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14. How long will $1.00 be in gaining as much as $895.00 in 5 mo. ? A. 4,475 mo. 15. How long will $1.00 be in gaining as much as $272.00 in 7 mo., and $336.00 in 6 mo. ?

Ans. $272.00 for 7 mo.=$1.00 for 1,904 mo., and $336.00 for 6 mo. $1.00 for 2,016 mo. Then, $272.00 for 7 mo.+$336.00 for 6 mo. $1.00 for 1,904 mo. +2,016 mo.=3,920 mo. Ans.

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16. How long will it take $1.00 to gain as much as $700.00 in 3 mo., $800.00 in 4 mo., $900.00 in 6 mo., and $500.00 in 12 mo. ? A. 16,700 mo.

Then, it is plain that if I keep $1.00, 16,700 mo., I gain as much interest as I should by keeping the several sums in the example, the times specified. The amount of those sums is $2,900, for 700+800+900+500=2,900. Then,

17. How long may I keep $2,900.00 so as to gain as much interest as I gain on $1.00 in 16,700 mo.? I may keep it as long. A. 5 mo. 222 d.

Now, as $2,900.00 is the amount of all the sums in ex. 16, if we consider them debts due at the times mentioned, it is plain that the whole might be paid in 5 mo. 2222 d., without loss on either side. Hence, the two questions might have been expressed in one, as follows:

18. If I owe several debts, due as in example 16, at what time may I pay the whole together, without loss to me or to my creditor? A. In 5 mo. 2223 d.

For $700 for 3 months is the same as $1 for 3X700 2,100 mo.
$300 for 4 months is the same as $1 for 4X800 3,200 mo.
$900 for 6 months is the same as $1 for 6X900=5,400 mo.
$500 for 12 months is the same as $1 for 12X500=6,000 mo.

Then, $2,900 for these several times, is the same as $1 for 16,700 mo. But $1 for 16,700 mo. is the same as $2,900 for z of the time=5 mo. 222

d.

Hence, to find the time when several payments, due at different times, may fairly be made at once,

MULTIPLY EACH PAYMENT BY ITS TIME, AND DIVIDE THE SUM OF THE PRODUCTS BY THE SUM OF THE PAYMENTS.

The time thus found is called the EQUATED TIME.

When there are years and months, or years, months and days, it is best to reduce the months, or months and days, to a vulgar or decimal Fraction. A decimal is to be preferred.

19. Find the equated time for the following payments; viz. $100 due in 6 mo., $120 in 7 mo., and $160 in 10 mo.

A. 8 mo. 20. Find the equated time for the following payments; viz. $50 in 2 mo., $100 in 5 mo., and $150 in 8 mo. A. 8 mo.

21. Find the equated time for paying $900 due in 4 mo., $700 due in 1 yr. 3 mo., $1,200 due in 2 yrs., and $600 due in 3 yrs. 6 mo. 22. Find the equated time for paying $900 due in 9 mo., $1,500 due in 6 yrs., $1,800 due in 7 yrs. 9 mo., and $1,000 due in 3 yrs. 8 mo., 12 d.

It will be seen that the above rule supposes discount and interest to be equal, which is not true. But the error is too trifling to be regarded, in the common concerns of business. If any instance should occur, where the debts are very large, and the times very unequal, a rule for discovering the equated time may be found by an algebraic operation, but it would not be understood here, if given, and is therefore omitted.

§ LXXXVIII. COMPOUND INTEREST. 1. A owes B $600, on which he is under obligation to pay interest at the end of every year, until a final settlement is made. At the end of the first year, he finds it inconvenient to pay the interest, and B allows him to keep it, on condition that he will pay yearly interest on this likewise. For the second year, then, he is to pay interest on the original debt of $600, and also on the interest which he should have paid at the end of the first. What was the sum at interest the second year? A. $636.

For 600.06=36 the interest for 1st year, and 600+36= 636. 2. At the end of the second year, A is again unable to pay the interest, and the same agreement is made as before. What was the principal for the third year. A. $674.16. For 636.06=38.16 int. for 2nd year, and 636+38.16=674.16. 3. A settlement was made at the end of the 3rd year. What did A.pay; and how much for interest?

Ans. He paid $714.6096, for int. $114.6096. Interest, calculated thus on both principal and interest, is called COMPOUND INTEREST. Interest, calculated only on the principal, is called SIMPLE INTEREST. Though the only just interest is compound interest, it is not allowed by law. From the preceding examples, we derive the rule,

I. I. FIND THE AMOUNT FOR THE FIRST YEAR, AS IN SIMPLE INTEREST, AND MAKE IT THE PRINCIPAL FOR THE SECOND; ON THIS, FIND THE AMOUNT FOR THE SECOND, AND MAKE IT THE PRINCIPAL FOR THE THIRD, AND SO ON, UP TO THE NUMBER OF YEARS GIVEN. The last AMOUNT IS THE AMOUNT AT COMPOUND INTEREST.

u. SUBTRACT THE PRINCIPAL FROM THE AMOUNT, AND THE REMAINDER WILL BE THE INTEREST.

NOTE. When there are months and days, calculate first for the years, and on the amount thus found, compute the interest for the months and days, which add to the amount previously obtained.

4. Required the amount of $100.00 for 3 years, at compound interest. A. $119.1016.

5. Required the compound interest on $300.00 for 4 years. A. $64.6519

In obtaining amounts, we may multiply the principal by the rate, with a unit added. (§ LXXIV. AND NOTE ex. 9. § LXXVI.) Hence, to find an amount at Compound interest, we may multiply the given principal, by the rate with a unit added, as many times successively as there are years. But this, (§ XII.) is the same as though we should multiply the principal, at once, by the product of all these successive multipliers; and as all the multipliers are alike, being the rate a unit, we have the following rule.

II. I. ADD A UNIT TO THE RATE, AND MULTIPLY THE SUM BY ITSELF CONTINUALLY, UNTIL IT IS TAKEN AS A FACTOR, AS OFTEN AS THERE ARE

YEARS.

II. MULTIPLY THE PRINCIPAL BY THE LAST PRODUCT, AND THE RESULT WILL BE THE AMOUNT AT COMPOUND INTEREST.

6. Find the amount of $256, at compound interest, for 3 yrs. A. $304.900-+

7. Find the amount of $1,000.00 for 4 yrs. at 7 per ct. compound interest. A. $1,310.796+ 8. Find the amount of $425.00 for 4 yrs. at 5 per ct. compound interest. A. $516.590+ 9. Find the amount of $2,000.00 for 3 yrs. 6 mo. at compound interest.

10. Find the amount of $1,900.25 for 9 yrs. 7 mo. 7d.
On the principles of the last rule, the following table has been prepared.
TABLE OF MULTIPLIERS,

FOR FINDING COMPOUND INTEREST, ON ANY PRINCIPAL, FOR ANY NUMBER
OF YEARS, FROM 1 TO 24 INCLUSIVE, AT 5 AND 6 PER CENT.

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For

In calculating compound interest by this table, take the multiplier opposite the number of years and under the given rate, multiply the principal by it, and the product will be the amount. most cases it will be sufficient to employ four decimal places in the multiplier, increasing the last figure by 1, if the figure omitted is 5

or more.

NOTE. If the time be greater than the extent of the table, take any two or more multipliers, whose times, added together, will equal the given time, and multiply by them successively; that is, multiply the principal by one of them, and that product by another, and so on-thus,

11. Find the amount of $1,000.00 at 5 pr. ct. Compound interest,

for 36 years.

Multiplier, by table 3.22509, for 24 years. Then 1,000×3.22509 =3,225.09 and 3,225.09×1.79585 (multiplier for 12 years)=$5,791. 7778+ Ans.

NOTE. The pupil will understand why we multiply successively by these multipliers, if he considers that the product $3,225.09 is the amount for 24 years. This, by the rule for compound interest, must be made the principal for the succeeding time; and, as 12 years remain, the question becomes, what is the amount of $3,225.09 for 12 years. We must therefore multiply $3,225.09 by the multiplier corresponding to 12 years.

12. Find the amount of $7,500.00 at 6 pr. ct. Compound interest for 4 years. Ans. $9,468.529.

NOTE. This answer is strictly correct, as will be seen y performing the example by rule I. The table will give $9,468.536. Thus it will be seen that, in ordinary cases, the error by table is of no importance.

13. Find the amount of $5,000.00 for 18 yrs. 6 mo. at 5 pr. ct. Compound interest. Ans. $12,333.876.

14. Find the amount of $1,108.092 at 6 pr. ct. Compound in- · terest for 11 years.

15. Find the amount of $63,700.00 for 19 years, at 5 pr. ct. Compound interest.

16. Find the amount of $47,753.18 for 17 yrs. 6 mo. 9 d. at 5 per ct. Compound interest.

If a principal be multiplied by a number taken from the table, an amount is obtained. Of course, if this amount be divided by the same number, the quotient will be the principal again. In calculating present worths, therefore, at compound interest, on sums due at a future day, we may divide the debt by the number in the table, corresponding to the given time. The present worth, thus obtained, being subtracted from the whole debt will give the discount.

17. What is the present worth of $304.900096 due 3 years from the present time, when compound interest is allowed at 6 per ct. ? A. $256.

18. Find the present worth of $1,593.30, for 20 yrs., compound interest being allowed at 5 per cent. A. $600.50- Of $1,925.881, for the same time, at 6 per cent. A. $600.50

19. Find the discount at 5 per cent. on $607.7531, for 4 years. A $107.7531. 20. Find the discount on $1,642.992 for 21 years, at 4 per cent. A. $921.992. This divisor must be calculated, as it is not in the table.

21. Find the present worth of $875.00 for 6 years at 5 per cent. Of $947.00 at 6 per cent. for 9 years. Of $1,785.00 for 16 years at 5 per cent. Of $8,963.25 at 5 per cent. for 11 years. At simple interest, a sun is 16 yrs. 8 mo. in doubling itself. At compound, it is 11 yrs. 10 mo. and between 21 and 22 days. But as simple interest only doubles the original principal, every 16 yrs. 8 mo., and compound interest doubles its amount, every 11 yrs. 10 mo. and 21 or 22 d., it is evident that a sum will increase vastly more rapidly at compound, than at simple interest. In a few centuries the increase of a sum of money at compound interest becomes almost incredible.

ENDORSEMENTS.

§ LXXXIX. TO CALCULATE INTEREST ON NOTES WHEN THERE ARE We have hitherto used the term NOTE without defining it. A NOTE is a written promise to pay a sum of money, either on demand (that is, when it is required,) or after a certain space of time has elapsed. Notes are given with or without interest. Unless the words, " with interest” are expressed, a note is understood to be without interest. If a note without interest given for a specified time, be not paid when it is due, it draws interest afterwards, till paid.

Payments in part, or partial payments are sometimes made, and, in this case, a written acknowledgement is made on the note, called

an ENDORSEMENT.

As the debtor is not obliged to make payments in part, before payment is demanded, or before the note falls due, it is evident that he ought to be allowed interest on the payment, up to the time of settlement, if the note is given on demand, or till it is due, if given for a specified time. For if he had kept the money in his own hands, he might have had the use of it for that t me. And, after he has paid it, the creditor has the same advantage. Hence, the fol.

lowing rule,

1. FIND THE AMOUNT OF THE PRINCIPAL OF THE NOTE FOR THE WHOLE

TIME.

II. FIND THE AMOUNT OF EACH PAYMENT, FROM THE TIME IT WAS

PAID, TILL THE TIME OF SETTLEMENT.

III. ADD THE AMOUNTS OF THE PAYMENTS, AND DEDUCT THE SUM FROM THE AMOUNT OF THE PRINCIPAL.

NOTE When the note is for a specified time, it should be recollected that the payments ought to draw interest no longer than till the note is due.

$500.00

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Hartford April 1, 1828. 1. On demand I promise to pay PACKARD AND BUTLER, or order, five hundred dollars; with interest; value received. E TIMOTHY BOOKWORM,

On this note were the following endorsements.

Jan. 16, 1829, rec'd $150.00

April 1, 1829,

دو

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$150.00

$100.00

What was due Aug. 1, 1830.

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