536 ARITHMETIC tional j-arts. them under each other. Multiply the product of all the second terms by the third term, and divide the result by the product of the first terms. Ex. If 36 men, working 10 hours a day, perform three- fifths of a piece of work in 17 days, how long must 25 men work daily to do the rest of it in 1 6 days ? Men, 25 : 36 :: 10 Fifths of the work, 3 : 2 Days, 16:17 36x2x17x10 51 = = IQi hours a day. 25x3x16 5 The length of the day will be greater the fewer the men and the fewer the days are, and less the less the work is ; we therefore state as above. 4 L It not unfrequently happens that ratios have to be compounded, or other reductions made, before we can state the proportion which will give the required result. We give an example or two of this. Ex. 1. Though the length of my field is one-seventh greater than that of my neighbour s, and its quality is one- ninth better, yet, as the breadth of mine is one-fourth less, his is worth five guineas more than mine. What is my field worth 1 The length of his field is to that of mine as 1 to li, i.e., as 7 to 8. The other ratios are 9 : 10 and 4 : 3 (see 16 ad fin.}. Therefore the values of the fields are as 7 x 9 x 4 to 8 x 10 x 3, i.e., as 21 to 20, and the difference of these values being 5, 5s., we have the proportion 1 : 20 :: 5. 5s., which gives 105 as the worth of my field. Ex. 2. If 9 men or 15 women, working 10 hours a day, could reap a field in 8 days 6 hours, in how many days of 10^ hours each could 10 men and 12 women reap a field one-fourth larger ? Since 15 women do as much as 9 men do in the same time, 12 women will do 7^ times a man s work, for 15: 12:: 9: 74. Therefore 10 men and i r -ill i ~ i 12 women will do 17^ times a mans work. From this the stating in the margin follows. The result is 56 hours, i.e., 5 days 3| hours, the day being 10 hours long. Ex. 3. A dealer who has bought 9 oxen and 5 sheep for 186, 2s. 6d., would lose 2 by exchanging 2 oxen for 1 1 sheep. What is the price of an ox ? The price of 2 oxen being the price of 11 sheep and 2 more, the price of 9 oxen will be (from the ratio 2 : 9) 49^ times the price of a sheep and 9 more. Hence from the data, 54 times the price of a sheep and 9 more will amount to 186, 2s. 6d. ; i.e., 54 times the price of a sheep is 177, 2s. 6d., and therefore a sheep cost 3, 5s. Also, since the price of 2 oxen is that of 1 1 sheep and 2 more, 2 oxen cost 35, 15s. +2, i.e., 37, 15s.; there fore 1 ox cost 18, 17s. 6d. 45. Proportional Parts. To divide a number or quan- tity i n to parts in proportion to given numbers, state and work out the proportions, As the sum of the given num bers is to each of them in succession, so is the number to be divided to the several parts required. If, for example, a bankrupt owes A 580, B 935, C 675, and D 770, and his assets amount to 999, the stating 2960: 580: 999 gives 195, 15s. as A s share, and the others are found similarly. Here, too, there may be a compound proportion, as when different sums are invested for different times. The divi sion in those cases must be in proportion to the amounts invested, and also to the time ; each amount is therefore to be multiplied by its time. Applications of Proportion. 46. In commercial and financial transactions frequent use is made of proportion ; and very often, when it is not directly employed, compu . 71 . , _ i-Tr.iO . 5 r tations are performed according to formulae or rules which rest on this as a basis. Advantage is very generally taken of the convenience of 100 as a standard of reference or comparison, proportional relations being stated as at such and such rates per cent. This occurs continually in the calculation of interest, discount, stock-exchange operations, &c., as well as in the expression of mercantile losses and sains. 47. Interest is the allowance given by the borrower to Interest, the lender for the use of money lent. It is usually com puted at a rate agreed upon of so many pounds for every hundred lent for a year ; this is called the rate per cent. The interest of 564, for instance, for 3 years 4 months at 3| per cent, per 100 : 564 :: 3| annum, is to be found by a compound 1 : 3J proportion, the meaning being If the interest of 100 for 1 year be 3|, what will the interest of 564 be for 3 years] The result (70, 10s.) may be obtained by the general rule based on this and similar proportions Multiply the amount lent (called the prin cipal) by the rate per cent, and by the number of years, and divide the product by 100. When the time is given in days, the fraction of a year is taken that the days amount to. Money is laid out at Compound Interest, when at the end of a year or other assigned period the interest that has accrued is not paid to the lender, but is put to interest along with the amount originally lent, Here the simple interest has to be computed for each successive year or period, and added to the principal or former amount. 48. Commission is the allowance paid to an agent for Commit, transacting commercial business, and usually bears a fixed sion - proportion or percentage, as may be agreed on, to the amount of value involved in the transactions. Brokerage BrokcrajJ is the allowance paid to a broker for buying or selling shares in the public funds, or bargaining otherwise with reference to money investments. Insurance Premiums insurant are payments in return for which the owner of the pro perty insured is entitled to receive the assured value of it in the event of its being destroyed. In all these the rates are commonly stated at so much per cent., and the com putation is similar to that for interest, but simpler, as the element of time does not enter into it. 49. Discount is a deduction allowed for a payment Disco-air, being made at a date prior to the time when the full amount is exigible. The true discount on a money payment duo on the expiration of a certain time is the excess of the amount over its present value the present value being the sum which, laid out at interest, would in the given time amount to the given sum. Suppose, for instance, it is required to find the true discount on 664 due 10 months hence at 4| per cent. Here the interest of 100, being 4| in 12 "months, will in 10 months be 3f, i.e., 100 would amount to 103f in 10 months, and the discount of 103| due 10 months hence is therefore 3f. Hence the stating 103| : 664 ::3|, which gives as the dis count 24. The present value is seen from this to be 640. A banker or merchant, in discounting a bill, charges interest instead of discount, and would in this in stance gain the interest of 24, since the 24, the true discount of 664, is evidently the interest of 640. 50. In calculations relating to mercantile Profit and Profit Loss, which are also effected by proportion, it must be loss. carefully remembered that the percentage of gain or loss is always reckoned on the buying price, unless the contrary is expressly stated. Thus, let it be required to find the cost of goods that are sold for 448, 17s. 6d. at a loss of 6^ per cent. Since goods that cost 100 are sold for 93|, we have the stating 93J : 448 ::100, which gives
478, 16s. as the cost price. (o. M A.)