Without performing long division, state whether $\frac{13}{3125}$ has a terminating or a non-terminating repeating decimal expansion. Write the expansion if it terminates.

Step-by-step solution

1. A fraction in its lowest terms has a terminating decimal expansion exactly when its denominator can be written in the form $2^n{5}^m$, where $n$ and $m$ are whole numbers.
2. Factorise the denominator: $3125=5^5$.
3. This is of the form $2^0\times 5^5$, so the decimal expansion terminates.
4. To write it out, turn the denominator into a power of ten by multiplying numerator and denominator by $2^5=32$.
5. $\frac{13}{3125}=\frac{13\times 32}{5^5\times 2^5}=\frac{416}{100000}=0.00416$

Concept tested: Condition for a terminating decimal expansion

Related questions

• 1 Check whether 6^n can end with the digit 0 for any natural number n.
• 2 Explain why 7 × 11 × 13 + 13 is a composite number.
• 3 Find the HCF and the LCM of 306 and 657, and verify that HCF × LCM = product of the two numbers.
• 4 Prove that √5 is an irrational number.
• 5 Determine whether the pair of equations 5x - 2y = 7 and 10x - 4y = 20 are consistent, inconsistent, or represent the same straight line. Justify your answer.
• 6 Three bells toll at intervals of 9, 12 and 15 minutes respectively. If they all toll together at 8:00 a.m., at what time will they next toll together?