4. Reduce 16 to a fraction whose denominator is 9.

5. Reduce 75 to a fraction whose denominator is 13. 6. Reduce 3 to a fraction whose denominator is 342. 7. How many fifteenths are there in 74?

8. How many eighths of a dollar in $647 ? 9. Reduce 36 to an improper fraction.

36 #

In this example, we add the 4-sevenths to the sevenths produced by the multiplication of 36 256 Ans. 256 by 7, and thus obtain 256. 10. Reduce 25 to an improper fraction. 11. Reduce 615234 to an improper fraction. 12. How many sixteenths of a dollar in $541?

CASE III. To reduce an improper fraction to a whole number, or a mixed number.

RULE. Divide the numerator by the denominator, and the quotient will be the whole, or mixed number. 13. Reduce 362 to a whole, or mixed number.

8)362

452=451

Since are equal to 1 unit, there are as many units in 362 as there are times 8 in 362.

14. Reduce 1993 to a whole, or mixed number. 15. How many units are there in 45315?

16. How many dollars in 262 of a dollar?

CASE IV. To reduce a compound fraction to a sim ple, or single fraction.

RULE. Multiply all the numerators together for a new numerator, and all the denominators for a new denominator: then reduce the new fraction to its lowest terms. When any numerator is equal to any denominator, the operation may be abbreviated by rejecting both.

If part of the compound fraction be an integer, or a mixed number, it must first be reduced to an improper fraction

17. Reduce of 3 of 3 of 6 to a simple fraction.

18. Reduce

19. Reduce 20 Reduce

=

Here the common term, 3, is omitted in the multiplication.

of to a simple fraction.

of

of of to a simple fraction.

15

of of 2 to a simple fraction.

21 Reduce of of of 5 to a simple fraction.

CASE V. To reduce a fraction from one denomination to another.

RULE. Multiply the proposed denominator by the numerator of the given fraction, and divide the product by the denominator of the given fraction; the quotient will be the numerator of the proposed denominator.

22. Reduce to a fraction whose denominator shall be 14 or, in other words change 5-sixths to fourteenths.

14

5

is equal to of 14, and 1⁄2 is 5 times as much: we therefore find 5 times 14-fourteenths and take of this product for the 14 required fourteenths. 23. How many fifths are there in 7?

6)70

11/

Ans. 11

24. is equal to how many twenty-fourths? 25. Reduce & to a fraction whose denominator is 4. 26. How many twelfths of 1 shilling in 4 of 1s.?

CASE VI. To reduce the lower denominations of a compound number to the fraction of a higher denomination.

RULE. Reduce the given quantity to the lowest denomination mentioned, and this number will be the numerator. then reduce a unit of the higher denomination to the same denomination with the numerator, and this number will be the denominator.

27. Reduce 7 oz. 18 dwt. 13gr. to the fraction of a pound.

We find, that 7 oz. 18 dwt. 13 gr. when reduced to grains, gives 3805 for the numerator; and 1 pound when reduced to grains, gives 5760 for the denominato Therefore, 357 is the fraction required.

3805

1152

28 Reduce 4s. 9d 3qr. to the fraction of £1.

29. Reduce 3 inches to the fraction of a yard. 30. What fraction of a hogshead is 9 gal. 2 pt.? 31. Reduce 5 cwt. 8 lb. 4oz. to the fraction of a ton.

CASE VII. To reduce the fraction of a higher denomination to its value in whole numbers of lower denomination. RULE. Multiply the numerator by that number of the next lower denomination which is required to make a unit of the higher, and divide the product by the denominator; the quotient will be a whole number of the lower denomi nation, and the remainder will be the numerator of a fraction. Proceed with this fraction as before, and so on.

It will be readily perceived, that the fraction of a higher denomination is reduced to the fraction of a lower, by multiplying the numerator by the number of units of the lower, required to make a unit of the higher. Thus, of a bushel is 4 times as many fifths of a peck; that is, 12 of a peck. Again, 12 of a peck is 8 times 12-fifths, that is, of a quart; and again, 26 of a quart is 2 times. 96-fifths, that is, 122 of a pint. If the denominator be multiplied, instead of the numerator, the effect is the reverse, and the fraction is reduced to a higher denomination. Thus, of a pint, (the 5 being multiplied by 2,) becomes 1 of a quart; of a quart, (the 10 being multiplied by 8,) becomes of a peck; and 3 of a peck, (the 80 being multiplied by 4,) becomes of a bushel. 32. Reduce of a gallon to its value in quarts, &c. We find by multiplication, that 1 of a gallon is 14 of a quart; and, by division, that of a quart is 3qt. and

3

11

4

12)44

3 8

2

12)16

1 4
4

3

12)16

11

80

8

of a quart.

We then find, that of a qt. is 19 of a pint; and, that 19 of a pt. is 1 pt. and of a pt. And thus, by finding the units of one denomination at a time, we finally obtain the whole answer, which, denoted as a compound number, is 3qt. 1 pt. 1gi.

33. Reduce of £ 1 to its value in shillings &c.

34. Reduce

of a yard, to its value in feet, &c.

35. In of Icwt. how many quarters, pounds, &c.' 36. Reduce of a bushel to pecks, quarts, and pints.

CASE VIII. To reduce fractions to a common denominator; that is, to change two or more fractions which have different denominators, to equivalent fractions, that shall have the same denominator..

RULE 1st. Multiply each numerator into all the denominators except its own, for a new numerator. Then multiply all the denominators together for a new denominator, and place it under each new numerator.

RULE 2nd. Find the least common multiple of the given denominators for the common denominator; then divide the common denominator by each given denominator and multiply the quotient by its given numerator; the several products will be the several new numerators. (See PROBLEM X, page 24.)

The 1st. of the above rules is convenient when the terms of the fractions are small numbers, but the 2nd. is otherwise to be preferred, as it always gives a denominator which is the least possible. Other methods of finding a common denominator will occur to the student, after further practice.

If any of the fractions to be reduced to a common denominator be compound, they must first be simplified. 37. Reduce, 11, 14 and 13 to a common denomi

nator.

In this example, the least common denominator is found to be 840. Then the several numerators of the common denominator are found as follows.

840 8 105, and 105 X 5=525. 840÷12=70, and 70 X 11=770. 840÷14 60, and 60 X 9=540. 84015

=

56, and 56 × 13=723.

2

38 Reduce, and to a common denominator.

15

10

39. Reduce, 19, and to a common denominator. 40. Reduce and to a common denominator.

27

41. Reduce

and 3 of 4 to

common denominator.

CASE IX To reduce a complex fraction to a simple fraction.

RULE. If the numerator or denominator, or both, be whole or mixed numbers, reduce them to improper fractions: multiply the denominator of the lower fraction into the numerator of the upper, for a new numerator; and multiply the denominator of the upper fraction into the numerator of the lower, for a new denominator.

9

42. Reduce 3 to a simple fraction.

43. Simplify each of the following complex fractions.

4.1. 51.31. 21. 6. 21.

3

53 4 7 1

ADDITION OF FRACTIONS.

Fractions are added by merely adding their numerators, but they must be of the same integers; we cannot immediately add together of a yard and of an inch, for the same reasons that we cannot immediately add together 5 yards and 3 inches. They must, also, be of the same denomination; we cannot immediately add together fourths and fifths.

RULE. Reduce compound fractions, (if there be any), to simple fractions, and reduce all to a common denomi nator; then add together the numerators, and place their sum over the common denominator. If the result be an improper fraction, reduce it to a whole or mixed number. 44. Add together, 37, §, 83 and 3.

5

By operations not here denoted, we find the common denominator to be 360; and also find the several new numerators. The sum of the fractions is =237, which, added to the whole numbers, gives the total sum, 13271