Question

Which of the following have a non terminating decimal expansion?

• A. 77/210
• B. 23/8
• C. 17/80
• D. 35/50

Answer 1

Answer: (A), (B)

The correct options are
A

77/210

B

23/8

If $\frac{p}{q}$ is a rational number with terminating decimal expansion where $p$ and $q$ are coprimes, then the prime factorisation of $q$ will be in the form of $2^n{5}^m$, where n and m are non - negative integers.

Consider the fraction $\frac{17}{210}$.
Here 210 = 2 $\times$ 3 $\times$ 5 $\times$7
Hence 210 is not in the form of $2^n{5}^m$.

Consider the fraction $\frac{23}{9}$.
Here 9 = 3 $\times$ 3 = $3^2$
Hence 9 is not in the form of $2^n{5}^m$.

Consider the fraction $\frac{17}{80}$.
Here 80 = 2 $\times$ 2 $\times$ 2 $\times$ 2 $\times$ 5 = $2^4\times 5$
Hence 80 is in the form of $2^n{5}^m$ with n = 4 and m = 1.


Consider the fraction $\frac{35}{50}=\frac{7}{10}$.
Here 10 = 2 $\times$ 5 = $2^1\times 5^1$
Hence 10 is in the form of $2^n{5}^m$ with n = 1 and m = 1.

Therefore, $\frac{17}{210}$ and $\frac{23}{9}$ will have non - terminating decimal expansion.