PROOF. $80X.09 $7.20, the interest of $80 for 1 year at 9 per cent., and $7.20 X5=$36.00, the interest for 5 years, which is equal to the sum paid. Hence, 422. To find the rate per cent. when the principal, interest, and time are given. Divide the given interest by the interest of the principal at 1 per cent. for the given time, and the quotient will be the required per cent. Or, find the interest of the principal at 1 per cent, for the given time; then make the interest thus found the denominator and the given interest the numerator of a common fraction; reduce this fraction to a whole or mixed number, and the result will be the per cent. required. (Art. 196.) 2. If I loan $500 for 2 years, and receive $50 interest, what is the rate per cent.? Ans. 5 per cent. 3. A man borrowed $620 for 8 months, and paid $24.80 for the use of it: what per cent. interest did he pay? 4. At what per cent. interest must $2350 be loaned, to gain $57 in 4 months? 5. At what per cent. interest must $1925 be loaned, to gain $154 in 1 year ? 6. A man has $12000 from which he receives $900 interest annually what per cent. is that? 7. A man deposited $2600 in a savings bank, and received $143 interest annually: what per cent. was that? 8. A man invested $4500 in the Bank of New York, and received a semi-annual dividend of $157.50: what per cent. was the dividend? 9. A man paid $16250 for a house, and rented it for $975 a year: what per cent. did it pay ? 10. A hotel which cost $250000, was rented for $12500 a year: what per cent. did it pay on the cost? 11. A capitalist invested $500000 in manufacturing, and received a semi-annual dividend of $12500: what per cent. was his dividend? QUEST.--422. When the principal, interest and time are given, how is the rate per ct. found! PROBLEM III. To find the PRINCIPAL, the interest, the rate per cent., and the time being given. 12. What sum must be put at interest, at 6 per cent., to gain $75 in 2 years? Analysis. The interest of $1 for 2 years at 6 per cent., (the given time and rate,) is 12 cents. Now 12 cents interest is of its principal $1; consequently, $75, the given interest, must be of the principal required. The question therefore resolves itself into this: $75 is of what number of dollars? If $75 is 12, is 12 of $75, which is $64; and 100=$64×100, which is $625, the principal required. Or, we may reason thus: Since 12 cents is the interest of 1 dollar for the given time and rate, 75 dollars must be the interest of as many dollars for the same time and rate, as 12 cents is contained times in 75 dollars. And $75÷.12=625. Ans. $625. PROOF.-$625X.06=$37.50, the interest for 1 year at the given per cent., and $37.50 X 2-$75, the given interest. Hence, 423. To find the principal, when the interest, rate per cent., and time are given. Divide the given interest by the interest of $1 for the given time and rate, expressed in decimals; and the quotient will be the principal required. Or, make the interest of $1 for the given time and rate, the numerator, and 100 the denominator of a common fraction; then divide the given interest by this fraction, and the quotient will be the principal required. (Art. 234.) 13. What sum must be put at 7 per cent. interest, to gain $63 in 6 months? 14. What sum must be put at 5 per cent. interest, to gain $90 in 4 months? 15. What sum must be invested in 6 per cent. stock, to gain $300 in 6 months? QUEST.-423. When the interest, rate per cent., and time are given, how is the princi pal found? 16. What sum must be invested in 7 per cent. stock, to gain $560 in one year? 17. A man founded a professorship with a salary of $1000 a year: what sum must be invested at 7 per cent. to produce it? 18. What sum must be put at 6 per cent. interest to pay a salary of $1200 a year? 19. What sum must be invested in 5 per cent. stock to make a simi-annual dividend of $750 ? 20. A man bequeathed his wife $1250 a year: what sum must be invested at 6 per cent. interest to pay it? PROBLEM IV. To find the TIME, the principal, the interest, and the rate per cent. being given. 21. A man loaned $200 at 6 per cent., and received $42 interest: how long was it loaned? Analysis. The interest of $200 at 6 per cent. for 1 year is $12. (Art. 404.) Now, since $12 interest requires the principal 1 year at the given per cent., $42 interest will require the same principal 42 of 1 year, which is equal to 3 years. (Art. 196.) Or, we may reason thus: If $12 interest requires the use of the given principal 1 year, $42 interest will require the same principal as many years as $12 is contained times in $42. And $42 $12-3.5. Ans. 3.5 years. Hence, 424. To find the time, when the principal, interest, and rate per cent. are given. Divide the given interest by the interest of the principal at the given rate for 1 year, and the quotient will be the time required. Or, make the given interest the numerator, and the interest of the principal for 1 year at the given rate the denominator of a common fraction; reduce this fraction to a whole or mixed number, and it will be the time required. OBS. If the quotient contains a decimal of a year, it should be reduced to months and days. (Art. 348.) QUEST.-424. When the principal, interest, and rate per cent. are given, how is the time found? Obs. When the quotient contains a decimal of a year, what should be done with it? 22. A man loaned $765.50, at 6 per cent., and received $183.72 interest: how long was it loaned? 23. In what time will $850 gain $29.75, at 7 per cent. per annum ? 24. A man received $136.75 for the use of $1820, which was 6 per cent. interest for the time: what was the time? 25. In what time will $6280 gain $471, at 5 per cent. interest? 26. How long will it take $100, at 5 per cent., to gain $100 interest; that is, to double itself? Operation. The interest of $100 for 1 year, at 5 per cent., $5)$100 is $5. (Art. 404.) 20 Ans. 20 years. PROOF.-$100X.05 X 20-$100, the given principal. (Art. 404.) TABLE, Showing in what time any given principal will double itself at any rate, from 1 to 20 per cent. Simple Interest. 27. How long will it take $365 to double itself, at 6 per cent.? 28. How long will it take $1181 to double itself, at 7 per cent.? 29. In what time will $2365.24 double itself at 7 per cent.? 30. In what time will $5640 double itself, at 10 per cent.? 31. How long will it take $10000 to gain $5000, at 6 per cent. interest? 32. A man hired $15000, at 7 per cent., and retained it till the principal and interest amounted to $25000: how long did he have it? 33. A man loaned his clerk $25000 to go into business, and agreed to let him have it, at 5 per ct., till it amounted to $60000: how long did he have it? COMPOUND INTEREST. 425. Compound Interest is the interest arising not only from the principal, but also from the interest itself, after it becomes due. OBS. Compound Interest is often called interest upon interest. When interest is paid on the principal only, it is called Simple Interest. Ex. 1. What is the compound interest of $842 for 4 years, at 6 per cent.? $842X.06= Operation. 50.52 Int. for 1st year. $892.52X.06= 53.55 Int. for 2d year. $1002.83X.06= 60.17 Int. for 4th year. 842.00 Prin. deducted. Ans. $221.00 Compound int. for 4 years. 426. Hence, to calculate compound interest. Cast the interest on the given principal for 1 year, or the specified time, and add it to the principal; then cast the interest on this amount for the next year, or specified time, and add it to the principal as before. Proceed in this manner with each successive year of the proposed time. Finally, subtract the given principal from the last amount, and the remainder will be the compound interest. 2. What is the compound interest of $600 for 5 years, at 7 per cent. ? Ans. $241.53. 3. What is the compound int. of $1260 for 5 yrs., at 7 per cent.? 4. What is the amount of $1535 for 6 yrs., at 6 per cent. compound interest? 5. What is the amount of $4000 for 2 yrs., at 7 per cent., ble semi-annually? QUEST.-426. How is compound interest calculated? paya |