2. What principal, at 6%, will amount to $655.20 in 8 months? Ans. $630. 3. What principal, at 5%, will amount to $106.855 in 5 years 5 months and 9 days? Ans. $84. 4. What sum, put at interest at 51%, for 8 years 5 months, will amount to $1897.545? Ans. $1297.09+. 5. What sum, at 7%, will amount to $221.075 in 3 years 4 months? Ans. $179.25. 316. The principal, time, and interest being given, to find the rate per cent and amount. 1. I lent $450 for 3 years, and received for interest $67.50. What was the rate per cent? OPERATION. $ 4.50 3 SOLUTION. Since at 1% $450, in 3 years, will gain $13.50 interest, the rate per cent at which the same principal, in the same time, will gain $67.50, must be equal to the number of times we obtain 5, the required $13.50, Int. of $450 for 3 yr. at 1 per cent. $67.50 $13.50 5%, Rate. $450+$67.50 $517.50, Amt. $13.50 is contained in $67.50; dividing, rate per cent. Adding the principal and interest, we obtain $517.50, the amount. = RULE. -I. Divide the given interest by the interest on the principal for the given time at 1%, and the quotient will be the rate per cent required. II. Add the principal and interest to obtain the amount. 2. If I pay $45 interest for the use of $500 for 3 years, what is the rate per cent? Ans. 3. 3. The interest of $180 for 1 yr. 2 mo. 6 da. is $12.78. What is the rate per cent? Ans. 6. 4. A man invests $2000 in bank stock, and receives a semi-annual dividend of $75. What is the rate per cent? 317. The principal, interest, and rate per cent being given, to find the time and amount. 1. In what time will $360 gain $86.40 interest, at 6%, and what will this principal amount to in this time at the given rate? $ 360 .06 OPERATION. $21.60, Int. of $ 360 in 1 yr. at 6 per cent. $86.40 $21.60 4 yr., Time. $360 +$86.40 $446.40, Amt. = SOLUTION. Since in 1 year $360, at 6%, will gain $21.60, the number of years in which the same principal, at the same rate, will gain $86.40, will be as many as $21.60 is contained times in $ 86.40; Adding the prin dividing, we obtain years, the required time. cipal and interest, we obtain $ 446.40, the amount. RULE. -I. Divide the given interest by the interest on the principal for 1 year, and the quotient will be the time required in years and decimals. II. Add the principal and interest to obtain the amount. The decimal part of the quotient, if any, may be reduced to mouths and days. 2. The interest of $325 at 6% is $58.50. What is the time? Ans. 3 years. 3. B loaned $1600 at 6% until it amounted to $2000. What was the time? Ans. 4 years 2 months. 4. How long must $204 be on interest at 7%, to amount to $217.09? Ans. 11 months. 5. Engaging in business, I borrowed $750 of a friend. at 6%, and kept it until it amounted to $942. How long did I retain it? Ans. 4 years 3 months 6 days. 6. How long will it take $200 to double itself at 6% simple interest? Ans. 16 years 8 months. 7. In what time will $675 double itself at 5% ? The time in years in which any sum will double itself may be found by dividing 100 by the rate per cent. COMPOUND INTEREST. 318. Compound Interest is interest on both principal and interest, when the interest is not paid when due. The simple interest may be added to the principal annually, semi-annually, or quarterly, as the parties may agree; but the taking of compound interest is not legal. $200 × .06 1. What is the compound interest of $200, for 3 years, at 6% ? $212 x .06 = $224.72 x .06 = EXAMPLES. OPERATION. $200 = 12 Principal for 1st year. $212 Principal for 3d year. 1.06 x 1.06 x 1.06 = 1.191 SECOND OPERATION. $200 × 1.191 = $238.20 Amount for 3 years. 200.00 Principal. $38.20 Compound interest. SOLUTION. The amount for the first year will be 106% of the principal; for the second year 106% of this, and for the third year 106% of the last result. Multiplying 106 % × 106 % × 106 %, we have 119%% 1.191, the third amount expressed decimally. Multiplying $200 by this amount, we have $238.20, and subtracting the principal from the amount, we find the compound interest to be $38.20. RULE.-I. Find the amount of the given principal at the given rate for one year, and make it the principal for the second year. II. Find the amount of this new principal, and make it the principal for the third year, and so continue to do for the given number of years. III. Subtract the given principal from the last amount, and the remainder will be the compound interest. Or, I. Find the amount expressed in per cent for the given time. II. Take this per cent of the principal. III. Subtract the given principal from the amount thus obtained, and the remainder will be the compound interest. 1. When the interest is payable semi-annually or quarterly, find the amount of the given principal for the first interval, and make it the principal for the second interval, proceeding in all respects as when the interest is payable yearly. 2. When the time contains years, months, and days, find the amount for the years, upon which compute the interest for the months and days, and add it to the last amount, before subtracting. 2. What is the compound interest of $500 for 2 years, at 7% ? Ans. $72.45. 3. What is the amount of $312 for 3 years, at 6%, compound interest? Ans. $371.59+. of $250 for 2 Ans. $31.37+. 4. What is the compound interest years, payable semi-annually, at 6% ? 5. What will $450 amount to in 1 year, at 7%, compound interest, payable quarterly? Ans. $482.33. 6. What is the compound interest of $236 for 4 years 7 months and 6 days, at 6%? Ans. $72.66+. 7. What is the amount of $700 for 3 years 9 months and 24 days, at 7% compound interest? Ans. $906.55+. 8. What is the compound interest of $640 for 4 years, at 5%? Ans. $137.924. 9. What will $250 amount to in 3 years, at 7% compound interest? Ans. $306.26. 10. Find the compound interest of $376 for 3 years 8 months 15 days, at 6%. Ans. $90.85. A more expeditious method of computing compound interest is by means of the following table: TABLE. Showing the amount of $1, or £1, at 3, 4, 5, 6, and 7 per cent, compound interest, for any number of years, from 1 to 20. Yrs. 3 per cent. 4 per cent. 5 per cent. 1 5 per cent. 7 per cent. 1.060,000 1.07,000 1.030,000 1.040,000 1.050,000 1.191,016 1.22,504 1.262,477 1.31,079 1.159,274 1.216,653 1.276,282 1.338,226 1.40,255 1.418,519 1.194,052 1.265,319 1.340,096 1.503,630 1.689,479 16 1.604,706 1.872,981 2.182,875 17 2.292,018 1.652,848 1.947,900 18 2.406,619 19 1.753,506 2.106,849 2.526,950 20 1.806,111 2.191,123 2.653,298 1.898,299 2.012,196 2.260,904 2.396,558 OPERATION. 2.540,352 2.692,773 2.854,339 3.025,600 3.207,135 1.50,073 1.60,578 1.71,818 1.83,845 1.96,715 2.10,485 2.25,219 2.40,984 2.57,853 2.75,903 2.95,216 3.15,881 3.37,293 3.61,652 3.86,968 11. What is the amount of $800 for 6 years, at 7% ? From the table $1.50073 = Amount of $1 at 7% for the time. 800 Principal. $1200.58400, Ans. 12. What is the compound interest of $120 for 15 years, at 5% ? Ans. $129.47+. 13. What is the amount of $.10 for 20 years, at 7%? Ans. $.38696. |