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2.. What is the present worth of 150l. payable in of a year, discount being at 5 per cent. ?
Ans. 1481. 28. 11 d.
3. Bought a quantity of goods for 150l. ready money, and sold them again for 200l. payable at of a year hence ; what was the gain in ready money, supposing dis-, count to be made at 5 per cent. ?
Ans. 421. 155. 5d.
4. What is the present worth of 1201. payable as follows, viz. 5ol. at 3 months, 5ol. at 5 months, and the rest at 8 months, discount being at 6 per cent. ?
Ans. 1171. gs. 5 d.
ent payment, according to the principles we have mentioned, is exactly the same as employing the whole sum at interest till the time the debt becomes due arrives ; for if the discount allowed for present payment be put out to interest for that time, its amount will be the same as the interest of the whole debt for the same time : thus, the discount of 105l. due one year hence, reckoning interest at 5 per cent. will be 5l. and 5l. put out to interest at 5 per cent. for one year, will amount to 5l. 55. which is exactly equal to the interest of 1051. for one year at 5 per cent.
The truth of the rule for working is evident from the nature of simple interest : for since the debt may be considered as the amount of some principal (called here the present worth) at a certain rate per cent. and for the given time, that amount must be in the same proportion, either to its principal or interest, as the annount of
other sum, at the same rate, and for the same time, is to its principal or interest,
DISCOUNT BY DECIMALS.
As the amount of 1l. for the given time is to il. 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.
i. What is the discount of 5731. 155. due 3 years hence, at 4 per cent. per annum ?
* 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 Worth OF MONEY PAID BEFORE IT IS DUE
AT SIMPLE INTEREST.
'045X3+1=1'135= amount of 11. for the given time.
And 573*75 X '045X3 = 7745625 = interest of the debt for the given time.
I.135 : 1 :: 7745625 :
OF DISCOUNTS TO BE ALLOWED FOR PAYING OF MONEY
BEFORE IT FALLS DUE AT SIMPLE INTEREST.
5 per cent.
2. What is the discount of 7251. 16s. for 5 months, at 35 per cent. per annum ?
Ans. 11l. 1os. 31d. 3. What ready money will discharge a debt of 13771. 135. 4d. due 2 years, 3 quarters and 25 daya hence, discounting at 4 per cent. per annum ?
Ans. 12261. 8. 8 d.
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.
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 upon a supposition, that the sum of the interests of the several debts, which are payable before the cquated 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 interest, and therefore the rule is not true.
1. A owes B 1901. to be paid as follows, viz. 50l. in 6 months, 601. in 7 months, and Sol. in 10 months ; what is the equated time to pay the whole ?
50 X 6=300
Ans. 8 months.
Although this rule be not accurately true, yet in most questions, ihat 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.
Id = first debt payable, and the distance of its term of 1
payment t. Let D = last debt payable, and the distance of its term T.
x = distance of the equated time.
The distance of the time and a
is = *--t. tween T and t
The distance of the time T and »
is = T--*.
Now the interest of d for the timc *--* is xmix dr ; and the
interest of D for the time T-x is 7-**Dr ; therefore x
x dr=T-XX Dr by the supposition ; and from this equation
DT+di * is found =D+ 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.