Question

Without actually performing the long division, state whether the following rational number will have the terminating decimal expansion or a non-terminating repeating decimal expansion $\frac{3125}{3\times 5^2\times 22}$

Answer 1

Definition:

$\frac{p}{q}$is terminating if

1. p and q are co-prime
2. q is of the form $2^n\times 5^m$
   

Co-primes:

A pair of numbers is called co-primes if those numbers have only one common factor, namely 1.

Example:

3,5 are co-primes as they have one common factor which is 1.

Explanation:

Checking Co-primes:

1. Factors of $3125=5\times 5\times 5\times 5\times 5$
2. Factors of $3\times 5^2\times 22=3\times 5\times 5\times 2\times 11$
3. $3125and3\times 5^2\times 22$ have common factors 5.
4. So, $3125and3\times 5^2\times 22$ are not co-primes.

Denominator:

1. Here, the denominator $3\times 5^2\times 22$ cannot be written in the form $2^n{5}^m$.

Conclusion:

Hence, $\frac{3125}{3\times 5^2\times 22}$ is a non-terminating repeating decimal number.