Chapter 1 · Question 5

When does the rational number $\frac{p}{q}$ have a terminating decimal expansion?

Answer Revealed

Direct Answer:

p

and

q

are coprime and the prime factorisation of

q

is of the form

2^m5^n

, then

\frac{p}{q}

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

A rational number terminates only when the denominator, after simplification, has no prime factors other than

2

and

5

For

\frac pq

in lowest terms, check the denominator

q

. If

q=2^m5^n

for non-negative integers

m,n

, then the decimal expansion terminates because the denominator can be converted to a power of

10

. If any prime other than

2

5

divides

q

, the decimal expansion is non-terminating recurring.

If $p$ and $q$ are coprime and the prime factorisation of $q$ is of the form $2^m5^n$ , then $\frac{p}{q}$ has a terminating decimal expansion.
Use the NCERT formula or theorem carefully.
Write units and final conclusion where applicable.

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