HOW TO DETERMINE IF A FRACTION IS TERMINATING OR REPEATING

How to Determine if a Fraction is Terminating or Repeating :

In this section, we will learn, how to determine if a fraction is terminating or repeating.

Let us consider the rational number p/q, where q ≠ 0.

If we can write the denominator in terms of 2 and 5, then the rational number p/q will be in the form as shown below.  

If p ∈ Z and m, n ∈ W, then the rational number will have a terminating decimal expansion.

Otherwise, the rational number will have a non-terminating and recurring decimal expansion.

That is, if the denominator can't be written in terms of 2 and 5, then the rational number p/q will have a non-terminating and recurring decimal expansion.

The following examples will illustrate the concept explained above.

How to Determine if a Fraction is Terminating or Repeating - Examples

Example 1 :

Without actual division, classify the decimal expansion of the following numbers as terminating or non-terminating and recurring.

7 / 16

Solution :

16  =  2⁴

7 / 16  =  7 / (2⁴ ⋅ 5⁰) 

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

Example 2 :

 13 / 150

Solution :

150  =  2 ⋅ 3 ⋅ 5²

13 / 150  =  13 / (2 ⋅ 3 ⋅ 5²) 

Since the given fraction 13/150 is not in the form p/(2^m ⋅ 5ⁿ), it has non terminating  and recurring decimal expansion.

Example 3 : 

-11 / 75

Solution :

75  =   3 ⋅ 5²

-11 / 75  =  -11 / (3 ⋅ 5²) 

Since the given fraction -11 / 75 is not in the form p/(2^m ⋅ 5ⁿ), it has non terminating  and recurring decimal expansion.

Example 4 :

17 / 200

Solution :

200  =   2³ ⋅ 5²

17 / 200  =  17 / (2³ ⋅ 5²) 

Since the given fraction 17/200 is in the form p/(2^m ⋅ 5ⁿ), it has terminating decimal expansion.

Example 5 :

 5 / 64

Solution :

64  =   2⁶ 

5 / 64  =  5 / (2⁶ ⋅ 5⁰) 

Since the given fraction 5 / 64 is in the form p/(2^m ⋅ 5ⁿ), it has terminating decimal expansion.

Example 6 :

11 / 12

Solution :

12  =   2² ⋅ 3

11 / 12  =  11 / (2² ⋅ 3) 

Since the given fraction 11 / 12 is not in the form p/(2^m ⋅ 5ⁿ), it has non terminating and recurring decimal expansion.

Example 7 :

27 / 40

Solution :

40  =   2³ ⋅ 5

27 / 40  =  27 / (2³ ⋅ 5) 

Since the given fraction 27 / 40 is in the form p/(2^m ⋅ 5ⁿ), it has terminating decimal expansion.

After having gone through the stuff given above, we hope that the students would have understood, how to determine if a fraction is terminating or repeating. 

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