Question

Which one of the following numbers is a non-terminating and repeating decimal?

• A. $\frac{39}{24}$
• B. $\frac{49}{56}$
• C. $\frac{78}{32}$
• D. $\frac{3}{11}$

Answer 1

The correct option is D

$\frac{3}{11}$

If $\frac{p}{q}$ is a rational number in its standard form (HCF of p and q is 1) whose decimal expansion is terminating, then $q=2^m\times 5^n$

(Decimal expansion of a number is terminating when the denominator is a power of 2 and/or 5, and the H.C.F of the numerator and the denominator is 1)

$\frac{39}{24}=\frac{13\times 3}{8\times 3}=\frac{13}{8}$
The denominator of $\frac{13}{8}$ is 8 which is $2^3$
Thus, the decimal expansion of $\frac{39}{24}$ is terminating


$\frac{49}{56}=\frac{7\times 7}{8\times 7}=\frac{7}{8}$
The denominator of $\frac{7}{8}$ is 8 which is $2^3$
Thus, the decimal expansion of $\frac{49}{56}$ is terminating


$\frac{78}{32}=\frac{39\times 2}{16\times 2}=\frac{39}{16}$
The denominator of $\frac{39}{16}$ is 16 which is $2^4$
Thus, the decimal expansion of $\frac{78}{32}$ is terminating

The denominator of $\frac{3}{11}$ is 11 which is ${11}^1$.
Thus, the decimal expansion of $\frac{3}{11}$ is recurring.