Again, divide of a pear into 6 equal parts.

If a whole pear were divided into 6 equal parts, each part would be expressed by But since the half of the pear was divided, each part will be expressed by of,

or 12.

In the division of fractions we should note the following principles:

1st. When the dividend is just equal to the divisor, the quotient will be 1.

2d. When the dividend is greater than the divisor, the quotient will be greater than 1.

3d. When the dividend is less than the divisor, the quotient will be less than 1.

4th. The quotient will be as many times greater than 1, as the dividend is greater than the divisor.

5th. The quotient will be as many times less than 1, as the dividend is less than the divisor.

CASE I.

159. To divide a fraction by a whole number. Divide the numerator or multiply the denominator by the whole number.

158. What does division of whole numbers explain? In division of fractions, may the divisor exceed the dividend? How will the quotient then compare with 1? If an apple be divided into 2 equal parts, what will express each part? If half an apple be divided into 4 equal parts, what will express one of the parts? What is one-half of one-half? What is one-sixth of one-half? What principles do you note in the division of fractions? When will the quotient be 1? When greater than 1? When will the quotient be less than 1? When greater than 1, how many times greater? When less than 1, how many times less?

160. To divide one fraction by another.

FIRST METHOD OF PROOF.

Let it be required to divide by 8.

The true quotient will be expressed by the complex fraction

24

Let the terms of this fraction be now multiplied by the denominator with its terms inverted: this will not alter the value of the fraction (ART. 122), and we shall then have,

It will be seen that the quotient is obtained by simply. multiplying the numerator by the denominator with its terms inverted. This quotient may be further simplified by cancelling the common factors 5 and 8, giving for the true quotient.

SECOND METHOD OF PROOF.

Let us first divide the dividend by 5. This is done by multiplying the denominator (ART. 120), which gives 12. But the divisor being but

of

OPERATION.

10÷5=12% 120X8=120

159. In how many ways may a fraction be divided by a whole number?

160. How do you divide one fraction by another? How may the quotient of one fraction divided by another be expressed? If any fraction be multiplied by the fraction which arises from inverting its terms, to what will the product be equal? In the second method of proof, after dividing by 5, is the quotient too small or too large, and how much? How then do you find the true quotient?

5, this quotient is 8 times too small, since the eighth of a number will be contained in the dividend 8 times more than the number itself. Therefore, by multiplying by 8, we obtain for the true quotient.

80

Hence, to divide one fraction by another,

Reduce compound and complex fractions to simple ones, also whole and mixed numbers to improper fractions: then multiply the dividend by the divisor with its terms inverted, and the product reduced to its simplest terms will be the quotient sought.

2. If of a dollar will pay for 10%. of nails, how much is the price per pound

Ans. $

3. If of a yard of cloth cost $3, what is the price per yard?

Ans. $51.

of 100 cents=4400=8,T Ans. 8

cents.

4. If $21 will buy 713 barrels of apples, how much are they per barrel? Ans. $225

5. If 4 gallons of molasses cost $25, how much is it per quart?

Ans.

6. If 11hhd. of wine cost $250, how much is the wine per quart?

7. If eight pounds of tea much is it per pound?

Ans. $1501-88-52cts. cost 7 of a dollar, how

Ans. 95cts.

8. In 8 weeks a family consumes 1653 pounds of butter: how much do they consume a week?

69

Ans. 1989 lb.

9. If a piece of cloth containing 1763 yards costs $375g, what does it cost per yard?

Ans.

10. If I pay dollar a pound for tea, how many pounds can I have for 4284 dollars? Ans. 48967b. 11. Bought flour at 79 dollars a barrel, and laid out 129 dollars for the article: how many barrels did I buy? Ans. 1633. 12. Paid 6662 cents for marbles at 6 cents apiece: how many did I buy? Ans. 111. 13. If raisins are 281 cents a pound, how much can I have for 17 cents?

Ans.

14. How many barrels of flour can I buy for 1613 dollars if I pay 143 dollars a barrel?

15. Divide 5205

what will each have?

dollars among

16. At 27 dollars an acre, how much

for of a dollar?

Ans. 111bar. of 90 persons: Ans. 720

53

land can I buy

Ans. 1 acre.

17. How many apples can I buy for 23 of of 2 cents, if I pay of 23 of cents apiece? Ans. 25.

18. Bought of a lot of land for 5040 dollars, and having sold of what was bought, I gave of the remainder to a charitable society, and divided the residue among 9 poor persons: what was the share of each?

Ans. 37.

-19. Of an estate valued 15000 dollars, the widow has , the oldest son of remainder, and the residue was divided among 9 children: what was the share of each of the 9 children? Ans. $37019.

DECIMAL FRACTIONS.

161. If the unit 1 be divided into 10 equal parts, the parts are called tenths, because each part is one-tenth of unity.

If the unit 1 be divided into one hundred equal parts, the parts are called hundredths, because each part is one hundredth of unity.

If the unit 1 be divided into one thousand equal parts, the parts are called thousandths, because each part is one thousandth of unity and we have similar expressions for the parts, when the unit is divided into ten thousand, one hundred thousand, &c., equal parts.

The division of the unit into tenths, hundredths, thousandths, &c., forms a system of numbers called Decimal Fractions. They may be written thus:

From which we see, that in each case the denominator gives denomination or name to the fraction; that is, determines whether the parts are tenths, hundredths, thousandths, &c.

162. The denominators of decimal fractions are sel dom set down. The fractions are usually expressed by means of a comma, or period, placed at the left of the

numerator.

161. When the unit 1 is divided into 10 equal parts, what is each part called? What is each part called when it is divided into 100 equal parts? When into 1000? Into 10,000, &c.? How are decimal fractions formed? What gives denomination to the fraction? 162. Are the denominators of decimal fractions generally set down? How are the fractions expressed? Is the denominator understood? What is it? What is the place next the decimal point called? The next? The third, &c.? Which way are decimals numerated?