22 per cent below par, promising him per cent. for his trouble; what did the whole cost me?

Ans. $5886,56 +

[NOTE. When the original or nominal value of a share is not mentioned, 100 is understood.]

6. Bought 56 shares in a New York state bank, at 71⁄2 per cent. advance, the original value being $75 per share; what will the whole cost me, allowing 3 per cent. to the broker, who made the purchase? Ans. $4530,75.

EQUATION OF PAYMENTS.

(ART. 105.) By this is meant the equitable time of paying several debts due at different times, so that neither party shall sustain any loss of interest.

To explain more fully, let us suppose that a person owes his neighbor as follows:

2 dollars, to be paid in 2 months;

2 dollars, to be paid in 3 months; and

2 dollars, to be paid in 5 months.

In what time shall he pay the whole 6 dollars, so that neither himself nor creditor shall lose interest? We analyze it thus:

2 dollars in 2 months will gain as much interest as 4 dollars in 1 month;

2 dollars in 3 months will gain as much interest as 6 dollars in 1 month;

2 dollars in 3 months will gain as much interest as 10 dollars in 1 month. Then

$6 for the several times will gain as much interest as $20 for 1 month.

But there are only 6 dollars to be paid, not 20; and as interest is always in proportion to the compound of money and time, we must find what number of months,

Q

multiplied by 6, will give the same product as 20, multiplied by 1; and that evidently is 20-3 months.

From this we may derive the following general

RULE. Multiply each payment by its time, and divide the sum of the several products by the sum of the payments, and the quotient will be the equated time for the payment of the whole.

EXAMPLES.

1.1f 600 dollars are now due, 600 dollars in 4 months, and 600 dollars in 8 months, 600 dollars in 12 months, what is the equitable time for paying the whole?

N. B. It is not necessary in this or any other problem, that we should use the sums of money actually given. It is sufficient that we use the same proportional parts. In this case we will use one dollar in place of 600, and the operation will stand thus:

3

2. If 750 dollars are to be paid, of it in 11 years, of it in 2 years, and the residue in 2 years, what is the equated time of paying the whole at once?

Ans. 23 months.

In this example, it is not necessary to use 750 dollars; we had better use 10.

3. A owes B 100 dollars, to be paid in 6 months; 120 dollars, to be paid in 10 months, and 160, to be paid in 14 months; what is the equated time for paying the whole? Ans. 101 months.

4. A merchant hath owing to him 300 dollars, to be paid as follows: 50 dollars at 2 months, 160 dollars at 5 months, and the residue at 8 months; and it is agreed to make one payment of the whole; when must that time be? Ans. 6 months. 5. Fowes H 2400 dollars, of which 480 dollars are

to be paid present, 960 dollars at 5 months, and the rest at 10 months; but they agree to make one payment of the whole, and wish to know the time?

Ans. 6 months.

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6. A merchant bought goods to the amount of 2000 dollars, and agreed to pay 400 dollars at the time of chase, 800 dollars at 5 months, and the rest at 10 months; but it is agreed to make one payment of the whole; what is the mean or equated time? Ans. 6 months.

N. B. To solve the following, we had better fall back on the general principle, and not attempt to apply the rule.

7. A owes B 600 dollars, to be paid in 2 years from the date of the note; but, at the expiration of 6 months, A agrees to pay 150 dollars, if B will wait enough longer for the balance, to compensate for the advance; how long ought B to wait?

Ans. 6 months after the 2 years. Here 150 dollars was paid 18 months before the time; how long shall the remaining 450 dollars remain after the expiration of the 2 years, to give the same interest. Statement: 150X18: 450X [] :: 1 : 1.

See Art. 80.

8. P owes Q 420 dollars, which will be due 6 months hence; but P is willing to pay him 60 dollars now, provided he can have the rest forborne a longer time; it is agreed on; the time of forbearance, therefore, is required? Ans. 7 months; that is, 1 month in addition to the 6.

9. A young gentleman has a legacy of 1500 dollars, to be paid him in 16 months; but, being in want of ready money, agrees to defer the payment of the balance the proper time, if he can have 500 dollars in hand; when shall the balance be paid? Ans. 2 years hence.

PROFIT AND LOSS PER CENT.

(ART. 106.) WHEN articles are bought and sold, it is often desirable to know the rate of gain or loss per cent., corresponding to any definite or assumed price. This of course can be done by proportion, Practice, and proportion. Interest and discount are all the arithmetical principles that can be brought into requisition, under this head; and practice, interest, and discount, are all resolvable into proportion; hence proportion alone is the rule.

EXAMPLES.

1. A merchant bought cotton cloth at 1 shilling 6 pence per yard, and sold it at 1 shilling 10 pence; what was his gain per cent.?

pence. pence. gain.

Statement: If 18 : 4 :: 100 : how much?

Ans. 223.

2. A grocer bought tea at 60 cents per pound, and sold the same at 75 cents; what was his profit per cent.?

Statement:

Ans. 25.

As 60 : 15 :: 100 : Answer or, 4 : 1 :: 100 : 25

3. I bought Irish linen at 56 cents per yard; what shall I sell it for, to gain 20 per cent.? Ans. 67.

Statement: As 100 120 :: 56: Answer.

4. A merchant sold broadcloth at 4 dollars 75 cents per yard, making a profit of 32 per cent.; what did the cloth cost him? Ans. $3,60, nearly.

Statement: 132, that he now receives, originally cost him 100; in that proportion, what did 475 cost? or, as 132 100 :: 475: Answer.

5. A merchant bought English broadcloth; the first cost, duties and transportation, amount to 14 shillings 8 pence per yard; what must be his price, in dollars and cents, to gain 20 per cent., the dollar being equal to 4 shillings 6 pence sterling? Ans. $3,91+

:

Statement: 100: 120 :: 143 price in shillings sterling. But to bring shillings, sterling money, into dollars, we must divide by 4, or multiply by 2 and divide by 9. The whole combined in one. cperation (Art. 24), stands thus:

6. If I purchase Irish linen at 2 shillings 4 pence per yard, sterling money, what must I sell it at per yard, in federal money, to gain 30 per cent.? Ans. 67.

7. A man bought 300 sheep, at 2 dollars 15 cents per head, and his expense in making the purchase, was 45 dollars 50 cents; he sold them at 3 dollars 20 cents per head; what was his whole gain, and what was his gain per cent.? Ans. Whole gain, $269,50; gain per cent.

39, nearly. 8. If I buy 12 hundred weight of sugar for 140 dollars, how much must it sell at per pound, to make 25 per cent.? Ans. 12 cents.

Solved as in Practice (Art. 88),

This cancels down to the answer.

9. Bought 126 gallons of wine for 150 dollars, and retailed it at 20 cents per pint; what was the whole gain, and what the gain per cent.?

Ans. Whole gain, $51,60; gain per cent. 343.

10. Bought a hogshead of molasses for 26 cents per gallon, and suppose it lost 5 per cent. in waste and leakage; how must I sell the remainder, per gallon, to gain 25 per cent.? Ans. 34 19 11. If, by selling 1 pound of pepper for 10 cents, there are 2 cents lost, how much is the loss per cent.? Ans. 16.