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4. Three merchants trading together lost goods to the value of $1860. A's stock was $2280, B's $11520 and C's $4800; what share of the loss must each man sustain?

Ans. A $288, B $1152, and C $480. 5. A ship valued at $25200 was lost at sea, of which i belonged to A, i to B, and the remainder to C; what is the loss on $1.00, and how much will each man sustain, supposing the owners effected an insurance of $18000?

A's $2400, B's $3600, and C's 1200. The pro-rata share on a dollar is 4.

CASE III.

When stocks have been put in trade for different periods of time, and settled with regard, both to stock and time.

RULE.—Multiply each man's stock and time, and then as the aggregate of products is to the whole gain, so is each man's stock to each man's share of the gain.

1. A, B, and C, join in company: A's stock is $100, for 12 months, B's 120 yards of cloth, for 8 months, and C's 240 bushels of wheat, for 7 months; they gained $1612, of which A had $100, B $512, and C $700; what was the value of B's cloth per yard, and C's wheat per bushel. Ans. B's cloth $1.60 pr yd, and C's wheat $1.25 pr bush.

2. A, B, and C, enter into partnership with a capital of $1100, of which A put in $250, B put in $300, and C $550; they lost by trading, 5 per cent. on their capital, what was each inan's share?

Ans. A's loss $12 50, B's $15, and C's loss $27.50.

In company accounts, when the times and payments are equal, the shares of gain or loss are evidently in proportion to their respective stock-and when the stocks are equal, the shares are in proportion to the times of payment. But when stocks and times are unequal, the shares are in proportion to the products of stock and time,

This may be clearly demonstrated thus:

Suppose $100 in trade 12 months, gain $20; $50 in trade in 6 months, will gain $5, and both together $25; for, as 100 X 12 : 50 x 6 : : 20 : 5 and 20 + ; again, by composition 100 X 12 + 50 x 6 : 100 x 12 : : 25 : 20; gain of $100 in 12 months, and 100 x 12 + 50 x 6 : 50 x 6 : : 25 : 5, gain of $50 in 6 months, from which the truth of the rule is evident.

3. A, B and C having traded together, gain $126.80— what is each man's share, allowing that A put in $50 for 4 months, B $100 for 6 months, and C $150 for 8 months?

Ans. A $12.68, B $38.04, and C $76.08.

EQUATION OF PAYMENTS Is a rule, for finding when any number of notes or bonds due at different times may be all paid at once, without loss to debtor or creditor.

RULE.--Multiply each payment by its time, divide the sum of the products, thence arising by the sum of all the payments, and the quotient will be the equated time required.

1. A. owes B. a bond for $100, due 2 months hence, and one for $500 due 18 months hence, what would be the equated time for paying them at once?

months.
Operation. $100 x 2 200
500 x 18

9000

mo.

6)00

6)9200( = 15Ans. 2. C. owes D. $550, of which $100 is to be paid at three months, $200 at 5 months, and $250, in 8 months, but have agreed to make one payment of the whole, at what time must it be paid?

Ans. 6 mos. 3. A man has owing to him $500 to be paid as follows, viz: $250 at 6 months, and $250 at 8 months, but it is agreed that the whole should be paid at one time, when must it be paid?

Ans. 7 mos. 4. A. owes B. 5 bonds, for $945 each, payable at 3, 9, 11, 19, and 29 months, what time might they be all paid at once?

Ans. 14} mos.

AVERAGE TIME OF SALES.

CASE I.

66

1. Sold merchandise at sundry times, and on different terms of credit, as per statement annexed. 1840 January 1st, $1500 at 3 months, due 1st April. February 10th, 250 at 2

66 10th · March 19th, 643 at 4

66 19th July. Septber 1st. 1400 at 6

“ 1st March. Required the average time of payment.

Ans. 21st August. Statement of the preceding question.

days. Due 1st April $1500 x 0 10th

250 X

9 19th July 643 x 109 1st March 1400 X 334

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3793 3793)539937(142 142 days from the 1st day of April which will make the average time fall on the 21st day of August. THEOREM and General Rule, to find the

average time that several bills of different dates, or different terms of credit, or both, become due.

D с B

TAPAUD In the above diagram let p, q, r, s, t, &c., be the several payments to be made, and B, C, D, E, F, &c., the different periods of time at which those payments are to be made, and o, the average point of time, then it is manifest (on the principal of Simple Interest, that p X BO + 9 x CO + r X DO t X FO + 3 XEO, and putting BO = x, we have p x + q (x— BC) + x (x BD) = 8 (BE- x) + t (BF — x) by transposition px + 2x + rx + sx + tx = q X BC + r X BD + 8 X BE + t X BF (p + q + r + s + t) x = x = q BC + r X BD + s x BE + t X BF.

p + q + r + s + t. Hence the following general rule: multiply the several payments to be made by the respective times from the first payment, add them together and divide that sum by the whole amount of the bills for the time sought, which is to be counted from the time on which the first payment falls due.

SOLUTION OF QUESTION I. CASE I. 334 1st March x 19th July. 10th April. 1st Ap.

Let x = average time from 1st April, then 1500

x + 250 (x-9) + 643 (x - 109) = 1400 (334 - x) then 1500 x + 250 x

– 2250 + 643 x - 70087 = 467600 — 1400 x. Again, 1500 x + 250 x + 643 x + 1400 x = 467600 + 70087

2250 + 70087 + 467600 + 2250 and x=

= 142 days

3793 after the 1st day of April, which agrees with the 21st day of August, 1840, as before.

SOLUTION OF CASE I. BY ANOTHER PRACTICAL METHOD.

0 = 9 =

25 X

to of the first bill is $150 X

000 2d

225
3d

64 x 109 = 6976
4th
140 x 334

= 46760

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66

379 379)53961(142 Note.—Agreeably to mercantile usage, a fraction less than one-half in dollars or days is omitted in the equation of payments, and when more than a half, it is considered as a unit.

CASE II.

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125 66

66

1. Sold merchandise at different times, and on various terms of credit. 1839 September 6th $100 for 1 month, due Oct. 6th.

14th
1

14th. October 10th 175. 66 2

Dec. 10th, November 14th 340 66 3

Feb. 14th.
January 14th 456 6 5

June 14th,
FIRST METHOD.'
SOLUTION.

days.
October 6th $100 X 0
14th

8
Dec'ber 10th 175 x 65
Feb’ry 14th 340 X 131
June 14th 456 x 251

125 x

66

1196 1196)171371(= 144 + 144 days from the 6th October will come to February 27, 1840, at which time a note for $1196 would be due.

SECOND METHOD.
October 6th $100 X 251

14th 125 x 243
Dec'ber 10th 175 x 186
Feb’ry 14th 340 X 120
June 14th 456 x 000

1196)128825(= 108 +

$1196 Due by average 108 days earlier than the 14th day of June 1840, which will be the 27th day of Feb. as above.

THIRD METHOD. 1839 September 6th $100 x 30 14th

38 October 10th

95 November 14th 340 X 161 January 14th 456 x 281

125 x 175 x

1196 1196)207251(174 + 174 days from the 6th September, which will make the average time fall on the 27th day of Feb. 1840, as before.

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