PROBLEMS IN INTEREST. ART. 322.—To find the principal, when the interest or amount and the rate and time are given. WRITTEN EXERCISES. 1. What principal will gain $90 in 2 years 6 months at 6%? Process.-Interest on $1 for 2 years and 6 months at 6% equals $.15; $90.15 = $600. Analysis. The interest of $1 for 2 years and 6 months at 6% being $.15, to give $90 interest, in the same time and at the same rate, it will take as many times $1 as $.15 is contained times in $90, or 600 times $1, equal to $600. Therefore, $600 will gain $90 in 2 years 6 months at 6%. 2. What sum will amount to $690 in 2 years 6 months at 6% ? Process.-Amount of $1 for 2 years 6 months at 6% equals $1.15; $690 1.15 = $600. Analysis. Since the amount of $1 for 2 years 6 months at 6% is $1.15, it will require a principal equal to as many times $1 as $1.15 is contained times in $690. $1.15 is contained 600 times in Therefore, $600 in 2 years 6 months at 6%, will amount to $690. $690. ART. 323.-Rule for finding the principal, when the interest or amount and the rate and time are given.-Di' vide the given interest by the interest of one dollar for the time at the given rate; or, Divide the given amount by the amount of one dollar for the given time and at the given rate. Formulas.-Principal = interest ÷ (rate % × time). Principal = amount ÷ [1 + (rate % × time) ]. What principal will gain— 3. $44.80 in 2 years 4 months, at 6% ? 4. $55.80 in 1 year 6 months 18 days, at 5% ? 5. $47.424 in 1 year 8 months 24 days, at 4% ? 6. What principal will amount to $271.05 in 2 years 7 months 12 days, at 6% ? 7. What principal will amount to $624.546 in 3 years 6 months 21 days, at 7%? 8. What principal will amount to $946% in 2 years 3 months 15 days, at 8% ? 9. What principal will amount to $163.359 in 3 years 3 months 16 days, at 9%? 10. Illustrate by an original problem the method of finding the principal, when the interest or the amount and the rate and time are given. ART. 324.-To find the time, when the principal, interest and rate are given. 1. In what time will $500 gain $30 interest at 4% ? Analysis. The interest of $500 for 1 Process. $500 ×:04 $20, the interest for 1 year. year at 4% is $20, and, to produce $30 $30 $20 = 1.5 = 1 yr. 6 mo. in the same time and at the same rate, it will require as many years as $20 is contained times in $30, which is 11⁄2 times. Therefore, $500 will gain $30 interest at 4% in 1 year 6 months. ART. 325.-Rule for finding the time, when the principal, interest and rate per cent. are given.-Divide the given interest by the interest of the principal for 1 year at the given rate, and reduce the decimal, if any, to months and days. Formula.-Time = interest ÷ ( principal × rate %) WRITTEN In what time will EXERCISES. 2. $160 gain $13.28 at 6% ? 8. $273 gain $71.162 at 8% ? 9. In what time will a sum of money double itself at 4%? at 5%? at 6%? at 7% ? at 8% ? at 10% ? at 12% ? 10. Illustrate by an original problem the method of finding the time, when the principal, interest and rate are given. ART. 326.-To find the rate, when the principal, interest and time are given. 1. At what rate will $450 gain $30 in 1 year 8 months? Process. $450 × 13 × .01 = $7.50 $30 $7.50 = 4 Analysis. The interest on $450 for 13 years, at 1% is $7.50: to produce $30 in the same time, it will require a rate equal to as many times 1% as $7.50 is contained times in $30, equal to 4 times 1%, or 4%. Therefore, $450 will gain $30 in 1 year 8 months at 4%. ART. 327.-Rule for finding the rate, when the principal, interest and time are given.-Divide the given interest by the interest of the principal for the given time at one per cent. Formula.-Rate = interest ÷ (principal × 1% × time). WRITTEN EXERCISES. At what rate will 2. $250 gain $52.50 in 3 years 6 months? 3. $3,400 gain $391 in 2 years 3 months 18 days? 4. $1,500 gain $151.083 in 1 year 5 months 8 days? 5. $185.80 amount to $313.03 in 7 years 7 months 9 days? 6. $2,500 amount to $3,873 in 6 years 10 months 12 days? 7. $1,350.50 gain $1,350.50 in 12 years 6 months? 8. A lady's investment of $1,600 brings her in a semiannual dividend of $36: what rate of interest does she receive on her investment? 9. A man built a hotel costing $120,000, and rented it for $6,600 per year: what per cent. interest did his money yield him? 10. A man who has $1,650 in a savings bank receives $41.25 every 6 months: what per cent. interest does he receive on his money? 11. A man who gave his note for $1,250, payable in 8 months, paid at its maturity $1,291.663: what was the rate of interest ? 12. A note for $320.25, dated February 20, 1886, was paid September 20, 1886, when the amount due was $333.33: what was the rate? 13. A capitalist who invested $28,000 in mining stock, drew a quarterly dividend of $700: what per cent. interest did his investment yield him? years? In 14. At what rate per annum will any sum double itself in 16 years 8 months? In 10 years? In 8 6 years ? 15. Illustrate by an original problem the method of finding the rate per cent., when the principal, interest and time are given. Compound Interest. ART. 328.-Compound Interest is interest on the principal and interest. When the interest is unpaid, the amount forms a new principal on which interest is computed for the next period. Compound interest on debts is illegal and its payment cannot be enforced. The interest on investments, however, may be compounded annually, semi-annually, quarterly, etc. WRITTEN EXERCISES. 1. What are the amount and compound interest on $480 for 3 years, 8 months at 6%. Analysis.-The amount of $480 for 1 year at 6% is $508.80; taking this as a new principal, it amounts in 1 year to $539.33; in the same manner, $539.33 in one year amounts to $571.69, which in the 8 months remaining, amounts to $594.56. Since $480 is the original principal, the compound interest is the difference between it and $594.56, which is $114.56. |