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3. The account-books of L. F. Wilson and S. P. Allen show that

Wilson owes Allen And that Allen owes Wilson $640 due Jan. 4, 1858.

$200 due Jan. 16, 1858. $900 due Feb. 6, 1858.

$1200 due Jan. 31, 1858. $500 due March 1, 1858.

$1000 due Feb. 10, 1858. $600 due March 20, 1858. $1000 due Feb. 28, 1858. What is the equated time for payment? 4. When was the balance of the following account due ? Dr. John H. Arnold in account with W. C. Burlingame. Cr. 1858.

1858. Jan. 9 To Mdse. 6 mo.* $253 25 Feb. 11 By Mdse. 5 mo. $500 00 Feb. 7 To Mdse. 4 mo.* 57387 Feb. 20 By Mdse. 4 mo. 326 27 Feb. 25 To Mdse. 4 mo. | 421 19 March 9 By Cash

349 20 March 1 To Mdse. 3 mo. / 147 00 March 23|By Mdse. 2 mo. 472 18

5. What was due on the following account July 1, 1858, interest being reckoned at 6 per cent?

Dr. Lewis Dexter in account with John Gorham. Cr. 1858.

I 1858. Feb. 3 To Mdse. 6 mo. $200000 Feb. 10 By Mdse. 4 mo. $800.00 March 1 To Mdse. 5 mo. 150000 Feb. 25 By Mdse. 6 mo. 100000 April 5 To Mdse. 2 mo. 180000 April 11 By Mdse. 4 mo. 250000 April 27 To Mdse. 4 mo. 1000.00 May 26 By Mdse. 2 mo. 190000 May 12 To Mdse. 3 mo. 120000 June 20 By Mdse. 1 mo. 1700 00 June 7 To Mdse. 2 mo. 3000100 June 30 By Mdse. 2 mo. 1000100

121. To find the Principal or the Interest when the

Amount, Rate, and Time are given. 1. What sum of money will amount to $367.65 in 1 year, at 74 per cent?

Solution. -- Since, at the given rate, the interest for 1 year must equal 77 15 3 semainin ivo = 200 = äs of the principal, the amount must equal 10%, or 3

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times the principal. Hence, the principal must equal 4, of the amount, or As of $367.65 = $342.

2. What principal will amount to $252.791 in 1 yr. 3 mo. 18 da. at 6 per cent, and what is its interest for that time ?

SOLUTION. - Since, at 6 per cent per year, interest for 1 yr. 3 mo. 18 da., or 468 days, is 768 = 1787 of the principal, the amount must equal 1,787 or 1878 times the principal. Hence, the principal must equal 1698, and the interest must equal 78, of the amount. 189. of $252.791 = $234.50 = principal; 78 of $252.791 = $18.291 = interest.

3. The amount of a certain sum for 1 yr. 5 mo. 15 da., at 5 per cent, is $1855.545. What is the principal and what the interest ?

SOLUTION. — Since, at 5 per cent per year, interest of any sum for 1 yr. 5 mo. 15 da., or 525 days, or 535 of a year, is 525 of 5o = of the principal, the amount must equal 1.2 or 3 times of the principal. Hence, the principal must equal 965, and the interest must equal os of the amount.

90 of $1855.545 = $1729.44 = principal; 7 of $1855.44 = $126.105 interest.

(a.) From these solutions, it appears that, to find the principal or the interest when the amount, rate, and time are given, we find the fraction which expresses what part interest for the given time at the given rate is of the principal, and divide the given amount by 1 plus this fraction.

Note. — The above-named fraction will always be the same part of the given annual rate that the given time is of 1 year, or of 360 days; or, if the rate is 6 per cent, it will equal the fraction expressing the part which the given time is of 200 months, or of 6000 days.

(b.) To find the interest, we subtract the principal, found as before, from the amount; or, we multiply the given amount by the fraction which shows what part interest for the given time and rate is of the amount. What principal will amount to

4. $342.40 in 1 yr. 2 mo. at 6 per cent?
5. $486,404 in 8 mo. 16 da. at 6 per cent?
6. $500.55 in 2 yr. 5 mo. 5 da. at 74 per cent?
7. $1834.24 in 11 mo. 15 da, at 4 per cent?

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(a.) Discount is an allowance made for the payment of money before it is due.

(b.) The PRESENT Worty of a debt due at a future time is the sum which it is supposed to be worth now.

(c.) According to the principles involved in simple interest, the present worth of a debt due at a future time is that sum of money which, put on interest now, will amount to the given debt at the time it becomes due. Hence, the given debt corresponds to the amount, the present worth to the principal, and the discount to the interest; and the problems in this article are merely additional applications of the principles involved in the last.

1. What is the present worth of $470.12, due in 1 yr. 4 mo. hence, at 9 per cent ?

Solution. — The present worth required is that sum of money which, put on interest at 9 per cent, will amount to $470.12 in 1 yr. 4 mo. Since, at 9 per cent per year, interest for 1 yr. 4 mo. or 16 mo. equals 16 of 9 per cent, equals 12 per cent of the principal, the amount must equal 112 per cent of the principal, and the principal, which is the present worth required, must equal 19. of the amount. But 11 of $470.12 = $419.75 = present worth.

PROOF. — The amount of $419.75 at 9 per cent. for 16 mo. = $470.12 = given debt. What is the present worth of —

2. $1453.20 due in 2 yr. 9 mo. 20 da. at 6 per cent?
3. $465.26 due in 1 yr. 4 mo. 12 da. at 6 per cent ?
4. $634.827 due in 2 mo. 15 da. at 8 per cent?
5. $600 due in 60 da. at 6 per cent?
6. $1000 due in 4 mo. at 9 per cent?

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11. What is the discount of $997.542 due in 2 mo. 12 da. hence, at 74 per cent?

Solution. The discount required is the interest of that sum of money which, at 77 per cent, will amount to $997.542 in 2 mo. 12 da. Interest for

1 74 3 2 mo. 12 da., or of a year, at 77 per cent, equals of = of the principal; hence, the amount must equal zo of the principal, and the discount must equal, is of the amount, equal to $14.742.

Note.—The discount might have been found by finding the present worth and subtracting it from the given debt.

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(a.) Merchants and business men are usually willing to allow a larger rate of discount than that obtained by the method of the preceding article. Sometimes they deduct the interest of the entire debt for the time it has to run, as is always done on notes discounted at bank (see 117), and sometimes they deduct a still larger sum.

1. What will be received on a debt of $1000 due in 2 mo., discount being allowed at the rate of 6 per cent per year?

ANSWER. --- $1000 minus the interest of $1000 for 2 mo. = $990.

2. How much money will be required to pay a note of $800 due in 3 mo., discount at 6 per cent per year being allowed ?*

3. Bought goods to the amount of $1200, for which I can pay either by giving my note payable in 6 mo., or by cash at a discount of 6 per cent per annum. How much shall I gain by borrowing enough money to pay for them now, and then paying the borrowed money with interest at the time when the note would have become due?

4. What is the difference between the present worth of a debt of $2000 due in 6 mo. and the sum which would be received on it, if discount be reckoned as in this article, money being worth 6 per cent per year?

5. I bought goods to the amount of $1500, payable in 4 mo.; but the seller offering to deduct 5 per cent of the debt for cash, I hired money enough to pay for them. At the end of the 4 mo., I paid the borrowed money with interest at 6 per cent. What did I gain by hiring the money?

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6. For how much can a debt of $695.34, due in 5 mo., be paid if a discount of 8 per cent of the debt be allowed for cash ?

7. I bought a note for $2500 due in 4 mo., at a discount of 5 per cent. Allowing that money was worth to me 6 per cent per year, and that the note was paid when due, how much did I gain by the transaction?

8. I sold my note for $1000 due in 2 mo., at a discount of 14 per cent per month. How much did I receive for it?

* Observe that grace is always to be allowed on notes on time.

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