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One half that prin. will produce in double that time;
os thrice that time; or Twice that principal will “ half
that time; Thrice that principal will “ a third of that time, &c.
For example, at any given per cent.
$1 for 3 years ; &c. The int. of $4 for 1 mo.
$1 for 4 mos.; The int. of $5 for 1 mo.
1 for 5 mos., &c. Hence, universally,
366. The interest of any given principal for 1 year, or 1 month, fc. is the same as the interest of 1 dollar for as many years, or months, fc. as there are dollars in the given principal.
Ex. 1. Suppose you owe a man $15 and are to pay him $5 in 8 months, and the other $10 in 2 monihs, at what time may both payments be made without loss to either party?
Analysis.--Since the interest of $5 for 1 month is the same as the interest of $1 for 5 months, (Art. 365,) the interest of $5 for 8 months must be equal to the interest of $1 for 8 times 5 months. And 5 mo. X8=40 mo. In like manner the interest of $10 for 1 month is equal to the interest of $1 for 10 months, and the interest of $10 for 2 months is equal to the interest of $1 for 2 times 10 months. And 10 mo. X2=20 months. Now 40 months added to 20 months make 60 months ; that is, you are entitled to the use of $1 for 60 months. But $1 is is of $15, consequently you are entitled to the use of $15, is part of 60 months, and 60 months 15=4. Ans. 4 months.
Proof. The interest of $5 at 6 per ct. for 8 mo. is $5 X.04=$.20 The interest of $10 (
2 mo. is $10 X.01= .10
Sum of both $.30 The interest of $5 at 6 per ct. for 4 mo. is 15 X.02=$.30
367. Hence, for finding the average time of two or more debts due at different times, we have the following
First multiply each debt by the time before it becomes due ; then divide the sum of the products thus obtained by the sum of the debts, and the quotient will be the average time required.
2. If I owe a man $20, payable in 4 months, $40 payable in 6 months, and $60 in 3 months, at what time may I justly pay the whole at once ?
“ $1 for 240
66 $1 for 180
20 3. A merchant bought three lots of goods, amounting to $300 ; for the first he gave $100, payable in 5 months ; for the second, $150, payable in 8 months ; for the third, $50, payable in 2 months : what is the average time of all the payments ?
4. A farmer has 3 notes ; one of $50 due in 2 months ; another of $100 due in 5 months; and the third of $150 due in 8 months: what is the average time of the whole sum ?
5. A merchant buys goods amounting to $1200, and agrees to pay $400 down, $400 in 4 months, and $400 in 8 months : he finally concluded to give his note for the whole : at what time must the note be made payable ?
6. A man borrows $600, and agrees to pay $100 in 2 months, 200 in 5 months, and the balance in 8 months : when can he justly pay the whole at once ?
Quest.–367. What is the rule for finding the average time of debts due at different times ?
7. A man buys a house for $1600, and agrees to pay $400 down, and the rest in 3 equal annual instälments : what is the average credit for the whole ?
8. I have $1200 owing to me, s of which is now due ;
of it will be due in 4 months, and the remainder in 8 months : what is the average time of the whole ?
9. A grocer bought goods amounting to $1500; for which he was to pay $250 cash in hand ; $300 in 4 months; and $950 in 9 months : when may
the whole at once ? 10. A young man bought a farm for $2000, and
agrees to pay $500 down, and the balance in 5 equal annual instalments : what is the average time of the whole.
368. Partnership is the associating of two or more individuals together for the transaction of business. (Art. 299.) The persons thus associated are called partners.; and the association is termed a company or firm. The money employed is called the capital or stock; and the profits or loss to be shared among the partners, the dividend.
CASE I. Ex. 1. A and B formed a partnership; A furnished $300 capital, and B $500 ; they gained $200 : what was each partner's share of the gain ?
Solution. Since the whole stock is $300+$500= $800, A's part of the stock was 300=, and B's part was 500=. Now since A put in of the stock, he must have f of the 'gain ; and $200 x}=$75. For the same reason B must have s of the gain ; and $200 x = $125.
Proof. $75+125=$200, the whole gain. (Art. 284. Ax. 10.) Hence,
Quest.-368 What is partnership? What are the persons thus associated called? What is the association called ? What is the money employed called? What is the dividend?
369. To find each partner's share of the gain or loss, when the stock of each is employed for the same time.
Make each man's stock the numerator, and the whole stock the denominator of a common fraction ; multiply the gain or loss by the fraction which expresses each man's share of the stock, and the product will be his share of the gain or loss.
Or, multiply each man's stock by the whole gain or loss ; divide the product by the whole stock, and the quotient will be his share of the gain or loss.
PROOF.- Add the several shares of the gain or loss together, and if the sum is equal to the whole gain or loss, the work is right. (Art. 284. Ax. 10.).
Obs. 1. This rule is applicable to questions in Bankruptcy, General Average, and all others in which there is to be a division of property in specified proportions.
2. The preceding case is often called Single Fellowship. But since a partnership is always composed of two or more individuals, it is somewhat difficult to see the propriety of calling it single.
2. A, B, and C entered into partnership ; A put in of the capital, B 3A, and C 2; they gain $4800 : what was each man's share of the gain ?
3. A, B, and C form a partnership; A furnishes $600, B $800, and C $1000; they gain $480 : what is each man's share of the gain ?
4. A Bankruptowes A $1200, B $2300, C $3400, and D$4500 ; his whole effects are worth $5600 : how much will each creditor receive ?
5. A, B, C, and D make up a purse to buy lottery tickets ; A puts in $30, B $40, C $60, and D $70; they draw a prize of $2000 : what is each man's share ?
6. A, B, and C freight a vessel with a cargo worth $30000; of which A owned $8000, B $10000, and C $12000; in a gale the master throws $ of the cargo overboard : what was each man's loss?
QUEST.-269. How is each man's share of the gain or loss found, hwen the stock of each is employed for the same time? How is the operation proved? Obs. To what is this rule applicable? What is it sometimes called ?
7. A and B formed a copartnership; A put in $300, and B $200. At the end of 2 months A took out his stock, while B’s was employed 6 months ; they gained $150: what was each man's just share of the gain ?
Note. It is obvious that the gain of each depends both upon the capital he furnished and the time it was employed. (Art. 364.)
Solution. Since A's capital $300 was employed 2mo., his share of the gain is the same as if he had put in $600 for 1mo.; (Art. 365;) for $300 x2=$600. Also, B's capital $200 being employed 6 mo., his share of the gain is the same as if he had put in $1200 for 1 mo.; for $200 x 6=$1200. The sum of $600 and $1200 is $1800. A's share of the gain must therefore be 100%=}. B's
“ 1208=3 Now $150 x }=$50, A's share. And $150 x}=$ 100, B's share. Hence,
370. To find each partner's share of the gain or loss, when the stock of each is employed for different periods.
Multiply each partner's stock by the time it is employed ; make each man's product the numerator, and the sum of the products the denominator of a common fraction ; multiply the whole gain or loss by each man's fractional share of the stock, and the product will be his share of the gain or loss. Obs. This Case is often called Compound, or Double Fellowship.
8. A, B, and Center into business together; A puts in $500 for 4 months, B $400 for 6 months, and C $800 for 3 months; they gain $340: what is each man's share of the gain?
9. A and B hire a pasture together for $60; A put in 120 sheep for 6 months; and B put in 180 sheep for 4 months : what should each pay?
Quest.–370. When the stock of each partner is employed for differ. ent periods, how is each one's share found ? Obs. What is this case sometimes called ?