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:

$\frac{5000}{70}$

Answer 1

Terminating decimal :

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

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

Where n and m are non-negative integers.

Co-primes:

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

Example:

3,5 are coprime as they have one common factor which is 1.

Explanation:

Checking co-prime:

Simplification of the given fraction:

$\frac{5000}{70}=\frac{500}{7}$

Here, $500and7$ have no common factors.

Hence, they are co-primes.

Denominator:

1. 7 can be written as $7=1\times 7$
2. Which is not in the form $2^n{5}^m$

Conclusion:

Hence, $\frac{5000}{70}$ is a non-terminating repeating decimal.