What

13. From a tract of land containing 1300 acres a farmer sold, at different times, to the amount of 9342 acres. quantity then remained in the tract?

Ans. 365 acres.

14. Out of the sum of 23457 dollars, which a person had collected, he paid 350 dollars for house rent, and 187 dollars for other expenses; what sum had he remaining?

Ans. 18087 dollars.

15. If two lots of ground, one of which was purchased for 1354 dollars, and the other for 800 dollars, should both be sold for 2579 dollars; what would be gained or lost by the sale?

Ans. 425 dollars gained.

16. A manufacturer put into one bale 200 yards of cloth, and into another 1873 yards. How many yards must be put into a third bale, so that the three together shall contain 500 yards?

Ans. 1125 yards.

bushels of wheat, bushels of it to B, How many bushels did C

17. A farmer who had raised 1000 sold 320 bushels of his crop to A, 2003 and the remainder of it to C. purchase?

Ans. 4797 bushels.

18. If a quantity of cloth be purchased for 3211 dollars, a quantity of silk for 137 dollars, and a quantity of linen for 937 dollars; what will be the gain or loss if the whole be sold for 600 dollars?

Ans. Gain 477 dollars.

19. Bought 350 acres of land for 43271 dollars. Having soid 137 acres for 1387 dollars, I desire to know how many acres remain, and what will be the gain or loss on the whole if the remainder be sold for 2300 dollars.

Ans. 212 acres; 640 dollars loss.

MULTIPLICATION OF FRACTIONS.

(74.) The Product from multiplying by a Fraction is that part of the multiplicand which is denoted by the multiplier.

The product of 20 multiplied by 2, for example, is 15, which is obtained by taking 3 fourths of 20, or 3 times 1 of 20, (26).

The product, 15, is less than the multiplicand, 20; because the multiplier is less than a unit.

(75.) A compound fraction is a fraction of a fraction; and is therefore equal to the product of the two fractions. Thus of is a compound fraction; and is equal to × (74).

RULE XVI.

(76.) For the Multiplication of Fractions.

1. Multiply the numerators together for a numerator, ana the denominators together for a denominator.

2. An integer and a fraction are multiplied together, by multiplying the numerator, or dividing the denominator, by the integer.

3. A mixed number may be taken in multiplication under the form of an improper fraction (64); or its fractional and integral parts may be taken separately.

EXAMPLES.

1. To multiply × 3; that is, to find 2 of 5.

One 4th of 1 seventh of any quantity is evidently one 28th of the quantity; then of 5 sevenths is five 28ths; and of 5 sevenths is 3 times five 28ths, which is fifteen 28ths.

The product is therefore obtained by multiplying the numerators together, and the denominators together.

2. To multiply by 6; that is, to find 6 times five 24ths.

4×6=30 = 11⁄2 = 1}.

Or, dividing the denominator by the integer 6 we obtain

= 1, as before.

Dividing the denominator multiplies the value of the frac tion, because the denominator is thus diminished.

3. To multiply 5 by 2; that is, to find twice 5, together with of 5.

We first say, twice 2 thirds is 4 thirds, equal to 1; set down; twice 5 is 10, and 1 makes 11.

Then, of 5 is 2, with 1 over; this 1 is, and added to the, makes 5 thirds; ofis (75).

The partial products 113 and 25 are added together, for the entire product.

This last Example may also be performed by reducing the given numbers to improper fractions (64); then multiplying the numerators together, and the denominators, and reducing the product to units (61).

1.

rate of

EXERCISES.

it should be paid for of a yard of linen, at the of a dollar per yard? If a yard of the linen costs cost of of a dollar; or of

of a dollar, of a yard will a dollar × 3 (75).

Ans. of a dollar

2. What should be paid for of a barrel of apples, if the whole barrel be worth 15 of a dollar? Ans. of a dollar. 3. A person owning of a tract of land, sells of his share to A; what part of the whole tract does A purchase? Ans. 6 of the whole. of a pound of tea, at the

4. What should be paid for rate of of a dollar per pound?

Ans. of a dollar.

5. A merchant owning of a ship, sells of his share to A, and the rest of it to B. What part of the ship does B pur

chase?

6. Bought

Ans of the ship. of an acre of ground, at the rate of 18 dollars

per acre; required the sum to be paid for it.

Ans. 10 dollars.

7. Sold 25 bushels of clover seed, at 71 dollars per bushel; what did it amount to? Ans. 181 dollars.

8. In how many days ought one man to accomplish a work equivalent to what 12 men performed in 71 days? Ans. 87 days.

9. What would be the profit on 75 barrels of flour, purchased at 33 dollars a barrel, and sold at 44 dollars a barrel? Ans. 56 dollars. 10. What would be the value of 3 pieces of cloth, each containing 25 yards, at 63 dollars per yard?

Ans. 511 dollars. 11. In how many days ought one man to accomplish an undertaking which 17 men could perform in 13 days?

Ans. 2273 days.

12. Find the distance a person would travel in 6 days, allowing him to proceed for 3 days, at the rate of 30 miles per day, and the rest of the time at the rate of 27 miles per day. Ans. 185 miles. 13. Find the entire cost of of a pound of pepper at of a dollar a pound, of a hundredweight of flour at 21 dollars a hundred, and 21 yards of cloth at 7 dollars a yard?

Ans. 161 dollars. 14. What would be the profit or loss on 357 yards of silk, purchased at of a dollar per yard, if 161 yards of it be sold at 14 dollars a yard, and the remainder at of a dollar a yard? Ans. Profit 5 dollars. 15. What should be paid for of a yard of cloth, at the rate of 8 dollars per yard?

A fraction is multiplied by its own denominator by merely canceling the denominator. In this example, the product of 8 × is at once known to be 5. Ans. 5 dollars.

16. What should be paid for 2 of of a yard of muslin, at the rate of of a dollar per yard?

The value to be found is evidently of off of a dollar.

In multiplying two or more fractions together, equal factors may thus always be canceled in the resulting numerator and denominator; for this divides the numerator and denoinator by the same number (65).

17. What should be paid for of the rate of of a dollar per pound?

of a pound of tea, at Ans. of a dollar.

of of

of a yard of silk, Ans. of a dollar

18. What should be paid for at the rate of of a dollar per yard?