Bryant and Stratton's Commercial Arithmetic

If compounded semi-annually, we should have the follow

We have seen heretofore that simple interest varies directly in the same ratio as the principal, time, and rate per cent. Compound interest likewise varies as the principal, but when the time or rate per cent. is increased, compound interest increases in a still greater ratio. e. g.

Doubling the principal doubles the compound interest.

Doubling the time more than doubles the compound interest. Doubling the rate more than doubles the compound interest, as may be seen from the following figures:

When the interest of $100 is

The interest of $200 would be

$5.

$10.

The compound interest of $100 for 2 years at 6% is $12.36

From the fact that compound interest (and therefore the compound amount) varies as the principal, the time and rate remaining the same, having ascertained the interest or amount for any one principal, the interest for any other may be found by a simple proportion. Tables have therefore been prepared giving the interest or amount of $1 for different intervals of time and at different rates per cent. The interest or amount for any given principal may then be found by simply multiplying the sum found in the table by the given principal. If the intervals are less than one year, as when the interest is to be compounded semi-annually or quarterly, tables computed with yearly intervals may still be used by reducing the rate per cent.

proportionably, and taking in the table the proper number of intervals.

For the time used in expressing any rate of interest is entirely arbitrary, and having fixed the ratio between the principal and interest at each compounding, the result depends upon the number of times the operation be repeated. Thus, if the interest be compounded a given number of times by adding to each respective amount 4% of itself, it matters not whether it be considered 4% per annum or 4% per minute, the result would be the same. If the interest is to be compounded quarterly, when the rate is said to be 8% per annum, 2% should be used at each compounding, though it would amount to more than 8% compounded annually.

Examples.

1. What is the compound interest and amount of $1000 for 5 yrs. at 6% per annum, payable annually?

Ans. $338.22 and $1338.22. 2. What is the compound amount of $2200 for 3 yrs. 2 mo. 12 d. at 6% per annum, payable annually?

Ans. $2651.67.

Note. After having computed the compound amount for the number of entire intervals at the end of which the interest is payable or to be computed, compute the simple interest on that amount for any remaining time before the settlement.

3. What is the compound interest of $1400 for 10 yrs. 8 mo. at 8% per annum, payable quarterly.

Solution.-$2.29724447 x 1400 $3216.1423= the comp. amount for 10 yrs. 6 mo., and $3216.1423 × 1.011-$1400the comp. interest for 10 yrs. 8 mo.= $1859.024.

4. If the population of a city containing 10,000 inhabitants should increase 10% annually, what would it amount to in 10 years? Ans. 25,937.

5. If a farmer beginning with one bushel of wheat should sow his entire crop each successive year, and the increase each year should be 1900%, what would he have at the end of 5 years? Ans. 3,200,000 bushels.

6. If a banker's rate in loaning money is 12% per annum, and he reloans all his capital every two months, what must have been the rate at simple interest to realize the same amount at the end of one year?

What, at the end of two years ?
What, at the end of eight years?
What, at the end of fifteen years?
What, at the end of twenty-five years?

Ans. 12% nearly.

Ans. 13% nearly.

Ans. 19% nearly.

Ans. 33% nearly.

Ans. 74% nearly.

ART. 102. In compound interest, as in simple interest, the four quantities, viz., principal, time, rate per cent., and interest or amount, bear such a relation to each other as that when any three of them are given, the fourth may be found. Hence four cases arise.

CASE I.

The principal, time, and rate being given, to find the compound interest and amount.

This case has already been presented, but the rule may be expressed in a more concise form.

RULE. Find from the table the amount of $1 for the given number of entire intervals, or times of compounding, at the proper rate for each interval, and multiply it by the given principal. Taking this product for a new principal, find the amount at simple interest for any fractional interval, if any, remaining before settlement. This will be the compound amount, and the compound interest may be found by subtracting from it the given principal.

CASE II.

The compound interest or amount, the time and rate being given, to find the principal.

RULE. Assume $1 for the principal; compute for the given time and rate its compound interest or compound amount, by which divide the given compound interest or compound amount, observing always to divide interest by interest and amount by amount. See Art. 97.

For illustration of Present Worth see Art. 99.

Examples.

1. What sum, in 17 yrs., at 6%, payable annually at compound interest, will amount to $1009.79 ? Ans. $375.

2. What sum, in 14 yrs., at 8%, payable semi-annually at compound interest, will amount to $10,795.34? Ans. $3600.

3. What principal will yield $3251.50 compound interest in 6 yrs. 2 mo. at 7%, payable semi-annually? Ans. $6150.

4. How much must a father, at the birth of his son, set apart for his benefit, so that with the interest at 7%, compounded semi-annually, it may amount to $10,000, when his son shall become 21 years of age? Ans. $2357 79.

5. What sum at 10%, payable quarterly, will produce $7197.22, compound interest, in 3 yrs. 6 mo. 9 d. ?

Ans. $17,280.

6. What is the present worth of $50,000, due 50 yrs. hence, at 9 per cent., payable annually? Ans. $672.43. How much greater would be the present worth at simple interest ?

CASE III.

The principal, time, and interest or amount being given, to find the rate. See Case IV.

CASE IV.

The principal, rate, and interest or amount being given, to find the time.

For the last two cases we have the following general

RULE.-Divide the given amount by the principal; the quotient will be the compound amount of $1 at the given rate for the required time or for the given time at the required rate. By reference to the table, the rate heading the column in which this quotient is found opposite the given time or number of intervals, will be the required rate; and the number in the left hand column opposite the quotient under the given rate will be the required time or number of intervals.

Examples.

1. At what rate will $7200 yield $12,665.02, compound interest in 15 yrs. ? Ans. 7 per cent.

2. At what rates will any sum of money double itself by compound interest in 8, 10, 15 yrs. payable semi-annually? Ans. 41%, 31%, 21%, respectively.

3. In what time will $5428 amount to $27157.31 at 5%, payable annually? Ans. 33 yrs. 4. In what time will any sum of money triple itself by compound interest at 4%, 7%, 8%, 10%, payable quarterly? Ans. 7 yrs., 4 yrs., 31 yrs., 3 yrs, nearly.

ART. 103. When partial payments are made on mercantile accounts which are past due, and on notes running only for a year or less, it is customary to use the

VERMONT RULE.

Compute the interest on the whole debt or obligation from the time it began to draw interest, and on each payment from the time it was made until the time of settlement, and deduct the amount of all the payments, including interest, from the amount of the debt and interest.

Note. When a partial payment is made on a note or obligation before it is due, no part is applied to the discharge of the interest, but the whole is used to reduce the principal in accordance with the above rule.

$600.

CLEVELAND, Nov. 18, 1856.

Ninety days after date, I promise to pay to the order of William Penn six hundred dollars, with interest, value received. WALTER JOHNSON. Indorsements.-Nov. 30, $100; Dec. 10, $250; Dec. 20, $100; Jan. 2, $80.

What was due at maturity?

$600, with interest for 93 days, amounts to

$609.30

$100,