Determine if the following rational numbers will produce terminating decimals without converting them to decimal form:

$$4 = 2^2$$

The only prime factor is 2.
Answer: Terminating

b) $\frac{22}{7}$

$$7 = 7^1$$

The prime factor is 7, which is not 2 or 5.
Answer: Non-terminating

c) $\frac{-3}{10}$

$$10 = 2 \times 5$$

The prime factors are 2 and 5.
Answer: Terminating

d) $\frac{4}{15}$

$$15 = 3 \times 5$$

The prime factors include 3, which is not 2 or 5.
Answer: Non-terminating

e) $\frac{-5}{22}$

$$22 = 2 \times 11$$

The prime factors include 11, which is not 2 or 5.
Answer: Non-terminating

f) $\frac{-11}{16}$

$$16 = 2^4$$

The only prime factor is 2.
Answer: Terminating

2. Express the following rational numbers as terminating decimals.

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

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

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

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

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) $\frac{2}{3}$

$$0.\overline{6}$$

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

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

$$-2.1\overline{6}$$

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

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

$$0.\overline{1}$$

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

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

$$0.\overline{428571}$$

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

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

$$\frac{-1}{3} \approx -0.333...$$ $$\frac{-3}{4} = -0.75$$

We need numbers between $-0.75$ and $-0.333...$.

The least common multiple of 3 and 4 is 12.

$$\frac{-1}{3} = \frac{-1 \times 4}{3 \times 4} = \frac{-4}{12}$$ $$\frac{-3}{4} = \frac{-3 \times 3}{4 \times 3} = \frac{-9}{12}$$

So we are looking for numbers between $\frac{-9}{12}$ and $\frac{-4}{12}$. Possible fractions include $\frac{-8}{12}, \frac{-7}{12}, \frac{-6}{12}, \frac{-5}{12}$.

A terminating rational number has a denominator whose prime factors are only 2s and/or 5s.
Let's choose $\frac{-6}{12}$.

$$\frac{-6}{12} = \frac{-1}{2}$$

The denominator is 2, so it's terminating. Its decimal value is $-0.5$.

$$-0.75 < -0.5 < -0.333...$$

Terminating rational number: $\boxed{\frac{-1}{2}}$ (or $\boxed{-0.5}$)

A non-terminating rational number has a denominator with prime factors other than 2s and 5s.
Let's choose $\frac{-7}{12}$.
The denominator is $12 = 2^2 \times 3$. Since it has a prime factor of 3, it is non-terminating.
Its decimal value is $\frac{-7}{12} \approx -0.5833...$.

$$-0.75 < -0.5833... < -0.333...$$

Non-terminating rational number: $\boxed{\frac{-7}{12}}$ (or $\boxed{-0.58\overline{3}}$)