5. A. B. and C. traded in company; A. put in $5760, B. put in $2400, and C. put in $1440; on settling their business, they found their gain to be $960; what was each one's share of the gain?

Ans. A.'s share, $576. B.'s share, $240. C.'s share, $144. 6. Three merchants entered into partnership; A. put in $6000, B. $4000, and C. $2000; they gained 12 per cent on the whole stock in trade; what was each one's share of the gain? Ans. A.'s gain was $750. B.'s $500. C.'s 250.

DOUBLE FELLOWSHIP.

Q. What is Double, or Compound Fellowship?

A. It is when the several shares of partners are continued in trade for an unequal term of time.

Q. What is the RULE for finding each partner's gain or loss? A. First multiply each partner's share, by the time it was continued in trade; then add all these products together, and by the rule of three, say: as the sum of these products is to the whole gain, or to the whole loss: so is each partner's particular product: to his share of the gain or loss.

EXAMPLES.

1. Three merchants joined their stocks in trade; A. put in $600, for 6 months; B. $500, for 9 months: and C. $300, for 12 months; by their speculations they gained $700; what was each man's share of the gain?

B.'s 66

mo.

Operation. A.'s share $600 × 6=3600 A.'s product. 500 x 9-4500 B.'s product. 300×12-3600 C.'s product.

C.'s 66

dolls.

Therefore say, as 11700: 700 ::

11700 sum of all the products. 3600 to $215,384+A.'s gain. 4500: to 269,23 B.'s gain. 3600 to 215,384+C.'s gain.

2. Three persons hired a large pasture, for $250, in which to pasture their cattle; on the first of May, each man turned in 40 head; on the first of June, A. turned in 20 head more, but B. took out 15 head of his; July 1st, B. turned in 20 head, and C. took out 20 head of his; August 1st, A. took out 25 head, B. put in 25, and C. put in 30 head; on the first of October, they give up the pasture, and settle the concern; what must each man pay for the use of the pasture?

Ans. A. must pay $84,55814. B. $91,91113. C. $73,529,7.

3. Four merchants entered into partnership; A. put in $800, for 5 months; B. $1200, for 7 months; C. $1920, for 8 months; and D. $2400, for 9 months; on closing their business, they find they have gained 1500 dollars; what must each partner receive for his share of the gain? Ans. A.'s share, $121,55+. B.'s, $255,26+ C.'s, $466,77+ D.'s, $656,40+.

4. On the first of January, A. began trade with $760; on the first of May, following, B. joined in company with him, with a capital of $540; on the first of August, following, they took in C. as a partner, with $800; at the end of the year they divided a gain of $872; what was each partner's share? Ans. A.'s share, $456. B.'s, $216. C.'s, $200.

5. A. and B. joined their stock in trade for 16 months; A. at first put in $1200, and 9 months after, put in $200 more; B. at first put in $1500, but at the end of 6 months, took out $500; at the expiration of the term, they found they had gained $772,20 cents; what was each partner's share of the gain? Ans. A.'s share, $401,70. B.'s, $370,50.

6. A. and B. join in company, and put into stock $600 each; but at the end of three months, A. takes out $150, and B. puts in $150 more; at the end of 6 months, A. puts in $500, and 3 months after, B. puts in $450; at the end of the year, they find they have gained $1500; how much is each man's part of the gain? Ans. A.'s part 708. B.'s, $792.

1. A. B. and C. freighted a ship with 68900 feet of boards; A. put in 16520 feet; B. put in 28750 feet; C. put in the residue; in a storm, the captain threw 26450 feet, overboard; how must the loss be divided among them?

Ans. A. must lose 6341+ feet. B., 11036+ C., 9071+; 2. Three men built a house that cost them $9500, and had it insured for three-fourths of the amount; the house caught fire, and was destroyed; how much must each man receive of the insurance, allowing A. to have owned 4, B., and C. the rest; and how much did each one lose?

{{

A. must receive $1781,25 and lose $593,75.

Ans. B.
C.

$2850,00
$2493,75

66

66

$950,00. $831,25.

3. A. B. and C. put into joint stock, $1200, in such proportion, that, as often as A. paid $3, B. paid 5, and C. 8; they gained $600; what was each man's stock, and share of the gain? A.'s stock $225, and his gain $112,50 cts. Ans. B.'s stock $375, and his gain $187,50 C.'s stock $600, and his gain $300,00

R

66

4. A ship valued at $20,000, was freighted with teas for A. valued at $10,400; coffee for B. at $9400; silks and crapes for C. at $5000; and the freight was reckoned at $6400; in a storm the captain was compelled to cut away the mast and rigging to preserve the ship, the repairs of which cost $1536; what is the loss on every 100 dollars, and how much must each party pay, as his share of the loss?

Average loss on $100 is $3. The ship must pay $600. Ans. Freight must pay $192. A. must pay $312.

B. must pay $282. C. must pay $150.

N. B. In averaging damages of vessels at sea, the value of the vessel, and the estimated value of the freight, pay their equal proportions.

5. A. B. and C. entered into partnership for 16 months; A. at first put into stock $7400, but at the end of 4 months, took out $2000, at the end of 12 months, he put in $3000, but at the end of 14 months, took out $850; B. at first put in $5900, and at the end of 3 months, put in $4300, but at the end of 9 months, took out $4000, and at the end of 12 months, put in $1500, but at the end of 14 months, took out $2000; C. at first put in $12000, but at the end of 6 months, took out $5000, and at the end of 9 months, put in $3200, but at the end of 12 months, took out $4000; on settlement of their accounts, they had gained 8000 dollars; what was each part. ner's share of it? A.'s. share $2219,3953278. Ans. B.'s. share $2634,870,620 C.'s. share $3145,7333 3658

3774

3774

6. A. and B. traded in company; A. put in $700 for 9 months; B. put in a certain sum for 12 months; they gained $500, of which B. took $150; what was B.'s stock? Ans. $225.

INVOLUTION.

Q. What is to be understood by Involution?

A. Involution is the raising of powers, and teaches to find the power of any number, or to raise any given number to any proposed power.

Q. How are the powers of numbers designated?

A. They are designated by numbers, as 1st, 2d, 3d, 4th, &c. Q. What is the first power of any number?

A. The number itself is called the first power, or root; as 5

is the first power of 5; and 9 is the first power of 9.

What is the second power of any number, and how is it

A. Any number, multiplied into itself, produces its second power, or square; as 5×5=25; the second power, or square of 5, marked 52.

Q. What is the third power of any number, and how is it produced?

A. Any number, multiplied into its second power, or square, produces its third power, or cube, as 5x5x5=125, the third power, or cube of 5, marked 53.

Q. What is the fourth power, and how produced?

A. Any number, multiplied into its third power, or cube, produces its fourth power, or biquadrate, as 5×5×5×5=625, the fourth power of 5, marked 54.

Q. What is the small figure that denotes the power, called? A. It is called the Index.

Q. What is the general RULE for finding the required power of any given number?

A. Multiply the given number into itself, till the number of multiplications is one less than the index of the required power, and the last product will be the power required.

Q. How do you raise a vulgar fraction to a required power? A. Raise the numerator to the required power, for a new numerator, and the denominator to the same power, for a new denominator.

Q. How do you raise a mixed number to any required power? A. Reduce it to an improper fraction, and raise the numerator and denominator as before, or, reduce the fraction to a decimal, and multiply the whole into itself, as the rule directs, and point off the decimals, as in multiplication of decimals.

6561 4th power, or answer.

2. What is the 4th power of??

3x3x3x3=81 4th power of the numerator.
5x5x5×5=625 4th power of the denominat

Therefore, is the 4th pe

623

3. What is the 2d power, or square of 21 ?
4. What is the 3d power, or cube of 4?
5. What is the 4th power of 1,05?
6. Raise to the 4th power.

7. What is the square of 4?

Ans. 484

25 or 19,36.

Ans. ,015625.

8. What fraction will express the 3d power of,25?

4096

EVOLUTION.

Q. What is Evolution?

A. It is the extraction, or finding the root of any power. Q. What do you understand by the extraction of the root of any power?

A. It is to find the root, or the original number, from which the given power was raised or produced; that is, to find such a number, as, being multiplied into itself, according to the index of the power, will produce the given power or number.

Q. Is there any number, from which a perfect power cannot be produced by Involution?

A. There is not; but there are many numbers of which an exact root can never be obtained.

Q. What are those numbers called, of which an exact root cannot be obtained?

A. They are called surd numbers.

Q. What are those called, whose roots can be exactly found? A. They are called rational numbers.

Q. How can you approximate to, or find, with a sufficient degree of exactness, the root of a surd power?

A. By the help or use of decimals, the root may be found to any necessary degree of exactness.

SQUARE ROOT.

Q. What is it to extract the square root of any number? A. It is to find such a number, as, being multiplied into it. self, will produce that number.

Q. What is the RULE for extracting the square root of any number, or rather what is the first process in the operation? A. Point off the given numbers into periods of two figures each, by putting a point over the unit, and then over every third figure, from the right hand to the left.

Why is it necessary to point off the figures into periods?

necessary, in order to know whether one or two