DIVISION OF FRACTIONS.

(77.) The Quotient from dividing by a Fraction, as well as by an integer, is the number of times that the dividend contains the divisor, or the part that the dividend is of the

The reciprocal of a Fraction is the fraction inverted; and

is equal to a unit divided by the fraction.

Thus the reciprocal of is; equal to 1 or 3 divided by 3.

What is the reciprocal of ? Of Ofg?

The reciprocal of a mixed number is that of its equivalent improper fraction; thus the reciprocal of 53, or 17, is 37.

What is the reciprocal of 24? Of 7? Of 101?

Mixed Fractions.

(78.) A mixed fraction is one which contains a fraction in one or both of its terms. It is reduced to a simple fraction by dividing its numerator by its denominator.

2

Thus numerator 2, denominator 3, is a mixed fraction. 3}'

By reducing the numerator to halves (63), and the denominator to halves, and dividing, we have

2÷÷3=1÷3, which is a simple fraction.

RULE XVII.

(79.) For the Division of Fractions.

1. Divide the numerator of the dividend by the numerator of the divisor, and the denominator by the denominator; or multiply the dividend by the reciprocal of the divisor.

2. A fraction is divided by an integer, by dividing the numerator, or multiplying the denominator, by the integer. 3. A mixed number may be taken in division under the form of an improper fraction (64); or its integral and fractional parts may be divided separately.

1. To divide by 3.

EXAMPLES.

As we cannot divide numerator by numerator, and denominator by denominator, without remainders; we multiply the dividend by the reciprocal of the divisor; thus

The correctness of this method may be shown thus the dividend is equal to

Dividing the numerator of this dividend by the numerator of the divisor, and the denominator by the denominator, we find the quotient 9x3 or 5 X 3.

35 X2

2. To divide 1632 by 5.

3

5)1633

32.

We say, 5 in 16, 3 times and 1 over; 5 in 13, twice and

3 over; this 3 and the

make 33, equal to ; then of

is

(75).

The division in this case might also be performed, by reducing the dividend to an improper fraction (64), and multiplying this fraction by the reciprocal .

From principles which have been established it follows, that, in all cases,

(80.) Dividing a given number is equivalent to multiply. ing it by the reciprocal of the divisor

EXERCISES.

1. How many yards of calico, at of a dollar per yard, may be purchased for of a dollar?

The number of yards is the number of times that tained in.

is con

Ans. 3 yards.

2. How many weeks would a family be in consuming 19 barrels of flour, at the rate of of a barrel per week?

Ans. 263 weeks. 3. How many yards of silk could be purchased for 12 dollars, at the rate of of a dollar per yard?

Ans. 142 yards.

4. How many hours would a person be in walking 175 miles, at the rate of 3 miles per hour? Ans. 583 hours.

5. A merchant laid out for broadcloth 5727 dollars, paying 5 dollars per yard. How many yards did he purchase? Ans. 996 yards.

6. A farmer purchased a farm for 4379, dollars, paying 161 dollars per acre. How many acres did he purchase? Ans. 259acres.

7. How many barrels of wine are there in 2753 gallons, allowing 31 gallons to make one barrel ?

Ans. 87 barrels.

8. What quantity of salt may be purchased for of a dollar, at of a dollar per bushel?

of a dollar will buy the same part of a bushel that of a dollar is of of a dollar, which will be found by dividing by (77). Ans. of a bushel.

9. What quantity of iron may be purchased for 22 dollars, at the rate of 45 dollars per ton? Ans. of a ton. 10. What quantity of land may be purchased for 153 dollars, at the rate of 30 dollars per acre ?

Ans. 2 of an acre.

11. If a person could accomplish a certain work in 251⁄2 days, what part of it could he perform in 3 days?

Ans. of the work.

12. A laborer agreed to work 30 days for a certain sum of

money; having worked but 17 lated sum ought he to receive?

days, what part of the stipu

Ans. of it.

13. A mason having undertaken to build a wall of specified dimensions in 62 days, what part of the wall ought he to accomplish in 61 days? Ans. of it.

14. What is the price of cloth per yard, when of a yard costs 5 dollars ?

5 dollars

If 2 thirds of a yard costs 5 dollars, 1 third of a yard costs of 5 dollars, and 3 thirds of a yard, which is a whole yard, costs of 5 dollars, which is 5 dollars X (74), or (80). Ans. 7 dollars. 15. What is the price of hay per ton, when of a ton costs 10 dollars? Ans. 13 dollars. 16. What should be paid for an acre of ground, when of an acre is sold for 211 dollars? Ans. 34 dollars.

17. What should be paid for a ton of coal, when of a ton is purchased for 183 dollars? Ans. 23 dollars.

18. If of an acre of ground produce 19 bushels of wheat, what is the produce per acre? Ans. 279 bushels.

19. A railroad car ran 37 miles in of an hour; at what rate was that per hour?

20. Allowing a person to walk 34 miles at what rate does he walk per hour?

Ans. 633 miles.

in % of an hour,

Ans. 3 miles.

21. A company of laborers have finished of a certain work in 43 days. In what time will the whole work be accomplished at that rate? Ans. 62 days.

22. What is the price of flour per barrel, when 51⁄2 barrels cost 49 dollars?

The price per barrel will be found by dividing 49 dollars by 5, because that price × 51 must produce 49 dollars; the product 491 and one of its factors being given, to find the other factor (37). Ans. 9 dollars. 23. What is the price of silk per yard, when 3 yards are purchased for 43 dollars? Ans. 11 dollars. 24. If a man travels 273 miles in of a day, at what rate does he travel per day? Ans. 36 miles. 25. What is the price of land per acre, when 13 acres are Ans. 5 dollars. 26. What is the price of coal per ton, when of a ton costs 9 dollars? and what would 101 tons amount to?

disposed of for 714 dollars?

Ans. 16 dollars; and 166 dollars. 27. If 25 cords of wood sell for 683 dollars, what is the price per cord? and what should be paid for å of a cord? Ans. 23 dollars; and 21 dollars.

28. If a man walk 623 miles in 18 hours, at what rate will he walk per hour? and how many miles would he go in 20 hours at the same rate?

Ans. 34 miles; and 67,43 miles. 29. If a railroad car run 230 miles in 10 hours, what will be its rate per hour? and how far would it run in 23 hours at the same rate?

Ans. 2218 miles; and 5164 miles.