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2. What is the discount of 7251. 16s. for 5 months, at 33 per cent. per annum ? Ans. 11. 10s. 34d.

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

Ans. 12261. Ss. 8d.

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.

EXAMPLES.

*This rule is founded upon 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, that 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 inter est, and therefore the rule is not true.

Although

EXAMPLES.

1. A owes B 190l. to be paid as follows, viz. 5ol. in 6 months, 6ol. in 7 months, and 8ol. in 10 months; what is the equated time to pay the whole ?

50× 6=300

60X 7=420

80 X 10800

50+60+80=190)1520(8

1520

Ans. 8 months.

2. A

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 T.

x = distance of the equated time.

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Now the interest of d for the time x-t is x

-ix dr

; and the

interest of D for the time T-x is T-xx Dr ; therefore

xdr=T―xx Dr by the supposition; and from this equation

DT+dt

xis found Dd which is the rule. And the same might

be shewn of any number of payments.

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

2. A owes B 521. 7s. 6d. to be paid in 4 months, Sol. 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 240l. to be paid in 6 months, but in one month and a half pays him 6ol. and in 4 months after that Sol. more; how much longer than 6 months should B in equity defer the rest? Ans. 3 months

at 5 months, and the rest

4. A debt is to be paid as follows, viz. at 2 months, at 3 months, at 4 months, at 7 months; what is the equated time to pay the whole ? Ans. 4 months and 18 days.

EQUATION OF PAYMENTS BY DECIMALS.

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

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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.

2. Multiply

* No rule in arithmetic has been the occasion of so many disputes, as that of Equation of Payments. Almost every writer upon this subject has endeavoured to shew the fallacy of the methods made use of 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.

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2. Multiply twice the first payment by the rate, and call this the second number.

3. Divide

The rule, given above, is the same as Mr. MALCOLM's, except that it is not encumbered 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 forward, 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 equal 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 that

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 be equal to the discount of the second sum for the time it is paid before due; because, in that case, the gain and loss will be equal, and consequently neither party can be the loser.

Now, to find such a time, let a first payment, b = second, and = time between the payments; r = rate, or interest of 11. for one year, and x = equated time after the first payment.

Then arx interest of a for x time,

btr-brx

and

1+tr-rx

discount of b for the time t-x.

But

3. Divide the first number by the second, and call the quotient the third number.

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Then it is evident that n, or its equal is greater than

n2 m2, and therefore will have two affirmative values, the

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mand n-n-m being both positive.

quantities nn. -m

But only one of those values will answer the conditions of the question; and, in all cases of this problem, x will be n

For suppose the contrary, and let x = n +n2—m]2.

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I

have from the first of these equations,t2-2/n=

bt-at X

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But

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