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Ex. 1. What is the compound interest of $ 360 for 5 years 6 months and 24 days?

Ans. $ 138.14.

OPERATION.

Amount of $ 1 for 5 years,

$ 1.3 38 226 Principal,

3 60 80 2 9 3 5 60

40146780 Amount of $ 360 for 5 years,

481.7 613 60 Amount of $ 1 for 6mo. 24d.,

1.034 19 270 45 440 144 5 2 8 4080

4817 61 360 Amount of $360 for 5y. 6mo. 24d., 498.1 41 246 240 Principal,

3 60. Comp. int. of $ 360 for 5y. 6mo. 24d., $138.1 4 Ans.

We find the amount of $ 1 for 5 years in the table, and, multiplying it and the number denoting the given principal together, obtain the amount of the $ 360 for 5 years. On this amount as a new principal we find the amount for the remaining 6 months and 24 days, by multiplying by the number denoting the amount of $ 1 for the same time. From the last amount subtracting the original principal, we have left the compound interest required. Hence,

Msultiply the amount of $1 for the given time and rate, as found in the table, by the number denoting the given principal. The product will be the required amount, from which subtract the given principal, and the remainder will be the COMPOUND INTEREST.

NOTE. — When the given time includes not only the regular periods at which interest becomes due, but also a partial period, as a succession of periods of a year each, followed by one containing months or days, or both, after finding the amount for the regular periods, multiply that amount by the amount of $1 for the remaining time or partial period, and the product will be the required amount for the given time. In like manner, when the number of successive periods exceeds the limits of the table, make the computations for a convenient length of time by means of the table, and on the amount thus found make another computation by means of the table, and so on.

In making computation for a succession of periods shorter or longer than one year each, use the numbers in the table the same as if the periods were those of one year each.

EXAMPLES.

2. What is the compound interest of $ 1200 for 11 years at

Ans. $ 1325.822.

7

per cent. ?

per cent. ?

per cent. ?

3. What is the compound interest of $300 for 10 years 7 months and 15 days ?

Ans. $ 257.401. 4. What is the compound interest of $ 5 for 50 years at 7

Ans. $ 142.285. 5. What is the amount of $ 480 for 40 years, at compound interest?

Ans. $ 4937.144. * 6. What is the compound interest of $ 40 for 4 years, at 7

Ans. $ 12.431. 7. What is the compound interest of $ 100 for 100 years ?

Ans. $ 33830.20. 18. What is the difference between the simple and the compound interest of $ 1000 for 33 years and 4 months ?

9. To what sum will $ 50, deposited in a savings bank, amount, at compound interest for 21 years, at 3 per cent., payable semiannually ?

Ans. $ 173.034. (10.) $100.

Boston, September 25, 1853. For value received, I promise to pay J. D. Forster, or order, on demand, one hundred dollars, with interest, after six months.

ALLEN T. DAWES.

On this note are the following indorsements:- June 11, 1854, received fifty dollars; September 25, 1854, received fifty dollars.

What was due, reckoning at compound interest, August 25, 1855 ?

Ans. $ 2.247.

(11.) $ 1000.

St. Paul, January 1, 1850. For value received, I promise to pay Stephen Howe, or bearer, on demand, one thousand dollars, with interest at 7

per

cent.

Wilson GOODHUE. Indorsements:- - June 10, 1850, seventy dollars; September 25, 1851, eighty dollars; July 4, 1852, one hundred dollars; November 11, 1853, thirty dollars; June 5, 1854, fifty dollars.

At 7 per cent. compound interest, what remains due April 1, 1855 ?

Ans. $ 1022.34.

378.

To find the PRINCIPAL, the compound interest, the time, and the rate being given.

Ex. 1. What principal at 6 per cent. compound interest will produce $ 2370 in 10 years ?

Ans. $ 3000.

OPERATION.

We find the compound in$ 2370 = .790 $ 3000 Ans. terest of $ 1 for the given time,

and at the given rate; and proceed as in like cases in simple interest (Art. 359).

RULE. - Divide the given compound interest by the compound interest of $ 1 for the given time at the given rate.

EXAMPLES

*2. What principal, at 7 per cent. compound interest, will produce $ 205.90 in 6 years and 6 months ? Ans. $ 372.16.

3. What sum of money, at compound interest, will produce $ 1026.54 in 3 years 2 months and 12 days? Ans. $ 5000.

4. What sum of money must be invested at compound interest at a semiannual rate of 33 per cent. to produce $857.25 in 151 years?

Ans. $ 450.

379. To find the RATE PER CENT., the principal, the interest, or the amount, and the time being given.

Ex. 1. At what rate per cent. must $ 500 be at compound interest to become $ 703.55 in 7 years ? Ans. 5

per cent.

OPERATION

$ 703.55 · 500 $ 1.4071, which for 7 years, in the table,

denotes a rate of 5 per cent.

Since $ 500 becomes $ 703.55 in 7 years at the required rate, $ 1 in the same time at the same rate will amount to go as much, or $1.4071. Corresponding to this amount of $ 1 for the given time, we find in the table (Art. 377) 5 per cent., the rate required.

Rule. - Divide the amount by the principal, and the quotient will be the amount of $1 for the given time and the required rate ; and in the table, over this amount, may be found the rate per cent. required.

NOTE. — If the given time contains a part over an exact number of periods, look in the table, against the number denoting the whole periods or years in the given time, for that amount of $ 1 which comes the nearest to the one found by dividing. Then see if the approximate amount, increased by its rate of interest for the fractional period, will equal the other amount; if so, the rate corresponding to the approximate amount will be the rate per cent. required; if not, the rate of the approximate amount will be as much greater or smaller than the required rate, as the interest added to the approximate amount is greater or smaller than that required to produce the amount found by dividing. The rule can only be well applied when the rate per cent. sought is within the limits of the table.

EXAMPLES.

2. At what rate per cent. will $ 400 amount to $ 640.405, at compound interest, in 12 years ? Ans. 4 per cent.

3. At what rate per cent. must $ 2500 be loaned, to produce $2096.147 compound interest, in 9 years? Ans. 7 per cent.

At what rate per cent. will any sum of money double itself at compound interest in 11,80 years? Ans. 6 per cent.

5. At what rate per cent. will $10,000 amount to $ 31479.70 in 19 years and 8 months ?

Ans. 6 per cent.

4.

380.° To find the time, the principal, the compound interest, and the rate per cent. being given.

Ex. 1. In what time will $ 500, at 7 per cent. compound interest, amount to $ 655.398 ?

Ans. 4

years.

OPERATION.

$ 655.398 ; 500 = $ 1.310796, which, at the given rate in

the table, denotes 4 years' time.

Since $ 500 amounts to $ 655.398 at 7 per cent. in the required time, $ 1 at the same rate, in the same time, will amount to do as much, or $1.310796. Corresponding to this amount of $1 at the given rate, we find in the table (Art. 377) 4 years, the time required.

RULE. Divide the amount by the principal, and the quotient will be the amount of $ 1 at the given rate for the required time; and in the table, against this amount, may be found the time required.

Note. — If the required time cannot be found exactly in the table, the number against that amount of $1 which under the given rate is next less than the amount found by dividing, will denote the whole periods or years. Then, find the fractional period or part of a year, by dividing 1 whole period or year by the ratio of the difference between the amount corresponding to the whole periods or years and that found by dividing to the difference between the former of the amounts and that next larger in the table; and the value of the fraction obtained as the result may be expressed in months or days, or both.

EXAMPLES.

Ans. 12 years.

2. In what time will $ 400 amount to $ 640.405 at 4 per cent. compound interest ?

3. In what time will $ 6000 amount to $ 9021.78 at 7 per cent. compound interest ?

14. In what time will any sum double itself at 5 per cent. compound interest ?

Ans. 14y. 2mo. 13d. 5. In what time will any sum double itself at 6 per cent. compound interest ?

Ans. 1ly. 10mo. 20+d. 6. A gentleman has deposited $ 450, for the benefit of his son, in a savings bank, at compound interest at a semiannual rate of 31 per cent. He is to receive the amount as soon as it becomes $ 1781.661. Allowing that the deposit was made when the son was 1 year old, what will be his age when he can come in possession of the money?

Ans. 21 years.

DISCOUNT AND PRESENT WORTH.

381. DISCOUNT is an allowance or deduction made for paying money before it is due.

382. The PRESENT WORTH is the amount of ready money that will satisfy a debt before it is due. It is equivalent to a principal which, being put at interest, will amount to the debt at the time of its becoming payable. Thus, $ 100 is the present worth of $ 106 due one year hence at 6 per cent.; for $ 100 at 6 per cent. will amount to $ 106 in that time; and $ 6 is the discount.

383. In discount, the rate per cent., the time, and the sum on which the discount is made, are given, to find the present worth, which corresponds precisely to the rate per cent., the time and the amount being given, in either simple or cornpound interest, to find the principal.

384. The interest or percentage of any sum cannot properly be taken for the true discount; for we have seen (Art. 382) that the interest for one year is the fractional part of the sum at interest, denoted by the rate for the numerator, and 100 for the denominator ; and the discount for one year is the fractional part of the sum on which discount is to be made, denoted by the rate for the numerator, and 100 plus the rate for the de

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