Reduce to their smallest, or lowest terms: 4. 44.

9. 188. 14. 448
10. 378

15. 1424
5. 9.
11. $18.

17. 3199
7.
13. 1428

18. 3434

7 20

20.

6. 305

16. 1702

35
3808
5 6 43
5940
1710
14364•
50 +0
17160
46 9 2
1619 76

21. 22.

1886

810
280
30 g

308

8. 490

144

630

11

CASE III.

163. To reduce integers or mixed numbers to improper fractions.

1. How many halves are there in 1 apple? In 4 apples? In 6 apples?

2. How many thirds are there in 1 orange? In 3 oranges? In 5 oranges?

3. How many fourths are there in 2? In 3? In 4?
4. How many fifths are there in 3? In 4? In 6?
5. How many fourths are there in 14? In 28?
6. How many thirds are there in 23? In 3} ? In 63?

WRITTEN EXERCISES.

are 7

PROCESS.

are

164. 1. Reduce 8 to sevenths.

ANALYSIS.—Since in 1 there sevenths, in 8 there 8 times 7

sevenths, or 56; and in 8+ there are 83 = 56 + = 5,2 59+, or 5

RULE.—Multiply the integers by the given denominator, to this product add the numerator of the fractional part, if there be any, and write the result over the given denominator.

2. Change 5% to fourths. 4. Reduce 131 to sixths.
3. Change 15 to fifths. 5. Reduce 183, to elevenths.

6. Change 54 to ninths. To eighteenths.
7. Change 6 to twelfths. To twenty-fourths.
8. Change 83 to fourteenths. To forty-seconds.
9. Reduce 9 to twentieths. To sixtieths.
Reduce to improper fractions:

14. 254. 10. 13

22. 86748

18. 42113. 11. 124 15. 2967.

19. 54013 23. 90434 12. 186 16. 3714 20. 76351

24. 314141 13. 235. 17. 5621 21. 419.

25. 721128

CASE IV.

165. To reduce improper fractions to integers or mixed numbers.

1. How many days are there in 6 half-days? In 8 halfdays? In 14 half-days?

2. How many yards are there in 9 thirds of a yard? In 15 thirds? In 18 thirds?

3. If a boy pick ļ bushel of peaches per hour, how many bushels can he pick in 10 hours? How many are 10 halves ?

4. If a man can earn į of a dollar per hour, how much can he earn in 12 hours? How many are 12 fourths?

5. How many units are there in 15? 36? 3? 61? 91? 6. How many dollars are there in $37? $4,7? $418? $109?

WRITTEN EXERCISES.

PROCESS.

166. 1. Reduce 123. to a mixed number.

ANALYSIS.—Since 7 sevenths equal 143 = 123 = 7=174

1 unit, 123 sevenths are equal to as

many units as 7 sevenths are contained times in 123, or 174 times. Therefore 143=174.

RULE.—Divide the numerator by the denominator.

167. To reduce dissimilar fractions to similar fractions.

1. How many fourths are there in 4 of an orange?
2. How many sixths of a field are there in ļof a field ?
3. How many eighths in ? How many ninths in z?
4. Express each of the fractions į , 1 , and as twelfths.
5. Express each of the fractions and as twentieths.
6. If į is divided into 3 equal parts, how large is each part?
7. If į is divided into 2 equal parts, how large is each part?

8. When į and } are divided into equal parts, what parts are common to both ?

9. When į and į are divided into equal parts, what parts are common to both ?

10. What equal parts are common to both and ??

11. When 1, 1 and š are divided into equal parts, what parts are common to all?

12. Change }, 1, 1, to equivalent fractions having the same fractional unit. Express the resulting fractions in equivalent fractions having their least common denominator.

Reduce to fractions having the same fractional unit: 11. 4 and 5 14. ^ and 11:

17. and 15. 12. f and 15 15. and

18. and 12 13. and 4 16. , and 12. 19. and 15.

5

168. Similar Fractions are those that have the same fractional unit.

169. Dissimilar Tractions are those that have not the same fractional unit.

170. Similar fractions have a common Denominator.

171. When similar fractions are expressed in their smallest terms they have their Least Common Denominator.

172. PRINCIPLES.—1. A common denominator of two or more fractions is a common multiple of their denominators.

2. The least common denominator of two or more fractions is the least common multiple of their denominators.

WRITTEN EXERCISES.

PROCESS.

3 X 8

X8

5 X4

173. 1. Reduce 1 and 3 to similar fractions.

ANALYSIS.-Since similar fractions have a common denominator, to make these fractions similar we must change them to equivalent fractions having a common denominator. Since a

common denominator of two or more fractions

a common multiple of their denominators (Prin.), we find a common multiple of the denominators 4 and 8, which is 32.

We then multiply the terms of each fraction by such a number as will change the fraction to thirty-seconds.

2. Reduce s, and a to similar fractions having their least common denominator.

PROCESS.

8

ANALYSIS.—The least common denomina

tor of several fractions is the least common ž=*4=

multiple of their denominators (Prin. 2); =4*4=13 therefore we find the least common multiple ==10 of 3, 4, and 6, which is 12.

We then multiply the terms of each fraction by such a number as will change it to twelfths, or to a fraction whose denominator is 12. Or,

Since 1 is equal to 13, j is equal to } of 19, or 12, and į are equal. to 2 times 12, or 182, etc.

RULE.—Find the common, or least common multiple of the denominators for a common, or least common denominator.

Divide this denominator by the denominator of each fraction and multiply both terms of the fraction by the quotient.

Reduce all mixed numbers to improper fractions and all fractions to their smallest terms.

Change the following to similar fractions having their least common denominator:

3. , , 1z. 7. 4, 5, f. 11. 35, 13, 14. 4. 4, 11, 18 8. ,, 12. 13, 14, 36 5. , , 18 9., 11, 12 13. ii, 13, 1. 6. 4, 16 10., 18, 6. 14. 18, 5, 16

5 27 18

174. 1. James has 2 fifths of a dollar, and his brother has 4 fifths of a dollar. How

many

fifths have both ? 2. George spent $on Monday, and $2 on Tuesday. How much did he spend in both days? How many sevenths are &

and ?