Solve the Equations worksheet featuring eight fraction problems to be solved, with instructions to provide answers as fractions or mixed numbers.

Let's solve each equation step by step. We are given proportions (equations with fractions), and we need to find the unknown variable in each case. We'll use cross-multiplication to solve them.

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1. $\frac{2}{7} = \frac{x}{15}$

Cross-multiply:

$$
2 \cdot 15 = 7 \cdot x \Rightarrow 30 = 7x
$$

Now divide both sides by 7:

$$
x = \frac{30}{7}
$$

As a mixed number: $4\frac{2}{7}$

✔ Answer: $\boxed{\frac{30}{7}}$ or $\boxed{4\frac{2}{7}}$

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2. $\frac{2}{p} = \frac{9}{4}$

Cross-multiply:

$$
2 \cdot 4 = 9 \cdot p \Rightarrow 8 = 9p
$$

Divide both sides by 9:

$$
p = \frac{8}{9}
$$

✔ Answer: $\boxed{\frac{8}{9}}$

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3. $\frac{1}{p} = \frac{6}{16}$

First, simplify $\frac{6}{16} = \frac{3}{8}$

So:
$$
\frac{1}{p} = \frac{3}{8}
$$

Now cross-multiply:

$$
1 \cdot 8 = 3 \cdot p \Rightarrow 8 = 3p
$$

$$
p = \frac{8}{3}
$$

As a mixed number: $2\frac{2}{3}$

✔ Answer: $\boxed{\frac{8}{3}}$ or $\boxed{2\frac{2}{3}}$

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4. $\frac{2}{5} = \frac{1}{n}$

Cross-multiply:

$$
2 \cdot n = 5 \cdot 1 \Rightarrow 2n = 5
$$

$$
n = \frac{5}{2}
$$

As a mixed number: $2\frac{1}{2}$

✔ Answer: $\boxed{\frac{5}{2}}$ or $\boxed{2\frac{1}{2}}$

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5. $\frac{y}{8} = \frac{1}{28}$

Cross-multiply:

$$
y \cdot 28 = 8 \cdot 1 \Rightarrow 28y = 8
$$

$$
y = \frac{8}{28} = \frac{2}{7}
$$

✔ Answer: $\boxed{\frac{2}{7}}$

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6. $\frac{20}{9} = \frac{8}{c}$

Cross-multiply:

$$
20 \cdot c = 9 \cdot 8 \Rightarrow 20c = 72
$$

$$
c = \frac{72}{20} = \frac{18}{5}
$$

As a mixed number: $3\frac{3}{5}$

✔ Answer: $\boxed{\frac{18}{5}}$ or $\boxed{3\frac{3}{5}}$

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7. $\frac{1}{7} = \frac{k}{13}$

Cross-multiply:

$$
1 \cdot 13 = 7 \cdot k \Rightarrow 13 = 7k
$$

$$
k = \frac{13}{7}
$$

As a mixed number: $1\frac{6}{7}$

✔ Answer: $\boxed{\frac{13}{7}}$ or $\boxed{1\frac{6}{7}}$

---

8. $\frac{5}{k} = \frac{9}{10}$

Cross-multiply:

$$
5 \cdot 10 = 9 \cdot k \Rightarrow 50 = 9k
$$

$$
k = \frac{50}{9}
$$

As a mixed number: $5\frac{5}{9}$

✔ Answer: $\boxed{\frac{50}{9}}$ or $\boxed{5\frac{5}{9}}$

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✔ Final Answers:

| Equation | Answer |
|--------|--------|
| $\frac{2}{7} = \frac{x}{15}$ | $\boxed{\frac{30}{7}}$ or $\boxed{4\frac{2}{7}}$ |
| $\frac{2}{p} = \frac{9}{4}$ | $\boxed{\frac{8}{9}}$ |
| $\frac{1}{p} = \frac{6}{16}$ | $\boxed{\frac{8}{3}}$ or $\boxed{2\frac{2}{3}}$ |
| $\frac{2}{5} = \frac{1}{n}$ | $\boxed{\frac{5}{2}}$ or $\boxed{2\frac{1}{2}}$ |
| $\frac{y}{8} = \frac{1}{28}$ | $\boxed{\frac{2}{7}}$ |
| $\frac{20}{9} = \frac{8}{c}$ | $\boxed{\frac{18}{5}}$ or $\boxed{3\frac{3}{5}}$ |
| $\frac{1}{7} = \frac{k}{13}$ | $\boxed{\frac{13}{7}}$ or $\boxed{1\frac{6}{7}}$ |
| $\frac{5}{k} = \frac{9}{10}$ | $\boxed{\frac{50}{9}}$ or $\boxed{5\frac{5}{9}}$ |

Let me know if you'd like a visual explanation or steps for any specific one!