is equal to

4x7 28'

= ; but this fraction must be taken 3 times to form the first fraction; hence of =3×4 of 3×1, a result which is obtained by multiplying together the numerators and denominators of the compound fraction. 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 numerator and all the denominators together for a denominator: their products will form the fraction sought.

Ex. 2. Reduce of of to a simple fraction. Ans.

Here

3. Reduce of of to a simple fraction.

Here ××90=10=4 by dividing the numerator and denominator first by 9 and then by 2, as shown in Case III.

Or,

x=4, by cancelling the 3's and 6's in the numerator and denominator.

By cancelling or striking out the 3's we only di vide 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 deno

minator.

4. Reduce of off to a simple fraction. Here ××==&=‡ Ans.

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

Ans. 12-1027.

6. Reduce 5 of of of 6 to a simple fraction. Ans. 14=24.

CASE VI.

130. 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.

Ex. 1. Reduce, }, and to a common denominator.

1×3×5 15 the new numerator for

and 2×3×5=30, the common denominator.

Therefore, 8, 38 and 3, are the equivalent fractions.

It is plain, that this reduction does not alter the values of the several fractions, since the numerator nd denominator of each are multiplied by the same imber.

118.

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

Thus, = 28, 18, 18.

7.

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.

Ex. 3. Reduce 24, and of to a common denominator.

,, and; the answers.

4. Reduce 5, of, and 4, to a common denominator.

tor.

Ans. 1,, and .

5. Reduce 7, 5, and 37, to a common denomina

1080

Ans. 15, 10, and 22200. 6. Reduce 4,,, to a common denominator. Ans.,, and 1550.

NOTE. 131. 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.

7. Let it be required to reduce and to a common denominator.

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 hym that they will have a common denominator.

The two fractions will be reduced to 3 a

8. Reduce and to a common denominator.

Here,; and ×=.

QUESTIONS.

Ans. and .

123. What is Reduction of Vulgar Fractions? 124. When is a fraction said to be in its lowest terms? 125. How do you reduce an improper fraction to its equivalent whole or mixed number?

§ 126. How do you reduce a mixed number to its equiva lent fraction? Will the fraction be proper, or improper?

127. How do you reduce a fraction to its lowest terms? Will the value be altered by the reduction? Why?

$128. How do you reduce a whole number to a fraction having a given denominator?

§129. How do you reduce a compound fraction to a sim. ple one? When you have the same multiplier in the nume. rator and denominator, what do you do? Does this alter the value of the fraction?

§ 130. How do you reduce fractions to a common denominator? Does this reduction change the values of the seve ral fractions? Why not?

§ 131. What is a second method of reducing fractions to a common denominator?

REDUCTION OF DENOMINATE
FRACTIONS.

132. We have seen §71, 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, that this unit has been divided into 3 equal parts, and that 2 of these parts are taken in the fraction.

The fraction of a shilling is also a denominate 3

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

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

§ 133. Reduction of denominate fractions consists in changing their denominations without altering their values. For example, to reduce of a £. to the denomination of shillings, or of a shilling to the denomination of pounds.

CASE I.

134. To reduce a denominate fraction from lower to a higher denomination.

RULE.

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

II. Then consider how many units of the second denomination 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.

III. Connect all the fractions together, forming a compound fraction; then reduce the compound frac tion to a simple one by Case V.

of

13

Ex. 1. Reduce of a penny to the fraction of a £. Here of£. The given fraction is of a penny. But one penny is equal to shilling: hence of a penny is equal to of a shilling. But one shilling is equal to

of a

of