Identify whether the following rational numbers will make terminating decimals or not, without writing the decimal form.

Here are the solutions to the problems from the image.

1. Identify whether the following rational numbers will make terminating decimals or not, without writing the decimal form.
A rational number $\frac{p}{q}$ (in simplest form) has a terminating decimal expansion if the prime factors of its denominator $q$ are only 2s and/or 5s. Otherwise, it is a non-terminating repeating decimal.

a) $\frac{1}{4}$
The denominator is $4 = 2^2$. The only prime factor is 2.
Answer: Terminating

b) $\frac{22}{7}$
The denominator is $7 = 7^1$. The prime factor is 7.
Answer: Non-terminating

c) $\frac{-3}{10}$
The denominator is $10 = 2 \times 5$. The prime factors are 2 and 5.
Answer: Terminating

d) $\frac{4}{15}$
The denominator is $15 = 3 \times 5$. The prime factor 3 is present.
Answer: Non-terminating

e) $\frac{-5}{22}$
The denominator is $22 = 2 \times 11$. The prime factor 11 is present.
Answer: Non-terminating

f) $\frac{-11}{16}$
The denominator is $16 = 2^4$. The only prime factor is 2.
Answer: Terminating

2. Express the following rational numbers as terminating decimals.

a) $\frac{-10}{4}$

$$\frac{-10}{4} = \frac{-5}{2} = -2.5$$

Answer: $\boxed{-2.5}$

b) $\frac{5}{16}$

$$\frac{5}{16} = 0.3125$$

Answer: $\boxed{0.3125}$

c) $\frac{31}{25}$

$$\frac{31}{25} = 1.24$$

Answer: $\boxed{1.24}$

d) $\frac{-21}{15}$

$$\frac{-21}{15} = \frac{-7}{5} = -1.4$$

Answer: $\boxed{-1.4}$

3. Express the following rational numbers as non-terminating decimals. Mark the repeating decimal digits with a bar over them.

a) $\frac{2}{3}$

$$\frac{2}{3} = 0.666... = 0.\overline{6}$$

Answer: $\boxed{0.\overline{6}}$

b) $\frac{-13}{6}$

$$\frac{-13}{6} = -2.1666... = -2.1\overline{6}$$

Answer: $\boxed{-2.1\overline{6}}$

c) $\frac{1}{9}$

$$\frac{1}{9} = 0.111... = 0.\overline{1}$$

Answer: $\boxed{0.\overline{1}}$

d) $\frac{3}{7}$

$$\frac{3}{7} = 0.428571428571... = 0.\overline{428571}$$

Answer: $\boxed{0.\overline{428571}}$

4. Write a terminating rational number and a non-terminating rational number between $\frac{-1}{3}$ and $\frac{-3}{4}$.

First, convert to decimals: $\frac{-1}{3} \approx -0.333...$ and $\frac{-3}{4} = -0.75$.
We need numbers between $-0.75$ and $-0.333...$.
Using a common denominator of 12: $\frac{-3}{4} = \frac{-9}{12}$ and $\frac{-1}{3} = \frac{-4}{12}$.

Terminating rational number:
Choose $\frac{-6}{12}$. This simplifies to $\frac{-1}{2}$. The denominator is 2, so it's terminating.

$$\frac{-1}{2} = -0.5$$ $$-0.75 < -0.5 < -0.333...$$

Answer: $\boxed{\frac{-1}{2}}$ (or $\boxed{-0.5}$)

Non-terminating rational number:
Choose $\frac{-7}{12}$. The denominator is $12 = 2^2 \times 3$. Since it has a prime factor of 3, it is non-terminating.

$$\frac{-7}{12} \approx -0.5833...$$ $$-0.75 < -0.5833... < -0.333...$$

Answer: $\boxed{\frac{-7}{12}}$ (or $\boxed{-0.58\overline{3}}$)