Cross Multiplication

Solve for $x$: $\frac{5}{8} = \frac{x}{24}$

Set up cross multiplication: $5 \times 24 = 8 \times x$ = Cross products are equal

Multiply the known values: $120 = 8x$ = $120 = 8x$

Divide both sides by 8: $\frac{120}{8} = \frac{8x}{8}$ = $15 = x$

Verify by substituting back: $\frac{5}{8} = \frac{15}{24} = \frac{5}{8}$ ✓ = Both ratios equal $\frac{5}{8}$

Answer: $x = 15$

Finding a Scale Factor

If 3 pencils cost 75 cents, how much do 7 pencils cost?

Write as a proportion: $\frac{3 \text{ pencils}}{75 \text{ cents}} = \frac{7 \text{ pencils}}{x \text{ cents}}$ = Set up equal ratios

Cross multiply: $3 \times x = 75 \times 7$ = $3x = 525$

Solve for x: $x = \frac{525}{3} = 175$ = $x = 175$ cents

Convert to dollars: $175 \div 100 = 1.75$ = 1 dollar 75 cents

Answer: 7 pencils cost 1 dollar 75 cents

Using the Scale Factor Method

Solve: $\frac{12}{15} = \frac{20}{x}$

Find the scale factor from 12 to 20: $20 \div 12 = \frac{20}{12} = \frac{5}{3}$ = Scale factor is $\frac{5}{3}$

Apply the same scale factor to 15: $15 \times \frac{5}{3} = \frac{75}{3} = 25$ = $x = 25$

Verify with cross multiplication: $12 \times 25 = 300$ and $15 \times 20 = 300$ ✓ = Cross products are equal

Answer: $x = 25$

Real-World Application

On a map, 2 cm represents 50 km. If two cities are 7.5 cm apart on the map, what is the actual distance?

Set up the proportion: $\frac{2 \text{ cm}}{50 \text{ km}} = \frac{7.5 \text{ cm}}{x \text{ km}}$ = Map distance to real distance

Cross multiply: $2 \times x = 50 \times 7.5$ = $2x = 375$

Solve for x: $x = \frac{375}{2} = 187.5$ = $x = 187.5$ km

State the answer with units: The cities are 187.5 km apart = Real distance: 187.5 km

Answer: The actual distance between the cities is 187.5 km

Mistake: Cross multiplying incorrectly: $\frac{3}{4} = \frac{x}{12}$ becoming $3 \times 4 = x \times 12$

Why: Cross multiplication means diagonal multiplication: numerator of one fraction times denominator of the other.

Correct: Correct: $3 \times 12 = 4 \times x$, so $36 = 4x$, giving $x = 9$

Mistake: Setting up the proportion with mismatched units

Why: Both ratios must compare the same quantities in the same order (e.g., pencils to cost, not cost to pencils).

Correct: Keep consistent: $\frac{\text{pencils}}{\text{cost}} = \frac{\text{pencils}}{\text{cost}}$ or $\frac{\text{cost}}{\text{pencils}} = \frac{\text{cost}}{\text{pencils}}$

Mistake: Forgetting to verify the answer

Why: Without checking, you might not catch arithmetic errors.

Correct: Always substitute your answer back into the original proportion and verify both ratios are equal.

Recipe Scaling

Chefs and home cooks use proportions to adjust recipe quantities for different serving sizes.

A cookie recipe makes 24 cookies using 3 eggs. To make 40 cookies: $\frac{3}{24} = \frac{x}{40}$, so $x = 5$ eggs.

Map Reading

Cartographers use scale ratios to represent real distances on maps.

If a map scale is 1:100,000, then 1 cm on the map equals 1 km in reality. A 5.5 cm road on the map is actually 5.5 km long.

Medicine Dosage

Pharmacists calculate proper medication doses based on patient weight using proportions.

A medication is prescribed at 5 mg per kg of body weight. For a 60 kg patient: $\frac{5}{1} = \frac{x}{60}$, so $x = 300$ mg.

Q: Why does cross multiplication work?

A: Cross multiplication is a shortcut for multiplying both sides of the equation by the product of the denominators ($bd$). This clears the fractions and gives us $ad = bc$.

Q: Can I solve proportions without cross multiplication?

A: Yes! You can use the scale factor method (find what multiplier transforms one ratio into the other) or multiply both sides by one denominator at a time.

Q: What if the unknown is in the denominator?

A: Cross multiplication still works! For $\frac{6}{x} = \frac{3}{5}$, cross multiply: $6 \times 5 = x \times 3$, so $30 = 3x$, giving $x = 10$.