Question 1

A rational number in its simplest form has a terminating decimal if its denominator's prime factors are only 2 or 5.

Step 1 — Analyze $\frac{7}{20}$

First, we examine the fraction $\frac{7}{20}$. The numerator is 7 and the denominator is 20. This fraction is already in its simplest form. Let's find the prime factors of the denominator 20.

$20 = 2 \times 10$

$= 2 \times 2 \times 5$

$= 2^2 \times 5^1$

The prime factors are only 2 and 5. So, its decimal expansion will be terminating. Now, let's perform the long division to check.

$7 \div 20 = 0.35$

$\boxed{\frac{7}{20} \text{ is terminating and equals } 0.35}$

Step 2 — Analyze $\frac{4}{15}$

Next, we consider the fraction $\frac{4}{15}$. The numerator is 4 and the denominator is 15. This fraction is also in its simplest form. Let's find the prime factors of the denominator 15.

$15 = 3 \times 5$

The prime factors include 3. This factor is not 2 or 5. So, its decimal expansion will be repeating. Now, let's perform the long division to check.

$4 \div 15 = 0.2666...$

$\boxed{\frac{4}{15} \text{ is repeating and equals } 0.2\overline{6}}$

Step 3 — Analyze $\frac{13}{250}$

Finally, we look at the fraction $\frac{13}{250}$. The numerator is 13 and the denominator is 250. This fraction is in its simplest form. Let's find the prime factors of the denominator 250.

$250 = 2 \times 125$

$= 2 \times 5 \times 25$

$= 2 \times 5 \times 5 \times 5$

$= 2^1 \times 5^3$

The prime factors are only 2 and 5. So, its decimal expansion will be terminating. Now, let's perform the long division to check.

$13 \div 250 = 0.052$

$\boxed{\frac{13}{250} \text{ is terminating and equals } 0.052}$

Answer

(i) $\frac{7}{20}$ will have a terminating decimal expansion, which is 0.35. (ii) $\frac{4}{15}$ will have a repeating decimal expansion, which is 0.26̅. (iii) $\frac{13}{250}$ will have a terminating decimal expansion, which is 0.052.