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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 belonged to A, 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 $400, 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.

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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 man'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;

20 + 5;

100 × 12

for, as 100 × 12: 50 × 6 :: 20: 5 and again, by composition 100 × 12 + 50 × 6 : :: 25: 20; gain of $100 in 12 months, and + 50 × 6: 50 × 6 :: 25: 5, gain of $50 in 6 months, from which the truth of the rule is evident.

100 x 12

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?

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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 lows, viz: $250 at 6 months, and $250 at

it is agreed that the whole should be paid

when must it be paid?

8

paid as fol

months, but

at one time,

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.

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.

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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.

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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 + q × CO + r × DO = t × FO + s × EO, and putting BO = x, we have p x + q (x— BC) + r (x BD) = 8 (BE— x) + t (BF-x) by transposition

px + qx + rx + sx + tx = q x BC + rx BD + $ × BE + t × BF (p + q + r + s + t) x = x q × BC + rx 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.

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April, then 1500 x+250 (x-9)

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

=

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- 2250 + 643 x

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-70087

1500 x

+ 2250 and x =

= 467600 - 1400 x. Again,

250 x + 643 x + 1400 x =467600 +70087 225070087 + 467600

3793

= 142 days

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.

of the first bill is $150 × 0

000

25 X 9 = 225

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64 × 109 = 6976

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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.

1. Sold merchandise at different times, and on various

terms of credit.

1839 September 6th $100 for 1 month, due Oct. 6th.

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

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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 × 30

66

14th 125 × 38

October 10th 175 X 95
November 14th 340 x 161
January

14th 456 x 281

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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