We will now solve the following questions by means of the above table. 4. What is the amount of $790, for 13 years, at 6 per cent.? From our table we find the amount of $1, for 13 years, at 6 per cent. to be $2.132928; this multiplied by the principal, $790, gives $1685.013, for the amount required. 5 5. What is the compound interest of $49, for 20 years, at cent.? per In this example, we find from the table, that the amount of $1, for 20 years, at 5 per cent., is $2.653298, which, multiplied by $49, gives $130.012 for the amount of $49, from which, if we subtract $49, we get $81.012, for the compound interest required. 6. What is the compound interest of $100, for 17 years, at cent.? Ans. $169.277. 6 per 6 7. What is the compound interest of $375, for 20 years, at per cent.? $827.676. 8. What is the amount of $875, for 12 years, at 6 per cent., compound interest? Ans. $1760.672. 9. What is the amount of $625, for 18 years, at 5 per cent., compound interest? Ans. $1504.137. 10. What is the amount of $379, for 30 years, at 3 per cent., compound interest? 11. What is the amount of $4000, for 27 years, at 4 per cent., compound interest? NOTE. When the interest is compounded half-yearly, we must take the amount of $1, for half a year, and raise it to a power denoted by the number of half-years in the whole time; this power multiplied by the principal will give the amount. We must proceed in a similar way for any other aliquot part of a year. 12. What is the amount of $100, for 3 years, at 6 per cent. per annum, when the interest is added at the end of every 6 months? In this example, we change the 6 per cent. to 3 per cent., and the 3 years to 6 years; we then find the tabular number to be $1.194052, which, multiplied by 100, gives $119.405, for the amount required. 13. What will £600 amount to in 6 years, at 8 per cent. compound interest, supposing the interest to be receivable half-yearly? Ans. £960 12s. 4 d. 14. What will $890 amount to in 5 years 4 months, at 9 per cent. per annum, compound interest, the interest being added at the end of every 4 months? Ans, $1428.188. 15. What will $3705 amount to in 3 years and 3 months, at 12 per cent. per annum, compound interest, the interest being added at the end of every 3 months? 16. What will $378 amount to in 7 years and 6 months, at 6 per cent. per annum, the interest being compounded halfyearly? 17. What will $1000 amount to in 15 years, at 8 per cent. per annum, the interest being compounded half-yearly? COMPOUND DISCOUNT. 70. Compound discount is an allowance made for the pay. ment of money before it is due, on the supposition that the money draws compound interest. The present worth of a debt payable at some future period without interest, is such a sum as being put out at compound interest, will, in the given time, at the given rate, amount to the debt. Hence, the finding the present worth, resolves itself into the following: Given the amount at compound interest, the time, and the rate per cent., to find the principal. Under compound interest it was shown, that the amount was equal to the principal multiplied by the amount of $1, for one year, raised to a power whose exponent is the number of years. Hence, we have the following rule to find the principal, or present worth : RULE. Divide the amount by the amount of $1, for 1 year, raised to a power whose exponent is equal to the number of years. Examples. 1. What is the present worth of $1685, due 13 years hence, allowing discount according to 6 per cent. compound interest? From the table under Art. 69, we find that the amount of $1, for 1 year, at 6 per cent., raised to the 13th power, is $2.132928; ... dividing $1685 by $2.132928, gives $789.994, for the present worth required. 2. How much money must be placed out at compound interest to amount to $1000 in 20 years, the interest being 5 per cent. ? Ans. 376.889. 3. What is the present worth of $3525, due in 3 years, discounting at 6 per cent., compound interest? Ans. $2959.658. 4. What is the present worth of $350, due 5 years hence, discounting at 6 per cent., compound interest? Ans. $261.54. The present worth of a given sum of money, discounting at compound interest, is easily obtained by the following table. This table gives the PRESENT WORTH of $1, or £1, for any number of years, from 1 to 30, at 3, 4, 5, and 6 per cent., compound discount. 23 24 0.390121 0.310068 25 26 27 0.506692 0.405726 0.325571 0.261797 0.246979 0.232999 0.207368 5. What is the present worth of $1000, due 27 years hence, discounting at 3 per cent., compound interest? From the above table we find the present worth of $1, for 27 years, at 3 per cent., to be $0.450189; this, multiplied by 1000, gives $450.189, for the present worth required. 6. What is the present worth of $375, due 17 years hence, discounting at 4 per cent., compound interest? Ans. $192.515. 7. What is the present worth of $672, due 13 years hence, discounting at 5 per cent., compound interest? 8. What is the present worth of $400, due 19 years hence, discounting at 6 per cent., compound interest? 9. What is the present worth of $111, due 29 years hence, discounting at 3 per cent., compound interest? ANNUITIES. 71. An annuity is a fixed sum of money which is paid periodically, for a certain length of time. CASE I. To find the amount of an annuity, which has been forborne for a given time. It is obvious that the last year's payment will be simply the annuity without any interest; the last but one will be the amount of the annuity for one year; the last but two will be the amount of the annuity for two years, and so on; and the sum of all these partial amounts, will give the total amount due. Now we discover that these partial amounts, or payments, form a geometrical progression, whose first term is the annuity, the ratio is the amount of $1 for 1 year, and the number of terms is equal to the number of years; therefore, the amount of an annuity is found by summing the terms of a |