Without performing long division, determine which of the following rational numbers will have terminating decimals and which will be repeating: $\frac{7}{20}$, $\frac{4}{15}$ and $\frac{13}{250}$. Then check your answers by explicitly performing the long divisions and expressing these rational numbers as decimals. [Page No. 61]

Rule: $\frac{p}{q}$ (in lowest form) terminates iff prime factors of $q$ are only 2s and/or 5s.

$$\frac{7}{20}: 20 = 2^2 \times 5 \rightarrow \text{only 2 and 5} \rightarrow \text{Terminating.}$$

$$\text{Verify: } \frac{7}{20} = \frac{35}{100} = 0.35$$

$$\frac{4}{15}: 15 = 3 \times 5 \rightarrow \text{contains } 3 \rightarrow \text{Repeating.}$$

$\text{Verify}: \frac{4}{15} = 0.2666... = 0.2\overline{6}$

$\frac{13}{250}: 250 = 2 \times 5^{3} \rightarrow \text{only } 2 \text{ and } 5 \rightarrow \text{Terminating.}$

$$\text{Verify: } \frac{13}{250} = \frac{52}{1000} = 0.052$$