Question

Without performing long division, determine which of the following rational numbers has a terminating decimal expansion. Also write its decimal expansion.

$\frac{7}{15}, - \frac{12}{840}, \frac{3}{80}, \frac{15}{220}$ .

Accepted Answer

The decimal expansion of a rational number $\frac{p}{q}, where p, q$ are integers, $q > 0, p, q$ have no common factors (except $1$ ) is:

(i) Terminating if $q$ can be expressed as $q = 2^{m} 5^{n}$ , where $m, n$ are whole numbers.

(ii) Non-terminating recurring if $q$ has a prime factor other than $2 or 5$

We have:

considering $\frac{7}{15}$ , in this rational number, the denominator is $15 = 3 \times 5$ , clearly, $\frac{7}{15}$ is not terminating as denominator cannot be written as $2^{m} \times 5^{n}$

considering $- \frac{2}{840}$ , in this rational number, the denominator is $840 = 2^{3} \times 3 \times 5 \times 7$ , clearly, $\frac{- 2}{840}$ is not terminating as denominator cannot be written as $2^{m} \times 5^{n}$ .

considering $\frac{3}{80}$ , in this rational number, the denominator is $80 = 2^{4} \times 5$ , which is a terminating as denominator can be written as $2^{m} \times 5^{n}$

considering $\frac{15}{220}$ , in this rational number, the denominator is $220 = 2^{2} \times 5 \times 11$ , clearly, $\frac{15}{220}$ is not terminating as denominator cannot be written as $2^{m} \times 5^{n}$ .

Hence, $\frac{3}{80}$ is a terminating and its value is $\frac{3}{80} = 0.0375$ .

M L Aggarwal Solutions for Chapter: Number System, Exercise 7: Chapter Test

M L Aggarwal Mathematics Solutions for Exercise - M L Aggarwal Solutions for Chapter: Number System, Exercise 7: Chapter Test

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