Q. How can you subtract one mixed number from another? A. If the numerator of the lower fraction, or subtrahend, be less than the numerator of the upper one, or minuend, subtract the lower numerator from the upper, and write the difference over the common denominator; then subtract the whole number as usual.

Q. When the lower fraction is greater than the upper one, how do you proceed?

A. Subtract the lower numerator from the common denominator, and add this difference to the upper numerator, and write the sum over the denominator, for the fraction, and always remember to carry one to the unit figure of the whole number, in the subtrahend, and proceed to subtract as usual. Q. When the minuend has a fraction, and the subtrahend has none, how do you proceed?

A. Bring down the fraction of the minuend, to the right hand, and proceed to subtract, as if there had been no fraction. Q. When the subtrahend has a fraction, and the minuend has none, how do you proceed?

A. Subtract the numerator from the denominator, and write the difference over the denominator, for the fraction, and al ways remember to carry one to the first figure of the whole number in the subtrahend, and proceed as in whole numbers.

9. From 100, take 87,3, how much remains? 10. From 973, take 4958, how much remains? 11. From 5716, take 28, how much remains? 12. From 19, take 24, how much remains? 13. From 4, take 37, how much remains? 14. If I purchase of a factory, and sell what proportion of the whole factory do I sell, portion do I still own?

I sell or

Ans. 16.

Ans. 343. of my share, and what proof the whole.

Ans. I still own 'e of the whole.

15. If I buy of a ship, and sell portion of the whole do I still own?

of my share, what proAns. of the whole.

15. A man who owned 13 of a manufacturing establishment, agreed to sell of the whole factory to A.; soon after he sold of of the whole, to B.; and to C. he sold of the whole; and D. wishing to purchase the residue of his share, he wants to know what proportion of the establishment he can sell him. Ans. He can sell him .

16. From of of 5 dollars, take § of 96 cents, added to of $13. Ans. 96 cents, 9 mills. 17. From £4, take of a shilling. Ans. 4s. 11d. 18. A man built a factory which cost him 35,000 dollars, and sold to A. of the whole; soon after, he sold to B. of the whole establishment; A. also sold to C. of his share; and B. sold to D. of his share; I wish to know what proportion of the establishment each man now owns, and how much each man's present share cost him, allowing each to purchase at the original cost? Ans. The builder still owns of the whole, at the cost of $1312,50 cts; A. still owns 3, at the cost of $13125; B. owns, at a cost $8750; C. owns 3, at a cost of $6562,50, and D. owns, at a cost of $5250.

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19. A gentleman bought at one time, of a township of land, ten miles squaré, and soon after he bought § of the residue, and at the third purchase, he bought the remaining portion; I wish to know what part of the whole he bought at each purchase, and how many acres each purchase contained? Ans. His first purchase,, contained 24,000 acres ; his second purchase, contained 25,000 acres ; his third purchase,, contained 15,000 acres.

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MULTIPLICATION OF VULGAR FRACTIONS.

Q. Will any number be increased or diminished by being multiplied by a vulgar fraction?

A. It will be diminished in the same proportion as the multiplier is less than an unit or 1.

Q. How can you demonstrate, that multiplying by a fraction decreases the number multiplied?,

A. By the self-evident fact that the greater the multiplier is, the greater will be the product; and of course, the smaller the multiplier, the less will be the product. Thus, 8×4=32. Now if the multiplier, 4, be reduced one half, or to 2, then 8x2=16, which is one half what it was, when multiplied by 4. Now let this last multiplier, 2, be again reduced one half, which will make 1, then 8x1=8, just one half what it was, when multiplied by 2. Again, if this last multiplier, 1, be still reduced one half, which will leave it, it must be evident that

P

the product arising from it, must be only one half what it was when multiplied by 1. Thus, 8×4, and so on, in propor. tion as the multiplier is less than 1.

CASE FIRST.

Q. What is the first case in Multiplication of Fractions? A. To multiply a whole number by a fraction, or a fraction by a whole number.

Q. What is the RULE in this case?

A. If the numerator of the fraction be one, as 1, 1, 1, &c. divide the whole number by the denominator of the given frac tion, and the quotient, thence arising, will be the proper pro. duct or answer.

Q. When the numerator of the fraction is more than 1, as ,,, &c. how do you multiply by it?

A. Multiply the whole number by the numerator of the giv en fraction, and divide this product by the denominator, and the quotient will be the product or answer.

Q. When it is required to multiply a whole number by a compound fraction, how do you proceed?

A. Reduce the compound fraction to a simple one, and this fraction to its lowest term, then proceed to multiply as before directed.

1.

EXAMPLES.

2.

3.

Multiply 144 by. Multiply 96 by . Multiply 136 by . Operations and answers.

144÷3-48 Ans. 96-8-12 Ans. 136÷4-34 Ans.

6. What cost 136 yards of cloth, at 3 of a dollar a yard?

7. What cost 75 pounds of coffee, at

8. What cost 138 yards of cloth, at per yard?

9. If a ship cost 15460 dollars, what is

Ans. $51. of a dollar per lb. ? Ans. $14 of of of a dollar Ans. $45,281 m.+ § of her worth? Ans. $9662,50 cts.

10. If I buy of a ship, which cost 18000 dollars, and sell of my share to A., and A. sells of his share to B., and B. sells of his share to C., what does C. pay for his share, and what proportion of the ship does each now own?

Ans.

{

I own of the ship. A. owns of it. B. owns.
C. owns at the cost of $900.

11. What cost 4 quarter casks of wine, each containing 31 gallons, at of a dollar per gallon?

Ans. $783.

CASE SECOND.

Q. What is the second case in Multiplication of Vulgar Fractions?

A. To multiply a whole number by a mixed one, or a mixed number by a whole one.

Q. What is the RULE in this case?

A. Multiply by the fraction as in case first; then multiply by the whole number as usual, and add the two products to. gether for the answer; or, if the multiplicand be a mixed num. ber, multiply the whole numbers together as before, then multiply the given multiplier by the fraction of the multiplicand, and add the products together as before, for the answer.

EXAMPLES.

1. What must be given for 137 yards of cloth, at $5 per yard? Ans. $702,12}. 2. What is the weight of 7 loads of hay, each weighing 241 Ans. 171 cwt.

cwt.?

3. What must I give for 4 lots of land, each containing 241 acres, at the rate of 13 dollars an acre? Ans. $1283,75.

4. Seven children, after dividing their father's farm equally among them, found they had 75 acres, and of of of of an acre, each; how many acres did the whole farm contain? Ans. 526 ac. 1 rood, 37

rods. 5. Six men drew a prize, which being divided among them, each had 46 and of of dollars; what was the whole prize? Ans. $277,963 cts. 6. What is the product of 378 multiplied by 14? Ans. 6613. 7. What is the product of 576 multiplied by 1 and of of

8. What is the product of 47 and of $28?

9. Multiply of of of, by 5.

Ans. $802. of, multiplied by Ans. 131723. Ans.

CASE THIRD.

Q. What is the third case in multiplication of fractions?
A. To multiply a mixed number, by a mixed number.
Q. What is the RULE in this case?

A. Reduce each term to an improper fraction, and multiply the two numerators together, and make their product a dividend; then multiply the two denominators together, and make their product a divisor; divide, and the quotient will be the proper product.

Then 157×7-5×2=1221 Ans.

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3. How many square feet, in a room 163 feet long, and 94 feet in width? Ans. 162 feet. 4. How many square yards, in a garden 37 yards square? Ans. 1415 yards, 5 feet, 1104 inches. 5. What is the product of $254, multiplied by $378?

Ans. $950,031.

6. How many square miles, in a town 123 miles in length, and 9 miles in breadth?

7. A man bought 45 sold 26 acres, at $29

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Ans. 124 sq. miles, and 272 sq. a.

acres of land, at $37§ per acre, and per acre, and the residue he sold at

$48 per acre; did he gain or lose, and how much?

CASE FOURTH.

Ans. He lost $2,871 m.

Q. What is the general RULE for the multiplication of frac tions, whether proper, improper, compound, or mixed?

A. Reduce compound fractions to simple ones, mixed numbers to improper fractions, and those of different integers to those of the same; then multiply all the numerators together, for a new numerator, and all the denominators together, for a new denominator.