Math worksheet for finding equivalent ratios, including problems to find missing numbers, solve for variables, and write equivalent ratios.

Problem: Finding Equivalent Ratios

We are tasked with solving a series of problems involving ratios. Let's break it down step by step.

---

#### Part 1: Find the missing number in the ratio

1. 12 : ____ = 3 : 8
- To find the missing number, set up the proportion:
\[
\frac{12}{x} = \frac{3}{8}
\]
- Cross-multiply:
\[
12 \cdot 8 = 3 \cdot x \implies 96 = 3x
\]
- Solve for \( x \):
\[
x = \frac{96}{3} = 32
\]
- Answer: \( 32 \)

2. 42 : ____ = 6 : 5
- Set up the proportion:
\[
\frac{42}{x} = \frac{6}{5}
\]
- Cross-multiply:
\[
42 \cdot 5 = 6 \cdot x \implies 210 = 6x
\]
- Solve for \( x \):
\[
x = \frac{210}{6} = 35
\]
- Answer: \( 35 \)

3. 56 : ____ = 8 : 7
- Set up the proportion:
\[
\frac{56}{x} = \frac{8}{7}
\]
- Cross-multiply:
\[
56 \cdot 7 = 8 \cdot x \implies 392 = 8x
\]
- Solve for \( x \):
\[
x = \frac{392}{8} = 49
\]
- Answer: \( 49 \)

4. ____ : 8 = 9 : 2
- Set up the proportion:
\[
\frac{x}{8} = \frac{9}{2}
\]
- Cross-multiply:
\[
x \cdot 2 = 9 \cdot 8 \implies 2x = 72
\]
- Solve for \( x \):
\[
x = \frac{72}{2} = 36
\]
- Answer: \( 36 \)

5. 48 : ____ = 6 : 7
- Set up the proportion:
\[
\frac{48}{x} = \frac{6}{7}
\]
- Cross-multiply:
\[
48 \cdot 7 = 6 \cdot x \implies 336 = 6x
\]
- Solve for \( x \):
\[
x = \frac{336}{6} = 56
\]
- Answer: \( 56 \)

6. ____ : 36 = 10 : 9
- Set up the proportion:
\[
\frac{x}{36} = \frac{10}{9}
\]
- Cross-multiply:
\[
x \cdot 9 = 10 \cdot 36 \implies 9x = 360
\]
- Solve for \( x \):
\[
x = \frac{360}{9} = 40
\]
- Answer: \( 40 \)

7. 24 : ____ = 4 : 7
- Set up the proportion:
\[
\frac{24}{x} = \frac{4}{7}
\]
- Cross-multiply:
\[
24 \cdot 7 = 4 \cdot x \implies 168 = 4x
\]
- Solve for \( x \):
\[
x = \frac{168}{4} = 42
\]
- Answer: \( 42 \)

8. 7 : ____ = 28 : 44
- Simplify the ratio \( 28 : 44 \) first:
\[
\frac{28}{44} = \frac{7}{11} \quad (\text{dividing both terms by 4})
\]
- Set up the proportion:
\[
\frac{7}{x} = \frac{7}{11}
\]
- Since the numerators are equal, the denominators must be equal:
\[
x = 11
\]
- Answer: \( 11 \)

---

#### Part 2: Find the unknown variable

9. 15 : j = 3 : 5
- Set up the proportion:
\[
\frac{15}{j} = \frac{3}{5}
\]
- Cross-multiply:
\[
15 \cdot 5 = 3 \cdot j \implies 75 = 3j
\]
- Solve for \( j \):
\[
j = \frac{75}{3} = 25
\]
- Answer: \( j = 25 \)

10. 27 : 30 = x : 10
- Set up the proportion:
\[
\frac{27}{30} = \frac{x}{10}
\]
- Simplify \( \frac{27}{30} \):
\[
\frac{27}{30} = \frac{9}{10} \quad (\text{dividing both terms by 3})
\]
- Set up the proportion:
\[
\frac{9}{10} = \frac{x}{10}
\]
- Since the denominators are equal, the numerators must be equal:
\[
x = 9
\]
- Answer: \( x = 9 \)

11. 24 : y = 8 : 9
- Set up the proportion:
\[
\frac{24}{y} = \frac{8}{9}
\]
- Cross-multiply:
\[
24 \cdot 9 = 8 \cdot y \implies 216 = 8y
\]
- Solve for \( y \):
\[
y = \frac{216}{8} = 27
\]
- Answer: \( y = 27 \)

12. 5 : 2 = 30 : l
- Set up the proportion:
\[
\frac{5}{2} = \frac{30}{l}
\]
- Cross-multiply:
\[
5 \cdot l = 2 \cdot 30 \implies 5l = 60
\]
- Solve for \( l \):
\[
l = \frac{60}{5} = 12
\]
- Answer: \( l = 12 \)

---

#### Part 3: Write any two equivalent ratios of the given ratios

13. 8 : 7
- Multiply both terms by 2:
\[
8 \cdot 2 : 7 \cdot 2 = 16 : 14
\]
- Multiply both terms by 3:
\[
8 \cdot 3 : 7 \cdot 3 = 24 : 21
\]
- Answers: \( 16 : 14 \), \( 24 : 21 \)

14. 4 : 7
- Multiply both terms by 2:
\[
4 \cdot 2 : 7 \cdot 2 = 8 : 14
\]
- Multiply both terms by 3:
\[
4 \cdot 3 : 7 \cdot 3 = 12 : 21
\]
- Answers: \( 8 : 14 \), \( 12 : 21 \)

15. 11 : 12
- Multiply both terms by 2:
\[
11 \cdot 2 : 12 \cdot 2 = 22 : 24
\]
- Multiply both terms by 3:
\[
11 \cdot 3 : 12 \cdot 3 = 33 : 36
\]
- Answers: \( 22 : 24 \), \( 33 : 36 \)

16. 4 : 13
- Multiply both terms by 2:
\[
4 \cdot 2 : 13 \cdot 2 = 8 : 26
\]
- Multiply both terms by 3:
\[
4 \cdot 3 : 13 \cdot 3 = 12 : 39
\]
- Answers: \( 8 : 26 \), \( 12 : 39 \)

---

Final Answers:

\[
\boxed{
\begin{array}{ll}
1. & 32 \\
2. & 35 \\
3. & 49 \\
4. & 36 \\
5. & 56 \\
6. & 40 \\
7. & 42 \\
8. & 11 \\
9. & 25 \\
10. & 9 \\
11. & 27 \\
12. & 12 \\
13. & 16 : 14, 24 : 21 \\
14. & 8 : 14, 12 : 21 \\
15. & 22 : 24, 33 : 36 \\
16. & 8 : 26, 12 : 39 \\
\end{array}
}
\]