HOW TO TELL IF A FRACTION IS TERMINATING OR REPEATING

Terminating decimals result when the rational number has a denominator which has no prime factors other than 2 or 5.

Problem 1 :

1/2

Solution :

2 = 2¹

1/2 = 1/(2¹ ⋅ 5⁰)

Since the given fraction 1/2 is in the form p/(2^m ⋅ 5ⁿ), it has terminating decimal expansion.

Problem 2 :

3/4

Solution :

4 = 2²

3/4 = 3/(2² ⋅ 5⁰)

Since the given fraction 3/4 is in the form p/(2^m ⋅ 5ⁿ), it has terminating decimal expansion.

Problem 3 :

3/5

Solution :

5 = 5¹

3/5 = 3/(2⁰ ⋅ 5¹)

Since the given fraction 3/5 is in the form p/(2^m ⋅ 5ⁿ), it has terminating decimal expansion.

Problem 4 :

17/50

Solution :

50 = 2¹ ⋅5²

17/50 = 17/(2¹ ⋅ 5²)

Since the given fraction 17/50 is in the form p/(2^m ⋅ 5ⁿ), it has terminating decimal expansion.

Problem 5 :

2/3

Solution :

3 = 3¹

2/3 = 2/(2⁰ ⋅ 3¹ ⋅ 5⁰)

Since the given fraction 2/3 is not in the form p/(2^m ⋅ 5ⁿ), it has non terminating and recurring decimal expansion.

Problem 6 :

2/9

Solution :

9 = 3²

2/9 = 2/(2⁰ ⋅ 3² ⋅ 5⁰)

Since the given fraction 2/9 is not in the form p/(2^m ⋅ 5ⁿ), it has non terminating and recurring decimal expansion.

Problem 7 :

9/40

Solution :

40 = 2³ ⋅5¹

9/40 = 9/(2³ ⋅ 5¹)

Since the given fraction 9/40 is in the form p/(2^m ⋅ 5ⁿ), it has terminating decimal expansion.

Problem 8 :

5/6

Solution :

6 = 2¹ ⋅ 3¹ ⋅ 5⁰

5/6 = 5/(2¹ ⋅ 3¹ ⋅ 5⁰)

Since the given fraction 5/6 is not in the form p/(2^m ⋅ 5ⁿ), it has non terminating and recurring decimal expansion.

Problem 9 :

7/8

Solution :

8 = 2³ ⋅ 5⁰

7/8 = 7/(2³ ⋅ 5⁰)

Since the given fraction 7/8 is in the form p/(2^m ⋅ 5ⁿ), it has terminating decimal expansion.

Problem 10 :

17/80

Solution :

80 = 2⁴ ⋅5¹

17/80 = 17/(2⁴ ⋅ 5¹)

Since the given fraction 17/80 is in the form p/(2^m ⋅ 5ⁿ), it has terminating decimal expansion.

Problem 11 :

37/125

Solution :

125 = 2⁰ ⋅5³

37/125 = 37/(2⁰ ⋅ 5³)

Since the given fraction 37/125 is in the form p/(2^m ⋅ 5ⁿ), it has terminating decimal expansion.

Problem 12 :

5/12

Solution :

12 = 2² ⋅ 3¹

5/12 = 5/(2² ⋅ 3¹  ⋅ 5⁰)

Since the given fraction 5/12 is not in the form p/(2^m ⋅ 5ⁿ), it has non terminating and recurring decimal expansion.

Problem 13 :

3/20

Solution :

20 = 2² ⋅5¹

3/20 = 3/(2² ⋅ 5¹)

Since the given fraction 3/20 is in the form p/(2^m ⋅ 5ⁿ), it has terminating decimal expansion.

Problem 14 :

4/11

Solution :

11 = 2⁰ ⋅ 3⁰ ⋅ 5⁰ ⋅ 11¹

4/11 = 4/(2⁰ ⋅ 3⁰ ⋅ 5⁰ ⋅ 11¹)

Since the given fraction 4/11 is not in the form p/(2^m ⋅ 5ⁿ), it has non terminating and recurring decimal expansion.

Problem 15 :

11/25

Solution :

25 = 2⁰ ⋅5²

11/25 = 11/(2⁰ ⋅ 5²)

Since the given fraction 11/25 is in the form p/(2^m ⋅ 5ⁿ), it has terminating decimal expansion.

Problem 16 :

7/9

Solution :

9 = 3²

7/9 = 7/(2⁰ ⋅ 3² ⋅ 5⁰)

Since the given fraction 7/9 is not in the form p/(2^m ⋅ 5ⁿ), it has non terminating and recurring decimal expansion.

Problem 17 :

7/90

Solution :

90 = 2¹ ⋅ 3² ⋅ 5¹

7/90 = 7/(2¹ ⋅ 3² ⋅ 5¹)

Since the given fraction 7/90 is not in the form p/(2^m ⋅ 5ⁿ), it has non terminating and recurring decimal expansion.

Problem 18 :

7/99

Solution :

99 = 2⁰ ⋅ 3² ⋅ 5⁰ ⋅ 11¹

7/99 = 7/(2⁰ ⋅ 3² ⋅ 5⁰ ⋅ 11¹)

Since the given fraction 7/99 is not in the form p/(2^m ⋅ 5ⁿ), it has non terminating and recurring decimal expansion.

Problem 19 :

7/999

Solution :

999 = 2⁰ ⋅ 3³ ⋅ 5⁰ ⋅ 37¹

7/999 = 7/(2⁰ ⋅ 3³ ⋅ 5⁰ ⋅ 37¹)

Since the given fraction 7/999 is not in the form p/(2^m ⋅ 5ⁿ), it has non terminating and recurring decimal expansion.

Problem 20 :

7/9999

Solution :

9999 = 2⁰ ⋅ 3² ⋅ 5⁰ ⋅ 11¹ ⋅ 101¹

7/9999 = 7/(2⁰ ⋅ 3² ⋅ 5⁰ ⋅ 11¹  ⋅ 101¹)

Since the given fraction 7/9999 is not in the form p/(2^m ⋅ 5ⁿ), it has non terminating and recurring decimal expansion.