24. In 12 how many thirds?
25. In 7 how many fifths?
26. In 103 how many eighths?
27. In 9 how many twelfths?
28. In 15 how many fourths?

29. Reduce 203 to an improper fraction. 30. Reduce 87 to an improper fraction. 31. Reduce 711 to an improper fraction. 32. Reduce 87 to an improper fraction. 33. Reduce 253 to an improper fraction. 34. How do you reduce a whole number to a fraction having a required denominator?

35. How do you change a mixed number to an improper fraction?

44. 1. In 16 how many times 1?

ANALYSIS. Since equal 1, 16 are as many times 1 as are contained times in 16, which are 4 times.

16 are 4 times 1, or 4.

2. In 25 how many times 1?
3. In 42 how many times 1?
4. In 72 how many times 1 ?
5. In 120 how many times 1?
6. In how many times 1?
7. In 100 how many times 1?

Therefore

9

11

2.

?

8. How many times 1 in 24? in 36? in 64 9. How many times 1 in 32? in 45? in 70? 10. How many times 1 in 40? in 48? in 12? 11. How many times 1 in 125? in 150? in 146? 12. How many times 1 in 116? in 205? in 125? 13. How many times 1 in 22? in 102? in 42? 14. How many times 1 in 2? in 114? in 100?

30

7

9

40

11

12

50

15. How many times 1 in 5? in $4? in 112? 16. How do you reduce an improper fraction to a whole or mixed number?

5

17. Reduce to an equivalent whole number. 18. Reduce 104 to an equivalent mixed number. 19. Reduce to an equivalent mixed number.

20

84
9

6

60

20. Reduce 10 to an equivalent mixed number. 21. Express the value of 120 in a whole number. 22. Express the value of 208 in a mixed number. 23. Express the value of 25 in a mixed number.

150

45. 1. In of 1, how many sixths?

ANALYSIS. Since in 1 there are §, in § of 1 are or &, and in 3 of 1 are 2 times &, or . Therefore in

of 8, of

14. In of 1

20

15. In 14 of 1

how many thirtieths?

16. In of 1 how many eighths?
17. In of 1 how many fourteenths?

how many fortieths?
how many eightieths?

how many ninetieths?

how many one hundred twentieths?

25. How many sixths in ?

ANALYSIS. Since in 1 there are 18, in † of 1 are of ; and is contained in 1, 5 times. Therefore are 5 sixths.

18, or

in

?

? in 14?

26. How many eighths in? in 18?
27. How many fourths in 15? in?
28. How many thirds in 28? in 14
29. How many sevenths in
30. How many twelfths in
31. How many ninths in ? in 2g?
32. How many sixths in 35? in 18?
33. How many eighths in 2? in ?

64

? in 14?

80

34. How many fourteenths in 18? in 28? 35. How many fifths in ? in 28?

36. How many sevenths in

37. How many fourths in

38. How many sixths in 39. How many halves in

? in 34?

? in 1?

8? in

66

?

28

?

in 42?

in 2 ?

43. How many sixteenths in 44? in 15?

64

120

44. How many twentieths in 36 ? in 42?
45. How many twenty-fourths in 4? in 4?

46. NOTE.

1. Reduce to its lowest terms.

A fraction is said to be in its lowest terms when no number greater than I will exactly divide both numerator and denominator.

2. Reduce 9 to its lowest terms.

18

12. Reduce to its lowest terms. 100

13. Reduce 45 to its lowest terms.

90

120 56 24

19. Reduce to its lowest terms.

20. Reduce 81 to its lowest terms.

27

21. Reduce 12 to its lowest terms.

132

22. Reduce 130 to its lowest terms.

23. How do you reduce a fraction to its lowest terms?

REMARKS. 1. If the numerator be multiplied by any number, the value of the fraction will be increased as many times as there are units in the multiplier.

2. If the numerator be divided by any number, the value of the fraction will be diminished as many times as there are units in the divisor.

3. If the denominator be multiplied by any number, the value of the fraction will be diminished as many times as there are units in the multiplier.

47. Fractions are said to have a common denominator when their denominators are alike. Any number that can be divided by each of the denominators of the given fractions may be taken for the common denominator.

1. Reduce and to fractions having a common denominator.

ANALYSIS. 12 is exactly divisible by 4 and 3, and may therefore be taken for a common denominator. Since in 1 there are 1, in of 1 there must be 1 of 12, or 12, and in of 1 there must be of 12, or. Therefore

are equal to
2. Reduce
denominator.

and

and

and to fractions having a common

3. Reduce and

4. Reduce and 5. Reduce and

to a common denominator.

to a common denominator.
to a common denominator.

6. Reduce and to a common denominator.
7. Reduce and g to a common denominator.
8. Reduce 5 4 and to a common denominator.

69 99

9. Reduce,,
1 and 3

14 to a common denominator. 10. Reduce 12 and to a common denomi

nator.

159

14. Reduce 11, 21 and to a common denomi

nator.

4. If the denominator be divided by any number, the value of the fraction will be increased as many times as there are units in the divisor. 5. If both numerator and denominator be multiplied by the same number, the value of the fraction will not be changed.

6. If both numerator and denominator be divided by the same number, the value of the fraction will not be changed.