n : I : 11 :: 1 that is, the ratio of the given interest to the interest for 1 year will be the number of years. 2nd. To find the rate. The rate evidently varies as the interest, when the principal and time remain the same; hence if Il be the interest for the given time at 1 per cent. ; I : 11 :: R : 1 that is, the ratio of the given interest to the interest at 1 per cent. is the number of pounds in the rate. 3rd. To find the principal. The principal evidently varies as the interest, when the time and rate are the same; hence if 11 be the interest on £l for the given time at the given rate, I : 11 :: P 1 that is, the ratio of the given interest to the interest on £l is the number of pounds in the principal. If the amount be given, instead of the interest, since the principal varies directly as the amount, if M, M1 be the given amount, and that of £1, M : Mi :: P : 1 that is, the ratio of the given amount to the amount of £1 is the number of pounds in the principal. : Prop. 59.--To explain the Rule for finding the true dis count on a sum of money. If A owe B a sum of money, which he is to pay at the end of a certain time, he will derive from the delay of payment an advantage equal to the interest of the debt for the given time. If then he pay the money at once, he ought to receive such a compensation as will leave him, at the end of the time, without loss. The question then is, what sum of money must A receive in order that, at the end of a given time, he may have gained a benefit from it equal to the interest on his debt? In other words, what is the sum which, put out to interest for the given time, will produce the interest on the debt? Thus the question of finding the discount becomes a question merely of finding the principal, when the amount, time, and rate are given : and hence the discount is the ratio of the interest on the debt to the amount of £1 for the given time. Again: the creditor ought not to receive the whole of the debt, but such a sum as, put out to interest for the given time, will amount to the debt, ė.e. the present worth of the debt. Hence the discount is equal to the difference between the debt and its present worth. Prop. 60.-To prove the Rule for finding the amount at Compound Interest of a given principal; and conversely. The amount at simple interest varies directly as the principal, the rate and time being the same: hence if r be the rate of interest per £1 for one year, Mi, Mii, Mii, &c. be the amounts at the end of the 1st, 2nd, 3rd, &c. years, since the amount of one year is the principal of the next, we have Mi 1+n P 1 1 :: Mii : 1 :: : Mi : whence, compounding the proportions, and expelling common means and extremes, Miji : (1+r)3 :: P : 1 which gives Miii = P(1 + r) 3 and a similar formula may be obtained for any number of years. Conversely, the principal required to produce a given amount at simple interest varies directly as the amount, the rate and time being the same; hence if Pii, Pii, Pi, be the principals required to produce the amounts at the end of the 3rd, 2nd, and 1st years, since the principal of one year is the amount of the year before, we have pii : 1 : 1tr pii : 1 Più : 1tr Pi 1 рії : 1+po whence pi : 1 M : (1+r)3. now (1 + r)3 is the amount of £1 for 3 years, whence it appears that the ratio of the given amount to the amount of £1 is the number of pounds in the principal. :: M :: Prop. 61.--To explain the Rules for the several cases of Stocks. These are all applications either of Simple or Compound Proportion. Let P stand for the price of £100 stock, M the amount of stock, S the cost of the stock, I the income; and let P1, M?, S?, 14, stand for the same in another kind of stock. First, let it be required to find the cost of a given amonnt of stock at a given price per cent. Evidently the cost varies directly as the amount to be purchased, if the price remain the same. Hence the cost required is found by a Simple Proportion, as this, 100 M P : S. M XP 100 Secondly, let it be required to find how much stock can be purchased for a given sum at a given price. Here the amount of stock, which can be pur : : chased, varies as the sum given for it, if the price remain the same; therefore the amount can be found by a Simple Proportion, as this P : S :: 100 : M. Thirdly, let it be required to find what rate of interest may be obtained by purchasing in stock bearing a given rate of interest. Here we have given the interest on the price of £100 stock, and the interest on £100 is required. Since the interest varies as the sum invested, other circumstances being the same, the solution of this question is one of Simple Proportion, thus P : 100 :: : R, r, R, being the rates of interest on £100 stock, and £100 sterling. Fourthly, let it be required to find in which of two kinds of stock it is more advantageous to invest. Evidently the answer is, that in which the higher rate of interest is obtained. The question therefore might be solved by finding these rates in the two cases, and comparing them. But since, from last case, it appears that R X 100, it follows that the rate of interest is greater in that kind of stock, in which the ratio r : P is greater; so that it is requisite only to compare these ratios; whence the Rule. Fifthly, let it be required to find what annual income will be realized by investing a given sum in stock at a given rate. The income of course varies directly as the sum invested, other circumstances remaining the same; hence, since the interest on £100 stock is the income arising from the investment of the price of £100 stock, we have the proportion, P : S ; I Sixthly, let it be required to find what sum must be invested to produce a given income. The sum varies directly as the incoine, other circumstances remaining the same: hence, since the price of £100 stock is the sum required to produce an income equal to the interest on £100 stock, we have the proportion, : I P S Seventhly, let it be required to find how much of one kind of stock can be purchased with the proceeds of the sale of a given amount of other stock. Here we consider that the amount of stock, which can be purchased with a sum of money, varies inversely as the price of the stock, therefore this question may be solved by a Simple Proportion thus, pi : P :: M : M1 Eighthly, let it be required to find the difference in a person's income caused by transference of money from one stock to another. Here it is to be considered, that the income derivable from the investment of a given sum varies directly as the rate of interest, and inversely as the price of the stock. Hence the solution of the question is obtained by the proportion, : : 11 p1 I yol XP 11 I 11 -I XI which gives the actual difference. Prop. 62.- To explain the Rules of Commission, Brokerage, and Insurance, Commission and Brokerage are allowances made to agents for assisting in the transference of property, These allowances are at the rate of so much for every £100 of value of the goods disposed of, or bought; except in the case of “Stock," where the calculation is made on the nominal value of the Stock. Insurance (or rather the premium for insuring) is a sum paid by one person to another, or to a company, in consideration of which the latter agrees to make good the loss of the former to a stipulated amount. The premium is calculated at so much for every £100 insured. It is evident therefore that the Commission, Brokerage, or Insurance on a given sum varies as the sum. Hence the amount of each of these is obtainable by a proportion thus :As £100 : the given sum £P :: the rate per cent. £R : the ans. £A. whence 100 : P R А 100 : R :: P А £P £A £A = {£P XR • 100. Whence the Rule. Sometimes an insurance is effected to an amount sufficient to cover not only the value of the goods, but also the cost of insurance. To find what sum must be insured to effect this, we consider that the insurance of £100 will actually recover in the event of loss only the excess £100 over the cost of insurance. Also the sum which must be insured varies as the sum to be covered ; hence the solution of the question is obtained by a proportion thus : :: or : : As the sum covered by £100 : the whole sum to be covered :: £100 : ans. Prop. 63.— To explain the several cases in Profit and Loss. 1st. If it be required to find the profit or loss per cent. made in a given transaction, it is to be considered that, if the circumstances under which Q answer. answer. the transaction is effected remain the same, but the amount of the transaction be doubled, the profit or loss would also be doubled. Hence the profit or loss varies as the value of the goods on which it is made: so that the gain or loss per cent. will be obtained by the proportion. As the price of the goods : £100 :: the given gain or loss : 2nd. If it be required to find at what price an article must be sold to produce a gain or loss of so much per cent. it is to be considered, that the produce of a given sum varies as the sum, the rate of gain or loss being the same. For instance, if £5 be gained or lost on every transaction of £100, then £10 will be gained or lost on a transaction of £200, so that £100 will produce £105 or £95, and £200 will produce £210 or £190, which is double the produce of £100. Hence the solution of this question also is by Simple Proportion thus: As £100 : the cost of the article :: the produce of £100 : 3rd. If the prime cost of an article be required, it being given that by selling it at a certain price a given gain or loss per cent is made, it is to be considered that the prime cost varies as the produce, the rate of gain or loss being the same ; for it was shewn in the last case that the produce varies as the prime cost. Hence this question is solved by a proportion thus : As the produce of £100 : the given price :: £100 4th. If it be given that by selling goods at a certain price a given gain or loss per cent. is made, and it be required to find the selling price at which another gain or loss will be made, then we know from the last two cases that the ratio of the produce of £100 to the selling price of an article is equal to the ratio of £100 to the prime cost. Hence this last ratio being the same, when the rate of gain or loss is altered, it follows that if the produce of £100 be doubled the selling price must also be doubled, i.e. the selling price varies as the produce of £100. Hence the solution of the question is effected by a Simple Proportion thus : As the 1st produce of £100 : the 2nd :: the 1st selling price : the 2nd. 5th. If it be given that by selling goods at a certain price a certain gain or loss is made, and it be required to find what will be the gain or loss when the goods are sold at another given price, it is considered that since, as was shewn in the last case, the selling price varies as the produce of £100, therefore also the produce of £100 varies as the selling price. Hence a proportion stated thus : As the 1st selling price : the 2nd :: the 1st produce of £100 : the 2nd. will give the produce of £100 in the event of the goods being sold at the second given price. The difference between this and the second price will be the gain or loss per cent required. : answer. |