100 Ex. 1.–Find the simple interest of £420 for 3 years at 5 per cent. £ 420 5 2100 3 £63:00 Ans.: £63. . Reason for this process— £100 gains £5 in 1 year, and the question is to find the gain upon £420 in 3 years. Hence, proceeding as in Double Rule of Three, we have 100 x 1: 420 x 3 :: £5 : interest required. .. interest required £420 x 5_X_), which is exactly as stated in the rule. Ex. 2.–Find the simple interest of £352. ls. 8d. from March 16 to August 21, 1873, at 4 The following process will be easily understood : £ d. 352 1 8 4 £14:08 6 8 20 1.66s. 12 8:00d. .:. £14. ls. 8d. is the interest for 1 year. Now, from March 16 to August 21 are 158 days; lience we have— 365 days : 158 days :: £14. 1s. 8d. : Interest required. .: interest required = £6. ls. 11 d. per cent. 8. Ex. XVII. Find the simple interest of 1. £350 for 4 years at 5 per cent. 2. £295. 2s. 1d. for 3 years at 4 per cent. 3. £375. 8s. 4d. for 2 years at 41 per cent. 4. £160 from Feb. 1 to June 12, 1872, at 74 per cent, 5. £48 for 7 months at 14 per cent. per month. 6. £219. 4s. 2d. for 6 years at 1% per cent. In the six following examples, understand simple interest. 7. At what rate per cent. will £129. 8s. 4d. gain £6. 3s. 51d. in 21 years? 8. A certain sum amounts in 3 years at 74 per cent. to £289. 16s. 3 d. ; find the original sum. 9. In what time will £175. 6s. 3d, amount to £192. 7s. 101d. at 2 per cent ? 10. What sum will amount in 2 years 9 months at 4 per cent. to £427. 7s? 11. If £320 gain £9 in 13 months, in what time will £480 gain £6 at the same rate? 12. Find the interest of £29. 7s. 5d. for 6 months at 516 per cent. on. Compound Interest. 44. RULE 1.-Find the interest for one year as in Simple Interest, and add it to the principal; then find the interest for one year upon this amount reckoned as principal for the second year, and add it to the second year's principal, and so Subtract the original principal from the amount so obtained for the given number of years, and the result will be the compound interest required. RULE. 2.-Divide the given rate per cent. by 100, putting the result in a decimal form, and place unity before the decimal point. Raise the number thus obtained to a power corresponding to the given number of years, multiply the principal by the result, and we get the amount for the given number of years. Thus, supposing 5 to be the rate per cent., we have, dividing by 100 and placing unity before the result, the number 1.05. Then, if the given number of years be 4, and £162 the principal, we have, according to rule, amount = £162 x (1.05). The first rule requires no explanation; the second rule may be explained thus : 9 Ex.—Find the compound interest of £360 for 3 years at 4 per cent. Now, interest for £100 for 1 year £4, £1 £.04; hence, amount of £1 £1 + £.04 £1.04. We thus see that the amount of £1 for 1 year at 4 per cent. is 1•04 times the original sum. It therefore follows that Amount of the £1:04 for 1 year = 1.04 times £1:04, ... amount of £1. 2 years = £(1.04 x 1.04) £(1•04)" ; and so, amount of £1 3 years = 1.04 times £(1•04) = £(1:04)", hence, amount of £360 360 times £(1•04) £360 < (1•04)". The compound interest is then found by subtracting from this the original principal. Ex. 1. Find the compound interest of £570 for 3 years at 5 (We shall work this by Rule 1, and for convenience shall keep our quantities in a decimal form.) £ 5 ; 570 5 interest 5 628.425 £659.84625 amount at end of third year,, therefore 570 = original principal ; | subtracting; £89.84625 compound interest for 3 years. per cent. Ex. 2.-Find the compound interest of £327. 123. 6d. for 4 years at 3 per cent. Now, £327. 12s. 6d. = £327:625. Hence, by Rule 2– Amount = £327.625 x (1•03)* = £368. 14s. 10 d. nearly. Ex. 3.-—What sum of money, if put out for 2 years at 4 per cent., will amount to £324. 9s. 71d., compound interest being reckoned? By Rule 2, the principal may be found by dividing the amount by (1•04). Now, given amount = £324. 9s. 7d. £324:48. Hence principal required £324.48 ; (1•04) £300. Ex. XVIII. Find the compound interest of 1. £284 for 2 years at 4 per cent. 2. £312. 12s. 7}d. for 3 years at 5 cent. per cent. 5. What is the difference between the simple and compound interest of £150 for 2 years cent. ? 6. Find the amount of £381. 1 florin 3 cents 5 mils for 3 years at 5 per cent. (£1 = 10 florins, 1 florin = 10 cents, 1 cent = 10 mils.) 7. Find the amount of £250 for 2 years at 4 per cent. per annum, interest being payable half-yearly. 8. What sum will amount in 3 years at 41 per cent. compound interest to £200 ? 9. A town has 200,000 inhabitants, and it increases at the rate of 5 per cent. per annum ; find the number of inhabitants at the end of 3 years. 10. Find the difference in amount of £350 for 3 years at 4 per cent. simple interest, and £420 for 2 years at 5 per cent. compound interest. 11. How much would a person who lays by. £50 a year at 5 per cent. compound interest, draw out at the end of 4 years? 12. A person expects to receive £450 in 3 years; what present sum is equivalent to this, reckoning compound interest at 4 per cent. ? Discount. 45. When money is paid before it is due, the payee may, of course, put out the money at interest for the rest of the term, and thereby increase it. It therefore follows that the amount which ought to be paid for the discharge of an account before its proper time should be such a sum that, if put out at interest for the remainder of the term, will just amount to the original sum in question. Thus £102. 10s. (interest being reckoned at 5 per cent. per annum) payable 6 months hence, would be fully discharged by paying £100 at once. For £100 in 6 months at 5 per cent. per annum would amount to £102. 10s. Hence, the payee ought to remit £2. 10s. from the full account. The amount remitted is called discount. It will be seen, therefore, that the discount on £102. 10s. due 6 months hence at 5 per cent. is £2. 10s. Bankers, however, are in the habit of charging interest instead of discount. The banker's discount, therefore, on £100 due 6 months hence at 5 per cent. is £2. 10s. Hence, the true discount on £102. 10s. due 6 months at 5 per cent. is the same as the banker's discount on £100 under the same circumstances; and bankers' discount on any given sum is in excess of the true discount. Tradesmen's bills are legally due three days after the term for which they are drawn is completed. This extension of time is called three days of grace. When a bill falls due on a Sunday, it is usual in England to meet it on the previous Saturday. Ex. 1.–Find the difference between the true discount and the banker's discount on £306 due 4 months hence at 6 per cent. Now, £100 would in 4 months gain } of £6, or £2. Hence, the true discount on £102 due four months hence at 6 per cent. is £2, and therefore we have £102 ; £306 :: £2 : true discount required. £6. Again, proceeding according to the rule for simple interest £306 x 6 x } Banker's discount = £1:3= £6. 2s. 4;d. 100 102 |