APPLICATION OF ARITHMETIC TO COMMERCIAL CALCULATIONS. PARTNERSHIP. 120. PARTNERSHIP is the method of dividing any quantity into any proposed number of parts, having a given ratio to one another. By it the gains or losses of partners in trade are adjusted, the effects of bankrupts are divided amongst creditors, and contributions are levied. When two or more partners invest their money together, and gain or lose a certain sum, it is evident that the gain or loss ought not to be divided equally among them all, unless each partner contributed the same sum. Suppose that P contributes 3 times as much as Q, it is evident that P's share of the gain or loss ought to be 3 times as much as the share of Q; hence dividing the gain or loss into 4 equal parts, P must receive or pay 3 of these parts, and Q one of them. Ex. 1. A ship is to be insured, in which P has ventured £2500; Q, £3500; and R, £4800. The expense of insurance is £495. 10s.; how much must each pay of it? The entire amount of money risked is £10800, and if the expense of insurance be divided into 10800 equal parts, each of them will express the expense of insurance for one pound of capital; consequently the sum that each must pay will be expressed by × 2500; £495. 10s. × 3500; and £495. 10s. X 4800. £495. 10s. 10800 These results are furnished by the following proportions, in which the first term is the sum of the money risked by all the partners, viz. £10800. Ex. 2. Three persons, A, B, C, have a pasture in common, for which they are to pay £30 per annum, into which A put 7 oxen for 3 months, B put 9 oxen for 5 months, and C put 4 oxen for 12 months; how much must each person pay of the rent? The principle in questions of this kind is, that the same sum should be paid for the keep of one ox for one month or one year by each person. Now since A put in 7 oxen for 3 months, he might have had an equal share of the pasture by putting in 7 oxen x 3, or 21 oxen for 1 month. In like manner, B might have put in 9 oxen x 5, or 45 oxen for 1 month, and C might have put in 4 oxen x 12, or 48 oxen for 1 month. Hence, if we divide £30 into 7 × 3 +9 × 5+ 4 × 12, or 114 equal parts, A must pay 7 x 3 or 21, B must pay 9 x 5 or 45, and C must pay 4 × 12 or 48 of those parts. The three persons A, B, C, must therefore pay respectively, or £5. 10s. 6d. P, £11. 16s. 10d. Fr, and £12. 12s. 74d. To. This question may consequently be resolved in the following manner : £. £. S. d. 7 x 3 114: 45: 30: 11 6 114: 48:30:12 The reasoning employed above may be conducted in a somewhat different manner; thus, suppose one pound were charged for the pasturage of one ox for a month, it is obvious that A would have to pay 7 pounds for having 7 oxen for 1 month, and consequently £7 x 3, or £21, for having 7 oxen in the pasture for 3 months. In like manner B would have to pay £9 × 5, or £45, for having 9 oxen at pasture for 5 months, and C, £4 × 12, or £48, for having 4 oxen at pasture for 12 months. Hence it is evident that the rent must be divided amongst them in such a manner, that if it be divided into 114 equal parts, A must pay 21, B 45, and C 48 of these parts. INTEREST. 121. INTEREST is the sum of money paid for the use of other money, and is always estimated at so much for £100 during a year. Thus, if £100 are lent at 4 per cent., it must be understood to mean 4 per cent. per annum, that is, that £4 are paid annually for the use of £100. Principal is the money lent; the rate per cent. is the interest of £100 for a year; and the amount is the interest and principal together. Simple interest is only the interest of the principal for the whole time it is lent, and compound interest is not only the interest of the principal for the whole time it is lent, but if the interest is not paid at the stated intervals it is considered as principal as soon as it is due, and then the original principal, together with the unpaid interest, forms a new principal, the interest of which becomes due at the next stated time of payment. Ex. 1. Find the interest of £355. 12s. 6d. for 4 years at 4 per annum. £. £. 8. 100: 355 12 per cent. In practice, it is usual to multiply the principal by the rate per cent., and by the number of years; then to divide by 100, as in the margin. If the interest be required for any number of days, we must find the interest for one year, or 365 days, and then by the Rule of Three find the interest for any given number of days 122. To find the interest of any sum at compound interest, it is necessary to find the amount of the principal and interest at the end of the first year, because it is this amount on which interest must be charged at the end of two years. Ex. 2. Find the interest and amount of £400 for 3 years at 5 per cent. per annum, compound interest being allowed. As £5 is of £100, the interest of any sum at 5 per cent. per annum is found by dividing that sum by 20; hence we have, 123. But the best way of performing calculations in interest is to find the amount of one pound for any given number of years at the proposed rate of interest, and then to multiply the result by the number of pounds in the given sum. Thus in the last example, the interest of £1 for 1 year, at 5 per cent., is 1 x 5 £105, or £105 of a £. or 05 of a £., and the amount is Again, the interest of 05 of a £. is ⚫05 X 5 or 0025 of a £., and the interest of £1 being 05 of a 100 = £. 1053 = £. 1.157625 400 £463 050000 20 pound as before, the interest of £1.05 is 05 of a £. + 0025, or (05) of a £.; hence the amount at the end of two years in pounds is 1 +05 × 2 + (·05), or (1·05)2. In like manner, the amount in pounds at the end of three years is (1·05)3, and if this be the amount of £1 in three years, the amount of £400 will be 400 times £(1.05). The operation by this method will be as in the margin; but when the number of years is very great, the labour is so enormous that recourse must be had either to tables of interest already calculated for Amount is £463. ls. £1, or to logarithms. 1'00s. If a certain sum of money has at simple interest amounted to £750 in 4 years at 5 per cent., the original sum may be found in the following manner. Take £100, and find what that sum would amount to in 4 years at 5 per cent., simple interest. The interest for one year is £5, the interest for 4 years is £20, and the amount is £120. Now £100 will have the same ratio to £120 which the original sum has to £750; hence, inversely, £. £. £. = £625, the original sum. 120 124. Commission and Brokerage are charges made by persons acting as agents or brokers, as a remuneration for their skill and trouble in executing business confided to their management, and are calculated at a certain rate per cent. on the amount of the transactions. 125. Insurance is a per centage paid to those who engage to make good to the payers any loss they may sustain by accidents from fire, or storms, etc., up to a certain amount named in the agreement. The sum of money paid for insurance is called the premium, and it is reckoned by a per centage upon the amount insured. All questions relating to commission, brokerage, and insurance, are solved by the Rule of Three, as examples in simple interest. PUBLIC FUNDS, OR STOCKS. 126. The Public Funds, or, as they are sometimes called, Stocks, are the debts contracted by the Government, and are transferable at pleasure from one person to another. The exigencies of a country often compel the governing body to negociate a loan with some monied persons, or great capitalists, who contract with the Government to supply the required sum on condition of receiving a certain interest or annuity until the money is repaid. The contractors do not advance the whole of the money themselves, but bring the stock into the market, and if they dispose of it for more or less than the contracting price, they gain or lose accordingly, and in this way some persons amass immense fortunes, whilst others are ruined. The business in the public funds is transacted at the Stock Exchange, and is confided to the agency of stock-brokers, who usually charge or 2s. 6d. per cent. on the amount of stock bought or sold. Similar contracts are made by large commercial companies, such as the Bank of England, the East India Company, and some Railway Companies, and their Stocks are denominated accordingly Bank Stock, East India Stock, etc., but we shall only allude to one or two of the Government stocks. Consols is a description of stock bearing 3 per cent. interest, and it is so named from several annuities being consolidated together, and their dividends are now chargeable on the Consolidated Fund, that is, the permanent taxes. Reduced Annuities are those which have been reduced from a higher to a lower rate of interest. Long Annuities are stock which terminate in 1859 or 1860, and are quoted in the price of stocks at so many years' purchase; thus, if the quotation be 8, then for an annuity of £30, until 1860, there must be paid 84 × 30, or £255. Exchequer Bills are a kind of promissory notes issued from the Exchequer, and entitle the holder to receive the sums for which they are drawn, with interest, when they are advertised to be paid off, which is usually about 12 months from the time they are issued. Exchequer Bills are issued for £100, £200, £500, and £1000, which bear an interest of 1d. per cent. per diem, and sometimes 2d. or more, according to the state of the money market. These bills are much sought after by bankers, as a steady investment, being generally quoted at a premium. Thus, if the premium be 17 shillings, we must pay £100. 17s. for £100 Exchequer Bills. Ex. Bought in the 3 per Cent. Consols, £540 at 91; how much was paid for it, and what was the broker's charge at per cent.? 127. DISCOUNT is an allowance made for advancing money on securities before they are due, at a certain rate per cent.; and when the discount is subtracted from any proposed sum, the remainder is termed the present worth. Suppose that A owes to B £625, to be paid at the end of 4 years, but that A is desirous of paying the debt immediately. Now if A paid £625 to B, he would lose, and B would gain, 4 years' interest. We must therefore find what sum A must pay to B so that, with 4 years' interest, it may amount exactly to £625. Now if interest be reckoned at 5 per cent., then (123), £. £. 120 or £520. 16s. 8d. Hence, £104. 3s. 4d., the difference between £625 and £520. 16s. 8d., must be allowed in consideration of present payment, and this is the discount of £625 payable at the end of 4 years at 5 per cent., while £520. 16s. 8d. is the present worth or value of £625 due 4 years hence, discount being at 5 per cent. Since £20 is the discount of £120 due 4 years hence, at 5 per cent., it is evident that the discount of £625 will be found by the following proportion As bills or promissory notes are usually made payable in a few months, it is customary for bankers, and those who discount bills, to consider the discount the same as the interest of the money for the short time specified. Thus the discount of £250 for 3 months at 4 per cent. would be considered the interest of the same sum for the given time and rate, viz., £2. 10s. The true discount would be found thus £. 101 £. £. 250 and therefore the difference is in favour of the party who discounts the bill. |