Math worksheet for solving ratios and proportions, featuring ten algebraic equations to find the value of x.

Let’s solve each problem one by one. We’ll use cross-multiplication to solve for x in each proportion.

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Problem 1:
\[\frac{4}{x - 8} = \frac{8}{2}\]

First, simplify the right side:
\(\frac{8}{2} = 4\)

So now:
\(\frac{4}{x - 8} = 4\)

Multiply both sides by \(x - 8\):
\(4 = 4(x - 8)\)

Divide both sides by 4:
\(1 = x - 8\)

Add 8 to both sides:
\(x = 9\)

✔ Check: Plug back in → \(\frac{4}{9-8} = \frac{4}{1} = 4\), and \(\frac{8}{2} = 4\) → Correct!

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Problem 2:
\[\frac{x - 5}{x + 8} = \frac{2}{7}\]

Cross-multiply:
\(7(x - 5) = 2(x + 8)\)

Expand both sides:
\(7x - 35 = 2x + 16\)

Subtract 2x from both sides:
\(5x - 35 = 16\)

Add 35 to both sides:
\(5x = 51\)

Divide by 5:
\(x = \frac{51}{5} = 10.2\)

✔ Check: Left side: \(\frac{10.2 - 5}{10.2 + 8} = \frac{5.2}{18.2} ≈ 0.2857\); Right side: \(\frac{2}{7} ≈ 0.2857\) → Correct!

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Problem 3:
\[\frac{x + 10}{x - 7} = \frac{8}{9}\]

Cross-multiply:
\(9(x + 10) = 8(x - 7)\)

Expand:
\(9x + 90 = 8x - 56\)

Subtract 8x:
\(x + 90 = -56\)

Subtract 90:
\(x = -146\)

✔ Check: Left: \(\frac{-146 + 10}{-146 - 7} = \frac{-136}{-153} = \frac{136}{153}\). Simplify: divide numerator and denominator by 17 → \(\frac{8}{9}\) → Correct!

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Problem 4:
\[\frac{6}{x - 1} = \frac{9}{7}\]

Cross-multiply:
\(6 * 7 = 9 * (x - 1)\)

→ \(42 = 9x - 9\)

Add 9:
\(51 = 9x\)

Divide by 9:
\(x = \frac{51}{9} = \frac{17}{3} ≈ 5.666...\)

✔ Check: Left: \(\frac{6}{\frac{17}{3} - 1} = \frac{6}{\frac{14}{3}} = 6 * \frac{3}{14} = \frac{18}{14} = \frac{9}{7}\) → Correct!

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Problem 5:
\[\frac{x}{12} = \frac{10}{2}\]

Simplify right side: \(\frac{10}{2} = 5\)

So: \(\frac{x}{12} = 5\)

Multiply both sides by 12:
\(x = 60\)

✔ Check: \(\frac{60}{12} = 5\), \(\frac{10}{2} = 5\) → Correct!

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Problem 6:
\[\frac{40}{25} = \frac{x}{20}\]

Simplify left side: \(\frac{40}{25} = \frac{8}{5}\)

So: \(\frac{8}{5} = \frac{x}{20}\)

Cross-multiply:
\(8 * 20 = 5 * x\)

→ \(160 = 5x\)

Divide by 5:
\(x = 32\)

✔ Check: \(\frac{40}{25} = 1.6\), \(\frac{32}{20} = 1.6\) → Correct!

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Problem 7:
\[\frac{40}{24} = \frac{20}{x}\]

Simplify left side: \(\frac{40}{24} = \frac{5}{3}\)

So: \(\frac{5}{3} = \frac{20}{x}\)

Cross-multiply:
\(5x = 60\)

Divide by 5:
\(x = 12\)

✔ Check: \(\frac{40}{24} = \frac{5}{3}\), \(\frac{20}{12} = \frac{5}{3}\) → Correct!

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Problem 8:
\[\frac{11}{10} = \frac{x}{11}\]

Cross-multiply:
\(11 * 11 = 10 * x\)

→ \(121 = 10x\)

Divide by 10:
\(x = 12.1\)

✔ Check: Left: \(\frac{11}{10} = 1.1\), Right: \(\frac{12.1}{11} = 1.1\) → Correct!

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Problem 9:
\[\frac{x - 3}{x} = \frac{9}{10}\]

Cross-multiply:
\(10(x - 3) = 9x\)

Expand:
\(10x - 30 = 9x\)

Subtract 9x:
\(x - 30 = 0\)

Add 30:
\(x = 30\)

✔ Check: Left: \(\frac{30 - 3}{30} = \frac{27}{30} = \frac{9}{10}\) → Correct!

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Problem 10:
\[\frac{x}{x - 3} = \frac{2}{3}\]

Cross-multiply:
\(3x = 2(x - 3)\)

Expand:
\(3x = 2x - 6\)

Subtract 2x:
\(x = -6\)

✔ Check: Left: \(\frac{-6}{-6 - 3} = \frac{-6}{-9} = \frac{2}{3}\) → Correct!

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Final Answer:
1. 9
2. 10.2
3. -146
4. 17/3
5. 60
6. 32
7. 12
8. 12.1
9. 30
10. -6