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.

`19/3750`

Answer 1

To determine if the fraction `19/3750` has a terminating decimal expansion without performing the division, we need to look at the prime factorisation of the denominator after simplification.

A rational number `p/q` in simplest form has a terminating decimal expansion if and only if the prime factorisation of q contains only 2s and/or 5s.

Let’s analyse the denominator: 3750 = 2 × 3 × 5⁴.

Since there is a factor of 3 which is neither 2 nor 5, the denominator is not purely powers of 2 and 5.

However, we must check if 19 and 3750 have any common factors to simplify the fraction first 19 is a prime number and does not divide 2, 3 or 5.

Thus, the fraction `19/3750` is in simplest form and the denominator contains a prime factor 3 besides 2 and 5.

Therefore, `19/3750` will have a non-terminating recurring decimal expansion.