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Practice is an expeditious way of performing questions in Compound Multiplication and Proportion.
RULE. — Assume the price at some unit higher than the given price; that is, if the price be pence, or pence and farthings, assume the price at a shilling a yard, or pound, fc.; if the price be in shillings, or shillings and pence, c., assume the price at a pound a yard, &c.; then take the aliquot parts of a pound.
EXAMPLES. 1. What will 368 yards of ribbon cost, at 6 pence a yard ?
Ans: 9£. 4s.
We assume the price at a shilling 6d. = })368s.
a yard, and then say, if 368 shillings 20) 184
be the price at a shilling a yard, at
6 pence it must be half as much, 9.£. 4s. viz. 184 shillings. We then reduce
the shillings to pounds. 2. What will 4785 yards of cotton cost, at 8 pence a yard?
Ans. 159£. 10s. OPERATION.
Having found the 6d. = })47855. = price at 1 shilling. price at 6d. as be2d. = 4)2392s. 60. = price at 6d.
fore, we find it for 797s. 6d. = price at 2d.
the 2d. by saying
that 2d. is of 6d. 20)3190s. Od.
159£. 10s. = price at 8d. 3. What is the interest of $368, at 15 per cent. ?
Ans. $ 55.20. OPERATION. 10 per cent. = 10)368 5= })36.80
10 per cent. is to of the principal. 18.40 and 5 per cent. is 1 of 10 per cent.
$ 55.20 4. What is the value of 17 acres 3 roods 35 rods of land, at $ 80 per acre ?
Ans. $ 1437.50.
EQUATION OF PAYMENTS.
When several sums of money, to be paid at different times, are reduced to a mean time for the payment of the whole, without gain or loss to the debtor or creditor, it is called Equation of Payments.
EXAMPLES 1. A owes B $19, $5 of which is to be paid in 6 months, $ 6 in 7 months, and $8 in 10 months. What is the medium time for the payment of the whole ? OPERATION.
By analysis. $ 5 for 6 months $5 x 6 = 30
is the same as $1 for 30 months ; $6 x 7 = 42
and $ 6 fór 7 months is the same $8 x 10 = 80
as $1 for 42 months ; and $8 for 1919)152(8 months. 10 months is the same as $1 for 80
months ; therefore $1 for 30 +42
+80 = 152 months is the same as $ 5 for 6 months, $ 6 for 7 months, and $8 for 10 months ; but $5, $ 6, and $8 are $19; therefore $1 for 152 months is the same as $19 for of 152 months, which is 8 months, as before. Hence the propriety of the following
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 true time required.
2. A owes B $ 300, of which $ 50 is to be paid in 2 months, $100 in 5 months, and the remainder in 8 months. What is the equated time for the whole sum ?
Ans. 6 months.
* This is the rule usually adopted by merchants, but it is not perfectly correct; for if I owe a man $ 200, $ 100 of which I was to pay down, and the other $ 100 in two years, the equated time for the payment of both sums would be one year. It is evident, that, for deferring the payment of the first $ 100 for 1 year, I ought to pay the amount of $ 100 for that time, which is $ 106; but for the other $ 100, which I pay a year before it is due, I ought to pay the present worth of $100, which is $94.33%), whereas, by Equation of Payments, I only pay $ 200. Strict justice would therefore demand that interest should be required on all sums from the time they become due until the time of payment, and the present worth of all bums paid before they are due. The better rule would be, to find the present worth on each of the sums due, and then find in what time the sum of these present worths would amount to the payments.
3. There is owing to a merchant $1000 ; $ 200 of it is to be paid in 3 months, $ 300 in 5 months, and the remainder in 10 months. What is the equated time for the payment of the whole sum ?
Ans. 7 months 3 days. 4. A owes B $150, $ 50 to be paid in 4 months, and $100 in 8 months. Bowes A $ 250 to be paid in 10 months. It is agreed between them that A shall make present payment of his whole debt, and that B shall pay his so much sooner as to balance the favor. I demand the time at which* B must pay the $ 250.
. Ans. 6 months. 5. A merchant has $144 due him, to be paid in 7 months, but the debtor agrees to pay one half ready money, and one third in 4 months. What time should be allowed him to pay the remainder ?
. Ans. 2 years 10 months. 6. There is due to a merchant $ 800, one sixth of which is to be paid in 2 months, one third in 3 months, and the remainder in six months; but the debtor agrees to pay one half down. How long may the debtor retain the other half so that neither party may sustain loss ?
Ans. 8 months. 7. I have purchased goods of A. B. at sundry times and on various terms of credit, as by the statement annexed. When is the medium time of payment ?
Jan. 1, a bill amounting to $ 375.50 on 4 months' credit. 66 20.
168.75 on 5 months' credit. Feb. 4, 66
386.25 on 4 months' credit. March 11, "
144.60 on 5 months' credit. April 7,
386.90 on 3 months' credit.
FORM OF STATEMENT.
June 20, 168.75 X 50 = 843750
$1462.00 ...... )6224150(424635da. Ans.
83750 The medium time of payment will therefore be 42167 days, that is, 43 days from May 1, which will be June 12.