What is the difference between the interest of £50. 19 for 7 years at 3 per cent, and for 8 years at 2½ per cent? _Answer_, 10_s._ (2½)_d._ What is the interest of £157. 17. 6 for one year at 5 per cent? _Answer_, £7. 17. 10½. Shew that the interest of any sum for 9 years at 4 per cent is the same as that of the same sum for 4 years at 9 per cent. 250. In order 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 every year; because in this case (248) it is the amount of both principal and interest at the end of the first year, upon which interest accumulates during the second year. Suppose, for example, it is required to find the interest, for 3 years, on £100, at 5 per cent, compound interest. The following is the process: £100 First principal. 5 First year’s interest. --- 105 Amount at the end of the first year. (249) 5 . 5 Interest for the second year on £105. -------- 110 . 5 Amount at the end of two years. 5 . 10 . 3 Interest due for the third year. ------------ 115 . 15 . 3 Amount at the end of three years. 100 . 0 . 0 First principal. ------------ 15 . 15 . 3 Interest gained in the three years. When the number of years is great, and the sum considerable, this process is very troublesome; on which account tables[54] are constructed to shew the amount of one pound, for different numbers of years, at different rates of interest. To make use of these tables in the present example, look into the column headed “5 per cent;” and opposite to the number 3, in the column headed “Number of years,” is found 1·157625; meaning that £1 will become £1·157625 in 3 years. Now, £100 must become 100 times as great; and 1·157625 × 100 is 115·7625 (141); but (221) £·7625 is 15_s._ 3_d._; therefore the whole amount of £100 is £115. 15. 3, as before. [54] Sufficient tables for all common purposes are contained in the article on Interest in the Penny Cyclopædia; and ample ones in the Treatise on Annuities and Reversions, in the Library of Useful Knowledge. 251. Suppose that a sum of money has lain at simple interest 4 years, at 5 per cent, and has, with its interest, amounted to £350; it is required to find what the sum was at first. Whatever the sum was, if we suppose it divided into 100 parts, 5 of those parts were added every year for 4 years, as interest; that is, 20 of those parts have been added to the first sum to make £350. If, therefore, £350 be divided into 120 parts, 100 of those parts are the principal which we want to find, and 20 parts are interest upon it; that is, the principal is £(350 × 100)/150, or £291. 13. 4.