15. Given log. 125 = 2.096910, find log. 128 without the Tables. 16. What are the numbers whose logs. = 8, 1, spectively? - 2, 1, -, re 17. If x = (9.035)5 1 find x by the Tables. 18. Find x in each of the following equations :— 6; 2 = 19; (1.03) = 3.421763; 9.41-* = 23r × 27. = 3y 1}; 8+1,1251 − = = 24 +7.5*+1; 22o − 9.2o + 8 = 0. 4* = 6y+1) 19. Find, without the Tables, Log., (125)3, log. 8, log.10.00001, log.10 (赤) CHAPTER XX. INTEREST, ANNUITIES, &C. 183. Interest is the sum paid for the use of money. The money lent is called the Principal. In estimating the Interest due on any Principal, it is necessary to know the interest on a given sum for a given time. When the interest on £100 for one year is given, that quantity is called the Rate per Cent. Thus, if the interest of £100 for one year be £3 10s., the rate per cent. is 34. When the Principal remains constant, it is said to be at Simple Interest; but when, at the end of every year or other fixed period of time, the Principal is increased by the Interest accumulated during that period, it is said to be at Compound Interest. When the Principal and Interest for any time are added together, the sum is called the Amount for that time. Discount is an allowance made for paying money before it becomes due. 184. To find the Simple Interest of a sum of money for any number of years. Let P = the principal in pounds. r = the interest of £1 for one year in pounds. n = the number of years. 185. To find the Compound Interest of a sum of money for any number of years. Take the same notation as in (184). Then Or, Or, Or, Pthe first principal. P(1+r). (1+r) amount at end of second year P(1+r)2 = = the third principal. P (1 + r)2. (1 + r) = amount at end of third year. P(1+r)3 = amount at end of third year. And, by continuing this process, it will appear that 186. To find the interest and amount of a sum of £P, for n years, when the interest is paid q times a year; each 187. To find the present worth and discount of a sum due at the end of any time. 188. To find the amount of an Annuity left unpaid for any number of years. Let A the annuity in pounds. Now £4 becomes due at the end of the first year. This sum therefore bears interest for n 1 years Also £4 becomes due at the end of the second year. This sum therefore bears interest for n 2 years. second th = A (1+r); The whole amount therefore equals the sum of the above series, or 189. To find the present value of an Annuity left unpaid for n years. In the last Article the amount for n years was found to be And the present value of this must be, Art. (187), 190. To find the present value of an Annuity to begin at the end of p years and to continue q years. The value required is the present value of the Annuity for (p + q) years, diminished by its present value for p years, or, adopting the notation of the previous Articles, 1. Find the interest and amount of £400 for 3 years at 4 per cent., Simple Interest. 2. Find the interest and amount of £400 for 3 years at 4 per cent., Compound Interest. Note. The difference in the results of (1) and (2) shows the advantage of Compound Interest over Simple. 3. At what rate per cent. will a principal of £a double itself in a years, at Compound Interest? Whence (1+r) can be found from the Tables of Logarithms, and thus r becomes kuown. 4. Find at what rate per cent. a sum will double itself at compound interest in 14 years. 5. Find the interest on £250 put out to interest at 5 per cent. per annum for 10 years, compound interest, the payments being made half-yearly. |