When the denominators of two or more fractions are alike, (as in the foregoing examples,) they are said to have a common denominator. The parts are then in the same denomination, and, consequently, of the same magnitude or value. It is evident, therefore, that they may be added or subtracted, by adding or subtracting their numerators, that is, the number of their parts, care being taken to write under the result their proper denominator. Thus, +=}; } ~ }

6. A boy, having an orange, gave of it to his sister, and of it to his brother; what part of the orange did he give away?

4ths and 8ths, being parts of different magnitudes, or value, cannot be added together. We must therefore first reduce them to parts of the same magnitude, that is, to a common denominator. are 3 parts. If each of these parts be divided into 2 equal parts, that is, if we multiply both terms of the fraction by 2, (¶ 46,) it will be changed to §; then § and are f. Ans. of an orange,

7. A man had of a hogshead of molasses in one cask, and of a hogshead in another; how much more in one cask than in the other?

Here, 3ds cannot be so divided as to become 5ths, nor can 5ths be so divided as to become 3ds; but if the 3ds be each divided into 5 equal parts, and the 5ths each into 3 equal parts, they will all become 15ths. The will become 1%, and the will become; then, taken from + leaves 15, Ans.

¶ 60. From the very process of dividing each of the parts, that is, of increasing the denominators by multiplying them, it follows, that each denominator must be a factor of the common denominator; now, multiplying all the denominators together will evidently produce such a number.

Hence, To reduce fractions of different denominators to equivalent fractions, having a common denominator,-RULE: Multiply together all the denominators for a common denominator, and, as by this process each denominator is multiplied by all the others, so, to retain the value of each fraction, multiply each numerator by all the denominators, except its own, for a new numerator, and under it write the common denominator.

EXAMPLES FOR PRACTICE.

1. Reduce, and to fractions of equal value, having a common denominator.

3 X 4 X 560, the common denominator.

2 × 4 × 5 — 40, the new numerator for the first fraction. 3 × 3 × 545, the new numerator for the second fraction. 3 X 4 X 4 = 48, the new numerator for the third fraction.

The new fractions, therefore, are 48, 5, and 3. By an inspection of the operation, the pupil will perceive, that the numerator and denominator of each fraction have been multiplied by the same numbers; consequently, ( 46,) that their value has not been altered.

240

2. Reduce, and to equivalent fractions, having common denominator. Ans. 18, 18, 19, 118. 3. Reduce to equivalent fractions of a common denominator, and add together, 3, 3, and 4.

Ans. 33 +38+z8=78=1, Amount.

14. Add together and 4.

5. What is the amount of ++++++÷?

Amount, 11.

Ans. 43127.

=

6. What are the fractions of a common denominator Ans. 1 and 22, or and 1.

equivalent to and ?

We have already seen, (T 59, ex. 7,) that the common denominator may be any number, of which each given denominator is a factor, that is, any number which may be divided by each of them without a remainder. Such a number is called a common multiple of all its common divisors, and the least number that will do this is called their least common multiple; therefore, the least common denominator of any fractions is the least common multiple of all their denominators. Though the rule already given will always find a common multiple of the given denominators, yet it will not always find their least common multiple. In the last example, 24 is evidently a common multiple of 4 and 6, for it will exactly measure both of them; but 12 will do the same, and as 12 is the least number that will do this, it is the least common multiple of 4 and 6. It will therefore be convenient to have a rule for finding this least common multiple. Let the aumbers be 4 and 6.

It is evident, that one number is a multiple of another, when the former contains all the factors of the latter.

L

The

factors of 4 are 2 and 2, (2 × 24.) The factors of 6 are 2 and 3, (2 x 3 = 6.) Consequently, 2 × 2 × 3 = 12 contains the factors of 4, that is, 2 × 2; and also contains the factors of 6, that is, 2 × 3. 12, then, is a common multiple of 4 and 6, and it is the least common multiple, because it does not contain any factor, except those which make up the numbers 4 and 6; nor either of those repeated more than is necessary to produce 4 and 6. Hence it follows, that when any two numbers have a factor common to both, it may be once omitted; thus, 2 is a factor common both to 4 and 6, and is consequently once omitted.

¶ 61. On this principle is founded the RULE for finding the least common multiple of two or more numbers. Write down the numbers in a line, and divide them by any number that will measure two or more of them; and write the quotients and undivided numbers in a line beneath. Divide this line as before, and so on, until there are no two numbers that can be measured by the same divisor; then the continual product of all the divisors and numbers in the last line will be the least common multiple required.

Let us apply the rule to find the least common multiple of 4 and 6.

2)4

6

2 3

4 and 6 may both be measured by 2; the quotients are 2 and 3. There is no number greater than 1, which will measure 2 and 3. Therefore, 2 X 2 X 3: 12 is the least common multiple of 4 and 6.

If the pupil examine the process, he will see that the divisor 2 is a factor common to 4 and 6, and that dividing 4 by this factor gives for a quotient its other factor, 2. In the same manner, dividing 6 gives its other factor, 3. Therefore the divisor and quotients make up all the factors of the two numbers, which, multiplied together, must give the common multiple.

7. Reduce †,†, and to equivalent fractions of the least common denominator. OPERATION.

2)4.2

3 6

Then, 2 X3 Xx 2 = 12, least common denominator. It is evident we need not multiply by the 1s, as this would not alter the number.

To find the new numerators, that is, how many 12ths each fraction is, we may take 2,, and of 12. Thus :

Ans.,, and .

8. Reduce,, and to fractions having the least common denominator, and add them together.

Ans. 14+2+2=#=14, amount. 9. Reduce and to fractions of the least common denominator, and subtract one from the other.

Ans. 1=1, difference. 10. What is the least number that 3, 5, 8 and 10 will measure? Ans. 120. 11. There are 3 pieces of cloth, one containing 72 yards, another 13 yards, and the other 15ğ yards; how many yards in the 3 pieces.

Before addii.g, reduce the fractional parts to their least common denominator; this being done, we shall have,

72=728 131⁄2 = 1330 157=1521

Ans. 37.

Adding together all the 24ths, viz. 18+ 20 +21, we obtain 59, that is, 52 = 211. We write down the fraction

under the

other fractions, and reserve the 2 iegers to be carried to the amount of the other integers, making in the whole 371, Ans. 12. There was a piece of cloth containing 343 yards, from which were taken 123 yards; how much was there

left?

348

34

123 = 1234

Ans. 211 yds.

:

We cannot take 16 twenty-fourths (1) from 9 twenty-fourths, (;) we must, therefore, borrow 1 integer, 24 twenty-fourths, (,) which, with makes ; we can now take from , and there will remain ; but, as we borrowed, so also we must carry 1 to the 12, which makes it 13, and 13 from 34 leaves 21. Ans. 211. 13. What is the amount of of of a yard, of a yard, and of 2 yards?

Note. The compound fraction may be reduced to a sim. ple fraction; thus, of ; and of 2; then, + }+&=38=12 yds., Ans.

¶ 62. From the foregoing examples we derive the following RULE:-To add or subtract fractions, add or subtract their numerators, when they have a common denominator; otherwise, they must first be reduced to a common denominator.

Note. Compound fractions must be reduced to simple fractions before adding or subtracting.

EXAMPLES FOR PRACTICE. 1. What is the amount of , 4 and 12? 2. A man bought a ticket, and sold of the ticket had he left?

3. Add together,, 4, 5, and 1. 4. What is the difference between 14

5. From 1 take 2.

6. From 3 take .

Ans. 17H

of

of it; what part Ans. §.

Amount, 238.

and 16?

Ans. 118. Remainder, *.

Rem. 23.

Rem. 988.

Rem. .

9. Add together 112, 311, and 1000%. 10. Add together 14, 11, 4,

and

11. From take. From 7 take .

12. What is the difference between and ? and †? and? and ? and ? and ?

1 t?

1

1.

13. How much is 1 2- 2-4? 24-47 34 1000? 2–号?

REDUCTION OF FRACTIONS.

63. We have seen, ( 33,) that integers of one denomination may be reduced to integers of another denomination. It is evident, that fractions of one denomination, after the same manner, and by the same rules, may be reduced to fractions of another denomination; that is, fractions, like integers, may be brought into lower denominations by multiplication, and into higher denominations by division.