**How to determine if a fraction is terminating or repeating :**

Here we are going to see how to determine if a fraction is terminating or repeating.

If a rational number p/q, q ≠ 0, can be expressed in the form

where 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.

The following examples will illustrate the concept given above.

**Example 1 :**

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

7 / 16

**Solution :**

16 = 2^{4}

7 / 16 = 7 / (2^{4 }⋅ 5^{0})

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

**Example 2 :**

13 / 150

**Solution :**

150 = 2 ⋅ 3 ⋅ 5^{2}

13 / 150 = 13 / (2 ⋅ 3 ⋅ 5^{2})

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

**Example 3 : **

-11 / 75

**Solution :**

75 = 3 ⋅ 5^{2}

-11 / 75 = -11 / (3 ⋅ 5^{2})

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

**Example 4 :**

17 / 200

**Solution :**

200 = 2^{3} ⋅ 5^{2}

17 / 200 = 17 / (2^{3} ⋅ 5^{2})

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

**Example 5 :**

5 / 64

**Solution :**

64 = 2^{6}

5 / 64 = 5 / (2^{6} ⋅ 5^{0})

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

**Example 6 :**

11 / 12

**Solution :**

12 = 2^{2} ⋅ 3

11 / 12 = 11 / (2^{2} ⋅ 3)

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

**Example 7 :**

27 / 40

**Solution :**

40 = 2^{3} ⋅ 5

27 / 40 = 27 / (2^{3} ⋅ 5)

Since the given fraction 27 / 40 is in the form p/(2^{m }⋅ 5^{n}), 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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