 22. Sent my agent in St. ITyacinthe 87G5 to purchase a quantity of maple sugar, his commission is 2/', which he is to deduct on the money sent. What is his commission and how much does he spend for sugar ? Ans. 8750 for sugar, 815 Commission. 23. A Factor sold goods for 80,202, and received 8105, which in- cluded a charge for freight of 88.45. What rate of commi.-sion did he receive ? Ans. .0248. 24. A Sherbrookc merchant wishes to draw on New York for an amount which will leave him the sum of 80141.39, after paying a of 4% for negotiatino; and exchansie. For how much must premu bed raw Ans. 80387.05. IMAGE EVALUATION TEST TARGET (MT-3) 1.0 I.I 141 IIIIM ||2.5 us iU: m 2.2 m 11: m ZO 1.8 1.25 1.4 1.6 •a 6" ► Hiotographic Sciences Corporation 23 WEST MAIN STREET WEBSTER, N.Y. 14580 (716) 872-4503 ^e^^ ^^ o % .V 6^ ^ ''%" f^ s.° MP s \ ifii] 1 1 1 ■ Ft ' 1 i''j ^ 1 ll k I I 92 Commission, Brokerage and Stocks. 25. Sold goods at 2'% commission, which I invested in sugars, and sold them at a profit of 15%, realizing a gain of $240. How much commission did I receive, and how much did the goods sell for ? Ans. 831.31 Commission, $208.69 goods sold. 26. My agent in Minneapolis informs me of the purchase of 4,000 bushels of Indian corn at 80 cents per bushel, and desires me to remit a cheque on Chicago which he can sell to a broker at f % premium ; what should the amount of the cheque be, his commission being 3% ? Ans. $3271.464. 27. What must be paid for 75 shares of C. P. Railway of $100 each at 25% premium ? 7500 x 1.25 = $9375. 28. I own $10,400 stock of the Bank of Montreal, and sell out @ 75% premium. What do I receive ? Ans. $28700. 29. What is the value of 20 shares in the Eastern Townships Bank at 12^% discount, the original value being $100 each? Ans. $1750. 20 X 100=2000 original value. 1.00— 12^ = .875. 2000 X .87') = $1750 present cost. 30. Bought 87 shares in a certain corporation at 12% below par, and sold the same at 19|% above par ; what sum did I gain, the origi- nal shares being $175 each, Ans. $4795.872- 31. I charge a broker to invest $17,700 in " La Banque Nationale " when the market value is 12% bdow par, giving him ^ per cent, for his services. How many shares can he buy, the original value of the shares being $50. Ans. 4U0 shares. Operation. — 1.00— .12 = .88 market value. .88 +.005 = .885 present cost. 17700.000 -h .885 = 20000, or 400 shares of $50 each. Since the stock is 12% below par, the market value will be 88% only ; adding to the market value the rate of brokerage, we obtain .885 present cost. Hence for $17,700, the broker can purchase for $20000 of stock, original value, or 400 shares. 32. If I remit to my agent $17500 with instruction to deduct his brokerage at 1J%, and invest the remainder in the Great North- Western Telegraph Company, then selling at 7% premium, what amount of stock do I receive ? Ans. $16153.22. Commission, Broktmge and Stocks. 33. What amount can we obtain by investing $10,260 in La Banque du Peuple 6% bonds, purchased at .95% ? $10260 H- 95 = 010800 stock bought. 10800 X .06 = $648 annual income. 34. The C. P. Kailway declares a dividend of 15%; what shall I receive for 65 shares of $200 each. Ans. $1950. 35. In buying stock in the Asbestos Mines of Thetford for the value of $10425 at $500 per share, each producing $36 dividend, Arthur secured a revenue of $540 . Required tlie market value of the stock per share, and at what rate his money was invested ? Ans. $695 a share. M+"/^. 36 . Rex owns 150 shares in " La Canadienne Insurance Company ; " I order my agent to buy them when they will rate at 5^% premium ; how much will the 1 50 shares cost me, knowing that the agent will, charge me f% brokerage . Ans. $15956.25 i * k STORAGE. Storage is the placing of goods in some depository or warehouse until required. Storage means also a certain price paid per barrel, bale, box, etc.. according to regulations made by the Chambers of Commerce, or any association of warehousemen. A " Bonded Warehouse" is a place where Customs OflBcers store imported goods on which duties have to be paid. Any other place for storage is called freight-house, storehouse, or warehouse. A warehouse receiving or shipping grain is called an "Elevator." All goods stored are subject to one month's storage. In some places, if there remain any part of a month, a full month is charged. In others, after the first month, if taken out within 15 days, a half month is charged. The owners of the goods pay for putting them in store, stowing away, and the expenses of delivery. When goods are received and delivered at the pleasure of the Con- signor, the price to be paid for storage is usually determined by an average. How to compute storage. ^.-k L.i'dL ?"ir ' Mi I; H Storage. t! 1 '•' 111 H ■i 1 i' 1' : What is the cost of storing apples @ 3 cents per barrel, received and delivered at the following dates : Received January 8th, 300 barrels ; February 5th, 400 barrels ; March 18th, 900 barrels. Delivered Feb- ruary 25th, 500 barrels ; March 28th, 600 barrels ; April 20th, 300 barrels ; May 29th, 150 barrels. What is the storage on this account June 5th, and how many barrels are on hand ? Opeuation. Dr. Received. Date $ Total. Days. AiDounts. January 8th, 300 February 5th, 400 March 18th, 900 June 5th, 30 Ox 700x 1600x 27 i3 77 = 8100 = 30100 =123200 161400 days N. B. — We reckon 30 days to a month. Delivered Cr. Date $ Ti.tal. Days. 33 22 39 6 Amounts. February March April May -Tiinp 25th, 500 28th, 600 20th, 300 29th, 150 5tli, 500x llOOx 1400x 1550x = 1G500 = 24200 _ 54600 = 9300 104600 days June 5tli 50 barrels Dr = 161400 days Cr = 104600 " 1894 03 on hand. 30 56800 " $56.82 Ans 1893^ months. Explanation. — If none had been delivered you would have been charged for storage 161400 days, and have in warehouse 1600 barrels ; but upon the number delivered you are entitled to a credit of the storage of one barrel for 104600, days which being subtracted from the debit side leaves a balance of storage against you of 56800 days, or 1894 months. 1894x.03=$56.82, amount of storage to be paid. There remain 50 barrels in the warehouse. storage. 2ND MUTHOD. Received January 8th, 6300x27 = " February 5th, 400 8100 Delivered February 25th 700x20 = 500 14000 Received March 18th, 200x23 = 900 = 4600 Delivered " 28th, 1100x10= 600 = 11000 " April 20th, " May 29th, 500x22 300 200x39 150 = 11000 = 7800 50x 6= 300 On hand June 5th, 56800 da 30 56800 1894 0^ 1893J ^56.82 Ans 95 Explanation, — From January 8th to February 5th, 300 barrels were kept ; from February 5th to February 25th, 700 bbls. were stored ; from February 25th to March 18th, 200 barrels were warehoused ; from March 18th to March 28th, 1100 bbls. were kept ; from March 28th to April 20th, 500 bbls. were stored ; from April 20th to May 29th, 200 bbls. were kept ; from May 29th to June 5th, 50 remained in the warehouse. 1. What will be paid for storage if flour at 5 cents per barrel,per month, is received and delivered as follows : — Received July 1st, 400 bbls. " " 15th, 350 " •* " 2fith, 4fi0 " '»! I w 96 Storage. Delivred July 12th, 200 bbls. " '* 20th, 400 " August 1st, 200 ** « « 8th, 400 " Ans. 824.10. 2. What will be paid for the storage of grai"> at 6 cents per bushel, received and delivered as follows : — Eeceived, May 1st, 1000 bbls. " " 26th, 2000 " Delivered" 16tli, 500 " " June 1st, 1000 *' " " 12th, 1100 " " July 2nd, 400 " Ans. 8114. INTEREST. When any one borrows money from a pevson, he is not only bound to return it, but he should also indemnify that person for all gains that he would have realized by keeping his money. This is called Interest. Interest is then a compensation for the use of money or value. The legal Rate of interest in Canada is 6%. The Principal is the sum lent. The Rate per cent, is the interest paid for the loan of money. The Amount is principal and interest added together. Simple Interest is the sum paid for the use of the principal only. Compound Interest is the interest on the interest and principal. My sole aim in publishing this arithmetic was to afford young men short, sure and easy methods to do all business calculations. Interest in the various forms under which it accrues, has so large a place in everv day business transactions, that rapid and accurate methods of computing it are among the most indispensable items of business knowledge. Interest. 97 1st CASE. To find the interest for one, two, three, four, five, etc., years. Rule — For one year, multiply the Principal by the Hate. " two years, double the Kate, and multiply by Principal. '* three *' multiply Rate by 3 and then by Principal. « four " " "4 " " *' five " " ** 5 " *' etc., etc., etc., etc. and the result will be the interest. Ex. 1. What is the interest on 8790 for 1 year ® .03Vo ^ Ans. $23.70. Explanation. — Multiply the Principal $790 by .03%, and point $790 x .03 = $23.70 interest, off two decimals in the answer. Ex. 2. What is the interest of $150 for 2 years at .OSVq ? .05x2 =.10 450 X .10 ^ $45 interest. Explanation. — Double the rate .05x2 - .10, rate in two year.*, multiply Principal $450 by .lOVo- ^^^ point off two decimals. Ex. 3. What is tho interest of $565 for 3 years at .055Vo ? Explanation. — Multiply the .055 x 3 - .1(35 rate by 3, and then the Princi[>al $565 x .165 = $93 225. by the rate for 3 years or .165Vo. Point off' three decimal figures. Ex. 4. What is the interest on $875.75 for 4 years, at .06"/„ ? Ans. $210.18. .06 X 4 = .24 $875.75x24 = $210.18. Explanation. — Multiply the rale by 4, and then by Principal. Point off 4 decimal figures. 1. What is the interest of 2. .02iVo ''' Ans. $23,955. 8693.80 " 3 years, fS) .03Vo? Ans. $62.44. $490.85 '« 4 years, tS) M°/^? Ans. $78,536. $219.25 " 6 years, /© .04iVo ? Ans. $49.33. $495.50 '* C years, 'S) .OSVo? Ans. $148.65. $189.99 " 3 years, ^ .OS^Vo ? Ans. $31,348. $256.50 "4 years, m .06%? Ans. $61.56. $335.60 " 5 years, ^ .06^% ? Ans. $109.07. $391.50 " 6 years, ^ .07% ? Ans. $164.43. $589.4.^i " 7 years, fS) .07^% ? Ans. §309.463. $555.60 " 3 years, fS> .08% ? Ans. $133,344. $990.30 " 4 years, ® .08^% ? Ans. $336.70+. $898.80^ " 5 years, ^ .09% ? Ans. $404.46. $775.75 " 5 years, '©.09^%? Ans. $368.48+. $639,101 '< 2 years, fd) .10% ? Ans. $127.82. $569.18 "3 years, 'S) .01^%? Ans. $::5.61+. $698.25 " 4 years, fS> 02% ? An.^ $65.86. u I 2DdCASE. To find the interest of any sum for years and months. EuLE. — Reduce the years and months to months, and place 12 under as a fraction of a year. Interest. 99 Ex. — What ia the interest of S210.50 for 2 years, months, at .04%? Ans. $21.05. 2 X 12 = 24 months. 24 + 6 = 30 mouths iii all. 30 , — X 04. 210.50x10 = 21.05. 12 » Explanation. — In two years you have 24 months, 24 and 6 = 30 months. •'"? x .04 = h~P, Avhich being reduced by 2 and by 6 will give li' 12 12 1 • Multiply the Principal by .10%, and point off 4 doeiujals. Or again, 2 years and 6 months are the same as 2J yearn. Multiply 2.5 by .04 you will obtain .100+. Annex one cipher to your principal, as you have already two, and point off 4 decimals. Note. — You could proceed the same way for 2 years and 3 months, 2 years and 8 mouths, 2 years and 9 months, as they make 2J years, 2§ years, 2f years. 1. What is the interest of ^49.50 for 2 years, 7 mos., ^ .03% ? Ans. $3.83. $26.90 « 3 years, 4 mos., ^ .03^% ? Ans. $3.13. $85.75 " 3 years, 10 mos., (a) .04% ? Ans. $13.14. $96.10 " 3 years, 8 mos., ^ .04^% ? Ans. $15.85. $105.15 " 4 years, 5 mos., ^ .05% ? Ans. $23.22. $135.10 " 4 years, 9 mos., fS) .05^% ? Ans. $35.29. $150.25 *' 5 years, 10 mos., ^ .06% ? Ans. $54.68. $165.18^" 5 years, 6 mos., ^ .06 J% ? Ans. $59.05. $192.10 " 2 years, 4 mos., ^ .07% ? Ans. $31.37. $209.09 J " 2 years, 3 mos., ^ .07p ? Ans. 835.28. $236.76 " 2 years, 11 mos., f^ 08% ? Ans. $55.24. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. (( t( i< (I (I (( « it If it fi f r 100 Interest. K M i t !'■ li « II II i< 12. What is the interest of 8249. lor (I years 8 luos., ffi) .00/, ? AtiH. 89!>.()(). 0250.04 for 3 yrs., 9 ujos., C'!) .06^;; ? An.^. 3(J0. $275.00 " 4 yrs., 7 mos., @ .00^%? Aus. 88(».t:}5. 8290.15 " 4 jra., 2 luos.. @ .OC.i/: ? Ails. 874.72. 8300.104 •' 5yrs., 3 mos., @ .(k;,/ ? Alls. ^94.53. 8319.10 " 4yrs., 5 mos., (ai .05%? Ans. S70.4G. 8326.60 " 7yr.s., 7 mos., (T,- .07/.? An^. 8173.37. 8394 " 4yr.s., 4iun.v,^ .04%? Ans. 868.29. 8450. «' 5yrs., 5 djos., @ .05^%? Ans. 8134.06. 8312.50 " 6yrs., 4 mos., @ .06%? Ads. 8118.75. 3rd CASE. 18. 14. 15. 16. 17. 18. 19. 20. 21. (I u (I « (( To find tlie interest of any sum for years, months and days. Note. — In all our problems on interest, we reckon 30 days to a month and 360 days to a year, except otherwise mentioned. The time in expressing the rate is arbitrary, and as neither 360, 365, nor 366 is the exact number of days in all years, either civil or astro- nomical, does not the increased facility in computation and the perfect accuracy in the result warrant the adoption of 30 days to a month, and 360 days to the year ? Rule. — Reduci. the years, months and days into days, and put 360 under the number of days as a fraction of a year. Ex. — What is the interest on 8190 for 1 yr., 3 mos. and 10 days @.04%? . Ans. 89.71^. One year = 360 460 .46 3 months= 90 x .04= 10 days -^ 10 360 9 460 days 360 90 ;ii Interest, lol 8100. X .4t; .48 9 1140 760 9 87.40 S0.711 interos Reduco to clays, multiply by the rato, and Ciuu'i'l, you obtain .•♦5'',tliL'n uuiltiply 8190 by .4(! !Ui. l.'i Point off two decimals more without any reganl to the already f:;iven duoi. mils in your principal, and divide by 2. The result will be the interest for one luunt'i. Example. — What 'is the interest on 82(jO, for 8 years, 10 months, and 15 days at .06% ? Ans. 8138.45. Explanation. — In 8 years we have 96 munths plus 10 equal 106 months. i^ of 15 days is 5. lOG. 5 months. Now point off two deci- mals in 2.60 and divide by 2 2 | 2.60. Wo obtain 81.30 per month. If wc have 106.5 months, 1.3 the answer will be 106.5 x 1.30 = 8138.45. Note. — As .06 is the rate mostly employed, we can say that thi.s extraordinary short method will be of great use to Students and busi- ness men. EXAMPLES FOR PRACTICE. The rate being .06%, what will be the interest on : — 1. 8300 for 9 years, 11 months, 27 days ? 2. 8250 for 8 years, 10 months, 24 days ? 3. 8460 for 12 years, 2 months, 15 days ? 4. 8280 for 11 years, 6 months, 6 days ? 5. 8390 for 10 years, 8 months, 21 days? Ans. 8179.85. Ans. $133.50. Ans. 8336.95. Ans. 8193.48. Ans. 8250.965. ^ i ! r 1 ' 1 1 : l^*«f! ' i ■i Mi m ■« .■ 'i :!t iiill" 104 Interest. Ans. 8460.80. Ads. S39.21. Ans. $145.08. Ans. S364.21. Ans. $190.57. Ans. $1149.60. Ans. $297.37. C. $600 for 12 years, 9 months, 18 days ? 7. $120.30 for 5 years, 5 montb'., 6 flays ? 8. $370.60 for 6 years, 6 months, 9 day.s? 9. $621 for 9 years, 9 months, 9 days? 10. $292.30 for 10 years, 10 months, 12 days? 11. $1600 for 11 years, 11 months, 21 days ? 12. $908 for 5 years, 5 months, 15 days? Notes being generally given for one, two, three and four months, the following business method of computing interest at .06% will prove- very useful. Although notes bear generally 3 days of grace, it must be borne in mind that interest is not always charged ibr that extra length of time. Example.— What is the interest on $330 for 60 days at .06%? Point ofiF two decimals and your answer will be $3.30. Note. — When the time is more or less than 60 days, get, first, the interest for 60 days, and if the note is for 15 days take | of your answer. for 30 days take ^ of your Answer, for 90 " ^ of your Ans. and ndd it up. for 120 " double of your Answer. etc. etc. etc. EXERCISES. 1. What is the interest on $416 for 15 days at .06% ? Ans. $1 .04. Explanation.— The interest for 60 days is $4. 16 ; ^ of $4.16 = $1.04, the required answer. 2 . Wh.it is the intei-est on $360 for 15 days at . 06% ? Ans. 90 cents. " $640 for 15 days at .06%? Ans $160. " $824 for 15 days at .06%? Ans. $2.06. " $424 for 15 days at .06%? Ans. $1.06. " $1200 for 15 days at .06%? Ans. $3. " $414 for SOdaydat .06%? Ans. $2.07. 3. 4. 5. 6. 7. (( « it (( h ' I El';' Interest. 105 KxPLANATiON. — The interest for 60 days is 4.14; 4.14 x \ = 2.07, the required Answer. 8. AVhat is the interest on 3412 for 30 days at .06%? Ans. S2.06. 9. 10. (( u " 30 days at .00%? Ans. S3. 10. « " $120 " for 90 daysatOG%? Ans. a 1.80. Explanation ihe interest for 60 day.s is $1.20; 1.20 x ^ = .60. .$1.20+ 60 = $1.80, the required answer. 11. What is the interest on $620 for 90 days at .06%? Ans. $9.30. 12. 13. (( u (( $926 " 90 days at .03%? Ans. $13 89. " " $960 •• 120 days at .06%? Ans. $19.20. Explanation. — The interest for 60 days is $9.60 ; double that answer for 120 days. 14. What is the interest on $625 for 120 days at .06%? Ans. $12.50. 15. (t " " $493.10 for 120 days at .06%? Ans. $9.86. ■M EXACT METHOD OP COMPUTING INTEREST. In the preceding problems of interest we have taken 30 days to the month, and 360 days instead of 365 days to the year. However, certain banks in using their tables take the Calendar months with 365 days for the year, and 29 days for February in a leap year. The result obtained with those tables is certainly more accurate. Banks in the Province of Quebec use generally Matte's Tables of Interest, Those Tables are simply perfect with regard to accuracy and simplicity of comprehension. 106 Interest TO FIND THE DIFFERENCE OF TIME BETWEEN Rule. — Opposite the day of the month is written the number of whole number of days that have expired at the date required. January. February. March. Ap ril. May. June. 1 1 1 32 1 60 1 ~9r 1 121 1 152 2 2 2 33 2 61 •> 92 2 122 2 153 3 3 3 34 3 62 3 93 3 123 3 154 4 4 4 35 4 63 4 94 4 124 4 155 5 5 5 36 5 64 5 95 5 125 5 156 6 6 6 37 6 65 6 96 6 126 6 157 7 7 7 38 7 6G 7 97 7 127 7 158 8 8 8 39 8 67 8 98 8 128 8 159 9 9 9 40 9 68 9 99 9 129 9 160 10 10 10 41 10 69 10 100 10 130 10 161 11 11 11 42 11 70 11 101 11 131 11 162 12 12 12 43 12 71 12 102 12 132 12 163 18 13 13 44 13 72 13 103 13 133 13 164 14 14 14 45 14 73 14 104 14 134 14 165 15 15 15 46 15 74 15 105 15 135 15 166 16 16 16 47 16 75 16 106 16 136 16 167 17 17 17 48 17 76 17 107 17 137 17 168 18 18 18 49 18 77 18 108 18 138 18 169 19 19 19 50 19 78 19 109 19 139 19 170 20 20 20 51 20 79 20 110 20 140 20 171 21 21 21 52 21 80 21 111 21 141 21 172 22 22 22 53 22 81 22 112 22 142 22 173 23 23 23 54 23 82 23 113 23 143 23 174 24 24 24 55 24 83 24 114 24 144 24 175 25 25 25 56 25 84 25 115 25 145 25 176 26 26 26 57 26 85 26 116 26 146 ; 26 177 27 27 27 58 27 86 27 117 27 147 27 178 28 28 28 59 28 87 28 118 28 148 28 179 29 29 29 88 29 119 29 149 29 180 30 30 30 89 30 120 30 150 30 181 31 31 31 90 31 151 Interest. 107 THE DATES BY THE FOLLOWING TABLE. days of the year which have expired, subtract this number from the July. August. Sept. October. November. Dec. 1 182 1 213 1 244 1 274 1 305 1 335 2 183 2 214 2 245 2 275 2 306 2 336 3 184 3 215 3 246 3 276 3 .307 3 337 4 185 4 216 4 247 4 277 4 308 4 338 5 186 5 217 5 248 5 278 5 309 5 339 6 187 6 218 6 249 6 279 6 310 6 340 7 188 7 219 7 250 7 280 7 311 7 341 8 189 8 220 8 251 8 281 8 312 8 342 9 190 9 221 9 252 9 282 9 313 9 343 10 191 10 222 10 253 10 283 10 314 10 344 11 192 11 223 11 254 11 284 11 315 11 345 12 193 12 224 12 255 12 285 12 316 12 346 13 194 13 225 13 256 13 286 13 317 13 347 14 195 14 226 14 257 14 287 14 318 14 348 15 196 15 227 15 258 15 288 15 319 15 349 16 197 16 228 16 259 16 289 16 320 16 350 17 198 17 229 17 260 17 290 17 321 17 351 18 199 18 230 18 261 18 291 18 322 18 352 19 200 19 231 19 262 19 292 19 323 19 353 20 201 20 232 20 263 20 293 20 324 20 354 21 202 21 233 21 264 21 294 21 325 21 355 22 203 22 234 22 265 22 295 22 326 22 356 23 204 23 235 23 266 23 296 23 327 23 357 24 205 24 236 24 267 24 297 24 328 24 358 25 206 25 237 25 268 25 298 25 329 25 359 26 207 26 238 2G 269 26 299 26 330 26 360 27 208 27 239 27 270 27 300 27 331 27 361 28 209 28 240 28 271 28 301 28 332 28 362 29 210 29 241 29 272 29 302 29 333 29 363 30 211 30 242 30 273 30 303 30 334 30 364 31 212 31 243 31 304 31 365 11 i ll'Sfci ! It >i> T : i iii' ■If 108 Interest. A few words of explanation on the above Table, Suppose you wish to get the days between January 5th to May 16th ? Tiike the number opposite 16 in the column of May, 136, subtract from it 5 which is the number of days expired since the 1st of January, and you obtain 136- 5 = 131 days. How many days are there between February 19th to July 26th ? T:ike the number opposite July 26th 207 Take the number opposite February 19th 50 Exact number of days between the two dates = 157 Note. — The above Table must be used to find the exact time in all the followinir problems. Rule. — Multiply the Principal by the rate and the time together, and divide by 365, Always cancel if po.ssible. Ex. — What is the interest on $390.60 from February 5th to August 5th@.06Vo? Ans. S11.G2.+ ExPLAXATiON. — From February 5th to August 5th, by the table, we obtain 181 days, which we put as a fraction of a year by writing as follows !;51. Then put the rate next to the numerator ^1 x ,06. Being unable to cancel anything, we multiply 181 x. 06 = 10.86. Multiply the principal 390.60 by 10,86, and divide the result by 365. The quotient will be the answer. 390.60 10.86 234360 312480 390600 365|4241.9160| 11.6216+ 365 591 365 2269 2190 i i^^ 791 730 616 365 2510 2190 Interest. lUi> What is tlie interest on $375.50 from March 4th to October 18th @ .06 Vo. Ans. 314.07;} + . Explanation. — From March 4th to October 18th according to Tabic we have 228 days. From October 18th 291 days " March 4th 63 Exact number of days 228 "When no table can be obtained, we calculate the diiys as follows, taking calendar months : — As we have 4 days gone already in March we count March 27 days April 30 " May 31 '' June 30 " July 31 «' August 31 " Sept. 30 " Oct. 18 " 228 To obtain the exact number of d;iys in each month, u mechanical way can be employed. The following illustration will explain it : — CO All the months comina; on the extremities of the fingers are all of 31 days ; those coming between the fingers 30, except February, which has 28, and every fourth year 29 days. 11 rH' I llli i !i 'If""™' il ,1' ; -' ' i 1 , ^. r r I l\ »' ill'; ri i: 110 Interest. 1. What is the exact interest on $700, from August 16th to May 25th, at .07% ? Ans. $37.85. 2. What is the interest on $140.40, from Sept. 29th, 1890, to March 15th, 1891, at 087o? Ann. $5.13+. 3: What is tho interest on $425.50, from January 8th to Nov. 20th, at .05^% ? Ans. $20.26. 4. What is the interest on $670.40, from February 18th to De- cember 24th, at .04JV„ ? An». $25.54. 5. What is the interest on $345, from February 25th, 1886, to May 28th, 1891, at .06%? Explanation. — 5 years of 365 days each, from February 25th, 1866, to February 25th, 1891. Get by the Table the number of days between February 25th to May 28th = 94 days. Proceed as usual for the rest. Ans. $110.34+. How to find the Rate, wiien Principal, Time and Interest are given. Rule. — Get the interest on the principal at .OlV^ for the given time, and then divide the real interest by the interest of .01 per cent. Ex. — Required the rate per cent, if the interest on $680 during 4 years, 1 month, 15 days is $168.30? 30 days to a month. Ans. .06Vo. 4 years, 1 month, 1 5 days 12 48 months in 4 years 1 month 49 30 = 1485 Add 15 days. 680 .33 2040 2040 cancel 5 33 99 by 297 1485 X .01 — .33 Days = 3 3 360 8 72 24 8 8 224.40 28.05 Interest at .OlVo- Eeal Interest 28.051168.301.06 rate. 168.30 Interest ill 2. Eequired the rate per cent., if the interest on 61789 for 20 jcars, 1 month, 25 days is $574,775 + ? Ans. .015 +. 3. Required the rate percent., if the interest on 81G8.13 during 4 years, 6 months, 10 days is $45.G75 + ? Ans. .067,,. 4. Required the rate per cent., if the interest on §168. 13 for 8 year.s, 5 months, 3 days is $85,100 +? Ans. .06"/.,. 5. Required the rate percent., if the interest on $850 for 1 year, 7 months, 18 days is $97.18 ? Aus. .077,,: 6. Required the rate per cent., if the interest on $865.75 for 3 year.>;, 9 months, 24 days is $231,299 + ? Ans. .077o. 7. Required the rate per cent., if the interest on $500 for 1 y-ar, 10 months, 15 days is $56.25 ? Ans. .067„. 8. Required the rate per cent., if the interest on $143 for 2 years, 9 months is $31.46? Ans. .08Vo. 9. Required the rate per cent., if the interest on $52.50 for 4 years, 5 months, 10 days is $14 ? Ans. .067o. How to find the Principal when Interest, Time and Rate per cent. are given. Rule. — Divide the interest by rate multiplied by the time. Ex. 1. — What principal will, in 3 years, 6 months, 20 days, at .OG'/g give $50 interest ? Ans. $234,375 +. IS ■ Explanation. — 3 years x 360 = 1080 6 months x 30 = 180 20 days — = 20 1280 days. Cancel 360 and .06 by 6 ; 1280 and 60 by 10 ; 128 and 6 by 3. 64 1280 64 Q-. _ ^ Put a decimal point rjnr. ' ~ o before 64 to represent 6 the rate. 'JF !!!' f t l"'ii i! i! ■■ 1 1 ' li 'H i'i ■X ^' I ■■, ' 112 Interest. Divide 50 by V- Invert tho divisor. 50 X 3 = 150. .64 15000 234.375 128 220 192 280 256 240 192 480 448 320 320 2. What principal will in 2 years, 6 mnntlis, 12 days at .OBVq give SCO interest ? Ans. $394.73. 3. Wh.it interest will in 4 years, 6 months, 27 days, at .05Vo S'^^ 890 interest ? Ans. 8393.45. 4. What interest will in 2 years, 9 months, 20 days at .08Vo o^'^'*^ 8100 interest ? Ans. 8224.45. 5. What principal will in 6 years, 3 months at .OGVq give 856.25 ? Ans. 8150. 6. What principal will in 4 months, 18 days at .047,, give 813.20 interest ? Ans. 8860.87. 7. What principal will in 3 years, 8 months, 15 days at .06 Vq give S76.10? Ans. 8342.03. 8. What principal will in 4 years, 9 months, 18 days at .09Vo give 865.10? Ans. 8150.69+. ^-i To find the Time when Principal, Interest and Rate per cent, are given . Rule. — Divide tho given interest by the interest on the principal for one year at the given rate. Ex. 1.— In what time will 8340 produce 8190 interest at .06Vo ^ Interest. 113 it. are Explanation.— S340 x .06 = $20.40 interest, Eeal interest for one year. 20.40 I 19000 I 9 yrs., 3 mos., 22 days jf 18360 640 12 Boduce into month?. 7680 6120 1560 80 Reduce into days. 46800 4080 ifi 6000 4080 iQ9n 1920 16 2040 17 17 Cancel the above fraction by 10, and then by 12, the result will be |^, which we annex to the days . 2. In what time will ^26 at .06Vo give $1.95 interest? Ans. 1 yr., 3 mos. 3. In what time will $216 at .05Vo produce $150 interest ? Ans. 13 yrs., 10 mos., 20 days. 4. In what time will $180 at .06^Vo give $30 interest? Ans. 2 yrs., 6 mos., 24 days. 5. In what time will $250 at .07% give $160 interest ? Ans. 9 yrs., 1 mo., 22 days. 6. In what time will $280 at .08% give $140 interest ? Ans. 6 yrs., 3 mos. 7. In what time will $31 5.10 at .02% give $190 interest ? Ans. 30 years, 1 mo., 23 days. 8. In what time will $419.50 at .06% give $250 interest ? Ans. 9 yrs., 11 mos., 5 days. 8 114 Interest. ■1: « ' I' 1 1, f:\ PROIILEMS. 1. I have shares in the Canadian Pacific R.R. to the amount of $8750, and producing 82-5 interest. Wliat is the rate percent. ? Ann. .06'Vo. 2. A Havinps bank pays to depositors .03J"/o. How much shall I receive lor a deposit of S4910.5U ? Ans. §171.8G. 3. Anthony has deposited J' of his funds at .04Vo mid L at .05V„ ; every year lie draws as much as will pay the rent of a house $117. f)0. What is the amount of his funds? Ans. S28()0. 4. What is the exact interest on $420.50 at .06"/^ from February 17th to Nov. 26th ? Ans. 819.49+ . 5. A sum of money at simple interest has in 4 years, G months amounted to $735, the rate of interest being .05Vo ; what was the sum at first, and iu how many years more will it amount to $1140 ? 1st. $600; 2nd. 13 yrs., (J mos. G. Eichard lias accumulated enough of money by his savings to give him a n venue of $140 ; if the rate is .05Vo, what sum has he ? Ans. $2800. 7. Francis says that his gains while he carried on business amounted to the price of 3G5i) yards of velvet at $2.08 a yard ; what was his annual revenue, if his gains are placed at .05%'^ Ans. $380.53 + . 8. Camillus having raised a capital of $29G5.10 wishes to know iu what time he will receive $889.53 interest at .05°/o ? Ans. 6 years. 9. What will bo the interest on 3 notes, 1st at 6 months for $600 at .OGVo ; the 2nd at 9 months for $870 at .07% ; the 3rd at 12 months for $1270.75 at .08Vo' What amount of interest will be received for each ? Ans. 1st $18 ; 2nd $45.67 ; 3rd $101.66. 10. Alfred placed on interest at .04Vo a sum of money which pro- duced in 5 years a sum equal to 368 lbs. of tea ® 46J cents a pound. What was that sum ? Ans. $855.60. 11. Thomas has invested in the Montreal Sugar Refinery a sum of $21840 at 12^Vo gain ; but his health failing, he retires from business, and loans his money at .07fVo- How much will he lose in 2 years, 5 months, 10 days by the change ? Ans. $2535. 86§. 12. A grocer buys 85 barrels of flour ^ $6 a barrel, payable in 3 months. At that date, he pays $300 on account and he pays the rest Interest. 115 4 niontlirt later witli interest at .00"/^ Hinco tlio lirht imyiucnt. What will bo the interest ? Aiis. 84.20. 13. What will be the value at tlie end of 3 montiiH, capital aiitl in- terest to-other, of 67800 at .04i" o interest? Ann. 878.S7.7r). 14. What sum invested at .Ot)"/u for one year will produce 8.'Jl!0 ? Ans. 80000. 15. What must I pay for real estate, producing 8750 per year, that I uiay receive .00V«. on my investment ? Ans. 812500. K). A bouse which cost 84SO0 was rciituil for 8204 prr year. Wjiat per cent, does it piiy on the investment ? Ans. .0.'),'," „. 17. If I invest 8;{500 for 1 year, 2 mmitlis, and reeuive 8i!)0, what rate do 1 receive per year V Ans. 1 1:" ^ , 18. In what time will 85530.40 produce 830.42 interest at .OO'Vo? Ans. 9 yrs., 2 mcs., day. PARTIAL PAYMKNTS. Partial Payments are payments made at different times of a note, bond or other obligation, and should be indorsed upon t\i\i back of it, A note or bond is a written promise to pay a certain amount at a specified time. A note is generally given by a particular individual and for a short time. A bond is given by governments or corporations as surety for the borrowing of large sums of money, an 1 payable generally after 20, 30, 40 or 50 years. In computing the interest on partial payments, several methods arc used by mathematicians. Those rules are more or less long and em- barrassing for a student. Without pretending to have found the shortest method, I think, however, that the following, mostly used in Canada and the States, is the easiest. Rule. — Find the interest on the principal from the time it becomes due to the time of settlement, and add it to the principal. Find the interest on each payment and add it to the sum of the payments. Subtract the sum of the payments and interests from the amount of principal and interest together, and the difference will be the balance due. li J 116 Partial Payments. A nofo of 82000 ih dutod Fobruary Ist, 1887, with the following indorHumoutH : — March l.st, 1887 «200 July l8t, " 300 Oct. Ist, " 500 July l.sl, 1888 100 Oct. iHt, 1880 ' 200 Jau'y. 1st, 1800 000 What was due July 1st, 1890, the interest being at .00% ? Explanation. — Principal 82000 Int. from Fob. 1st, 1887, to July 1st, 1881), 41 months = 410 Amount of note to July Ist, 1890 = 2410 1st Indorsement, March 1st, 1887 8200 Int. March 1st, 1887, to July Ist, 1890—40 mos 40 2nd Indorsement, July 1st, 1887 300 Int. July 1st, 1887, to July 1st, 1890—36 mos. 64 3rd. Indorsement, Oct. 1st, 1887 500 Int. Oct. 1st, 1887, to July 1st, 1890— 33 mos. 82.50 4th Indorsement, July 1st, 1888 100 Int. July 1st, 1888, to July 1st, 1890—24 mos. 12 5th Indorsement, Oct. 1st, 1889 200 Int. Oct. 1st, 1889, to July 1st, 1890—9 mos. 9 6th Indorsement Jaimary 1st, 1890 600 Int. Jan. 1st, 1890, to July 1st, 1890—6 mos. 18 Amount of Indorsements and Interests. 82115.50 | 2115 50 Balance due July 1st, 1890. 8294.50 i 'i'ooo! : Sherbrooke, July 12th, 1889. For value received, on demandj I promise to pay to the order of Paul Bergeron, Six Hundred Dollars, with interest at .OBVq. Chables Ruddy. Partial Payments, ir 1NDOR8EMKNT8. August 12Ui, 1889 $100 November 12tli, •* S50 January 12th, 1890 120 How much was duo Feb. 12th, 1890? Ans. 8I46.G6 :1$1500. • Richmond, July 1st, 1887. On di nmnd, I promise to pny James Bddard or order, Fifteen Hun- dred Doihirs, at .00% ? Edmoni) Lecours, indorsements. July 1st, 1888 §50 January 1st, 1890 $1000 JIow much was due at tlie time of settlement, July 1st, 1890. 91 ."5150. Montreal, January 13th, 1888. For value received, I promise to pay Louis O'Neill or order, Four Hundred and Fifty Dollars, with interest at .06% ? A. J. Shea. 'INDORSEMENTS. Oct. tth, 1889 $125.10 Aug. 25th, 1890 225.35 How much remained due December 19th, 1890 ? if'::!!. I %'\, I 1 1 h i s: 118 Compound Interest, COMPOUND IN'IEEEST. Compound Interest is the interest on the interest. Compound interest is allowable when there is a cont/act to that ofFect, either express or implied, otherwise it cannot be legally allowed. When it is not allowed by law, or by a Court of Justice, it cannot be demanded in conscience either. Still, in America, the following opinion concerning compound interest seems to prevail everywhere. The law specifies that the borrower of money shall pay the lender a certain sum lor the use ol' 6100 for a year. Now, if he does not pay this sum at the end of the year, it is no more than just that he should pay interest for the use of it :is long as he shall keep it in his possession. The computation of compound interest is based upon this principle. TABLE. Showing the amount of One Dollar at Compound Interest, from ^ to 8 per cent., for any number of years not exceeding twenty. years h per cent. 1^ per cent. 2 piir cent. 2^ per cent. 1.025000 years 1 1.005000 1.015000 1.020000 1 2 1.010025 1.030220 1.040100 1.050625 •J 3 1.015075 1.045670 1.061208 1.076891 3 4 1.020150 1.061350 1 .082432 1.103813 4 5 1.025251 1.077270 1.104081 1.131108 5 6 1.030377 1.093429 1.126162 1.159693 6 7 1.035529 1.1098^0 1.148686 1.188686 7 8 1.041207 1.126479 1.171659 1.218403 8 9 1.046413 M4337r 1.195093 1.248863 9 10 1.051645 1.16052G 1.218994 1.280085 10 11 1.050904 1.177934 1.243374 1.312087 11 12 1.062118 1.195603 1.268242 1.344889 12 13 1.067499 1.213537 1.293607 1.378511 13 14 1.072836 1.231740 1.319479 1.412974 14 15 1.078199 1.250210 1.345868 1.448298 15 K) 1.083589 1.268969 1.372786 1.484506 16 17 1.089007 1.288003 1.400241 1,521618 17 18 1.094452 1.307323 1.428246 1.559659 18 19 1.099924 1.326932 1.456811 1.598650 19 20 1.105424 1.346835 1.485947 1.638616 20 Compound Interest. 119 years 3 per cent. 3^ per cent. 4 per cent. 4^ per cent. years 1 1.030000 1.035000 1.040000 1.045000 1 2 1.0G0900 1.071225 1.081600 1.092025 2 3 1.092727 1.108718 1.124864 1.141166 3 4 1.125509 1.147523 1.169859 1.192519 4 5 1.159274 1.187686 1.216653 1.246182 5 6 1.194052 1.229255 1.265319 1.302260 6 7 1.229874 1.272279 1.315932 1.360862 7 8 1.2G6770 1.316809 1.368569 1.422101 8 9 1.304773 1.362897 1.423312 1.486095 9 10 1.343916 1.410599 1.480244 1.552969 10 11 1.384234 1.459970 1.539454 1.622853 11 12 1.425761 1.511069 1.601032 1.695881 12 13 1.468534 1.563956 1.665074 1.772196 13 14 1.512590 1.618695 1.731676 1.851945 14 15 1.557967 1.675349 1.800944 1.935282 15 16 1.604706 1.733986 1.872981 2.022.370 16 17 1.652848 1.794676 1.947900 2.113377 17 18 1.702433 1.857489 2.025817 2.208479 18 19 1.753506 1.922501 2.106849 2.307860 19 20 1.806111 1.989789 2.191123 2.411714 20 years 5 per cent. 6 per cent. 7 per cent. 8 per cent. years I 1.050000 1.060000 1.070(100 l.OSOOOO 1 2 1.102500 1.123600 1.144900 1.166400 2 3 1.157625 1.191016 1.223043 1.259V 12 3 4 1.215506 1.262477 1.310796 1.360489 4 1.276282 1.338226 1.402552 1.469328 5 6 1.340096 1.418519 1.500730 1.586S74 6 7 1.407100 1.503630 1.605782 1.713824 7 8 1.477455 1.593843 1.718186 1.850930 8 9 1.551328 1.689479 1.838459 1.999005 9 10 1.628895 1.790848 1.067151 2.158925 10 11 1.710339 1.898299 2.104852 2.331639 11 12 1.795856 2.012197 2.252192 2.518170 12 13 1.885649 2.132928 2.409845 2.719624 13 14 1.979932 2.260904 2.578534 2.937194 14 15 2.078928 2.396558 2.759032 3.172169 15 18 2.182875 2.540352 2.952164 3.425943 16 17 2.292018 2.692773 3.158815 3.700018 17 18 2.406619 2.854339 3.379932 3.996020 18 19 2.526950 3.025600 3.616528 4.315701 19 20 2.653298 3.207136 3.869685 4.660957 20 1 i a 1 Bi ' if" 1 1 1 ; 1 ■!8 !^ i J :■ a-i ?! i f V '"^^f 1; .f li liilh 1 1 H. \ 120 Compound Interest. Ex. — What is the interest of $290 for 6 years, 4 months, 6 days, at .06% ? Ans. $107,593. OPERATION. Amount of $1 for 6 years per table 1.418519 Principal 290 127666710 2837038 Amount of principal for 6 years 411.370510 Interest of $1 for 4 months, 6 days, at .06%... .021 •21 4 mos. =120 days 126 X .06 = -fl or .021 41137051 6 days= 6 360 82274102 126 60 10 Amt. of interest for 4 mos., 6 days 8.63878071 Amt. of principal for 6 years '*11.37051 Amt. for 6 yrs., 4 mos., 6 days 420.00929071 Principal subtracted 290. Interest for given time. - $lb0.00929 EuLE. — Multiply the amount of $1 as per Table, for the given rate and time by the principal, and the product will be the amount for the years. Subtract the principal from the amount, and the remainder will be the compound interest. If there are months and days in the given time, find the interest for the months and days as in the above problem. 1. What is the compound interest of $884 for 7 years, at *» per cent. ? Ans. $279,283+. 2. AVhat is the interest on $721 for 9 years, at 5 per cent. ? Ans. $397.50. 3. What is the amount of $960 for 12 years, 6 months, at 3% ? Ans. $1389.26. 4. What is the amount of $25.50 for 20 years, 2 months, 12 days, at 7 per cent. ? Ans. $100,058+. 5. What is the amount of $12 for 6 months, at 6 per cent., the interest to be added each month? Ans. $12.36+. Comi^ound Interest. 121 6. What is the amount of 8100 for 6 years, 6 months, 10 days, at (>% ? Ans. 814H,34+ How to compute compound interest without the assistance of a table. EuLE. — Find the interest on the given principal to the time the interest becomes due, and add the principal. Then find the interest on this amount for the next year, and add as before, and so on for each succeeding year. Subtract the given principal from the last amount, and the remainder will be the compound interest lor years. When the time is also for mouths and days, before subtracting the principal from the last amount, find the interest for months and days at the given rate per cent., and add the principal just multiplied. Now subtract the original principal from the last amount, and the remainder will be the interest. When interest is payable oftenerthan once a year, find the amount for each interval in the same manner as when the interest is payable yearly. Ex. — What is the compound interest of $590 for 3 years, 7 months, 12 days at 6 per cent.? Ans. 8117.54+- OPERATION. Principal 8590 Interest of 81 for 1 year .06 Interest for 1st year ., 35.40 590 Amount for 1st year 625.40 .06 Interest for 2nd year 37.524 625.4 Amount for 2nd year 662.924 .06 Interest for 3rd year 39.77544 662.924 Amount for 3rd year 702.69944 WT \'i '-■ ft. N 1 1 1: 1 122 Compound Interest. Amount for 3rd year 702.69944 Interest of $1 for 7 months, 12 days .037 4.91889G0S 210809832 Interest for 7 montlis, 12 days 25.99987928 702.69944 Amount for 3 years, 7 months, 12 days 728.69931928 Principal subtracted, 590. Compound interest 8138.69931928 1. What is the compound interest on $970 for 2 years, 9 months, 24 days at .067,, ? Ans. §173.29+. 2. What is the compound interest on §500 for 3 years, 6 months, 12 days at .067,, ? Ans. 4114.50+. 3. A money dealer borrowed $1000 for 2 years at 6 per cent, inter- est; and loaned the sime in sucli a manner as to compound the interest every 6 months. What profit did he make in 2 years by this proceed- ing ? Ans. 8142.47+ jziiin. 4. Find the diiforence in compound interest on §5000 for 2 years at 4 per cent., jiccordiug as it is reckoned yearly or half-yearly? Ans. 8441.291 g.iin. 5. What is the difference between the compound interest on 840,000 for 2 years, and on 880,000 for 2 years, the rate in both cnses being 57, ? Ans. 8ilOO difference. 6. Xavier and Bullock lend eacli 8248 for 3 years at .03^7o) 0"g ^^ simple, the other at compound interest ; find the difference of the amount of interest which they respectively receive ? Ans. 92 cents difference. Ill W js ti_; ComjioiLnd Interest. 123 TABLE Showing in how many years a given principal will double itself. At At Compound Interest. Rate. Simple Compounded Compounded Compounded Interest. Yearly. Half Yearly. Quarterly. 1 100. 69.666 69.487 69.400 H 66.66 46.5o6 46.382 46.298 2 50. 35.004 34.830 34.743 2h 40. 28.071 27.899 27.812 a 33.33 23.450 23.278 23.191 ^ 28.57 20.150 19.977 19.890 4 25. 17.673 17.502 17.415 •^h 22.22 15.748 15.576 15.490 5 20. 14.207 14.036 13.946 5i 18.18 12.946 12.775 12.686 6 16.67 11.896 11.725 11.639 6i 15.38 11.007 10.836 10.750 7 14.29 10.245 10.075 9.989 ^ 13.33 9.585 9. 914 9.328 8 12.50 9 . 006 8.837 S.751 8^ 11.76 8.497 8.346 8.241 9 11 11 8.043 7.874 7.788 9^ 10.52 7.638 7.468 7.383 10 10. 7.273 7. + 7. + Rule. — Multiply the given sum by the rate and time mentioned in the table, and you will always get an interest equal to the given sum. TAXES. Taxes are sums of money raised on the property of a municipality for local improvement, payment of officers, school teachers, etc., etc. When raised on persons without regard to property, they are called "Capitation " Taxes. In levying taxes care should be taken that a complete inventory of the value of all the property, movable and immovable, in the city, town, ppp 124 Taxes. 11 I'M ':h vilLage, township, county, etc., in which the taxes are to be raised, should be made. Immovable Property or Real Estate includes houses, lands au'l manufactures. Movable Property includes furniture, tools, cattle, etc., etc. The tax imposed on hotelkeepers, licence-holders, travelling-clerks, carters, hackmen, itinerant-merchants, etc., etc., is a " Specific Tax." Manufactures, Railways, Mines, etc., are called " neutral '' property, and the money raised on them for school purposes is divided at the "pro rata " of tlie Catholic and Protestant children attending school in the municipality. The persons cliai-ged to estimate the value of the immovable proper- ty to be taxed are called Assessors. They are named by the Munici- pal Councillors. The tax on movable and tlie polls is established by municipal by- laws. Before assessing taxes, it is necessary to ascertain the sum to be raised, including expenses for collection, if collectors are employed, and the proportion which it is expected will be uncollectable. The payment of taxes is required before the privilege of voting is granted. Persons exempted, by special privilege from paying taxes, have no right to vote either. In the Province of Quebec, buildings of public utility, such as churches and parsonages, colleges and convents, school houses, govern- ment and municipal buildings are not taxable. To find what sum must be assessed to include collection and net amount. Rule. — Subtract the rate of collection from $1 and divide the net amount to be raised by the remainder on $1. The quotient will be the sum to be assessed. Ex. 1. What sum must be assessed to raise $57600 net, allowing .04Vo for collection ? Ans. $60,000- 1.00 - .04 = .96. 57600-r- .96=60000. Proof .—.04 of $60000 = 2400. 60000 - 2400=57600. 2. What sum must be assessed to raise $42750 net, allowing .05% for collection ? Ans. $45000. Taxes. 125 be raised, lands and ling-clerks, jific Tax." ' property, ded at the 5 school in ble propcr- le Munici- nicipal by- sum to be loyed, and voting is s, have no such as 3, govern- a and net e the net vill be the allowing S60,000- ing .05% 345000. 3. What sum must be assessed to raise $83386.05 net, allowing .03Vo for collection ? And what will be the amount I'or collection ? Amount S859G5. Collection $2578.95. 4. In a city, the real estate is valued at $3,000,000. There are 1320 polls at $1, and the specific tax amounts to $8750, Wishing to raise a uel amount of $46,070, what rate per cent, shall the city cliarge to pro- prietors? Ans .012Vo. Polls 1320 X $1 = 1320 Specific Tax 8750 10070 46070 - 10070 = 36000 amt. to be paid by proprietors. 36000.00-^3000000 = .012V« Rule. — Multiply the tax on each poll by the number of taxable polls, the product of which, added to the specific tax and subtracted from the whole sum to be raised, will give the sum to be assessed on the property. The sum to be raised on property, divided by the whole taxable property, will give the rate per cent, to be paid on each dollar of pro- perty taxed. Each man's taxable property multiplied by the rate on $1, and his poll tax added to the product, will give the amount of his tax. 5. In a certain town, a tax of $3900 is to be assessed. The real estute is valued at $840000 and the movable $210,000 ; there are 500 polls at $1.50 each. What is the rate percent, of taxation on all the pro- perty immovable and movable. Ans. .003Vo. 6. The real estate of the city of H, is $4590000. The taxable pro- perty is valued at $4100000. The non taxable property is valued at §490000. There are 650 polls at $2 each and a specific tax amounting to $9500. Now, if tho tax to be assessed is $80,500 ; what is the tax on $1 on the taxable property ? Ans. .Ol7Vo. 7. At the above rate, how much will Ivineo pay on a property worth $15,900, and for 5 polls ? Ans. $280.30. 8. A tax of $5130 is to be raised by a certain town; the property to be taxed is assessed at $430,000. There are 240 polls, each $1. What amount must, be levied, allowing SVq for collection ? What will be the tax on $1 ? What will be the commission of collection ? And what will be the tax of each of the following persons ? IF' \i ' if '*■ 12() Taxes. u Anthimian's property is valued at 86000, and he pays for 3 polls, liarney's property is valued at §5400, and he pays for 2 polls. Ch'ineiis' property is valued at 84H00, and he pays for 4 polls. Ans. Collection S270. Ifate per eent .012Vo- Amount to be raised 85400. Anthimian's tax 875. Barney's tax 8GG.80. Clemens' tax 801. GO. 9. The tax assessed on a certain town is 81485 ; its property is valued at 842,000, and it contains 300 polls at 75 cents each. What per cent, is the tax on the dollar, and how much is Kuoul's tax who pays for 3 polls, iind whose property is valued at 82250 ? Ans. Hate per cent. 03. lltioul's tax 8G0.T5. INSURANCE. Insurance or Assurance is a guarantee against losses or damages given for a compensation by tin insurance company. An Insurance Company is a corporate body that assumes all risks against fire, shipwreck or other calamities. The Insurance is on persons or on property, such as houses, chattels, moi'chandise, cargoes, vessels, etc., etc., etc. Tiie Insurer is the person who takes the risk. He is also called Underwriter, from the custom of writing his name with the amounts for which he will be responsible, under a description of the property insured. The Premium is the sum or percentage paid on the amount insured. The Policy is the Parelimcnt containing the contract, and describing the property or person insured, on the terms on which the insurance is etfected. There are several kinds of insurance on persons, the principal ones are : — 1. Life Insurance Policy is a contract in which a company stipul- ates to pay a certain sum of money on the death of the person insured, in consideration of payments made by the insured as specified in the policy. 2. A Term Policy is one in which a company agrees to pay to the party insured a specified sum at a certain age or time, or to his heirs, should his death occur before that age or time, on payment of an annual premium. Insurance. 12^ 3. A lion-forfeiting Policy is one in which even though the party insured should tail to pay his annual premiums after the first, the Com- pany agrees to pay an equitable amount of the sum insured on tlie maturity of the Policy. 4. A. Guarantee Insurance is one in which persons accepting a posi- tion involving money transactions should he insured. 5. An Insurance against accidents is one in which travellers or per- .-•"ons, on account of their work, are exposed to accidents, are insured. Mutual Jusuiancc Companies are those in which, ajiart i'rom a little annual percentage to cover the expenses of administration, tiio mem- bers are liable to pay only v>\\vn losses occur. To guard against fraud, property is not insured for its full value, and no more can be recovered than the amount of actual loss. Houses are generally insured from one-half to three-fourths their estimated value. Goods in transportation arc insun^d from 5% to 25% over cost in order to cover the expenses of freight and insur nee. Goods in store arc insured to cover their value ; but the goods are not usually specified until a fire has t)ccurred. An Insurance Company will generally take u.. a stock a risk ran- ging from $100 to $30,000, but not more. The rate for insuring varies according to the locution of the Ijuild- ing, the difficulty of preservation, or recovery when fires take place. Some property is so risky that companies will always refuse to insure it at any rate. I could here mention the numerous Charitable and Benevolent Societies forming also a class of Insurance. The members pay a monthly fee, and when they are sick, obtain from the society to which they belong a weekly pension, and at their death a certain sum is paid to their heirs. The rate per cent, on Life, Endowment or Term and non-forfeiting Policies is bused on the age of the person applying for a policy and on the number of years each person is expected to live after insuring, The " Expectation of Life " is the average number of years persons of the same age are expected to live. Tables of mortality showing, out of a given numbar of persons, how many complete each subsequent year and how many die m it till the whole are extinct, are used in computing premium on policies. Two tables are generally used both in England and in America, namely : — ipp '^ 128 Insurance. n 1. The *' Carlisle Table, " computed by Mr. Milne according to the mortality observed at Carlisle, Enj^land ; and 2. The " Wifr{j;lesworth Table," prepared by Dr. Wigj^lesworth from dates founded upon the mortality of the United States of America. Three-fourths of oui" Insurance Companies beini? British and Amer- ican, the following Tables arc the only ones used in Canada. Note. — The advantages to commerce from insurance are immense. Without its aid, few business men would be found disposed to risk their goo'is in a long and perilous voyage; by its means the capital of the merchant whose vessels are dispersed over every sea, and exposed to all the caprices of the ocean, is as secure as that of the farmer. To all who have limited means, Life Insurance offers a provision against the innumerable accideufcs of life. By paying u moderate annual premium, a person at his death can leave his family in comfort. t • i«-'" 11* I 1 "i- Insurance, 129 Tables siiowino the expectancy of life in years and iiundrkdths. to < Carlisle Table. < Carlisle Table. 1 .1; -J OH 38.72 28.15 32 33.03 29.43 64 12.30 1 44.68 36.78 33 32.36 20.02 ' 65 11.79 47.55 38.74 34 31.68 28.62 ' 66 11.27 3 49.82 40.0 1 35 31.00 28.22 1 6< 10.75 4 50.76 40.73 36 30.32 27.78 : 68 10.23 5 51.25 40.88 37 29.64 27.34 69 0.70 <5 51.17 40.69 38 28.96 26.01 70 0.18 7 5(1.80 40.47 30 28.28 26.47 ! 71 8.65 8 50.24 40.14 40 27.61 26.04 : 72 8.16 9 49.57 39.72 41 26.07 25.61 73 7.72 10 48.82 39.23 42 26.34 25.10 74 7.33 11 48.04 38.64 43 25.71 24.77 75 7.01 12 47.27 38.02 44 25.00 24.35 76 6.60 13 46.51 37.41 45 24.46 23.02 77 6.40 14 45.75 36.7'J 46 23.82 23.37 78 6.12 15 45.00 36.17 47 23.17 22.83 79 5.80 16 44.27 35.76 48 22,80 22.27 8'i 5.51 17 43.57 35.37 40 21.81 21.72 81 5.21 18 42.87 34.98 50 21.11 21.17 82 4.93 19 42.17 34.59 51 20.39 20.61 83 4.65 20 41.46 34.22 52 19.68 20.05 84 4.39 21 40.75 33.84 53 18.97 10.49 85 4.12 22 40.04 33.46 54 18.28 18.i>2 8(5 3.90 23 39.31 33.08 55 17.58 18.35 87 3.71 24 38.59 32.70 56 16.S9 17.78 88 3.59 25 37.86 32.33 57 16.21 17.20 89 3.47 26 37.14 31.93 i 58 15.55 16.63 90 3.28 27 36.41 31.50 59 14.92 16.04 91 3.26 28 35.69 31.08 i 60 14.34 15.45 02 3.37 29 35.00 30.66 61 13.82 14.S6 93 3.48 30 34.34 30.25 62 13.31 14.26 94 3.53 31 33.68 29.83 63 12., si 13.66 05 3.53 o 13.05 12.43 11.96 11.48 11.01 10.50 10.06 0.60 !).14 8.60 8.25 7.83 7.40 6.!t9 6.59 6.21 5.85 5.50 5.16 4 87 4.66 4.57 4.21 3.90 3.67 3.56 3.73 3.32 3.12 2.40 1.08 1.62 The Wigglesworth Table shows a smaller expectation of life than the Carlisle Table. This discrepancy is due to the habits, customs and climate of America. 9 1 ■' r> 130 Insurance. V, m illM From the above calculations, tables are arranged, sliowinj; the rates at which conipaniis will insure liven, such rates including all probable losses, expenses, etc., etc. LIFE TAHLE OP ANNUAL PREMIUM ON A POLICY OF 8100. Age at Payments PaynientH Pay men ts Payments Age at i.'^HUe. linriiig life. 1 cease at 65. To cease at 60. To cease at 50. isHue. 14 .f 1.4707 $1.4999 $1..5238 $1.6150 14 1.5 1.5105 1.5422 1.5683 i.6(;8i 15 10 l.55!(i 1.5861 1.6145 1.7240 16 17 1.51)40 1.6316 1.6625 1.7826 17 IH 1.11377 1.6786 1.7124 1.8444 18 11) 1.»;H2U 1.7275 1.7644 1.9096 19 20 1.72!)(; 1.7782 1.8186 1.9785 20 21 1.77HO 1.8310 1.8753 2.0516 21 22 1.H280 1.8S59 1.9.344 2.1292 22 2;{ 1.871)8 1.9431 , 1.9963 2.2118 23 24 l.!)335 2.0027 2.0012 2.3000 24 2r) l.llHiU 2.0648 2.1291 2.3944 25 2H 2.0470 2.1300 2.2007 2.4959 26 27 2.1071 2.1981 2.2761 Z.0054 27 28 2.10% 2.2695 2.3.555 2.7238 28 21) 2.234(1 2.:U44 2.4395 2.8525 29 30 2.3023 2.4230 2.5284 2.9928 30 HI 2.3728 2.5058 2.6226 3.1460 31 32 2.4464 2.5930 2.7228 3.3103 32 3.3 2.5232 2.6H51 2.8296 3.5044 33 34 2.6034 2.7824 2.9436 3.7142 34 35 2.6873 2 . 8856 3.0657 3.9503 35 :!() 2.7752 2.9951 3.1971 4.2182 36 37 2.8674 3.1117 3.3.387 4.5251 37 38 2.9641 3.2361 3.4919 4.8807 38 39 3 . 0658 3.3692 3.6584 5.2981 39 40 3.1729 3.5120 3.8402 5.7959 40 41 3.2856 3.0654 4.0393 41 42 3.4046 i 3.8311 4.2588 42 43 3.b.U)3 ! 4.0106 4.5021 43 44 3.6632 4.2055 4.7735 44 45 3 . mM) 4.4181 5.0782 45 46 3.9530 4.6512 5.4235 40 47 4.1111 4.9075 5.8180 47 48 4.2782 5.1902 6.2726 48 49 4.4.549 5.5038 6.80.32 49 50 4.6417 5.85.36 7.4317 50 51 4.8393 6.2470 51 52 5.0486 6.6935 52 53 5.2708 7.2061 53 54 5.5067 7.8017 54 55 5.7577 8.5048 55 Ex. — What must a person pay yearly to the " New York Life " for a life policy of 84000, his age being 25 years at the issuing of policy? Ans. $79,564. Insurance. 131 ■H Aj;t' lit M. 1>^HIU'. U 15 Iti 17 18 ID 20 21 22 23 24 25 26 27 28 29 30 31 H2 33 34 35 3(i 37 38 39 40 41 42 43 44 45 4G 47 48 49 50 51 52 53 64 55 Kxi'LANATioN. — Maltiply iiJ4000 by the tabic opposite 25 yours. .019891 X 4000 -^87().r)G40('0. Tho rati; must bo always txpresiod ducinmlly by adding u tu tho givuu tublo aud placiiii^ tho decimal point before that 0. TERM rOUCY TABLE. ANNUAL IMIKMII'M ON A POLICY OK ,*10(). A^;(' at 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 I'oliiT lueal40, $3,356 3.545 .3.752 3.978 4.228 4.504 4.812 5.156 'i 544 985 489 082 7.752 8.558 9.526 Policy hu'Ut 15. $2,475 2.587 2.707 2.835 2.!I37 3.122 3.283 3.458 3.648 3.855 4.083 4.333 4.611 4.92(1 6.265 5.6,i4 6.(196 6.6(11 7.185 7.866 8.673 9.641 I'nlioy line at 50. $2,113 2.197 2.285 2.379 2.47S 2.585 2.(;98 2. Hi 9 :i.9l9 3.081) 3.2.39 3.402 3.578 3.770 3.979 4.208 4.461 4.740 5.051 5.398 5.789 6.232 6.739 7.325 8.008 8.816 9.787 Pol oilov $1,868 1.935 2. ((04 2.077 2.153 2.234 2.320 2.410 2.506 2.608 2.717 2.832 2.956 3.088 3.231 3.384 3.549 3.728 3.923 4.135 4.368 4.624 o. 5. 5. (■). 6. 7. 8. 9. 10. 90(i 221 572 967 415 927 ,-.I8 207 022 000 Policy I'olicv .\;;i' at lie at (iO. due at 65. i.ssue. $1,704 14 1 . 759 15 1.816 $1,694 IG 1.876 1.716 17 I.!>39 1.799 18 2.004 1.855 19 2 . 073 1.914 20 2.145 1.974 21 2.220 2.03S 22 2.300 2.1(14 23 2. .384 2.174 24 2.473 2.247 25 2.567 2.323 2G 2 . 666 2.404 27 2.772 2.489 28 2.884 2.578 29 3.004 2.672 30 3.132 2.772 31 3.2(18 2.877 32 3.415 2.989 33 3.573 3.108 34 3.743 3.2.35 35 3.928 3.370 .36 4.128 "..515 37 4.347 .'.670 38 4.58(} 3.837 39 4.849 4.017 40 5.139 4.212 41 5.462 4.424 42 6.822 4.655 43 6.227 4.908 44 6.686 5.185 45 7.210 5.491 4G 7.813 5.830 47 8.515 6.208 48 9.343 6.630 49 10.332 7.105 50 11.536 7.645 51 8.265 52 8.983 53 9.82G 54 10.831 55 ' MVr 9 j'1 ■ 132 Insurance. m':'^m^ . N, 1 1!^ ^1 ,i.i I- Ex. — Zilox who is 30 years old takes a term policy payable at the af;o of 40. What will be his annual premium on ^10,000 ? Explanation. — The term policy table gives .09r)26V(,. .0952G x 10,000 = $952.60000. He will therefore pay yearly $952.60. 1. What will it cost to insure a house for $3500, at a premium of .0U%? Ans. $52.50. 2. What is the premium for insuring goods for $4500 at .02^% ? Ans. $112,50. 3. What premium must I pay for insuring a cargo of flour worth $4S,900, from Sherbrookc to Liverpool, at .03% ?; Ans. $1467. 4. A Montreal manufacturer paid 81450 premium for the insurance of a cargo of cotton, shipped to Japan by way of the Canadian Pacific R.R. at .02^ Vn- What is the value of the cargo ? Ans. $58,000. 5. AVh:it is the cost of insuring chattels worth $3000 at 80 cenu per $100 ? Ans. $24. 6. I paid $52.50 premium on goods valued at $35,00!) What was the rate of insurance ? Ans. .0015 '/„. 7. A cargo of apples was insured at .02^-% to cover f of its value ; the premium paid was $44. 07. The apples being worth 80 c^Mits per bushel, how many bushels were shipped ? Ans. 2 -!25 bushels. To find what sum u? iist be insured on property so that, if destroyed, its value and premium n.ay both be recovered. Rule. — Its being a fourth case of percentage, divide the value of property by $1, minus the rate. 8. For what sum must I insure my house worth $17450, so that if it be destroyed I may recover both the value of the house and the premium/rate being .046Vo ? Ans. $18291.404+. Explanation.— 1.000— .046. = .954. 17450-^.954 = 18291.404+. If I insure my house for $100 @ .046"/o and it is destroyed the 1st year, I receive only $95.40 towards my loss, since I paid $4.60 for insurance. Since[,'then the recovery of .854 cents requires $1 to be insured, the recovery of $17450 will require as many dollars to be in- sured as 0.954 is contained times in $17450. 9. A. Gendron ships $31360 worth of wheat to Halifax. For what must he get it insured at .02% so as to cover both the v;ilue of the wheat and the premium? As^s. $320.00 Discount. J 0'> 10. A steamer worth $33000 is insured @ .05|%. In case of its becoming a total wreck, for what sum must it be insured to cover both capital and premium? Ans. $35013. 2G|. 11. A merchant insured his goods worth $19800 in one ("(tinpany at .OliVo. and in another for .01^°/^. What is the premium in boih ? Ans. $5:;4.60. 12. My house was insured for $45,000 during 5 years. The fir-t year I paid $1.50 i'or policy, and ^% premium; every succeeding year, I paid ^Vo premium. The house having been destroyed the 5th year, what was the loss of the insurance ? Ans. 843817.-5. 13. A policy covering property and premium is taken for SI 2045, What is the value of the property insured, the rate being ^ per cent. ? Ans. $12000. DISCOUNT. Discount is the percentage off or allowa.ice made for the payment of nioncy before maturity. There are three classes of discount, viz. : — 1. Commercial Discount. 2. True Discount. 3. Bank Discount. COMMERCIAL DISCOUNT. The term discount, in this connection, is used without reference to time. It is the deduction made from Invoices or Bills in consideration for cash payment. The abatement ^rui. he credit price such as 5,6, 8, 10% off is termed discount. Rule. — '' -'iply the face of the Invoice or Bil' by the given rate, and the produci; w.ll be the Commercial Discount, Subract the Discount from the face of the lu.uice or Bill, and the difference will )e the net price or net proceeds. 1. What discount shall I obtain on an Invoice of goods amount- log to $13950, at 06Vo. Ans. $837. i']\PLANATioN.— $1395^ X .06-^^837 d:s'iuuut. 2. Bough^ $800 worth of goods ; l)y pi' V'iig 'mmediately I can obtain a reduction of .15Vo- Whal ill be the tet price ? Ans. $680. There is also the deducti')ii m,.dr by t - ;pa of a year. If A be disposed to pay off the debt at once, the sum whiuh hf juglit to pay should be such that, if put out at interest by B it will anount at the end of a year to $105. Suppose further that B can put out liis money at 5Vo interest, then if he put out $100 at interest, this is the sum which will amount at the end of a year to $105' Hence $100 is the sum which A ought to pay oA- once, and this is called the Present Worth of the debt, and is evidently such a sum as would, if put out at interest for the given time and rate, amount to the debt. The difference between the Debt and the Present Wortli, which is in the cnse under consideration $5, is called the true discount To find the Present Worth or Value. Divide the face of the note or amount of the debt by 100 plus ^1 e interest on $1.00 for the given time and rate. From the face of note or amount of debt subtract the Preaoi^ Worth, and the difference will be the true discount. Note. — We reckon 30 days to every month and 360 days for the year in true discount. Ex. 1. What is the present worth of $968, note due in 3 years () months, at .06 Vq? Ans. Explanation. — 3 years x 12 = 36 months. plus 6 " 42 1.00 + 21 = 1.21 in a 1142 1^1968.00 1800. 968 — X 06 r. ^9. ['■S-'800= leSdisct. I Discount 135 !S money on e holder of lebt and its t value from ntercst, has rth ; hence : the date of A of a year. ;h >.o )uglit will auount put JUt his I, this is the and this i,s d\ a sum as , amount to sent Worth, iQ discount. LOO plus ^; e ^ 1 the PrtioCii^ ^ lys for the in 3 years Ans, X 06 r: ,21 , 8 disct. S 2. What is the present worth of $1781.40 due 4 years hence at .05Vo interest ? Ans. $1484.50. 3. What is the discount on $373.75 paid 11 months before matu- rity at 6|V„ ? Ans. $20.99. 4. What is the present worth of $117. GO, payable in 1 year at 125/o? Ans. $105. 5. What is the present worth of $618, note due in 6 months at 6Vo? Ans. $600. 6. What is the discount on $1360 duo 6 years hence at 67„? Ans. $360. 7. What is the present worth of $25.44 due one year hence, dis- ccunt being 6Vo ? What is also the discount ? Ans. Present W. $24; Disct. $1.44. 8. What is the discount of $152.64 clue 1 year hence at 6Vo ? Ans. $8.64. 9. What is the present worth of $477.71 due 4 years hence @ 6Vo? Ans. $385.25. 10. What is the present worth of $8000.01 on 6 months credit @ 6°/„ disct.? Ans. $7767. 11. What is the discount on $195.10 payable in 13 months @ 4°/^ ? Ans. $8,10. 12. What is the present worth of the following note dated January 25th payable in 9 months, amounting to $7592, and discounted April 25th @ 8Vo? What is the disct. ? Ans. $7300 P. W. $292 Disct. Explanation. — Date January 25th Disct. April 25th Maturity October 25th As we have no risk to run on what is passed, we count the days from April to October, reckoning 30 days to a month. April 5 days remaining May 30 " days 180 .04 June 30 " — -.08 = .04Vo. July 30 *' 360 Aug. 30 " of a year 6 .9, — X .08 12 Sept. 30 " Oct. 25 '< 180 days or 6 months. 1(1 mm ,p i 1.00+ .04 = 1.04 Discount. 1.04'7 59200| 7300 present worth. 728 312 312 7592—7300 = 292 true discount. 13 What is the present worth of a note d.ited June 14th on 3 months credit, amounting to $1500.90, disct. August 2nd @ 6"/„ ? Ans. 81550.049+. ' t. Find the Discount on $1781.40 due 4 years hence @ 5"/„? Ans. 8290.90. 1 >. ''\ .lit is the present worth of a note dated Nov. 11th, amountiui: to $17t,ii , on 7 months credit, discounted May 4th @ tJVy ? Ans. 8174.225. IG. 1 sold a house, which cost me 82964.12 ready moni'y, for 83665,20, payable in 1 year, 6 months ; what will be ray gain, in ready money, by discounting @ 8"/,, ? Ans, 8308.38. 17. A store was offered for $25,000 cash, or for 812,000 payable in 6 months and 815,000 pa3'able in 15 months. Accepting the last con- dition, I would like to know how much I have gained or lost, money being worth lOVo? Ans. $238.10. 18. A merchant gave out two notes : the first, of $243.36, jiayable May 6th, 1867 ; the second, of 8178.64, payable Sept. 25th, 1867 ; what sum is required to pay the two notes Oct. 11th, 1866. Discount @7V3? Ans. $401.49+. 19. What is the disct. of $800, due 3 years, 7 months and 18 days hence @6Vo? Ans, 8143.19+. 20. Samuel Ht^roux has given his note for $375.75 dated Oct. 4th, 1852, payable to John Smith, or order, January 1st, 1854; what is the real value of the note for the given time @ 6Vo? Ans. 8349.69+. 21. Eought a chaise and harne>s of Isaac Morse for $125.75, for which I gave him my note, dated Oct. 5th, 1852, to be paid in 6 months ; what is the present worth of the note Jan. 1st, 1853 ? Ans. $123.81. 22. What is the present worth and discount of a note of $450 dated May 22nd payable in 6 months and discounted July 27th @ 8Vo ? Ans. P.W, $439,02. Disct. $10.98. Discount. 137 23. A note for 8800 dated March 19tl), due July 25th, and discounted May 21 St (o) 7"/ J What is the present worth ? Ans. 24. $360' 1011 Sherbrooke, February 15th, 1890. Three months after date, I promise to pay Joseph Shea or order One Hundred und Sixty Dollars, value received, @ 8Vo interest ? Georobs DesEuisseaux. What is the present worth if discounted immediately ? Ans, 25. $125 lOU Magog, March 12th, 1890. Sis months from date, I promise to pay Azaric Baron or order One Hundred and Twenty-Five Dollars @ 7"/,,, value received. James Keilly. Disct. June 30th. Wiiat is the discount? Ans, 26, $409-1^0^^ Richmond, August 6th, 1890. Four months after date, we jointly and severally promise to pay Joseph Valine or bearer, Four Hundred and Nine t'/o Dollars, @ 6"/o, value received, Blouin & Dallaire. Disct. Sept. 26th. What is the present worth ? BANKING AND BANK DISCOUNT. general information. Banks are chartered institutions for the employment of capital. Banks of ''Circulation and Deposit" have the use, under certain rules, of the capital paid in by the shareholders, the money belonging to depositors, and the notes of their own circulation. Since 1890, by an act of the Dominion Parliament, called the " Banking Act, " Canadian banks have to deposit in the hands of the Honorable Minister of Finance, a sum equal to 5 Vo on the average amount of their outstanding circulation. This deposit will bear 3 "/o interest, and if, during the course of time, the circulation of a bank is augmented, a sum equal to 5Vo shall be asked to be paid on that neio emission oi notes, after deducting, however, the interest accumulated- . i |bf ! f w 138 Banking. t^ > - ia mm !: m That deposit is kept as a reserve fund in order to help the liquidators of an iusolvent bank in redeeming its outstanding notes, if need there be. This law will be a guarantee to note-holders against any possible loss. In the United States of North America gold-bearing bonds must be deposited by the National Banks with the Treasurer of the U. S., as security for their circulating notes. BANKING. Banking is a general name given to all sorts of transactions with banks, viz. : Receiving deposits, loaning money, dealing in exchange, di.« jnting notes, etc., etc. A Bank is managed by a Board of Directors chosen by the Share- holders. liie Board of Directors select one of their own as President. The Board, being legally organized by the choice of a President, will immediately select an efficient staff of employees, such as: — A Cashier who superintends the bank accounts. A Paying Teller, who pays out money. A Keccivini;' Teller, who receives money, and several other clerks. The President and Cashier sign the bills issued. However, another employee than the Cashier might also sign the bills. Make your deposits early in the day and always with your bank book. For your own security have one particular person to do your h inking transactions. Write or stamp upon all cheques, etc., which you send to be deposited the words : — " for deposit to my credit^" and in doing so you will prevent their being used for other purposes. When cheques are deposited, banks will require them to be endorsed by the depositor when drawn to order. Keep your bank book under your own lock, and do not keep it runnmg too long without being balanced, and when returned by the bank compare it with your account. In filling up cheques do not leave space in which the amount could be increased, because ii any alteration would be made on that account the loss arising from it would be suffered by the Drawer. Adopt a special signature and adhere to the same style. Banking. BANK DISCOUNT. 139 Bank Discount is the simple interest of a note, draft or bill of exchange deducted from its face before maturity. The interest U computed, not only for the specified time but also for three days of grace. Thus, if a note is given for 90 days, the discount is computed for 93 days. The principal object in discounting is to help business men by short loans, and in times of activity to furnish capital for the shipment of produce and other morchandiso to places v\here they may bo in demand. Paper offered for discount is of two classes : — Home or Domeatic and Foreiijn. When an order is addressed to a person residing in the same country as the drawer, it is called a Draft or a Domestic Bill of Exchange. When the order is addressed to a person residing in a foreign country it is called a Foreign Bill of Exchange. Banks discount for short periods of time for the following reasons : — 1. A large portion of the capital invested in discounts is based upon deposits, which are subject to withdrawals, and their own circulation which is redeemable on demand. 2. By shortening the time, the risk arising from the varying cir- cumstances of the makers and indorsers is lessened. To find the Bank Discount and the Present Worth of a note. EuLE. — Find the interest on the note for the given rate and time, including three days of grace, and the interest is the discount. Subtract the discount from the sum discounted and the remainder is the present worth ; or Multiply the amount by \ the number of days, including the day of discount and the three days of grace, and in the product point off three decimals. This last rule gives the interest at 6Vo' For any other rate add or subtract in proportion as the rate is greater or less than 6Vo- w^ 140 Discount. EXAMPLES. k; \: \ . 1^^ Ill' r. i'l 1. Wliat is the bank discount on a note for S780 payable in 30 clays @GVo? OPERATION. 33 11 .11 Days with 3(1. of grace x .00 = 300 20 im 20 20 780 .11 780 780 8.5S0 4.29 Ans. Wc place a dot before 1 1 to represent the rate. Ans. 84.29. Days 6 33 5i or '. = ti _ 11 780 11 780 780 9 8580 4.290 Ans. Point off 3 decimals in the answer. ..amos Williamson, on Mjiy 2nd, 1891, offered the following note properly endorsed, for discount : — $525 100 Sherbrooke, March 29th, 1891. Sixty dnys after date, I promise to pay to Jame.l^ni"ly ^"'l truly [2] Mnwo or im- that I am ra I th<^ iiitiy. of till! Lfood.s montiniu'd in tlio invoico now produced by uic. (2) Swciir or " , iiiHrni, 'iH //i' and luirt'uuto ariMcxtMl and siiiucd by uiL', a ' ♦hat the said cdsi mil 'I ''"•..., , , r . . ciyi'hfowiKr, invoico is the true and only invoice rccou , or flOIIHlUlHM! '"■ 1 • I • n 11 1 1 • iriiiH.rt.T ; or ii wliicli oxpcct to rccoive 01 all thcgoodH im- iin'inbcr of tlm , . ,. • , . linn of (niviii« poi'leu as tliLToui .stated tor account or (4) ; iiiiiiu') the ow- .1 , ,1 . , , , , •! 1 • ,1 • 1 • iicr.i.coiisiniucH iliat tile Haiu o;o()ds are propeny described in tlie waiu in- inliemny !>,■. voice and in tills entry thereol, and that notninjj; has been Ih iMT>ioii or'iin'u ^"1 my part, nor to luy knowh.'dge on the part of any other or'uwut"ri-*!"^"' "^ person, done, concealed or suppressed, whereby ller Ma- jesty the Queen may be defraudid of any part of tlic duty lawfully dui; on the said jioods ; that any goods in- cluded in this entry as paying a lower rati; of duty fi>r a specific purpose than would otherwise bo chargeable upon the same arc to be, and will he, used for such specific purposes only ; and I do further solemnly and thfuUy (2) that the prices named in the said invoi 'he goods mentioned in this Bill of Entry now presented by ine are net jirices, and exhibit to my personal knowledge the fair market value of the said goods for consumption at the time and place of their exporttvtion to Canada, without any deduction or discount for cash, or because of the cxporta- " " iration, wliat- (5) aflirined tion thereot, or tor any ( Sworn or ever. So help me God. specu (5) before me this day of ISO •COLLECTOR. THE OATH OF THE AGENT. When the owner is prevented from claiming his merchandise himself, he may send his agent who will take the following oath : — Oath or Affirmation of an Agent or Attorney of the Owner, Consignee or Impnrtrr. "I (name of agent) do solemnly and trnly (swear or affirm) that I am the duly authorized agent and attorney of (name of the owner, con- signee or importer) and that I have means of knowing and do know Customs, H5 that tlio invoice now prosontotl by mo of the ^oodsrar'tit'onod in thlf^hill ot'iMitry is tiu' trim atul only invoice received l»y tlie said {name of the owner, consignee or importer) ofiill the j^oods imported as within stiite(l I'or (his r//' their) account; tlmt the said goods are properly described in the said invoice and entry, and that the suid invoice and entry ex- liibit the fair market value of the said goods at the time and place of their exportation to Canada, without any deduction or discount for cash, or because of theexpurtation thereof, or for any otht^r cause whatsoever, and that nothintr has been, on my part, nor to my knowledge on th(^ part of any other person, done, concealed or suppressed whereby Her Majesty the Queen may be defrauded of any part of the duty lawfully due on the said goods ; and I do further solemnly and truly (swear Customs. Ii7 STERLING TABLE at 9^ or Par of Exchange, POUND STG. e4.86§. 68 14 Pence Cts. £ S c. £ $ c. £ S c. 1 2 7 34 07 51 248 20 95 462 33 2 4 8 38 93 52 253 07 96 467 20 ;-i 6 9 43 80 53 257 93 97 472 07 4 8 10 48 67 54 262 80 98 476 93 5 10 11 53 53 55 267 67 99 481 80 (] 12 12 ii8 40 56 272 53 100 486 67 7 14 l:; 63 27 57 277 40 101 491 53 8 16 14 68 13 58 282 27 102 496 40 18 15 - 73 00 59 287 13 103 501 27 10 20 16 77 87 60 292 00 104 506 13 11 22 17 82 73 61 296 87 105 511 00 12 m 18 87 60 62 301 73 106 515 87 19 92 47 63 306 60 107 520 73 Shil's. 20 97 33 64 311 47 lOS 525 60 21 102 20 65 316 33 109 530 47 1 '^^ 107 07 m 321 20 110 535 33 2 48^ 23 HI 93 67 326 07 111 540 20 3 73" 24 116 80 68 330 93 112 545 07 4 97;^ 25 121 67 69 335 80 113 549 93 5 1 21f 26 126 53 70 340 67 114 554 80 6 1 46' 27 131 40 71 345 53 115 559 67 7 1 70^ "8 136 27 72 350 40 116 564 53 8 1 94f 29 141 13 73 355 27 117 569 40 9 2 19 30 146 00 74 360 13 118 574 27 10 2 43^ 31 150 87 75 365 00 119 579 13 11 2 67§ 32 155 73 76 369 87 120 584 00 12 2 92 33 160 60 77 374 73 121 588 87 13 3 16,^ 34 165 47 78 379 60 122 593 73 14 3 40f 35 170 33 79 384 47 123 598 60 15 3 65' 36 175 20 80 389 33 124 603 47 16 3 89^ 37 180 07 81 394 20 125 608 33 17 4 13| 38 184 93 82 399 07 126 613 20 18 4 38 39 189 80 83 403 93 127 618 07 19 4 62 40 194 67 84 408 80 128 622 93 20 4 86f 41 199 53 85 413 67 129 627 80 42 204 40 8() 418 53 130 632 67 £ $ c. 43 209 27 87 423 40 131 637 53 44 214 13 88 428 27 132 642 40 1 4 87 45 219 00 89 433 13 133 647 27 2 9 73 46 223 87 90 438 00 134 652 13 3 14 60 47 228 73 91 442 87 135 657 00 4 19 47 48 233 60 92 447 73 136 661 87 5 24 33 49 238 47 93 452 60 137 666 73 f) 29 20 50 243 33 94 457 47 138 671 60 148 Customs, TABLE OP Belgian, French and Swiss Francs, Spanish Pesetas, Grecian Drachmas, and Italian Livres. - !i: 1 M Customs value, each 19.3 cents. a o ft 3 u a 31 t 1 « . 1 o d a ea 01 '7S a en o B a n3 c ■"1 la o 1— 1 1 19 5 98 11 77 91 17 56 2 39 i 32 6 18 62 11 97 92 17 76 3 58 33 6 37 1 63 12 16 93 17 i>5 4 77 34 6 56 64 12 35 94 18 14 5 97 35 6 76 65 12 55 95 18 34 6 1 16 36 6 95 66 12 74 96 18 53 7 1 35 37 7 14 67 12 93 97 18 72 8 1 54 38 7 33 68 13 12 98 18 91 9 1 74 39 7 53 69 13 32 99 19 11 10 1 93 40 7 72 70 13 51 100 19 30 11 2 12 41 7 91 71 13 70 150 28 95 12 2 32 ! 42 8 11 72 13 90 200 38 GO 13 2 51 i 43 8 30 73 14 09 300 57 90 14 2 70 44 8 49 74 14 28 400 77 20 15 2 90 45 8 69 75 14 48 500 96 50 16 3 09 46 8 88 76 14 07 600 115 80 17 3 28 47 9 07 77 14 86 700 135 10 18 3 47 48 9 26 78 15 05 800 154 40 19 3 67 49 9 46 79 15 25 900 173 70 20 3 86 50 9 65 80 15 44 1000 193 00 21 4 05 51 9 84 81 15 63 2000 386 00 22 4 25 52 10 04 82 15 83 2500 482 50 23 4 44 53 10 23 83 16 02 3000 579 00 24 4 63 54 10 42 84 16 21 4000 772 (JO 25 4 83 55 10 62 85 16 41 5000 965 00 26 5 02 56 10 81 86 16 60 6000 1158 00 27 5 21 57 U 00 87 16 79 7000 1351 00 28 5 40 58 11 19 88 16 98 8000 1544 00 29 5 60 59 11 39 89 17 18 9000 1737 00 30 5 79 60 11 58 90 17 37 10000 1930 00 'i' Customs. 149 Table ot Florins of Austria Cus- toms value 33.6 cents. a u o 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 2G 27 28 29 30 CO u c3 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 9 9 9 10 o 34 67 01 34 68 02 35 69 02 36 70 03 37 70 04 38 71 05 38 72 06 39 73 06 40 74 07 41 74 08 o 31 32 33 34 35 36 37 38 39 40 41 42 4:} 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 'ji O 10 10 11 11 11 12 12 12 13 13 13 14 14 14 15 15 15 16 16 16 17 17 17 18 18 18 19 19 19 20 a o O 42 75 09 42 76 10 43 77 10 44 78 11 45 78 12 46 79 13 46 80 14 47 81 14 48 82 15 49 82 16 Table of Marks of German Em- pire Customs value 23.8 cents. : j' - i ^:'. ; f r ': ! !• 1 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 29 30 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 5 6 6 6 6 7 50 a O 24 48 71 95 19 43 67 90 14 38 62 86 09 33 57 81 05 28 52 76 24 47 71 95 19 43 Gi\ 90 14 03 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 04 55 56 57 58 59 60 CO 7 7 7 8 8 8 8 9 9 9 9 10 10 10 10 10 11 11 11 11 12 12 12 12 13 13 13 13 14 14 a o O 38 62 85 09 33 57 81 04 28 52 76 23 47 71 95 19 42 66 90 14 38 61 85 09 33 57 80 04 28 .*«'] 150 Customs. Table of Florins of the Netherlands Customs value 40.2 cents. if! W. |: a s Dollars. 42 r/2 a 2 p a 1 40 31 12 46 2 80 32 12 86 3 1 21 33 13 27 4 1 Gl 34 13 67 5 2 01 35 14 07 6 2 41 36 14 47 7 2 81 37 14 87 8 3 22 38 15 28 9 3 G2 39 15 68 10 4 02 40 16 08 11 4 42 41 16 48 12 4 82 42 16 88 13 5 23 43 17 29 14 5 63 44 17 69 If) 6 03 45 18 09 IG 6 43 46 18 49 17 G 83 47 18 89 18 1 24 48 19 30 19 7 64 49 19 70 20 8 04 50 20 10 21 8 44 51 20 50 22 8 84 52 20 90 23 Q 25 53 21 31 24 9 65 54 21 71 25 10 05 55 22 11 26 10 45 56 22 51 27 10 85 57 22 91 28 n 26 58 23 32 29 11 66 59 23 72 30 12 06 60 24 12 PROBLEMS. 1. What is the duty on 10 hoa^sneadsof molasses, each gauging 150 gallons, and the duty being .08 cents per gallon, specific ? Explanation. — 150 X 10 = 1500 gallons. 1500 X 08 = $120 Duty specific. Problems. 151 2. Imported 25 bags of seed, weighing 5920 lbs., the tare of which was 75 lbs., and paid $1 duty per busliel of CO lbs. Ans. 897.41+. 3. Imported from Manchester, N. H., 40 bales of cotton eucli 350 yds. @ 16 cents a yard. What is the duty at ISVo ^^ valorem ? Explanation. — 350 X 40 = 14,000 yds, 14,000 X 16 = 82240. price. $2240 X 15 = 8336 duty ad valorem. 4. Wliat is the duty on 1728 lbs. of copper invoiced @ 83200, duty being 20Vo ^^ valorem. Ans. 8640. 5. A mercliant imported from France 64 casks of wine, each containing 60 gallons @ 10 francs a gallon. Specific duty being 75 cts. per gal., and ad valorem 30V5 on the invoice ; what is the whole amount of duty ? Explanation. — 64 X 60 = 3840 gallons. 3840 X 10 = 38400 francs. According to Table a franc is worth 19 cts. of our money. Multiply 3S400 x 19 = 87206 of our currency. 3840 gals. X 75 = 82880. specific duty. 7296 X 30 = 2188.80 ad valorem. 2880 X 2188.80 = 85068.80 amt. of duty. 6. Imported from Italy 800 gallons of wine, amounting to 8000 livres, specific duty being 75cts. a gal. and ad. valorem 30Vo- What is the whole amount of duty ? Ans. 81063.20. 7. What is the specific duty @ 3^ cts. "per lb. on 5 hogsheads of sugar, each weighing 1347 lbs., allowing 6 lbs. per hundred for tare. Ans. 8221.58+ 8. What is the ad valorem duty @ 21% on an invoice of silk amounting to 817429.80 ? Ans. 83660.258+. 9. A portion ^/f the cargo of the ship " Brazilian," from Liverpool, Eng., to Montreal, was invoiced as follows: — 750 yards Blue Cloth @ 15 shillings per yard. 670 " Irish Lace @ 5 1280 " Carpet @ 16 3650 " Tweed @ 5 (( >( (( (( (I <( It (( « < lii'^ h fls- I 152 Problems. The duty on Blue Cloth was 30"/o I on Irish Lace, 30"/„ ; on Carpets, 15Vo; on Tweed, 25%; ^^^ ad valorem. What is the amount of duty? Ans. $2923,446+. 10. A merchant in Sherbrooke imported goods invoiced at 820950. On goods invoiced (a). 86500,35% duty was paid ; on $7560, 25"/^ duty was paid ; on $4900, no duty j on tlie balance, 30"/o. How much duty in all? Ans. 84702 11. What is the duty at 18Vo on 60 kegs of prunes, each weiiiliing 1 00 lbs., invoiced (a) 7f cts. per lb., tare (a) 3AV„ ? Ans. 880.77. 12. What are the net weight and duty, (a) 30% ad valorem, on 13 boxes of sugar, weighing gross 450 lbs. each, tare 15Vo, and the cost of the sugar being 8 cts. per pound ? Ans. 4972ilbs.net. Duty, 8119.34. PARTNERSHIP. GENERAL INFORMATION. A Partnership is a voluntary contract between persons who h;iv agreed to combine their property, labor or ability in some lawful busi- ness, and to share the profits or bear the losses arising therefrom. The partnership is sometimes called a compnny, a firm or a house. Each person forming the association is called a partner. The means contributed by the partners are called Capital, Slock Joint Stock, or Stock in Trade. A partner may either furnish money, or property, or his time, or his name only. When the partnership is formed and in operation, money, properly, notes and debts due to the firm are called Resotirces or Assets. Debts owed by the firm are called Liuhillties. When the Resources exceed the Liabilities, the difference is called a Net Capital, and if the Liabilities are greater than the Resources, then it is a Net Insolvency. The profits accumulated during a year are called the Gross Odin; and if, after deducting the yearly expenses of the firm, there still remain something, it is called a Net Gain. Tlie gains and losses of a partnership are divided between the associates according to agreement or contract between them. The division is seldom made according to the amount invested ; sums withdrawn by each partner during the year must bo deducted from Ma share of the net gain. Partnership. 153 Slock or his died a i, then (; and lomain n the sums •m Ms N. B. — For more information, vide Campbell's " Commercial Law," published for the pupils of our Business Class. EXAMPLES. Arthur and Gaudios commenced business with S5000 in cash, $3000 worth of g;oods in store, and owing to several persons $2000. At the end of the year they had $3000 in cash, $3500 in i,'oods, $3300 in notes, and they owed to various persons $1350. How much did they gain ? $5000 cash 3000 more $8000 Resources at commencing 2000 Liabilities $3000 cash 3500 mdso 3300 notes 9800 Resources at closing 1350 Debts at closing $0000 net capital $8450 Net Capital 8450 — 6000 = 2450 net gain — which, being divided equally between the two partners, would give to each $1225. Rule. — Find net capital or net insolvency at commencing and at closing, and subtract. The balance will either be a gain or a loss. Second Rule. — Find what per cent, of the entire stock the total profits or losses may be, then multiply each partner's stock or invest- ment by the rate per cent, expressed decimally. George and Gustavus started business with $4000 cash, 5000 worth of goods, and owing to several persons $1900. At the closing of the business they had $8900 casli, $7000 worth of stock in the Capelton Copper Mines, and they owed $2300. How much did they gain ? Ans. $5500 gain. 3. A, B and C enter into partnership in the lumber business for 3 years. A puts in $2,400; B, $3,000; C, $6,000. At the time of the firm the net gain was $4560. What is each partner's share of the gain ? OPERATION. 4560^11400 = 40%. Hence the profits = 40% of tlie invest- ment. A's share of profits = 2400 x 40 = $ 960 B's " " " = 3000 X 40 = 1,200 V/s " *' " = 6000 X 40 = 2,400 Whole gain... $4,560 154 Partner ^hijJ. Il ! : If li I V ; 't il^ i ' i ■•' 4. Throe merchants, A, B aud C, engaged in trade. A put iu $6,000; B, $9,000; C, $5,000. They gained $840. What is each merchant's sliare of the gain ? Aus. A'sgain, $252; B's, $378; C's, $210. 5. An insolvent mcrcliant owes Perron, $8,750 ; Dolan, $3,010 and Gagnier, $7,000. His effects, sold at auction, amount to $6,875 ; of this sum $375 are to be deducted for expenses, etc. What will each receive of the dividend ? Ans. Perron, $2,937.75+; Dolan, $1,212.03+; Gagnier, $2,350.20+. 6. Alfred, Bruno and Conrad formed a partnership. A invested, $3,000; B $2,500; and C $1,500. Their profits were $2,800. What was each partner's share of the profits ? Ans. A's $1,200; B's $1,000 ; C's $000. 7. A and B speculated in grain. A invested 1200 bushels of barley at $1.50 per bushel; B invested 2000 bushels of corn .-it 80 cts. per bushel. The gains and losses were shared in proportion to value of investment. Their net gains were $510. Whnt was the share of each? Ans. A's gain $270; B's $240. 8. A merchant, failing in trade, owes A $500, B $386, C $988, D $126. His effects are sold for $100. What will each receive ? Ans. A $25, B -^9.30, C $49.40, D $6.30. 8. A, B and C engaged in business. A put in $700, B put in $300, and C put in 100 bbls. flour. They gained $90, of which sum C took $30 for his pari ; what will A and B receive, and what was C's flour valued per bbl. ? ■ Ans. A receives 842, B $18, C's flour $5 per bbl. 2nd CASE. To divide the profits or losses when sums are invested for different times. Rule. — Multiply each investment by the number of days, months or years invested. Add products together and you have the total capital invested for one day, one mouth or one year. Then, find the proportionate parts of gain or loss as in simple partnership. EXAMPLES. 1. Coogan puts in $700 for 5 months ; Dufcuc, $800 for 6 months ; and Ingram, $500 for 10 months, the gain being $399. What is eacli man's share of the gain ? Ans. C's gain $105 ; D's $144; I's $150. in Partnership. loo OPEUATION. Interest — Invcptnient rate. 30!) -^13300 = .03%. C's investuicnt $700 x 5 niontliH —3500 for one day. D's " 800 X (5 " =4800 I's " 500 X 10 " =5000 u it Total averau;o invcstniont 13300 3500 X 03 = $105. C's share of the gnin. 4800 X 03 = 144. D'f, 5000 X 03 = 150. I's u (( Total gain S399 Proof. 2. A and B arc partners, gains or losses to bo divided according to average investment. A puts in Jan. 1st. $5000 " " Feb. 1st. 1000 B puts in Ji.n. 1st. $2000 " " April 1st. 3000 " " July 1st. 1000 " Sept. 1st. 2000 January 1st, one year from date of first investment, the books are closed, and the net gain ascertained to be $2720. What is cacli part- ner's share ? Ans. A's $1580; B's$1140. SOLUTION. Invest- Time. Average Average, iiieut. ciii)it!il. time. A invested Jan. 1st to Jan. 1st 5000 x 12 = 00000 1 month " " Feb. 1st " " 1000 X 11 =r 11000 1 " « " Sept. 1st " " 2000 X 4 = 8000 1 " A's average investment for the average time 79000 1 " B invested Jan. 1st to Jan. 1st 2000 x 12 = 2400i) 1 " " April 1st " " 3000 x 9 = 27000 1 " July 1st " " 1000 X 6 == 6000 1 B's average investment for the average time 57000 (( A's 79000 Total Capital of both for average time 136000 F' '' II 156 Partnership. You ciin do the last part by our systom of proportion. 130000 = 2720 67000 = X Ans. eiUO B's. 13G000 = 2720 79000 = X Ans. $1580 A's. 3. A, U an>l C engai^o in trade. A put in $300 for 7 months, B put in $500 for 8 months, and C put in $200 for 12 montlis ; they gain $85 j wliat share of the gain does each receive ? Ans. A $21, B $40, and C $24. 4. Duhamcl and Gaiilin arc partners; each invested $3500, and are to share the gains and losses C(i[ual]y. During the year Duhamcl drew out $710, and Gaulin $570. What were thcii" gains at the cud of the year, their resources and debts being as follows: — RESOURCES. Cash on hand $40011 Mdse. as per inventory 3750 Bills Receivable 4400 Debts due tlie firm 1900 Total resources $14050 DEBTS. Debts owed by the firm $2750 Bills Payable 1300 Total debts 8 4050 Net capital at closing SIOOOO Duliamel invested $3500 Has withdrawn 710 D's'cred it balance $2790 Gaulin invested $3500 Has withdrawn 570 G's credit balance $2930 Balance of investment $5720 Partnership's net gain 84280 Partnership. 157 5. Letondrc, Lccours and Lespcrancc are partners. Letcndrc in- vested 84500, Lecours $3780, Lesperance 85G00, and are to share the giiins and losses according to investment. Letendre drew out $370, Lecours 8150, and Lesperancc 8^35. What were their j^ains at the end of the year. Their resources and liabilities being as Ibllows : — Resources. Liabilities. Cash 85800 Debts 82750. Mdsc. G910 Bills Pay. 1781.75 Bills Rec. 3210 Debts due the firm 2700 Ans. Total net gain 811C3.25, Lctcndre's share of the gain 8371.70 Lecours' " '• 32i?452 \m \ Shcibrookc, January 2Sth, 1M)2. At siijhtpay to Josrph Michaud or order, Four Hundred and Fifty- Two l)ollars, value receivid, and charge the same to my account. To La Jianque Niitioualc, Montreal. ClIAKLES MollAZAlN. Ui'EiiATioN.— 1.00— .015 - .985. 452 X .985 = 8445.22. Ans. 5. ValCirc Olivier wishes to pay in Quebec 8080, exchaiigo bein02.S5. 6. What c:)st a bill (.u Granby, P. Q., for 84000 {<(} 2Vy„ di.s. count? Ans. 83!)00, 7. What must bo paiil in Toronto for a draft of 83500 for 3:} days, (o) G"/,„ exchange being 2^"/o premium? Ans. 8355G. OrERATION. 81.000 OOG disct. for 36 days @ GV... .994 net worth. .022 rate of premium of Exch 1.016 real value per dollar. 83500 X 1.016 = 8355G. The draft is for 3G days @ 6"/,,. As it is discounted immediately, you have a right to your di.sct. Subtract tlic aisct. from 81, and you get a balance of .9I)4cts. to be paid for every dollar. To that add the premium, but you would sub- tract if it \\as a discount. Then multiply by the face of the draft. 8, A merchant purchased a draft ou Halifax for 8840, j nyable 30 days after sight® 6%, exchange being UVo discount? Ans. 0822.78. y J'' K f i,i > r ■! 'lilif' C -■ 160 Exchange. OPERATION. 1.0000 .0055 disct. for 33 days @ 6%. .9945 net worth. 015 rate of discount. .9795 real value per dollar. $840 X .9795 = $822.78 price of the draft. FOEEIGN EXCHANGE. Foreign Exchange consists of liills principally issued by Banks and Bankers upon their Agents in diiferciit countries, to be used in settle- ment of accounts, without the necessity and risk of special gold remittances. Premium and Discount on Bills of Exchange are generally regulated by tlie greater or smaller demand, as any marketable commodity. A Bill of Exchange is at premium when against the country where it is issued, and at discount when in its favor. The value in gold of a £1 sterling is $-1.8666. This is equal to 9^Vo premium on the old par value of $1,444+ . Exchange quotations refer to the old par. When , therefore, exchange is quoted at about 9|"/„ premium, there is in reality no premium, but the true par of value has been attained. FOREIGN BILL OP EXCHANGE ON LONDON, ENG., PROM SHERBROOKE. No. 35. LA BANQUE NATIONALE. £990 %. SHERBROOKE, February 6th, 1892. Thirty days after Sight pay thi^ First of Exchange {second unpaid) to the order of OTUO BALDWIN, Nine Hundred and Ninety Pounds Sterling, Value reed. To W. Gahoury, Manager. The National BanJc of Scotland, Ltd., 37 Nicholas Lave, E.G., London, Eng. E. A. Duhnc, Acct. ri: Foreign Exchange. 161 No. 35. LA BANQUE NATIONALE. Janks and [ in settle- 3cial gold regulated dity. itry where equal to exchange iium,but IBROOKE. \h, 1892. My thi^ [order of lue recii^ tanager. ic, Acct. £990 Stg. SHERBROOKE, February Qtli, 1892. Thirty days after Sight pay this Second of Exchange (first unpaid) to the order of OTIIO BALDWIN, Nine Hundred and Ninety Founds Sterling, Value reed, To ^ The National Bank of Scotland, Ltd., \- 37 Nicholas Lane, E.G., London, Eng. j W, Gahoury, Manager. E. A. Duhuc, Acct, Apart from England you can pay any amount in any country of Europe, Asia and Africa, by drawing on Paris only. The international monetary system on standard (which is nothing else but the French system) is already adopted by a great number of nations all over the world. No. 1. Fr8. 37890 DRAFT ON PARIS. LA BANQUE NATIONALE. SHERBROOKE, 5 Fcvrier 1892. A 'presentation veuillez payer contre ce cheque (Duplicata ne Vdtant) a I'ordrc de JOSEPH L. DERBOMEL, la somme de trente-sept niille Jiult cent quatre-vingt-dix Francs, dont rous dSiterez notre conipte sidvant avis. MM. Griinehaum Frdres tt- Cie.,'^ Baiiquicrs, ! W. Gahoury, Gerant. 28 Boidevard Haussraann, [ Paris, France.) E. A, Duhuc, CompL 11 ^lk/i,iii|j w if IV (',,!' I;. i 1' ; I s 1^ f No. 2. LA BANQUE NATIONALE. Frs. 37890. SHEIlBIiOOKE, 6 Fevrier 1892. -A ■pvi^sentation veuillez imyer contre ce cheque (OHginal ne I'dant) d I'ordre de JOSEPH L. DERBOMEL, la somme de trente-seiit mille huit cent quatre-vingt-dix Francs, dont vous ddhiterez notre comyte suivant avis. MM. Go'unehaum Frdres S Cie.,'^ Banquiers, [ W. Gahoury, Gdrant 28 Boulevard Haussmann, [ Paris, France, j E. A. Didjuc, Compt. PROBLEMS, 1. J. C. St. Aubin, wishing to remit to London, Eng., £360, 10 shillings and 6 pence, exchange being 9|Vo premium, how much will he have to pay for the bill in Canadian Currency ? EXPLANATION. '-< p '■ $4,444 or ii x 1.095 = 43.800 -f- 9 = 4.8t)(i6 £360.525 X 4.8666 = $1754.314 + 10s. 6p. 2 I 10.5 .525 decimals of a pound The old par multiplied by 1, plus the rate of premium, will give the cost of one pound. Now reduce the shillings and pence into deci- mals of a pound, (i pence is equal to h or 5 decimals of a shilling. As there are 20 shillings in a pound we divide. 10.5 -f- 2 = .525, which we annex to the pounds. Rule. — Multiply the face of the bill by the cost of one pound at the given rate of exchan Aus.6184(i.!)4 4-. 6 What will a bill of exchange on Paris cost for 56245 francs (^ 5 tr. lb centimes per dollar ? ® "^ 56245.00 I 5.16 516 81090009+" 4645 4644 1000 516 4840 4644 196 8. How many Pounds, Shillings and Pence a O'o/ nr • v mS'i.-2^ Canadian Currency ? ® ' ^' ^'"'"^'""^ ^'^'' OPERATION. Find the cost of £1. as before - x 1.005 = 4 866 + ___i:866J 3982.250 | £818 7s. 7|d 38928 ~ " ~ 4866 If 1! f : ' I I, ip 11 40790 38928 Ans. £818 Ts. 8-1. i " i ■■ 164 Exchange. ii ii^^= m^ J I; • 'I .'■■•, 1862 Multiply by 20 for the shillings 20 37240 34062 Multiply by 3178 12 for the pence 12 38136 34062 Multiply by 4074 4 for the farthings 4 16296 14598 Eui J]. — Divide the given sura by the cost of one pound at the given rate of exchange, and the quotient will be the face of the bill in pounds, shillings and pence. 9. What will be the amount of a bill of exchange on London @ 8% for ^540.40 ? Ans. £123. 10. What would be the face of a bill of exchange on Edinburgh, Scotland, @ 10% premium for $350.50 ? Ans. £72 Os. 7d. PROFIT AND LOSS. Profit and loss are terms denoting the gain or loss arising from business transactions. The cost of goods is the price paid to the person from whom they were purchased, or the expense of producing them, and all charges, such as commissions, freight, packing, duties, exchange, insurance, drayago, etc., necessary to put the goods ready for sale. The total gains or losses on goods may be easily ascertained, when all are sold, by taking the difference between the cost and selling price ; but when part remains unsold, add the value of the merchan- dise unsold to the amount received for sales, and take the difference bet'^een the sum thus obtained and the cost of merchandise ; the ditfer- cnce will be either the gain or loss. Fro fit and Loss. 165 In estimating the value of goods unsokl, we use the invoice price ; but if the value of the goods have depreciated, or if it has a decided advance in prices, an allowance must be made accordingly. A safe rule is, if they are saleable goods, to estimate them at what it would cost to rcpliice them. In solving the following problems, follow the four rules of Percen- tage. PROBLEMS. 1. A. Kerr purchased a quantity of lumber for §2200. He paid for drayage §75, commission for selling $125. He gained 27% on the entire cost. IIow much was it sold for, and how much did he make ? Ans. Sold for $3048. Gain, $648. Solution.— 2200 + 75 + 125 = S2400 cost price. 2400 X 27 = $(>48, gain. 2400 + 648 = 3048, selling price. 2. Bought a cargo of wheat for $12500, and sold it at a profit of 16^%. How much did I gain? Ans. $2062.50. 3. How much do I gain per barrel, if I sell flour which cost me $11 per bbl. at a profit of 25% ? And what profit will I realize on 375 bbls ? Ans. Profit on one bbl. $2.75. Profit on 375 bbls. $1031.25. 5. Bought 100 bbls of sugar for $1500.75, which I sold at an advance of 12^%. How much did I gain, and how much did I receive? ^ Ans. Eec. $1688.34+. Gain $187.59, 6. Bought 40 yds. of broadcloth ^ $5.60 per yd. and I sell f of it @$6.50 per yd., and the remainder @ $7.25 per yd. What is my gain ? Ans. $43.50 7. Prosper Olivier bought a hogshead of golden syrup, containing. 100 gallons, @ 60 cts. per gallon ; but 30 gallons having leaked out, he disposed of the remainder @ 80 cts. per gallon. Did he gain or lose, and how much per cent. ? Ans. Lost 6§;/. Solution.— 100 x 60 = $60.00 ; cost price. 100 — 30 = 70 gals, remaining. 70 X 80 =: $5600, 60 — 56 = 4 net loss. 60 I 4 00 I .00§ The above problem is solved by second case of Percentage. 9. A merchant purchased for c:>sh 50 bbls. of flour @ $5 per bbl., and sold the same @ $5.98 per bbl. ; what did he gain per cent. ? Ans. .19^%. W'f ll 1" i-' t lliid tlio uiaturity of the invoices. The l.-t invoice hc'conies due Juiior)tli; the 2nd invoice July 25th, etc. 'I'iien, we take tlie first time when any invoice becomes due, for the prriod from wliich to compute tlie avcratic time. Now, since June 5tli is the period from which tlie average time is computed, no time will be reckoned on the first invoiee, but the time for the payment of tlie first invoice extends 50 days beyond June 5, and we multiply it by 50. I'roceedini; in like manner with the riiuainiiig invoices, we find the e((uated time to be 121 ilay.s nearly, and Oct. 4 the average time of payment. 2. Bought of I). I. Lemairc several bills of goods, at differe tit times, and on various terms of credit, as by the following statement. What is the equated time for the payment of the whole ? Jan. 1, a bill amounting to $oOO, on 4 months. Feb. 7, " " " " 185, on 5 months. March 15, " " " " 280, on 4 months. April 20, " " *' " 210, on G months. Ans. July 11th. 3. Purchased goods of W. Murray, iSherbrookc, as below: — January 8, 1892, an invoice of $375.25 on 3 months credit. (1 u (< 10. 15, 20, 29. (I K U ii u 562.25 250. 100. 225.75 322.50 When could I make but one payment ? 4. Sold Therrien &. Fortier several parcels of goods, at sundry times, and at various terms of credit, as follows : — Jan. 7, 1891, a bill amount, to $350 on 4 months credit. Ans. (I 20, 25, March 5, " 29, April 10, (I (( (> u a (( If 4(i0 150 560 436 620 (I (( (( (( a i( (< When could Therrien & Fortier make but one payment ? Ans. Compound Eqiuiiiou. 171 AVERAGING OF ACCOUNTS OR COMPOUND EQUATION. A C*8 of the use or interest of that sum to the credit side of the account, and a corresponding uain to the debit side. Now, as the settlement is re- quired to be one of equity, we find how long it will take the balance of the account, 6359, to gain the interest that S260 would gain in the 29 days. If it takes $260 to gain a certain interest in 29 days, it would take $1 to gain the same interest 260 times 29 days or 7510 days, and 8359 to gain the same amount of interest jJ^ of 7540 days, or 21 days' Hence, the balance became duo 21 days back of March 3, or on Feb- ruary 10. , In this example, the time was counted back from the average date of the larger amount, since it became dnejirst ; but when that amount become due last, the time is counted forward from its average time. Rule. — Find the average time of each side becoming due. Multiphj the amount of the amaller side hi/ the number of days betioeen the two average dates, and divide the product by a balance of the ac- count. The quotient will be the time of the balance becoming due, counted from the average date of the larger side, back wheii the amount of that side is due first, but forward lohen it is due LAST. Dr Jos. Michaud in account with Rosaire Dubrule. Cr. 1891 Time of credit. 1891 April 3 To Mdsc. $200 3 months. July 1 By Cash $200 00 May 1 u 125 5 Oct. 3 (( 150 00 " 15 ((■ 200 6 " Dec. 20 (( 300 00 June24 u 140 8 " July 1 a 190 9 " mi Compound Equation, Debitt, 17o Credits. Due, 1891 July 3, S220 x 00 = Oct. 1, 125 X 90 = ^'ov. 15, 200 X 135 = 1892 Feb. 24, 140 X 28ej = April 1, 190 X 272 = $875 ) 11250 27000 33040 51(J80 122970 141 da. Debits arc duo 141 days from July 3, which is Nov. 21. The above account thus equated will stand as follows : Dr. Cr Due, 1891 July 1, $200 X 00 _ Oct. 3, 150 X 94 — Dee. 20, 300 x 172 — $650 ) 14100 51000 (55700 101 da. Credits are duo 101 daysi from July 1, which is Oct. 10. Due, Nov. 21, 1891, $875 Difference in days Oct. 10 and Nov. 21 Due, Oct. 10, 1891, $650 Smaller amt. 650 42 42 days 1300 2600 27300 00 00 )0 00 m 00 Bigger amount 875 Smaller " 050 Balance 225 27300 -f- 225 = 121 days. Now, E. Dubrulc paid J. Michaud $650. November 21st, 1891, J. Michaud paid R. Dubrule $875, Then as J. Michaud had the use of $650 for 42 days, so sliould R. Dubrulo be entitled to the use of $225, the balance, until its interest equals the interest of $050 for 42 days, which is 121 days. Counting 121 days from Nov. 21st, we reach March 21st, 1892. Note. — In the above example, the time in the Answer is counted foricard because the larger amount becomes due first. AVhen an account is settled by cash, the true or cash balance of an account at a particular date may be found directly as follows : — PI t- -I fl: Iml 174 Com2)Ound Equation. What will be the balance of the following account June l>t, 1891, the (late of settlement, allowing that each item draws interest from its respective date, @ 6% ? Dr. Joim NouRSE in account xoith Jas. Key. Cr. 1891. 1891. March 5 To Mdse. 6160 00 March 7 By Cash $270 00 " 25 u a 440 00 '' 28 a u 200 00 April 11 a a 100 00 May 2 u a 440 00 19 (( u no 00 " 20 u u 720 00 May 1 ii a 330 00 4 a li 370 00 " 21 U (( 220 00 As the equated time of payment is June 1st, we count From March 5th to June 1st 88 days. C( 25th " 11 (( 68 (( April 11th " u a 51 a K 19th " u (( 43 (( May 1st " a <( 31 (( i( 4th " (( i( 28 (( a 21st " u (( 11 (( N. B. — Do likewise for the credits —March 7 to June 1st — 86, etc. OrERATlON. Credits. Dehits. L'lie, J Uays. March 5, $ 160 x 88 — 14080 25, 440 X 68 =z 29920 April 11, 100 X 51 — 5100 19, 110 X 43— 4730 May ., 330 X 31 — 10230 4, 370 X 28 — 10360 " 21, 220 X 11 = 2420 61730 6%)76840 612.807 Sum of debit items, 61730 " " credit " 1630 Balance of itemp, 6100 Due, Days. March 7, 6 270 x 86:'= 23220 May 28, 20, 200 X 65 = 13000 440 X 30*= 13200 720 X 12= 8040 61630 6%)58060 69.677 Interest of debit items, " " credit " Balance of interest, 612.807 9.677 1; 63.130 Compound Equation. 175 5270 00 200 00 440 00 720 00 True balance June 1, 8100 + S3.13 = $103.13. Explanation. — Since each account of tho debit was on interest from its date to the time of .settlciuent, the totnl intere.-^t of the several accounts of the debit equals the interest of i^l for 70840 days, wliich at 6%ii;ives §12.80 + . As the interest of $1 for G days is 1 niill, so the interest of 81 for 70840 d;iys will be found by dividin<>; 70840 by G, pointing off 3 decimals. The total interest of the several crrdit items will equal the interest of $\ for 580G0 days, which is 89.07 f . Now, instend of increasing e;ich ;Me of the account by its interest, and then finding the balance, this same result may be obtained by finding separately the balance of items and the balance of interests. If the two balances fall on the same side of the account, it is evident the true balance will be their sum ; if on different sides, their difference. Do the f jllowing problem or example on same principle. Br. MuRitAY & Co. in account with JoNES & liEILLY. Cr. 1891. Apri 10 'i 30 May IG u 24 June 1 u 10 (( 20 To Mdsc. it u (( 8150 400 90 100 300 340 200 1891. April 12 By Cash. May June 1 7 25 8250 180 400 5G4 Wnun will be the true balance of tlie above account July 1st, 1891, the Lime of settlement, allowitig that each item draws interest froti! its d.-'c, atG%? Ans. Cash Bal. 8189.03. ALLIGATION. Alligation (in French : Mdlangr) is a process employed in mix- ing articles of different qualities or values into a conipouad, whose value or (|uality will differ from that of its i..gredients. Alligation is divided into Alligation Medial and Alligation Alter- nate. ALLIG2VTION MKDIAL. Alligation Medial is the process of finding the average value or quality of a mixture, the quality and quantity of whose ingredients are known. This is nothing else but the application oi the principle of simple equation or average. ,„^.M:- ■1«i ii 176 Alligation Medial, EXAMPLES. 1, Arthur mixes together 50 bushels of oals @ 40 cents per bushel ; 30 bushels of barley @ 50 cents ; aud 25 bushels of corn @ 60 cents. What is a bushel of the mixture worth ? Ans. 48 cents. OPERATION. Cts. Eush. Cts. 40 X 50^2000 50 X 30 = 1500 60 X 25 = 1500 105 5000 420 800 735 47^3 65 Get the total cost, and divide by tlic number of bushels. As the fraction remaining is laiger by m tc than half the divisor, we add one cent to the answer . EuLE. — Find the value of each article, and divide their total sum by the sum of the articles. The quotient will be the average value of the mixture. 2. A grocer mixes together 15 lbs. of coffee at 18 cts. a pound ; 35 lbs. @ 16 cts. ; aud 40 lbs. at 14 cts. What is a pound of the mix- ture worth ? Ads. 15f cts. 3. A grocer mixes 25 gallons of wine at 90 cts. a gal. ; 40 gallons of brandy at 75 cts. a gal.; and 10 gallons of water. What is a gallon of the mixture worth? Ans. 70 cts. 4. A grocer mixes together 30 gallons of molasses @ 20 cts. a gal. ; 40 gallons @ 25 cts. ; 70 gallons @ 30 cts. ; and 80 gallons @ 40 cts. What is a gallon of tlie mixture worth ? Ans. 31 + cts . 5. A farmer mixe.>i together 4 bushels of oats at 40 cts ; 8 bushels of corn (a) 85 cts. ; 12 bushels of rye @ §1.00; and 10 bushels of wheat @ ^1.50 per bushel. What will one bushel of the mixture cost? Ans. SI .04+. Alligation Alternate. Ill 6. A dealer in liquor would mix 14 gallons of water with 12 gallons of wine at 80.75, 24 gallons at $.90, and 16 gallons at $1.10 ; how much is a gallon of the mixture worth ? Aus. 73 + cts. 7. A farmer mixes together 10 bushels of oats at 40 cents per bushel, 15 bushels of corn at 50 cents per bushel, and 25 bushels of rye at 70 cents per bushel ; what is the value of a bushel of the mixture? Ans. 58 cts. ALLIGATION ALTERNATP]. Alligation Alternate is the process of finding what quantities of certain ingredients may bo compounded to form a mixture of a required grade or value. 1st PROCESS. The prices of the ingredients being given, how to find the propor- tion in which they must be mixed in order that the compound may be worth a given price. 1. A grocer has teas worth 30, 40, 50, 70, 80 and 90 cents a pound. What proportions must be obtained to form a mixture worth 60 cents ? Given price 60 of the compound. prices OPERATION. differencesjdifferences 70 80 90 10 20 30 30 20 10 prices 30 40 50 60 given price of the com- pound. Prices Opposite Diff. 30 = 10 40 = 20 50 = 30 Prices Opposite Diff. 70 = 30. 80 = 20. 90 = 10. x\ns. 1 lb. @ 30 cts; 2 lbs. @ 40 cts ; 3 lbs. @ 50 cts ; 3 Ihs. @ 70 cts ; 2 lbs, @ 80 cts ; 1 lb. @ 90 cts. Explanation. — Draw five p(;rpeudicular lines, place the average price on each side of the group of lines, put the various prices of the ingredients in the outer columns, placing those greater than the price of the compound to the left and those less than it to the right. On the right we begin by the smallest number. In the middle column put the differences between the prices of the several ingredients and of the compound, writing each difference next to the number by which 12 178 Alligation Alternate. t,. A' t lii: n " MM it was obtained. And by placing next to each price the difference of the opposite uuuiber, we obtain the different proportions. The ])riuciple upon which this rule proceeds is that the excess of ingredients above the average is made to counterbalance or cover the deficiency of those less than tlie average. It is obvious that any e([uiniultiples of those quantities or ingre- dients would answer equally as well; hence a great number of answers, e(jually good, may be given to such a question . 2. Ill what pi'opurtion must a grocer mix coffee worth 15, 18, 20, 22 and 24 cents pei- pound that the mixture may be worth 21 cents per pound ? 21 22 1 () If) 24 3 3 18 24 3 1 20 21 N.B. — To complete an equation, we repeat the last number. Prices, lbs. 15 = ) 18 = 3 2U ^=: 3 24 being repeated, we take the sum 22 = 6 of the differences with which it is 24 =:= 4 connected. 3. Wishing to xuix sugar @ 20, 23, 26 and 28 cents per pound, so that the mixture may be worth 25 cts. per pound. How many pounds of each must I take ? Ans. 1 lb. @ 20 cts. ; 3 lbs. @ 23 cts. ; 5 lbs. @ 26 cts. ; 2 lbs. @ 28 cts. 4. A. Lanctot has coffee worth 10, 12, 14, i5, 16, 17 and 18 cents per po;ind, and wishes to form a mixture 13 cts. a pound. How many pounds of each must he use ? Ans. 1 lb. @ 10 cts. ; 14 lbs. @ 12 cts. ; 3 lbs. @ 14 cts. ; 1 lb. @ 15 cts. ; 1 lb. @ 16 cts. ; 1 lb. @ 17 cts. ; 1 lb. @ 18 cts. 5. C. Guay has sugar worth 5, 7, 12 and 13 cts. per pound. How many of each kind will form a mixture worth 10 cts. per pound ? Ans. 2 lbs. @ ') cts. ; 3 lbs. @ 7 cts. : 5 lbs. @ 12 cts. ; 3 lbs. @ 13 cts. 2nd PROCESS. When the quantity of one is given, to find the quantity of each of the others. AUigat'ioii Alternate. 170 00 10 40 40 !tO 10 20 (iO 80 90 10 5 75 0. A merchant lias teas worth 40, 00, 75 and 90 cents per pound. Wishinf]^ to put 20 lbs. @ 75 cents a pound, how many jwunds of each must he use to form a mixture worth SO cents? Ans. 10 lbs. for tho ist three kinds, and 130 lbs. @ 90 cts. Opeiiatiox. 90 10 40 40 40 =: 10 x 2 - 20 60 = 10 X 2 = 20 80 75 = 10x2 =20 90 = G5 X 2 = i;-{0 Expr.AXATiON. — Tho proportionnl quuntitios are 10 for the (ir.-«t three kinds; but as 1 want to put 20 lbs. (VV, 75 cts.. and 10, tho pro- portional (juanlity, being only ^ of tho iutonded amount, I shall mul- tiply each quantity by 2. 7. A grocer wishes to mix 100 lbs. of sugar @ 12 cts. with other qualities worth 10, 14, 18, 20 and 24 cents to make a compound worth 16 cents a pound. How many of each kind must he put? Ans. 50 lbs. @ 10 cts. 150 lbs. @ 18 cts. 100 " " 12 " 100 " " 20 " 200 " " 14 " 50 " " 24 " 8. How many pounds of sugar, worth 8 cts., 12 cts., and 24 cts. per pound, must be mixed with 40 lbs, worth 20 cts., that the mixture may be worth 22 cents per pound ? Ans. 40 lbs. for the 1st three kinds, 520 lbs. @ 24c. ? 9. How much aicohol worth 60 cents a gallon, and how much water, must be mixed with ISO gallons of rum worth 81..30 a gallon, that the mixture may be worth 90 cents a gallon ? Ans. 60 gallons each of alcohol and water. 10. How many acres of land worth 35 dollars an acre must be added to a farm of 75 acres, worth §50 an acre, that the average value may be $40 an acre? Ans. 150 acres. 3rd PEOCESS. To find the quantity required of each ingredient to make a compound of a given quantity, when the value of the ingredients is given. 11. A merchant of Sherbrooke received an order for 460 lbs. of sugar, at 15 cents per pound. He has 'n his store sugars wor*h 10, 12, 20 and 25 cents per pound. How much shall he take of each to fill the order ? IIP f;^,: m i;i m 180 Alligation Alternate. 20 5 5 10 15 25 10 3 12 cts. lbs, 10 = 5 X 20 = 100 15 12 = 10 X 20 = 200 i>0 = 5 X 20 .•= 100 25 = 3 X 20 = 60 — Amount Total 23. required 460 lbs. 460 -^ 23 = 20 12. A merchant wishes to fill a barrel which will hold 200 lbs. with sugar worth respectively 8, 12 and 14 cts. per pound, so that the mix- ture may be worth 13cts. How much of each must he take ? Ans. 25 lbs. for the 1st 2 kinds and 150 lbs. @ 14 cts. 1.3. A druguist is desirous of producing; from medicine at $1.00, SI. 20, SI. 60 and $1.80 per lb., 168 lbs. of a mixture worth SI. 40 per lb. ; how mucli of each kind must he use for the purpose ? Ans. :i81bs. at SI. 00, 561bs. at SI. 20, 561bs. at SI. 60, and 28lbs. at SI. 80. 14. A jeweller melted together gold 16, 18, 21 and 24 carats fine, so as to make a compound of 51 ounces 22 carats fine ; how much of each sort did he take ? Ans. 6 ounces each of the first three, and 33 ounces of the last. 15. I have teas at 25 cents, 35 cents, 50 cents and 70 cents a pound, with which I wisli to make a mixture of 1801 bs. that will be worth 45 cents a pound. How much of each kind must 1 take? Ans. 151bs. @ 25 cts., 75 lbs. @ 35 cts., 00 lbs. @ SOcts., 30 lbs. @ 70 cts. WEIGHTS AND MEASUEES. The weights and measures used by the different nations throughout the world have been derived from very imperfect and variable standards. Thus, a foot was the length of a king's foot, and, consequently, varied as a king with a long foot or a short foot happened to reign. Henry I, king of Great Britain, decided that the yard should be the length of his own arm, from the extreme end of the longest finger to the middle ol the breast, and that tho other measures should be computed upon this. The old English pound, which was the legal standard of weight from the time of William the Conquercr to that of Henry VII, was derived from the weight of grains of wheat gathered from the middle of the ear, and well dried, 24 grains making a pennyweight, or the weight of a penny, 20 pennyweight 1 ounce, and 12 ounces a pound. Henry VII introduced the Troy pound. Weights and Measures. 181 The Avoirdupoids pound was iati"oduced during the reign of Henry VIII. An acre was as much land as a yoke of oxen could plough in a day. America, England and France have endeavored to found their systems of weights and measures upon invariable or natural standards, The standard units of linear, superficial, and solid measures, of the Dominion of Canada, are identical with thoso of Great Britain. DOMINION STANDARDS. The Dominion Standards were constructed under the direction of the Commissioner of Inland Revenue. The Standard for determining the length of the Dominion standard yard is a solid square bar, 38 inches long and 1 inch square, in trans- verse section, the bar being of bronze or gun metal (known as Baily'a metal) ; near to each end a cylindrical hole is sunk (tlic distance between tlie centres of the two holes being 36 inches) to the d(^pth of half an inch ; at the bottom of each hole is inserted in a smaller hole a gold plug QT pin, about to of an inch in diameter, and upon the surface of each pin are cut, a fine line transverse to the axis of the bar, and two lines at an interval of about yja of an inch parallel to the axis of the bar ; the measure of length of the Dominion standard yard is given by the interval between the transverse line at one end and the transverse line at the other end, the part of each line which is employed being the point midway between the longitudinal lines, and the said points are the centres of the said gold plugs or pins. There are also, on the upper side of the bar, two holes for the insertion of the bulbs, of suitable thermometers, for the determination of the temperature. The Dominion Standard for determining the weight of the Dominion standard pound isof platinum-iridium, the form being that of a cylinder nearly I'.'^S inch in height and I'lo inch in diameter, with a grooveor channel round it, the middle of which is about 0'34 inch below the top of the cylinder, for insertion of the points of the ivory fork by which it is to be lifted ; the edges are caretully rounded off, and such standard pound is marked " A." The weight of this standard, in terms of the Imperial standard, is 6999*97694 grains when both are weighed In vacuo, and <')999'98387 grains when both are weighed in air at the tempera- ture of 62° of Fahrenheit's thermometer, the barometer being at 30 inches, and for which due allowance is to be made when comparing other standards. m '■ rm n m I It; .! "^I I 182 Weights and Measures. The Dominion standard for determining the weight of the Dominion standard Troy ounce is of platinum-iridium, the form being tlial of a truncati'd cone, with a knob nearly ]r of an inch in height, iiieludin « pl: Pj. 'Il «■ 184 Weights and Mcasiires. The standard gallon or Winchester contains 231 cubic inches. Tho imperial gallon of (Jreat Britain containing 277.271: cubic inches, and is equal to 1 more tlian tho Wiiiclioster gallon. The Imperial and Winchester are leij;al in Canada. In nieasurinj^ cist(!rns, resiirvoirs, vats, etc., the barrel is estimated at 31 J j:allons and the hny 4 without a remainder, except years whose number can be divided without a remainder by 100, but not by 400. A Sidereal Year is the time in which the earth revolves round the sun, and is 365d. 6h. 9m. 9j|sec. NoTR — The 12 calendar months, composing the civil year, are January, February, March, April, May, June, July, August, Sep- tember, October, November, December, and the number of days in each may be readily remembered by the following lines : — " Thirty days hath September, April, June, and November ; And all the rest have thirty-one, Save February, which alone Hath twenty-eight ; and this, in fine, One year in four hath twenty-nine. )) fi'l; •A'^ mm EXPLANATION OF SOME CALENDAR TEEMS. Lunary Cycle. — We call lunary cycle or metonic cycle a period of 19 years, comprising about 235 revolutions of the moon, after the lapse of which the new and full moon returns to the same day of the year. It was discovered by Meton, a Greek astronomer of the 5th century before Christ, r.nd engraved in gold letters on the temple of Minerva ; hence the custom to call the Golden Number the ordeal number of each year in the lunar cycle. This year, 1891, the golden number is 11, because this yoir is in the 11th of the lunar cycle ; hence in 19 years from now, i. e. in 1910, the golden number shall be the same as this year, and the new and full moon shall return to the same rates as this year, 1891. Solar Cycle. — The solar cycle is a period of 28 years, at the end of which time the days of the month return to the same days of the week. As in a year, there are 52 weeks and one day, it must then end on Weights and Measures. 189 the same day on whieh it has begun, and the next year must begin tlie following day. If there were no leap year, every seventh year, tho year would begin by the same day, and the days of the months would return to tho same days of the week ; but, on account of leap years coming every four years, the solar cycle hnppens only every 28th year — id est — 4 times 7 years. For instance, the number of days, months and weeks in 1891 is tho same as those of 1863, and will happen again 1919. Epact. — We call epact the moon's age at the end of the year, or the number of days by which the last new moon has preceded the beginning of the year. The epact raries every year, for there are 12 lunations and 11 days. Hence by adding 11 to the Epact of the preceding we have the Epact of the present year ; but if the sum would give more than 30, subtract 30, which is a lunation, from that sum, and the number of days remaining will be the Epact re(iuired. Now, the age of the moon, at any epoch of the year, can be obtained by adding together the Epact and the number of days passed since the first of January or the first of March if we arc after that date, to the date of the month in which we are. If the sum is more than 30, we subtract 30 from it, and the balance is the Epact. Suppose I wish to find the age of the moon on the 15th of June, 1891, I would add to 20, the Epact of this year, the number of months passed since the 1st of March r= 3, and 15 days in June = 20 + 3 + 15 = 38. Now in 38, 1 have a lunation of 30 days, so if I subtract 30 from 38, I have a balance of 8. 8 will be, on the 15th of June, 1891, the age of the moon. Bomhncnl Letter. — The Dominical Letter of a year is one of the seven following letters : A, B, C, D, E, P, Gr, serving to distinguish the Sundays in perpetual calendars — that of the Breviary for instance. The 1st of January is A, the 2nd B, the 3rd C, &c. Now this year, 1891, the first Sunday in January happened to be on the 4th consequently, the Dominical Letter is D, and all the 7th days of the year will be Sundays. Next year, 1892, being a leap year, there will be two letters, C and B. The first Sunday in January was on the 3rd, C must be used up to F'ebruary 25th and B for the rest of the year. Easter Ferlod. — The council of Nicea, held in the year of our Lord 325, has decided that the feast of Easter should be celebrated the first Sunday after the first moon following the 20th of March. ,* 190 Weif/hts and Measurefi. According- to that rule, Easter can be celebrated within the space of 35 days from the 22nd of March to the 25th of April inclusively. That space is called the Easter Period. Hence, if there were full moon on the 21st of March, and if the 22nd were a Sunday, Easter would be on that day. It is the earliest date for Easter. It happened in 1818. But if there wero full moon the 20th of March, Easter would come the followinir Sunday, after the full moon of April 18th, and if the 18th would fall on a Sunday, the following Sunday would be Easter (April 25th). It happened in 188G. In this present year 1891, the Epart is 20, i. c., the lunation has already 20 days passed on the 1st of January. January and February forming just two lunations (59 days), the lunation lias already 20 days before the 1st of March and ends the 10th of March. 15 days later, Wednesday 25th, there will be full moon, consetjuently, Easter will fall the following Sunday, March 29th. Roman In diction. — We call Koman Indiction a cycle of 15 year ., instituted in Home by Constantine tne Great, in connection with tho payment of an extraordinary tribute, and afterwards made a substitute for Olympiads in reckoning time. It was much used in the ecclesias- tical chronology of the middle ages, and is reckoned from the year 313 as its origin, so that, if any j-iven year, less 312, be divided by 15, the quotient will give the current period of the Indiction, and the re- mainder the current year of the period. This period has no astronomical character, but the Popes make use of it still in dating theirs bulls or letters to the Catholic world. Our present year 1892 is tho 5th ^ Roman Indiction of the lOGth Cycle ( commenced in 1888, and which shall j terminate in 1902. J 1892 312 15 j 1580 1^105 cycle 15 80 75 N.B. — The 1st of January of any year will always fall on the day corresponding to that of the 1st of May of the preceding year. For instance, May 1st, 1892, fiills on a Sunday, then, January 1st, 1893, will also fall on a Sunday. Weights and Measur, es. 191 MISCELLANEOUS TABLE. 14 pounds of Iron or Lead 100 100 100 100 196 200 240 280 5G 20 units 12 •' 144 " 12 gross (( >( ^™ °''F1«"»- 1 cental. Kiiisins 1 , 1 cask. 1 (luintal. 1 keg'. ;^ Pork, Beof; or Fish !."'Z'"''." ! 1 Z^, " Dry Fisli '• Nails " Flour Lime " Salt = 1 firkin of butter = 1 score. = 1 dozen. = 12 <^ or 1 = 1 great 2 Rqams 5 Bundles used by the paper am. 1 Bundle 1 xJale contams ^00 Quires or 4800 Sheets. BOOKS. Names and Sizes as classified hj PuhUshers I :ri^'" '''-''' ^"^ '-^- ^-«'"^'^^ sheet when manufactured. Name of book. Folio Quarto or 4to ^ Octavo or 8vo o Duodecimo or 12mo 12 16 "^0 "."'. 32 18" 18 24 ;; 24 ^^ " 32 When a sheet is folded into leaves. ^ loaves Contain. (I 4 pages. 8 u 16 24 64 36 48 64 « u 102 rf^'l m Valuable Rules, PRINTING AND TYPE SETTING. Printers generally charge for setting type, or the composition of matter, as it is technically termed, by the number of e^ns it contains, rated by the 1000 ems; an em is the square of the body of the type. VALUABLE RULES AND TABLES. MARKING GOODS. In many commercial establishments it is customary to use a private mark, attached to the goods, denoting their cost and selling prices. Various devices are used to render the cost and selling price marks from being understood by any except those employed in the establish- ment, whose duty it is to exhibit and sell goods. Any word or phrase containing 10 different letters is taken, the let- ters of which are written instead of figures. As an illustration, let us take the following : — Now be sharp 123 45 67890 You are required to write a tag or article showing the cost aud sel- ling price. Supposing the cost to be $3.70, and selling price $4.25, the proper mark will be : — 2chp — cost; hoe — selling price. It is customary to write the cost mark above the line, and the selling below, or vice versa, thus whp boe m Hieronymus, Black Horse, Cash Profit, Cumberland, JBlacksmith, Fair Spoken, are words and phrases thatcan be used. SUGGESTIONS TO MERCHANTS. Very often, merchants in buying by wholesale, buy a great many articles by the dozen, viz. : — overcoats, pants, shirts, boots, shoes, hats, etc., etc. The dealer knows generally what one of those articles will retail for in the place where he lives ; but unless he is " quick at figures " it will take him some time before he can say whether he can afford to buy those articles and make a profitable living on them in retailing them one by one. Valuable Rules. jgo Likewise, in buying his .„„d, by auction, a mcrol.ant ha, „„t ,i„„ make the caleulation before the «o„ds are cried off; ad b in fa d b d at raado.,, he might lo.«e the chance of making. «ooJ ba^r ' t thenbecomea a useful problem to determine iostantfy what ~; "t" he would g„,„ if he retailed the goods at a certain prifc '^ ' pfr;r;:;tCk!:!-^.?^:r'tC;"i:^-- for apiece to gam 20% on the cost. ^ ^^^ 17.50 1.20 »> 35000 1750 12 I 21.0000 We multiply by 1.20, because 1 represents the cost and 20 the gain. -TO , 1.75. selling price of one shirt ^Jf they cost 825.00 per do.c„, they should' L sold at 82 50 apiece, We take 20%, because we can determine the price instanflv on^ wt^out a residue, by simply removing the deeimaCt ^' °°' 1 ho following rules will prove very useful •_ To make 20y ..move the decimal point one place ,„ the left. « <( (1 « 80%, 60%, 50%, 44%, 40%, 35%, 32%, << « (t (( (( u <( u (( c( (( (( <( <( (( (i ti l( "and add J of itself u (( (( (( (( <( ^^If I buy 1 doz. hats for $M, what shall I retail them for", ,„ake. Bemove the point one place to 2.40 the left, and add J itself. 1.20 The retail price will be = $3.60 N,?l" ^'~^^^ ^^""^l '"^'' ^'' exclusively given for mental drUl Numerous exercises should be given to students. 13 !f^' 194 Valuable Rules, MEASUREMENT OF LUMBER. A standard board ia one that is 12 feet long, 1 foot wide, 1 inch thick, and therefore contains 12 square feet, in 1 inch thick. In timber for exportation, |- inch more is allowed for skrinkage, planing, etc. The length of boards, joists, beams, etc., is measured in even inches, odd inches not being counted. JX Fi.; v: TO MEASURE INCH BOARDS. Rule. — Multiply the length of the board in feet by its breadth in inches, and divide the product by 12. Example 1. How many square feet in a board 12 feet long and 14 inches wide ? 12 X 14 = 168 -^ 12 = 16 feet. 2. A board measuring 16 ft. by 13 inches; how many sq. ft. will it give ? Ans. 17^ feet. Note. — When the board is wider at one end than the other, add the width of the two ends together, and take half the sum for a mean width. 3. How many square feet in a board 10 feet long, 13 inches wide at one end, and 9 inches wide at the other end ? Ans. 9^ feet. (13 + 9) -^ 2 = 11, mean width. Then, multiply 10 x 11 = 110 -^ 12 = 9^ feet. Note. — Sawed lumber, as joists, plank and scantlings, are now generally bought and sold by board measure. The dimensions of a foot of board measure are 1 foot long, 1 foot wide, and 1 inch thick. to measure joists, beams^ scantling and plank. Rule. — Multiply the width in inches by the thickness in inches and the product by the length in feet, and divide by 12. 4. How many feet of lumber in 14 planks 16 feet long 18 inches wide, and 4 inches thick ? Process.— 16 x 18 x 4 == 1152 -^ 12 ^ 96 feet = contents of one plank. 96 ft. is the number of feet contained in one board ; but I have 14 of the same dimensions. We then multiply 96 x 14 = 1344 feet. Ans. Valuable Rules, 195 1 incli Limber g, etc. iucbcs, udth in mg and 16 feet. . ft. will 17i feet. , add the r a mean s wide at 9^ feet. X 11 = [are now lions of a thick. Iches and Is inches Intents of I have 14 Ans. TO FIND now MANY l^'EET OP LUMBER CAN BE SAWED PROM A LOQ. (Saw-cut \ of an inch). Rule. — From the diameter of the log in inches, subtract 4 (for slabs), one-fourth of tbe remainder squared and multiplied by the length in foot will give the correct amount of lumber that can be made from any log whatever. 5. How many feet of lumber can bo made from a log which is 40 inches in diameter and 15 ft. long? Ana. 1215 foci. Process.— From 40 subtract 4 (for slabs) = 3G, then divide 30 by 4 = 9, square it, 9 x 9 = 81 x 15 (length) = 1215 feet. GAUGING (in French : — Jaugcage.) Gauging is the process of finding the contents or capacity of casks and other vessels. Cflldge is tiie difforoncc between the actual contents of a vessel and its capacity, or that part which is empty. TO MEASURE CASKS OF WINE OR BEER. Rule. — Add the bung and head diameters in inches, and divide by 2 for the mean diameter. Then scjuare the mean diameter and multiply by the length in inches, and multiply again by .0034 for wine gallons and by .0028 for beer gallons. 6. How many gallons of wine in a cask, the bung diameter of which is 25 inches, the head diameter 23, and the length 34 inches ? Ans. 63 gallons. Operation. — 25 -t- 23 = 48 -r- 2 = 24 mean diameter. 24 x 24 == 576 X 32 = 18432. x .0034 = 66.66 + gallons. 7. How many gallons of beer in a barrel whose bung diameter is 20 inches, head diameter 18 inches, and length of barrel 26 inches ? Operation.— 20 -t-18 = 38-f-2=19. 19 X 19 = 361 X 26 = 9386 x .0028 = 26.28 gals. Note. — If the cask or barrel is not full, stand it on the head, and multiply by the height of the liquid, instead of the length of the barrel, for actual contents. When the cask is much rounded from the bung to the head, do as follows : To measure a cask hy four dimensions : 1st. Add the bung and head diameters in inches and the diameter in inclies between buns and head. 190 Valuable Rules. li tli' tiH ii:' 2 J. Divido their sum by 3 fur tlic uiean dianietcr. 3d. Multiply tho wjuare of tho mean diameter by the length of tho cask in inchcH. 4th. Multiply the last produet by .0034 for wine gallon.s, .0028 for beer f^allons. Example. — What arc tho contents in gallons of a cask, tho bung diameter of which is 24 inches, tho middle diameter 20 inches, the head diameter IG inches, and its length 40 inches? Wok K.— 24 + 20 + 16 == 60 -t- 3 == 20 (moan diameter), then 20 X 20 = 400 (s({uare of mean diameter) x 40 length = 16000 x .0034 == 54.4 giiUons. 1. The ale gallon contains 282 cubic inches. 2. The wine gallon contains 231 cubic inches. 3. Tlio bushel contains 2150.4 cubic inches. 4. A cubic foot of pure w;iter weighs 1000 ounces = 62i pounds avoirdupois. 5. To find what weight of water may be put into a given vessel. Multiphj the cubic feet by 1000 for the ounces or by 62^ for the pounds avoirdupois, 6. What weight of water can be put into a cistern 7^ feet square? Ans. 26,367 lbs., 3 oz. RULES FOR DETERMINING THE WEIGHT OF LIVE STOCK. For cattle of a girth of from 5 to 7 feet, allow 23 lbs. to the super- ficial foot. For cattle of a girth of from 7 to 9 feet, allow 31 lbs. to tho super- ficial foot. For small cattle and calves of a girth of from 3 to 5 feet, allow 16 lbs. to the superficial foot. For pigs, sheep, and all cattle measuring less than 3 feet girth, allow 11 lbs. to the superficial foot. Measure in inches the girth round the breast, just behind the shoul- der-blade, and the length of the back from the tail to the forepart of the shoulder-blade. Multiply tho girth by the length, and divide by 144 for the super- ficial feet, and then multiply the superficial feet by the number of lbs. Vdluahle Jiules. 197 the ) for bang , the en 20 .(1034 poundb id. for the quarc ? ., 3 oz. If c supcr- supiir- lallow 16 \\\, allow liic shoul- Irepart of \Q super- fer of lbs. allowt.-d for Tiivo Stdck of dilVureiit u'irths, ami the product will bo the number of lbs. uf hcej\ veal, or pnr/c in tin; four (|uarti'rH of the animal. KxA.Ml'LK — What is the estimated weij^ht of beef in a steer, whose Hirth is (! feet 4 inches, and Itiij^th f) foot ',) inches ? .Soi.U'iioN. — 7<) inclu'S ^irtli, x (!;5 inches length, = 4788 -~ 144 = 33J sijuiin' fW't, X 23 — 7'>4:/ lbs., or r)4g stoiu'. Ans. NoTK. — When tho aniniiil is but lialf fattened, a deduction of one lb. in every 20 lbs. must be made; and if very fat, one lb. for every 20 must be added. BRICKLAYERS' WORK. It i.'MueaPurcd by tin. .''00 bricks laid in the wall. Tlie following scale will give a fair av rage for estimating th«; (luantity of bricks rctjuired to build a "iven nmount of avuII : 4i in. ull, per ft. uperficial, (h brick) 7 bricks. 9 (( <( a {\ brick) 14 (( 13 (( (( a (lyj brick) 21 (( 18 (( (( (1 (2 bricks) 28 K 22 (( K i( (2i^ bricks) 35 U Note— Por each lialf l)rick ad.lcil to tho clilckiiogs of tho w.ill, ailil sevoii briciii<. A bricklaycT's hod meiiKuring 1 ft. 4 in. x in. x !> "n., c(juiils 1,29G inches in capacity, and will coutuin 20 bricks. A load of mortar measures 1 cubic yard, or 27 cubic foot; requires 1 cubic yard of sand and bushels of lime, and will fill 30 hods. PLASTERERS' WORK. Plasterers' work is principally of two kinds; namely, plastering upon laths, called ceiling, and plastering upon walls or partitions made of framed timber, called rendering. In plastering upon walls, no deductions are made except for doors and window ', because cornices, festoons, enriched mouldings, etc., arc put on after the room is plastered. In plastering timber partitions, in large warehouses, etc., where several of the braces and larger timbers project from the plastering, a fifth part is commonly deducted. Plastering between their timbers is generally called rendering between quarters. Whitening and coloring are measured in the same manner as plaster- ing ; and in timbered partitions, one-fourth or one-fifth of the whole 198 Valuable Rules. area is commonly added for the trouble of coloring the sides of the quarters and braces. Plasterers' work is measured by the yard square, consisting of nine square feet. In arches, the girt round them, mutiplied by the length, will give the superficies. Example 1. — If a ceiling be 59 feet 6 inches long, and 24 feet 6 inches broad, how many yards does that ceiling contain ? PAINTEES' WORK Is computed by the superficial yard ; every part is measured that is painted, and an allowance is added for difficult cornices, deep mould- ings, carved surfaces, iron railings, etc. Charges are usually made for each coat of paint put on, at a certain price per yard per coat. If you furnish the paint and the oil, charges will be so much a day by the painter. FOR MECHANICS. A stick of timber is carried by three men, one carries at the end, and the other two with a lever. How far should the lever be placed from the other end, that each man may carry equally ? Rule. — Divide the length of the stick by 4, and the quotient is the answer. There is a stick of timber, 30 foot long, to be carried by three men : one carries at the end, the other two carry by a lever ; how far must the lever be placed from the other end, that each may carry equally ? Ans. Y J feet from the end. GRAIN MEASURE. To find the quantity of grain in a bin or wagon. Rule. — Multiply the height, length, and breadth together, in inches, and divide by 2150.42: the quotient will be the number of bushels. To find the quantity of grain when heaped on the floor in the forw of a cone. Rule. — Square the depth and square the slant height, in inches ; take their difference and multiply by the depth, and this product by .0005 ; the result will be the contents, in bushels. Valuable Rules, jgg To find the quantity of grain heaped against a straight wall. ^^VLJi.~Square one-half the depth, and proceed as in the previous WEIGHT OP GRAIN PER BUSHEL, AS ESTIMATED AMONG SHIPPING AGENTS. W^eat 60 lbs. ( Oats, %«' 56 " Corn. Barley, 48 *' 35 lbs. 56 " THE METRIC SYSTEM. The Metric System of Weights and Measures is based upon the decimal system of notation. It is authorized to be used in the Uniteo' States, but not in Canada However, as those measures might be authorized, in Our Dominion" sooner or later, I thought it proper to introduce them in this arithl metic. The PHncipal Units, or those from which the others are derived are : — ' The Meter, which was intended to be, and is nearly, one ten- millionth of the distance on the earth's surface from the equator to the pole. ^ The Square Meter, which is the square whose side is one meter • and the Are (pronounced air), which is the square whose side is ten meters. The Cubic Meter, or Stere (pronounced stair), which is the cube whose edge is one meter; uud the Liter (pronounced leeter) which is one thousandth of a cubic meter. The Gram, which is the" weight, in vacuum, of a cubic centimeter of distilled water, at the greatest density. The Higher Denominations are expressed by prefixing to the name of the principal unit, ° Deka, 10; Hecto, 100; Kilo, 1000; Myria, 10,000; and the Lower Denominations by prefixin^r Deci, 10th ; Centi, 100th ; Milli, 1000th. Note. Kilo is pronounced kiVo, and Deci, des'6. 200 Metric System. Measures op Length. 10 mil'limefers (mm.) 10 centimeters, 10 decimeters, 10 meters, 10 dckametcrs, 10 hectometers, 10 kilometers. : 1 cen'timeter, 1 dcc'imeter, 1 METER, 1 dek'ametcr, 1 hoc'tometer. cm., : = .3937 inch. dcm.. 3.937 inches. me., 39.37 inches. dkm., 393.7 inclies. hm.. 328 ft. 1 in. km.. 3280 ft. 10 in. caym., 6.2137 miles. 1 kil'ometer, 1 myr'ianicter, mym., Note. 1. The meter is used as the unit of measure in measuring cloth, and common lengths and distances, and the kilometer in measur- ing long distances, as the length of roads and rivers, distances between cities, etc. Note 2. A meter is very nearly 3 feet 3 inches and 3 eighths of an inch in length ; and a Icilomcter is about 200 rods, or 3300 feet, or f of a mile. Measures of Surface. 100 sq. milUmeters (mm*.) =1 sq. centimeter, cm*., =•= .00155 sq. in. 100 sq. centimeters, 1 sq. decimeter, dcm*., .107G sq. ft. 100 sq. decimeters, 1 SQ. meter, m*., 1.19() sq. yd. Also, 100 cen'tiares (ca.), or sq. me., =1 are, ar., := 119. G sq. yd. 100 ares, I hectare, ha., 2.471 acres. Note 1. The square meter is used in general in measuring sur- faces. The are is the principal unit in measuring land ; and the hec- tare is also taken as a unit of the same kind of measurement. Note 2. A square meter is about 11 square yards, and an are nearly 4 square rods. Measures op Volume. 1000 cu. milUmeters (mm^), 1 cu. centimeter, cm\,=.06l cu. m. 1000 cu. centimeters, 1 cu. decimeter, dcm^, 61.022 on. in. 1000 cu. decimeters, 1 cu. .meter, m'., 1.308 cu. yd. Also, 10 dec'isteres (dcs.) = 1 stere, or cu. meter, st , =1.308 cu. yd. 10 steres 1 d^k'asterc, dks., 13.08 cu. yd. Metric System. 201 sur- yd. yd. And 10 mil'liliters (ml.) = 1 cen'dUfer, cl., = .338 fluid oz. 10 centiliters 1 dec'ilitcr, del., .845 llq. gill. 10 deciliters, 1 liter, It., 10 liters, 1 dek'aliter, dkl., 10 dekaliters, 1 hec'toUter, hi., 10 Iiectoliters, 1 kil'oliter, kl., Note 1. The cubic meter is used in general in measuring volume, and the stere in estimating wood. 1.0567 liq. qt. 2.6417 liq. gal. 2 bu. 3.35 pk. 1.308 cu. yd. Note 2. The liter is used in measuring liquids; it is about l^\ liquid quarts, or i of a dry quart ; 1000 liters are equal to 1 cubic meter. Note 3. The hectoliter, or for brevity hecto, is generally taken as the unit in measuring dry articles, such as grains^ salt, etc., and is about 2§ bushels, or I of a barrel. 10 mil'ligrams (mg.), 10 centigrams, 10 decigrams, 10 grams, 10 dekagrams, 10 hectograms, 10 kilograms, 10 myriagrams, 10 quintals, Weights. : 1 cen'tigram, 1 dec'i<>;ram, dcg., 1 Gram, gm., 1 dck'agram, dkg., 1 hec'togram, hg., 1 hiVogram, k., 1 myr'iagram, myg., 1 quin'tal, q., 1 foiineau, t.. = .1543 grains. 1.543 grains. 15.432 grains. .3527 av. oz. 3.5274 av. oz. 2.2046 av. lb. 22.0 iO av. lb. 220.46 av. lb. 2204.6 av. lb. Note 1. The gram is the unit in weighing gold, silver, and jewels, in mixing medicines, determining postage, etc. ; the Idlogram, or, for brevity, kilo, is the ordinary weight of commerce ; and the tonneau, or metric ton, is used in weighing coal, hay, and all very heavy articles. Note 2. A kilo is about 2i avoirdupois pounds, or 2.68 Troy pounds ; and a metric ton a little more than 2200 avoirdupois pounds. Equivalents of denominations of common weights and measures in metric denominations are exhibited in the followin"- 202 Metric System. Comparative Table. ■si ' i|^: An inch = 2.54 centimeters. A gallon — 3.786 liters. A foot 30.48 centimeters. A bushel .3524 hectoliter. A mile 1.6094 k iometers. A cu. inch. .01639 liter. A sq. inch .0006452 sq. meter. A cu. yard .7646 stere. A sq. foot .0929 sq. meter. A cord 3.625 stere. A sq. yd. .8362 sq. meter. A grain .0648 gram. A sq. rod .2529 are. A Troy lb. .373 kilo. An acre .4047 hectares. An av. lb. .4536 kilo. A sq. mile 259 hectares. A com. ton .9071 tonneau. The numbers in the tables are determined approximately, but are sufficiently exact for all ordinary purposes. In expressing Metric Weights and Measures the decimal point, as in Canadian money, is placed between the unit, and its subdivisions decimally written. ARITHMETICAL PROGRESSION. Quantities are said to be in Arithmetical Progression when they increase or decrease by a common difference. The Terms of a series are the numbers of which it is composed. The first and last terms are called the extremes, and the intermediate terms, the means. The common difference is the number' added or subtracted, in order to form each successive term. An Ascending Series is produced by adding the common difference to each term successively : 2, 4, 6, 8, &c. A Descending Series is produced by subtracting the common diflference from each term successively ; as, 8, 6, 4, 2. First Case. The first term, the common difference, and the number of terms being given, to find the last term. Example 1. — The first term of an ascending series is 3, the com- mon difference 4, and the number of terms 21 ; what is the last term ? Number of terms minus one = 21 — 1 =20 Common difference = 4 80 First term= 3 p*t Last term= 83 Arithmetical Progression. 203 RuLE.—Multiply the number of terms, less 1, by the common diflference, and add the first term, and you shall have the last term. 2. If the first term is 1, the number of terms 7, and the common difierence 6, what is the last term ? Ans. 37. 3. If a man travel 7 miles the first day of his journey, and 9 miles the second, and shall each day travel 2 miles further than the preced- mg, how far will he travel the twelfth day ? Ans. 29 miles. 4. If A set out from Sherbrooke for Boston, and travel 20J milec the first day, and on each succeeding day 1^ miles less than on the pre- ceding, how far will he travel the tenth day ? Ans. 6| miles. Second Case. To find the common difference, when the extreme and number of terms are given. Example l._The first term of a series is 3, the last term is 15, and the number of terms is 7 ; what is the common difference ? Ans. 2. Difference between the extremes 15 3=12 Number of terms minus one =7 — 1=6. 12-f 6= 2 2. The extremes of a series are 3 and 35, and the number of terms is 9 ; what is the common difference ? Ans. 4. OPERATION. 3 5—3 = 4, common difference. 9—1 3. If the first term is 7, the last term 55, and the number of terms 17, required the common difference ? Ans. 3 4. If the first term is 4, the last term 14, and the number of terms 15, what IS the common difference ? Ans. 4 5. If a man travels 10 days, and the first day goes 9 miles, and the last 17 miles, and increases each day's travel by an equal difference what IS the daily increase ? Ans. -« miles. ' EuLE.— Divide the difference of the extremes by the number of terms minus one. 204 Arithmetical Progression. m m 1^ lit. J. I : Third Case. To find the number of terms, when the extremes and common diiFerence are given. 1. If the extremes of a series are 4 and 44, and the common differ- ence 5, what is the number of terms ? Ans. 9. OPERATION. 4 4—4 + 1=9, number of terms. 5 Rule. — Divide the difference of the extremes by the common differ- ence, and increase the fjuotient by one. 2. The first term is 8, the last term 203, and the common difference 5 ; what is the number of terms ? Ans. 40. 3. A man going a journey travelled the first day 7 miles, the list day 51 miles, and each day increased his journey by 4 miles ; how many days did he travel ? Ans. 12. 4. The extremes are 2h and 40, and the common difference is Tj ] what is the number of terms ? Ans. 6. 5. Jn what time can a debt be discharged, supposing the first week's payment to be $1, and the payment of every succeeding week to increase by $2, till the last payment shall be $103 ? Ans. 52 weeks. 6. A man going a journey travelled the first day 8 miles, and the last day 47 miles, and each day increased his journey by 3 miles. How many days did he travel ? Ans. 14 days. Fourth Case. To find the sum of all the terms, when the extremes and the num- ber of terms are given. Example 1. — The extremes are 3 and 21, and the number of terms 8 ; what is the sum of the series ? Ans. 96. 2+211=24-^2=12 12x8=96. Rule. — Multiply half of the sum of the extremes by the number of terms. 2. The extremes of an arithmetical series are 3 and 19, and the number of terms 9 ; what is the sum of the series ? Ans. 99. Arithmetical Progression. 205 3 A man bought 16 acres of land, giving 81 for the first acre, and mi for the last acre; the prices of the successive acres form an arithmetical progression. How much did the 16 acres cost ? Ans. $976. 4. A gentleman wishes to discharge a debt in 11 annual payments such that the last payment shall be $220, and each pajn.ent 'reater than the preceding by $17 ; what is the amount of the debt, and the first payment? Ans. First payment, $50. 5. A merchant bought 20 pieces of cloth, giving for the first $2, and for the last $40, the prices of the pieces form an arithn.;,tical series ; how much did the cloth cost ? Ans. $^20 ANNUITIES. An Annuity is a sum of money to be paid annually, or at any other regular period, either for a number of years or for ever. The "Present Value" of an Annuity is a sum which, being placed at interest, will be suflicient to pay the Annuity. The "Amount" of an Annuity is the interest of all the payments added to their sum. ^ '' An Annuity is said to be in arrears when one or more payments are retained after they have become due. EXAMPLE.-Joseph bought a house for $5,000, and agreed to pay for It m 5 years, paying $1,000 yearly; but finding himself unable to make the annual payments, he agreed to pay the whole amount at the end of the 5 years, with the simple interest, @ 6%, on each pay- ment from the time it became due till the time of settlement ; what did the house cost him ? Ans. $5,600. Explanation— 5th payment will be $i qoo 4th pay. will be on interest 1 year and will amount to 1,060 3rd " " '' " <« 2 " <' '< u i'i20 2nd " " '< " «. 3 a u u a /jg^ 1st - - « «^ u 4 u u u .. j|24Q Amount... $5,600 w\ 'Hi 206 Annuities. The several Hums will form an arithmetical series, thus : — 1000, 1060, 1120, 1180, 1240 ; of which the 5th payment, or the annuity, is the first term, the interest on the annuity for one year the common difference, the time in years, the number of terms, and the amount of the annuity the sum of the series. 1000 + 1240 = 2240 -=- 2 = 1120 1120 X 5 = $5,600 2. What will an annuity of $250 amount to in 6 years @ 6% simple interest? Ans. $1,725. 3. )^hat will an annuity of $380 amount to in 10 years @ 5% simple interest ? Ans. $4,655. 4. An annuity of $825 was settled on Dick, January 1st, 1888, to be paid annually. It was not paid until January 1st, 1892; how much did he receive, allowing 6% simple interest ? Ans. $3,597* 5. Arthur let a house for 6 years @ $300 a year, the rent to be paid semi-annually, @ 8% per year simple interest. The rent, how- ever, was paid at the end of the 6 years only. What did Arthur receive ? Ans. 6. A certain Sherbrooke gentleman has a very fine house, which he rents @ $50 per mouth. His tenant omitting to pay until the end of the ysar, what sum should the owner receive @ 12% simple interest ? Ans. 7. What will an annuity of $450 amount to in 8 years @ 8% simple interest? Ans. COMPOUND ANNUITIES. An Annuity is at Compound Interest when compound interest is reckoned. Rule. — Multiply the amount of $1 for the given time and rate found in the following table by the annuity, and the product will be the required amount. liif I* ' ■ Compound Annuities. TABLE 207 SHOWING THE AMOUNT OP 81 ANNUITY AT COMPOUND INTEREST, PROM 1 YEAR TO 40. Years. 5 per cent. 6 per cent. Years. 5 per cent. 6 per cent. 1 1.000000 1.000000 1 21 35.719252 39.992727 2 2.050000 2.060000 22 38.505214 43.392290 3 3.152500 3.1836U0 23 41.430475 46.99582m 4 4.310125 4.374616 24 44.501999 50.815577 5 5.525631 5.637093 25 47.727099 54.864512 6 6.801913 6.975319 26 51.113454 59.156383 7 8.142008 8.393H38 27 54.669126 63.705766 8 9.549109 9.897468 28 58.402583 6S.528112 9 11.0265G4 11.491316 29 62.322712 73.639798 10 12.577893 13.180795 30 66.438847 79.058186 11 14. 206787 14.971643 31 70.7(50790 84.801677 12 15.917127 16.8b!)941 32 75.298829 90.889778 13 17.712983 18.8821-?8 33 80.063771 97.343165 14 19.598632 21.015066 34 85.066959 104.183755 15 21.578564 23.275970 35 90.220307 111.434780 16 23.657492 25.672528 36 95.836323 119.120867 17 25.840366 28.212880 37 101.628139 127.268119 18 28.132385 30.905653 38 107.709546 135.904206 19 30.539004 33.759992 39 114.095023 145.058458 20 33.065954 36.785591 40 120.799774 154.761966 EXAMPLES FOR PRACTICE. 1. What will an annuity of $378 amount to in 5 years, at 6 per cent, compound interest ? Ans. ^2130.821+. operation. Table 5.6 3 7 9 3 x 3 7 8 = $ 2 1 3 0.8 2 1+. 2. What will an annuity of $1728 amount to in 4 years, at 5 per cent, compound interest ? Ans. $7447.896+. 3. What will an annuity of $87 amount to in 7 years, at 6 per cent, compound interest ? Ans. $730.2634-. 4. What will an annuity of $500 amount to in 6 years, at 6 per cent, compound interest ? Ans. $3487.659+. 5. What will an annuity of $96 amount to in 10 years, at 6 per cent, compound interest ? Ans. $1265.356+. 6. What will an annuity of $1000 amount to in 3 years, at 6 per cent, compound interest ? Ans. $3183.60. INDEX TO TriE NEW PRACTICAL ARITHMETIC. Dedication ^^«*'8 Preface ■•••• ^ Preliminaries ^ Numeration '^ Spoken Numeration Written " ^ Arabic Notation " ^ Roman " ° Aritlimetical Signs *///^ \^ Addition-Expeditious and Comme^ciarMcthods!!!!!^///!!!^ J4 8 1st Method. 2nd " 3rd " 4th " i( 5th 6th Yth First Rule... Second Kule. u 14 15 17 1!) 19 19 20 22 Problems on Addition ..*] "^ Subtraction '^^ 1st Case 28 2nd '• 3ZZ*3 "^ Practical Applications ..*. ^^ Multiplication ' •- ^0 -.jr 1.. .. . „ ' Ql Multiplication Table ./ Problems 32 Practical Hints ^^ Suggestions Pratiques ^^ H ^^ -'■U m jj^?ii 210 hulex. PAdKH Division 41 Khort Division 42 Problems on Division 43 Memory Drills on Division 44 Vulgar Fractions 45 Gonoral Principles of Fractions 46 Cancellation 4G Drills on Cancellation 47 Greatest Common Divisor 48 Common Denominator 49 Least Common Multiple 49 Reduction of Fractions &0 Addition of Fractions 52 Subtraction " 55 Multiplication " ,. 57 Division " 59 Review of Vuljijar Fractions 01 Decimals and Decimal Fractions 63 Numeration of Decimals G3 Method by Zeros 64 Addition of Decimals 65 Subtraction of " 66 Multiplic, of " 67 Division of '' 67 Decimal Fractions 68 Money 68 Monetary Union 69 Proportion 71 Simple Proportion 72 Examples on " " 73 Compound Proportion.. 75 Problems on " " 76 Conjoined ** 78 Problems on" " 79 Percentage 80 Ist Case 81 2nd " 82 3rd " 83 4th " 83 w Index. 211 PAOEH HtwicnvoQ thu four ('Iihcs HR CoiQin'Lsaiou, Brukcnigc and Stockfl 88 Sturagi! 03 Iwt Mcthoa !)4 2Dd " or» Interest [Hi Ist Case .... 07 2ad " 08 .3rd " 100 The t) percent. Metliod 102 Kxact Method of Computing Interest 105 Tables lOG Partial Payments 115 Compound Interest 118 Tables 118 How to compute Compound Interest without tlie assistance of a table 121 Table showing in how many years a given principle will double itself 12:J Taxes 12:^ Insurance 12() Discount Ki;s Commercial Discount 133 True '« i:U Banking and Bank Discount 137 Banking 138 Bank DiL^couut > 130 Customs 142 Tables 146 I'roblcms 150 Partnership 152 Exchange 157 Domestic Exchange 158 Foreign " ItiO Problems H)2 Profit and Loss 1()4 Equation of Payments 167 Different Rules 167 Cojiipound Equation 171 c m ': 8 f^ yi m 212 Index. PAGES Alligation 175 Medial , 175 Alternate 177 Weights and Measures 180 Origin 180 Dominion Standards 181 Avoirdupois Weight 182 Liquid Measure 183 Troy or Mint Weigh. 184 Diamond Weight 184 Apothecaries' " 185 Dry Measure 185 Linear or ]iong Measure 186 Surface or Square Measure 187 Surveyors' Measure 187 Time Measure 187 Solar Year 187 Julian '' 187 Gregorian Year 188 Common '' 188 Sidereal " 188 Explanation of Some Calendar Terms 188 Lunary Cycle 188 Solar '' 188 Epact 189 Dominical Letter 189 E:ister Period 189 Roman Indiction 190 How to find on what day fills Jan. 1st, of any year 190 Miscellaneous Table 191 Paper and Books 191 The number of folds and pages in a single sheet when manu- factured 191 Printing and Type-Setting 192 Valuable Rules and Tables. 192 Marking Goods 192 Suggestions to Merchants 192 Measurement of Lumber 194 Several Rules 194 Index. 213 Gauging—Practical Rules ^^^^ll Kulos for detcrmiDing the Weight of Live' Stock ;*;.';*; V^f. Bricklayers' Work ^'^^ Plasterers' " ^^^ Painters' <' ^^^ Rule for Mechanics "."!.*!!.'.'.'.*.*.*.*.'! ^^^ Grain Measure .*, ^^^ The Metric System .S,S./S,/....,. ^^^ Measures of Length.."'""."'.'.* onn " " Surface :::;:;::;;;:z:;; Zl " "Volume ^00 Weights ;;;;; 200 Comparativf) Table.... " Arithmetical Progression '. ^^^ IstCase 202 2nd - .'";; 202 3rd " 203 4th « ..l.ZZ**"'"Z*"'''"* '^^^ Annuities, at Simple Interest" .*.'.'. l^^t " Compound « ^ Table ^ 206 207 <^l I mimi imfiUEE 4 [eiijs, COMMISSrONNAIRES-IMPORTATEURS, sLIbrairle, Papcterie et Objets Religleux, o CO eo riMAGERIE, VASES ET FLEURS POUR EOLISES, Fourqiture dt Classes et de Bureaus. B 375 St. Paul. :^ . . •• = V >f» .« r ontreal. BEBUIID, limiE'E 4 GEllllS, i StatiooGry, Religious Articles and Pictures VASES AND FLOWERS FOR CHURCHES, * Full Line of all Scliool Requisites. ">,