183. CASE VII.-To reduce fractions to a common denominator.

Fractions have a common denominator when their denominators are the same: asand, and

1. Reduce and to a common denominator.

Multiplying both terms of by 3, the denominator of the other fraction, we have. Now multiplying both terms of by 2, the denominator of the other fraction, we have }=. Hence, in the place of and we have & and, and these have a common denominator.

Ans. ?

4. and . Ans. & and 10.7. & and .

8. Reduce, 4, and to a common denominator.

EXPLANATION.-We multiply both terms

of by 7X8, the denominators of the other fractions, which gives = 22. We next multiply both terms of by 5 X 8, the denominators of the other fractions, and obtain%. Similarly we get

=

OPERATION.

105. Hence, instead of the given fractions we have their equals: 24, 28%, and 185, which have a common denominator.

80,

Hence, the

RULE.-Multiply both terms of each fraction by the product of all the denominators except its own.

NOTE.-Mixed and whole numbers, if any, must be reduced to improper fractions.

184. To reduce fractions to their Least Common Denominator.

RULE. Find the L. C. M. of the denominators; take it for a common denominator, and reduce each fraction according to Case III.

24. Reduce, §, and to their least common denominator.

EXPLANATION.-The L. C. M. of 8, 6, and 12 is OPERATION. 24, which we take for a common denominator. Now by Case III we say, 8 in 24 3 times, 3 × 3 = 9; hence, 4. In a similar manner we find =2, and=1

PRINCIPLE. To add two numbers, whether whole or fractional, they must be reduced, if not already so, to like units.

186. CASE I.-When the denominators, or fractional units, are alike.

1. Add together 4, 4, and 4.

EXPLANATION.-Since the numbers to be added have the same unit, viz: 1 seventh, we add as in whole numbers, and obtain 6 sevenths, or . Hence,

RULE.-Add the numerators and place the sum over the common denominator.

Ans. ?

5. Á, Á, Á, Â? Ans. 171. 9. $3, $17, $1?

187. CASE II.-When the denominators, or fractional

units, are unlike.

1. Add together and .

=

OPERATION.

EXPLANATION.-Since the numbers to be added, viz: 3 fourths and 2 thirds, are unlike, they cannot be added in their present form. Reducing them to a common denominator, or unit, by Art. 183, we obtain and 2, the sum of which, by Case I, is = 1. Hence, RULE. Reduce the fractions to a common denominator, and proceed as in Case I.

NOTE.

In addition and subtraction, the fractions should be written under each other after the manner of whole numbers.

2. What is the sum of and ? 3. What is the sum of 7 and }?

What is the sum of:

Ans. 31=111.

Ans.

11.

9.,, and g? Ans. 237.

Ans. 45. 10. 1, §, and ? Ans. 234.

4. # and ?

Ans. 1.

Ans.

.

11. §, &, and 7?

Ans. 2195.

7. & and ?

Ans. 7.

12. 4, 7, and 8?

Ans. 21.

8.

and?

Ans. 1.

13. 7, §, and? Ans. 2388.

14. Find the sum of 7 and 93.

EXPLANATION.-When there are mixed numbers, we add the fractions first, and then add their sum to the sum of the whole numbers. Adding and, we get 1}=1; put down the and carry to be added to the sum of the whole numbers, we get 17fz.

15. Add together 12 and 15.

16. Add together 23,

18%, and 321.

Ans. 275.

Ans. 7411.

17 A man paid $13 for a pair of pants, $17 for a coat, and $5 for a vest. What did he pay for all? 18. One boy weighs 642 pounds, another boy 56%, pounds, and the third boy 49 pounds. What is the total weight of the three boys? Ans. 17017 pounds.

19. A man planted 120 acres in corn, 75 acres in cotton, 32 acres in wheat, and 15 acres in oats. How many acres did he have in cultivation ?