When the compound fraction consists of more than two simple ones, two of them can be reduced to a simple fraction as above, and then this fraction may be reduced with the next, and so on. We therefore have the following

RULE.

I. Reduce all mixed numbers to their equivalent improper fractions by Case II.

II. Then multiply all the numerators together for a nume rator and all the denominators together for a denominator : their products will form the fraction sought.

2. Reduce of of to a simple fraction.

3. Reduce of of to a simple fraction.

Ans. 2

Here, by dividing the numerator and denominator of, first by 9 and then by 2, as shown in Case III.

Or, xx, by cancelling or striking out the 3's and 6's in the numerator and denominator.

By cancelling or striking out the 3's we only divide the numerator and denominator of the fraction by 3; and in cancelling the 6's we divide by 6. Hence, the value of the fraction is not affected by striking out like figures, which should always be done when they, multiply the numerator and denominator.

4. Reduce g of of to a simple fraction.

8

5

Herex=1030=1= Ans.

Or,

15

ส 15

××==} Ans.

Q. What is a compound fraction? How do you reduce a compound fraction to a simple one? When you find like figures in the numerafor and denominator, what do you do with them? Does this alter the value of the fraction? What is one-half of one-half? One-half of one-third? One-third of one-fourth? One-sixth of one-seventh ? Three-halves of one-eighth? Six-thirds of two-ones?

5. Reduce 21 of 6 of 7 to a simple fraction.

Ans. 12-102.

6. Reduce 5 of of of 6 to a simple fraction.

Ans. 30=24.

7. Reduce 63 of 71 of 634 to a simple fraction.

Ans. 106343

324

CASE VI.

§ 94. To reduce fractions of different denominators to equivalent fractions having a common denominator.

RULE.

I. Reduce compound fractions to simple ones, and whole or mixed numbers to improper fractions.

II. Then multiply each one of the numerators by all the denominators except its own, for the new numerators, and multiply all the denominators together for a common denominator: the common denominator placed under each of the new numerators will form the several fractions sought.

EXAMPLES.

1. Reduce,, and to a common denominator,

1x3x5 15 the new numerator of the 1st.

7×2×5=70

4x3x2=24

and 2×3×5=30, the common denominator.

15

2nd.
3rd.

Therefore, 18, 18, and 34, are the equivalent fractions. It is plain, that this reduction does not alter the values of the several fractions, since the numerator and denominator of each are multiplied by the same number.. (See Proposition V.)

When the numbers are small the work may be performed mentally.

Thus,

10

8, 18, 18.

Here we find the first numerator by multiplying 1 by 4 and 5; the second, by multiplying 1 by 2 and 5; the third, by multiplying 2 by 4 and 2; and the common denominator by multiplying 2, 4 and 5 together.

Q. What is the first step in reducing fractions to a common denominator? What is the second? Does the reduction alter the values of the several fractions? Why not? When the numbers are small, how may the work be performed?

3. Reduce 2, and of to a common denominator. 2; and of 14

2

4. Reduce 5, of, and 4, to`a common denominator. Ans 11, 4, and 6.

5. Reduce, 135, and 37, to a common denominator.

Ans. 525, 1080, and 22200 600' 600

6. Reduce 4, 31, 62, to a common denominator.

31

600

Ans. 200, 62, and 1550. 50" 器

7. Reduce 71, 11, 61, to a common denominator.

Ans. 1080, 248, and 900.

144 144

8. Reduce 41, 81, and 21 of 5, to a common denomiAns. 126 126 126

nator.

518 1026 1575

§ 95. NOTE 1. It is often convenient to reduce fractions to a common denominator by multiplying the numerator and denominator of each fraction by such a number as shall make the denominators the same in both.

EXAMPLES.

1. Let it be required to reduce and to a common minator.

we see at once that if we multiply the numerator and denominator of the first fraction by 3, and the numerator and denominator of the second by 2, that they will have a common denominator.

The two fractions will be reduced to & and 2

2. Reduce and to a common denominator

If we multiply both terms of the first fraction by 3 and both terms of the second by 5, we have

3. Reduce, and, to a common denominator.

Ans. 12, 14. 4. Reduce, 8 28,14, to a common denominator.

Ans. 1, 28, 28.

§ 96. NOTE 2. To reduce fractions to their least common denominator, we have the following

RULE.

1. Find the least common multiple of the denominators as in § 87 and it will be the least denominator sought.

II. Multiply the numerator of each fraction by the quotien? which arises from dividing the common multiple by the de

nominator, and the products will be the numerators of the required fractions; under which write the least common denominator.

nator.

Ans. 36, 40 and 5.

3. Reduce 145, 63 and 5, to their least common denominator. Ans. 15, 6 and 5. 4. Reduce, and to their least common denomiAns. 360 360 360°

nator.

6

60 72

320

5. Reduce 67, 4, 5, to their least common denomi

nator.

67

18 300

Ans. 120' 120' 120°

6. Reduce 1, 31, 41 and 8 to a common denominator.

7. Reduce 31, 412, 86, denominator.

8. Reduce,, 3, and common denominator. 9. Reduce,,, and common denominator.

605 450 800 100 100 100 100

Ans. 82 14176, to their least common

Ans. 450 624 1200 2079 144 144 144 144' to fractions having the least

6 8 9 10

Ans. 12, 12, 12: 12. to fractions having the least

Ans. 30, 90, 90, 90°

36 6050 63

10. Reduce,,, 7, 1, and 17 to equivalent fractions 'having the least common denominator.

42 16 36 40

33 34

Ans. 18, 48, 48' 48' 48' 48'

Q. How do you reduce fractions to their least common denominator? Does this reduction affect the values of the fractions?

REDUCTION OF DENOMINATE FRACTIONS.

§ 97. We have seen § 45, that a denominate number is one in which the kind of unit is denominated or expressed. For the same reason, a denominate fraction is one which expresses the kind of unit that has been divided. Such unit is called the unit of the fraction. Thus, of a £ is a denominate fraction. It expresses that one is the unit which has been divided.

The fraction of a shilling is also a denominate fraction, in which the unit that has been divided is one shilling. These two fractions are of different denominations, the unit of the first being one pound, and that of the second, one shilling.

Fractions, therefore, are of the same denomination when they express parts of the same unit, and of different denomi nations when they express parts of different units.

REDUCTION of denominate fractions consists in changing their denominations without altering their values.

Q. What is a denominate number? What is a denominate fraction? What is the unit called? In two-thirds of a pound, what is the unit? In three-eighths of a shilling, what is the unit? In one-half of a foot, what is the unit? When are fractions of the same denomination? When of different denominations? Are one-third of a £ and onefourth of a £ of the same or different denominations? One-fourth of a £ and one-sixth of a shilling? One-fifth of a shilling and one-half of a penny? What is reduction? many in £2? In 3? In 4? In 2s 8d? In 3s 6d? In 5s How many inches?

How many shillings in a £? How How many pence in is? In 2? In 3? 8d? How many feet in 3 yards 2ft.?

CASE I.

§ 98. To reduce a denominate fraction from a lower to a higher denomination.

RULE.

I. Consider how many units of the given denomination make one unit of the next higher, and place 1 over that number forming a second fraction.

II. Then consider how many units of the second denomi nation make one unit of the denomination next higher, and place 1 over that number forming a third fraction; and so on, to the denomination to which you would reduce.