Question

Without actually performing the long division, state whether the following rational number will have a terminating decimal expansion or a non-terminating recurring decimal expansion.

`7/200`

Answer 1

To determine whether the rational number `7/200` has a terminating decimal expansion or a non-terminating recurring decimal expansion without performing the division, we consider the prime factorisation of the denominator after simplifying the fraction (if necessary).

The rule is:

A rational number `p/q` in simplest form has a terminating decimal expansion if and only if the denominator q is of the form 2^m × 5ⁿ, where m and n are non-negative integers.

Otherwise, it has a non-terminating recurring decimal expansion.

For `7/200`, the denominator is 200. 

Factorize: 200 = 2³ × 5² 

Since the denominator is of the form 2^m × 5ⁿ, `7/200` will have a terminating decimal expansion.

Therefore, `7/200` has a terminating decimal expansion without performing the actual division.