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DISCOUNT BY DECIMALS.

RULE.*

As the amount of 11. for the given time is to 11. so is the interest of the debt for the said time to the discount required. Subtract the discount from the principal, and the remainder will be the present worth.

EXAMPLES.

What is the discount of 5731. 15s. due 3 years hence, at 41 per cent. per annum ?

045x3+1 1-135 amount of 11. for the given time.

=

And 573 75x ̊045×3=77°45625 interest of the debt for the given time.

Let m represent any debt, and n the time of payment; then will the following Tables exhibit all the variety, that can happen with respect to present worth and discount.

OF THE PRESENT ORTH OF MONEY PAID BEFORE IT IS DUE AT SIMPLE INTEREST.

The present worth of any sum m.

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2. What is the discount of 7251. 16s. for five months, at

37 per cent. per annum?

Ans. 111. 10s. 7d.

OF DISCOUNTS TO BE ALLOWED FOR PAYING OF MONEY REF RE IT BECOMES DUE AT SIMPLE NTS EST.

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3. What ready money will discharge a debt of 13771. 13s. 4d. due 2 years, 3 quarters and 25 days hence, discounting at 4 per cent. per annum?

Ans. 12261. 8s. 81d.

EQUATION OF PAYMENTS.

EQUATION OF PAYMENTS is the finding a time to pay at once several debts, due at different times, so that no loss shall be sustained by either party.

RULE.*

Multiply each payment by the time, at which it is due ; then divide the sum of the products by the sum of the payments, and the quotient will be the time required.

* This rule is founded on a supposition, that the sum of the interests of the several debts, which are payable before the equated time, from their terms to that time, ought to be equal to the sum of the interests of the debts payable after the equated time, from that time to their terms. Among others, who defend this principle, Mr. COCKER endeavours to prove it to be right by this argument; that what is gained by keeping some of the debts after they are due, is lost by paying others before they are due. But this cannot be the case; for, though by keeping a debt unpaid after it is due there is gained the interest of it for that time, yet by paying a debt before it is due the payer does not lose the interest for that time, but the discount only, which is less than the interest, and therefore the rule is not true.

Although this rule be not accurately true, yet in most questions, that occur in business, the error is so trifling, that it will be much used.

That the rule is universally agreeable to the supposition may be thus demonstrated.

d = first debt payable, and the distance of its term of

payment t.

Let D last debt payable, and the distance, of its term 7. = distance of the equated time.

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=rate of interest of 11. for one year.

EXAMPLES.

1. A owes B 190l. to be paid as follows, viz. 50l. in 6 months, 601. in 7 months, and 801. in 10 months; what is the equated time to pay the whole?

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2. A owes B 521. 7s. 6d. to be paid in 4 months, 801. 10s. to be paid in 3 months, and 761. 2s. 6d. to be paid in 5 months; what is the equated time to pay the whole?

Ans. 4 months, 8 days. 3. A owes B 2401. to be paid in 6 months, but in one month and a half pays him 601. and in 4 months after that 801. more; how much longer than 6 months should B in equity defer the rest? Ans. 27 months. 4. A debt is to be paid as follows, viz. at 2 months, at 3 months, at 4 months, at 5 months, and the rest at 7 months; what is the equated time to pay the whole?

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Now the interest of d for the time x-t is x-txdr; and the interest of D for the time T-x is T-xx Dr; therefore xxdr=T-xx Dr by the supposition; and from this equation DT+dt x is found: which is the rule. And the same might

D+d

be shown of any number of payments.

The true rule is given in equation of payments by decimals.

EQUATION OF PAYMENTS BY DECIMALS.

Two debts being due at different times, to find the equated time to pay the whole.

RULE.*

1. To the sum of both payments add the continual product of the first payment, the rate, or interest of 11. for one year, and the time between the payments, and call this the first number.

* No rule in Arithmetic has been the occasion of so many disputes, as that of Equation of Payments. Almost every writer on this subject has endeavoured to show the fallacy of the methods, used by other authors, and to substitute a new one in their stead. But the only true rule seems to be that of Mr. MALCOLM, or one similar to it in its essential principles, derived from the consideration of interest and discount.

The rule, given above, is the same as Mr. MALCOLM's, except that it is not incumbered with the time before any payment is due, that being no necessary part of the operation.

DEMONSTRATION OF THE RULE. Suppose a sum of money to be due immediately, and another sum at the expiration of a certain given time, and it is proposed to find a time to pay the whole at once, so that neither party shall sustain loss.

Now it is plain, that the equated time must fall between those of the two payments; and that what is got by keeping the first debt after it is due, should be cqual to what is lost by paying the second debt before it is due.

But the gain, arising from the keeping of a sum of money after it is due, is evidently equal to the interest of the debt for hat time.

And the loss, which is sustained by the paying of a sum of money before it is due, is evidently equal to the discount of the debt for that time.

Therefore it is obvious, that the debtor must retain the sum immediately due, or the first payment, till its interest shall ba equal to the discount of the second sum for the time it is paid Vol. I.

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