= Hence, amount of annuity of £1 for 4 years— = £1 + £1.05 + £1·1025 + £1.157625 £4.310125. .. amount of given annuity = £4.310125 × 300 = £1293. Os. 9d. 50. To find the present value of an annuity to continue for a given time. By the present value of an annuity to continue a given number of years is meant such a lump sum which, paid down at once, would, by accumulating at compound interest for the same time, amount to just the same sum as the annuity itself if it were allowed to accumulate. In the absence of algebraical symbols, we shall best illustrate the method of finding such a lump sum by an example. Ex.-Find the present value of an annuity of £50 for 3 years at 4 per cent. As in Art. 49, we find the amount of the annuity— = {£1 + £1·04 + £(1·04)2} × 50 = £156.08. Now, whatever be the present value required, we know that its amount in 3 years at 4 per cent. compound interest is found (Art. 44) by multiplying it by (1·04)3 or 1.124864. It, therefore, follows that if we know this amount beforehand we can find the present value by dividing it by 1.124864. Hence, present value nearly. = £156.08 1.124864 = £138.755 51. To find the present value of an annuity to continue for ever. It is evident that the sum we require is one which, put out at interest, will annually produce a sum equal to that of the annuity itself. The problem then is simply this Having given the interest for 1 year of a certain sum, and the rate per cent., to find the principal. We have therefore the following rule :— RULE. Divide the given annuity by the rate per cen.., and multiply by 100, and the result is the present value. Ex.-How much must a gentleman invest at 5 per cent. in order to endow a charity with £60 a year. Present value of the annuity of £60 to continue for ever— 60 X 100 1 5 = £1200. There are many other questions connected with annuities which are, however, best left till the student has a knowledge of Logarithms. Ex. XXI. Find the amount of an annuity of— 1. £120 for 3 years at 4 per cent. 2. £250 for 4 years at 4 per cent. 3. £321 for 5 years at 5 per cent. 4. What is the present value of an annuity of £80, to continue for six years, at 6 per cent. ? 5. A person who, according to the tables of mortality, is likely to live 10 years, wishes to insure an annual payment of £40 during life. What sum must he pay down, reckoning interest at 5 per cent.? (Give the result to four places of decimals.) 6. A house produces a clear rental of £30. How many years' purchase is it worth, interest being reckoned at 5 per cent.? 7. A gentleman invested a sum of money in the Three per Cent. Consols, in order that an annual payment of 7s. 6d. a year might be made in bread for ever. What sum did he invest? 8. Find the present value of a pension of £120 a-year, payable half-yearly for 5 years, interest being at the rate of 5 per cent. per annum. 9. A house, which ordinarily lets for £80 a-year, is leased for a term of four years, at a rent of £20, a certain sum being paid in addition at the time of letting. Find this latter amount. 10. What is the present value of a freehold which produces a clear rental of £50, but which cannot be entered upon for two years, reckoning interest at 5 per cent.? 11. Find the annuity which in four years, at 4 per cent., will amount to £100. 12. A corporation borrows a sum of £3000 at 4 per cent. What annual payment will clear off the debt in ten years? (Give the result correct to four places of decimals.) Profit and Loss. 52. All questions involving the loss or gain per cent. by any transaction belong to this rule, and may be generally worked by Proportion. Ex. 1.-A man buys goods at 5s. and sells them at 5s. 8d. Find his gain per cent. The actual gain upon 5s. is 8d., and we are required to find the gain upon £100. Now 5s. £100 :: 8d. : gain upon £100, = £100 X 8 £13}, 5 X 12 .. gain upon £100 = 133. Ex. 2.-By selling goods at 6s. 3d. there is a gain of 25 per cent. What will be the selling price to gain 10 per cent.? 25 Now, selling price of goods which cost £100, so as to gain per cent, is £125, and that to gain 10 per cent. is £110. Hence £125 £110: 6s. 3d.: selling price required; from which, selling price required = 5s. 6d. : Ex. 3.-Find the cost price when articles sold at 1s. 9d. entail a loss of 12 per cent. Now, articles which cost £100 when sold at a loss of 12 per cent. must sell for £87. Hence £87 1s. 9d.: £100: cost price required; from which, cost price 2s, Ex. XXII. 1. Find the cost price of goods which are sold at a loss of 10 per cent. for 4s. 10d. 2. Goods which are sold for 7s. 11d. entail a loss of 5 per cent. What should be the price to gain 30 per cent.? 3. A tradesman reduces his goods 7 per cent. What was the original price of an article which now fetches £1. 7s. 9d.? 4. In what proportion must tea at 4s. 2d. be mixed with tea at 6s. a pound, so that a grocer may sell the mixture at 5s. 6d. and gain by the sale 10 per cent.? 5. A quantity of silk, after paying a duty of 12 per cent., cost £54. Find the original cost price. 6. An innkeeper buys 37 gallons of brandy at 14s. a gallon, and adds to it sufficient water to enable him to sell it at the same price and gain 12 per cent. How much water does he add? 7. By selling goods at 8s. 2d. a tradesman gains 163 per What will be the gain or loss per cent. by selling at cent. 6s. 1 d. 8. A company has a capital of £750,000, and the working expenses for the year have been £42,123. 12s. 6d. What must have been the gross receipts in order that the shareholders may receive a dividend of 4 per cent.? 9. If stock which is bought at 91 is immediately sold at 91%, what is the gain per cent.? 10. A person buys goods at 6 months' credit and sells them for cash at the nominal cost price immediately. is his gain per cent.? (Interest 5 per cent.) What 11. Goods are marked at a ready-money price and a credit price allowing 12 months. The credit price is £4. 9s. 3d., what is the ready-money price? 12. Goods are now being sold at 10 per cent. loss. How much per cent. must be put upon the selling price in order that they may be sold at 20 per cent. gain? Square Root and Cube Root. 53. To avoid unnecessary repetition, the student is referred to the articles on Involution, Algebra, stage I., where the arithmetical principles and methods are explained. Estimates. 54. The following specimens will give the student an idea of what he may expect to meet with under the head of Estimates. It is usual, in ordinary transactions, to use certain abbreviations; as cub. for cubic measure, sup. for superficial measure, run. for running or lineal measure. Builders, too, are in the habit of calling twelfths of a foot-whether it be cubical, superficial, or lineal measure-by the name of inches. The names yards, feet, inches, are often written thus: yds.,," Ex. 1.-DIGGER, BRICKLAYER, AND MASON. |