Solving Proportions worksheet with ten algebraic equations to solve.

Let’s solve each proportion step by step. A proportion means two fractions are equal. To solve, we cross-multiply: multiply the top of one fraction by the bottom of the other, and set them equal.

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1)
$\frac{10}{8} = \frac{n}{10}$
Cross-multiply: $10 \cdot 10 = 8 \cdot n$ → $100 = 8n$
Divide both sides by 8: $n = \frac{100}{8} = 12.5$

✔ Check: $\frac{10}{8} = 1.25$, $\frac{12.5}{10} = 1.25$ → Correct.

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2)
$\frac{7}{5} = \frac{x}{3}$
Cross-multiply: $7 \cdot 3 = 5 \cdot x$ → $21 = 5x$
Divide by 5: $x = \frac{21}{5} = 4.2$

✔ Check: $\frac{7}{5} = 1.4$, $\frac{4.2}{3} = 1.4$ → Correct.

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3)
$\frac{9}{6} = \frac{x}{10}$
Simplify left side first: $\frac{9}{6} = \frac{3}{2}$
So: $\frac{3}{2} = \frac{x}{10}$
Cross-multiply: $3 \cdot 10 = 2 \cdot x$ → $30 = 2x$
$x = 15$

✔ Check: $\frac{9}{6} = 1.5$, $\frac{15}{10} = 1.5$ → Correct.

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4)
$\frac{7}{n} = \frac{8}{7}$
Cross-multiply: $7 \cdot 7 = 8 \cdot n$ → $49 = 8n$
$n = \frac{49}{8} = 6.125$

✔ Check: $\frac{7}{6.125} ≈ 1.142857$, $\frac{8}{7} ≈ 1.142857$ → Correct.

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5)
$\frac{4}{3} = \frac{8}{x}$
Cross-multiply: $4 \cdot x = 3 \cdot 8$ → $4x = 24$
$x = 6$

✔ Check: $\frac{4}{3} ≈ 1.333$, $\frac{8}{6} ≈ 1.333$ → Correct.

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6)
$\frac{7}{b+5} = \frac{10}{5}$
Simplify right side: $\frac{10}{5} = 2$
So: $\frac{7}{b+5} = 2$
Multiply both sides by $(b+5)$: $7 = 2(b + 5)$
Distribute: $7 = 2b + 10$
Subtract 10: $-3 = 2b$
$b = -\frac{3}{2} = -1.5$

✔ Check: Left: $\frac{7}{-1.5 + 5} = \frac{7}{3.5} = 2$, Right: $\frac{10}{5} = 2$ → Correct.

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7)
$\frac{6}{b-1} = \frac{9}{7}$
Cross-multiply: $6 \cdot 7 = 9 \cdot (b - 1)$ → $42 = 9b - 9$
Add 9: $51 = 9b$
$b = \frac{51}{9} = \frac{17}{3} ≈ 5.666...$

✔ Check: Left: $\frac{6}{\frac{17}{3} - 1} = \frac{6}{\frac{14}{3}} = 6 \cdot \frac{3}{14} = \frac{18}{14} = \frac{9}{7}$ → Matches right side. Correct.

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8)
$\frac{4}{m-8} = \frac{8}{2}$
Simplify right: $\frac{8}{2} = 4$
So: $\frac{4}{m-8} = 4$
Multiply both sides by $(m-8)$: $4 = 4(m - 8)$
Divide both sides by 4: $1 = m - 8$
$m = 9$

✔ Check: Left: $\frac{4}{9-8} = \frac{4}{1} = 4$, Right: $\frac{8}{2} = 4$ → Correct.

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9)
$\frac{5}{6} = \frac{7n + 9}{9}$
Cross-multiply: $5 \cdot 9 = 6 \cdot (7n + 9)$ → $45 = 42n + 54$
Subtract 54: $-9 = 42n$
$n = -\frac{9}{42} = -\frac{3}{14}$

✔ Check: Right side: $\frac{7(-\frac{3}{14}) + 9}{9} = \frac{-\frac{21}{14} + 9}{9} = \frac{-1.5 + 9}{9} = \frac{7.5}{9} = \frac{5}{6}$ → Matches left. Correct.

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10)
$\frac{4}{9} = \frac{r - 3}{6}$
Cross-multiply: $4 \cdot 6 = 9 \cdot (r - 3)$ → $24 = 9r - 27$
Add 27: $51 = 9r$
$r = \frac{51}{9} = \frac{17}{3} ≈ 5.666...$

✔ Check: Right: $\frac{\frac{17}{3} - 3}{6} = \frac{\frac{17}{3} - \frac{9}{3}}{6} = \frac{\frac{8}{3}}{6} = \frac{8}{18} = \frac{4}{9}$ → Matches left. Correct.

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Final Answer:
1) 12.5
2) 4.2
3) 15
4) 6.125
5) 6
6) -1.5
7) 17/3 or 5.666...
8) 9
9) -3/14
10) 17/3 or 5.666...