Without Actual Division, find which of the following rational numbers are terminating decimals? (i) 13/80 (ii) 7/24 (iii) 5/12 (iv) 31/375 (v) 16/125

Which Rational Numbers are Terminating Decimals?

Below, without using the long division methods, we can find out which of the following rational numbers are terminating decimals.

(i) 13/80

In the above digit the denominator is 80

80 = 2 × 2 × 2 × 2 × 5 = 2⁴ × 5

Consequently, the denominator only has the factors 2 or 5.

As a result, the number is non-terminating.

(ii) 7/24

In the above digit the denominator is 24

24 = 2 × 2 × 2 × 3 = 2³ × 3

Consequently, the denominator only has the factors 2 or 5.

As a result, the number is non-terminating.

(iii) 5/12

In the above digit the denominator is 12

12 = 2 × 2 × 3 = 2² × 3

Consequently, the denominator only has the factors 2 or 5.

As a result, the number is non-terminating.

(iv) 31/375

In the above digit the denominator is 375

375 = 5 × 5 × 5 × 3 = 5³ × 3

Consequently, the denominator only has the factors 2 or 5.

As a result, the number is non-terminating.

(v) 16/125

In the above digit the denominator is 125

125 = 5 × 5 × 5 = 5³

As a result, the denominator only has the factors 2 or 5.

As a result, the number is terminating.

Terminating Decimals

A decimal that has an end digit is known as the terminating decimal. This means that only a decimal with a finite number at the end is the terminating decimal. These are represented easily in the form of p/q. Some examples of these include 29.8, 71.5, 5.5, etc.

Summary:

Without Actual Division, find which of the following rational numbers are terminating decimals

The rational fractions 13/80 and 16/125 are terminating decimals without any actual division. To be a terminating decimal, the factors of the denominator for decimals be in the form of 2^m x 5ⁿ.

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