Let's solve each proportion using
cross multiplication. Cross multiplication is a method used to solve proportions (equations that state two ratios are equal). For a proportion:
$$
\frac{a}{b} = \frac{c}{d}
$$
We cross multiply:
$$
a \cdot d = b \cdot c
$$
Then solve for the unknown variable.
---
Problem 1:
$$
\frac{x}{6} = \frac{15}{18}
$$
Cross multiply:
$$
x \cdot 18 = 6 \cdot 15 \\
18x = 90
$$
Divide both sides by 18:
$$
x = \frac{90}{18} = 5
$$
✔ Answer: 5
---
Problem 2:
$$
\frac{2}{3} = \frac{16}{x}
$$
Cross multiply:
$$
2 \cdot x = 3 \cdot 16 \\
2x = 48
$$
Divide by 2:
$$
x = 24
$$
✔ Answer: 24
---
Problem 3:
$$
\frac{3}{5} = \frac{x}{15}
$$
Cross multiply:
$$
3 \cdot 15 = 5 \cdot x \\
45 = 5x
$$
Divide by 5:
$$
x = 9
$$
✔ Answer: 9
---
Problem 4:
$$
\frac{x}{6} = \frac{2}{12}
$$
Cross multiply:
$$
x \cdot 12 = 6 \cdot 2 \\
12x = 12
$$
Divide by 12:
$$
x = 1
$$
✔ Answer: 1
---
Problem 5:
$$
\frac{x}{5} = \frac{18}{20}
$$
Cross multiply:
$$
x \cdot 20 = 5 \cdot 18 \\
20x = 90
$$
Divide by 20:
$$
x = \frac{90}{20} = 4.5
$$
✔ Answer: 4.5 or $ \frac{9}{2} $
---
Problem 6:
$$
\frac{1}{2} = \frac{x}{8}
$$
Cross multiply:
$$
1 \cdot 8 = 2 \cdot x \\
8 = 2x
$$
Divide by 2:
$$
x = 4
$$
✔ Answer: 4
---
Problem 7:
$$
\frac{7}{8} = \frac{10}{x}
$$
Cross multiply:
$$
7 \cdot x = 8 \cdot 10 \\
7x = 80
$$
Divide by 7:
$$
x = \frac{80}{7} \approx 11.43
$$
✔ Answer: $ \frac{80}{7} $ or approximately 11.43
---
Problem 8:
$$
\frac{x}{2} = \frac{4}{8}
$$
Cross multiply:
$$
x \cdot 8 = 2 \cdot 4 \\
8x = 8
$$
Divide by 8:
$$
x = 1
$$
✔ Answer: 1
---
Problem 9:
$$
\frac{9}{4} = \frac{x}{12}
$$
Cross multiply:
$$
9 \cdot 12 = 4 \cdot x \\
108 = 4x
$$
Divide by 4:
$$
x = 27
$$
✔ Answer: 27
---
Problem 10:
$$
\frac{1}{3} = \frac{x}{3}
$$
Cross multiply:
$$
1 \cdot 3 = 3 \cdot x \\
3 = 3x
$$
Divide by 3:
$$
x = 1
$$
✔ Answer: 1
---
Problem 11:
$$
\frac{x}{2} = \frac{4}{8}
$$
Cross multiply:
$$
x \cdot 8 = 2 \cdot 4 \\
8x = 8
$$
Divide by 8:
$$
x = 1
$$
✔ Answer: 1
---
Problem 12:
$$
\frac{x}{2} = \frac{10}{20}
$$
Cross multiply:
$$
x \cdot 20 = 2 \cdot 10 \\
20x = 20
$$
Divide by 20:
$$
x = 1
$$
✔ Answer: 1
---
✔ Final Answers:
| Problem | Answer |
|--------|--------|
| 1 | 5 |
| 2 | 24 |
| 3 | 9 |
| 4 | 1 |
| 5 | 4.5 or $ \frac{9}{2} $ |
| 6 | 4 |
| 7 | $ \frac{80}{7} $ or ~11.43 |
| 8 | 1 |
| 9 | 27 |
| 10 | 1 |
| 11 | 1 |
| 12 | 1 |
---
🔍 Summary:
- Cross multiplication is a reliable method for solving proportions.
- Always check that the equation is set up correctly: $\frac{a}{b} = \frac{c}{d}$, then $a \cdot d = b \cdot c$.
- Simplify fractions when possible before or after solving.
Let me know if you'd like these answers in a printable format or explained with diagrams!
Parent Tip: Review the logic above to help your child master the concept of writing and solving proportions worksheet.
