9. What is the quotient of 1001, divided by of of ? Ans. 4004.

Note 7.-It was shown in the illustration, that a whole number might be divided by an improper fraction in the same manner as by a proper fraction. An improper fraction obtained by changing a mixed number to an improper fraction, is equal to the mixed number.

10. What is the quotient of 114, divided by 61 ?

Explanation.-In 6 there are 19 thirds, which is the divisor; in 114 there are 342 thirds.

11. If 2 yards of broadcloth make 1 coat, how many coats will 34 yards make?

Ans. 16. 12. How many square yards of floor will 27 yards of carpeting cover, if it require 1 yards of carpeting to cover 1 square yard of floor? Ans. 201.

What is the

13. A gentleman has $9000 deposited in a certain bank, which is of his whole estate. 31 amount of his property?

Ans. $21000. 14. Sold a quantity of cheese for 56 dollars, at of a dollar per pound; how many pounds of cheese were sold.

Ans. 532.

To divide a fraction by a fraction.

Illustration.-One is placed in the quotient every

time the divisor is taken from the dividend.

Let it be required to divide by . This may be done as follows, (3=3, for dividing the tor by 1 does not lessen it: can be taken from 3

I

numera

as many times as 1 unit can be taken from 3 units. When the divisor and dividend have a common denominator, and the numerator of the dividend can be divided by the numerator of the divisor without a remainder, we may divide as above and the quotient of the numerators will be the true quotient. When the denominators are equal, dividing the numerators is dividing several parts of the unit or quantity by another number of parts equally small, and the denominators may be entirely omitted in the operation. If it be required to divide by 3, the numerators may be considered as whole numbers, and divided as such, and the quotient would be 2, a whole number. If the divisor and dividend have not a common denominator, the quotient may be obtained without reducing them in the common way.

We reduce the fractions in this example to a common denominator for the purpose of showing more plainly how the numerators may be obtained without their denominators. =, and . After the fractions are reduced, all we have to do, is to divide the numerator of the dividend by the numerator of the divisor %, which gives a quotient of 2. To get 10 fiftieths, we multiply 1 the numerator of the divisor by 10 the denominator of the dividend; and to obtain 20 fiftieths, we multiply 4 the numerator of the dividend by 5 the denominator of the divisor. It will be seen that in dividing we have nothing to do

with 50 the common denominator. As the denomi nator is not used in dividing, all that is necessary to be done in this example, is to multiply each given numerator by the denominator of the other fraction, which gives the numerators the same as when the common denominator is found. The most convenient method of obtaining the numerators for dividing without the denominators, is, to let the terms of the divisor change places, that is, setting the denominator above the line and the numerator below it, (which is generally called inverting the divisor,) and then multiply the two numbers above the line together, and the two below the line. The two numerators in the last example may be found in this way as follows; X=20=2.

Multiplying 4 by 5, which is the denominator of the divisor, gives 20 the dividend, or the numerator of the dividend; and multiplying 1 by 10the denominator of the dividend, gives ten the divisor, but the dividend now stands in the place of the numerator, and the divisor in that of the denominator. 10, the denominator of 2, is the numerator of the fraction which is the divisor; but as that numerator is used as the divisor, it properly becomes the denominator of a fraction when the dividend is made the numerator of the same fraction, as the denominator always. represents a divisor. (See Illustration, page 145.) By putting the denominator of the divisor in place of the numerator, and the numerator in that of the denominator, then multiplying as in the last example, the products form a fraction having the numerator of the dividend uppermost, and that of the divisor in place of the denominator; and these two numbers will always be of the same denomination, for they are the two numerators, which would belong to the common denominator formed by the product of the two given denominators.

To obtain the quotient of the fractions in a single number, the numerator of the fraction, made by multiplying, must be divided by the denominator. If the numerator is less than the denominator, we do not

actually divide one fraction by the other, and can only represent the quotient by the fraction formed by multiplying.

The foregoing illustration contains the reasons for the following,

RULE.

Change the terms of the divisor, then multiply the upper numbers together for a numerator, and the under numbers for a denominator, the fraction thus obtained will express the quotient. If the numerator be equal to or greater than the denominator, divide the numerator by the denominator.

15. How many times can be subtracted from ?? 3, or by the rule x=-3 the Ans. 16. Divide by 2? Ans. 11. 17. What is the quotient of 4, divided by 1?

Explanation. is less than 1, and the quotient

will be less than an unit.

Ans.

18. At of a dollar per pound, how many pounds of coffee can be bought for of a dollar?

Ans. 31.

19. If of a bushel of corn cost of a dollar, what

will 1 bushel cost ?*

t

20. If of a yard of cloth cost of a dollar, what

is the cloth per yard?

22

Ans. $25. 21. If several men own equal shares of of a ship's cargo, and each man's share is, of the whole cargo, how many owners to the. Also, if each man's share be worth $500, what is the value of the whole cargo

?

*The price of a bushel must be as much larger than as 1 is larger than 2. Multiplying the numerator of by 4, while the other numerator is multiplied by 3, gives this increase in the quotient.

Note 8.-The meaning of the answers in the two last questions, is, that such a part of the divisor can be taken from the dividend. Only of can be taken from

28. How many times can be taken from 7?

Explanation.-As the numerator of the divisor is but 1, we have only to multiply the whole number by the denominator. When the numerator is more than 1, and the dividend a whole number, let 1 be used as a denominator to the whole number, which will carry the numerator of the divisior to its proper place. (See Illus. page 168.) Ans..28.

29. How many in 7?

30. How many times in 14?

Ans. 9. Ans. 651.

31. How many times can be taken from 1?

32. How many times in 7?

Ans. 34.

Explanation.-Reduce the mixed number to an improper fraction. Ans. 8.

33. At of a dollar per bushel, how many bushels of potatoes can be bought for $131? Ans. 2013. 34. A gentleman rode in a stage until his fare was 9 dollars, at 5 cents per mile. How many miles did he ride? Ans. 184

Remark.-5 cents equal of a dollar.