# Cross Multiplication Word Problems: 12 Real-World Examples with Step-by-Step Solutions

The hard part of a proportion problem is rarely the cross multiplication itself — it is translating the words into a fraction = fraction setup. 12 fully worked examples across recipes, maps, dosages, chemistry, sports, and more.

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- **Canonical URL:** https://dothecalculation.com/blog/math/cross-multiplication-word-problems
- **Category:** Math
- **Author:** Do The Calculation Team
- **Published:** 2026-08-03
- **Last updated:** 2026-08-03
- **Reading time:** 14 min read
- **Publisher:** Do The Calculation (https://dothecalculation.com)
- **Methodology:** https://dothecalculation.com/methodology

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## Cross Multiplication Word Problems: 12 Real-World Examples

Once you know the cross multiplication rule — if a/b = c/d, then a × d = b × c — the algebra itself is fast. The part that actually trips people up is everything before that: turning a paragraph about recipes, maps, or medication doses into a clean fraction = fraction setup. This guide skips the theory (see our full [Cross Multiplication Guide](/blog/math/how-to-cross-multiply-step-by-step-guide) for that) and goes straight to 12 fully worked real-world problems, one from each of 12 different domains.

- A 60-second recap of the cross multiplication rule
- Cross multiplication vs the unit rate method — solving the same problem two ways
- How to translate a word problem into a proportion, step by step
- 12 fully worked examples: recipes, maps, discounts, speed, currency, dosage charts, similar triangles, chemistry, fuel efficiency, sports stats, image scaling, and unit pricing
- 5 practice problems with verified answers

Tool: [Try the Cross Multiplication Calculator](https://dothecalculation.com/calculators/cross-multiplication-calculator) — Enter any proportion and get the solved variable with full step-by-step cross multiplication work.

## Quick Recap: The Cross Multiplication Rule

**Cross Multiplication**

```
If a/b = c/d, then a × d = b × c
```
- Only valid when the equation is already in the form fraction = fraction — see the full guide for why, and what to do when it isn't.

## Cross Multiplication vs the Unit Rate Method

Cross multiplication is not the only way to solve a proportion. The unit rate method finds "how much per one unit" first, then scales that up. Both are mathematically identical and always agree — the difference is workflow.

**Same Problem, Two Methods: A Recipe for 3 People Uses 1.5 Cups of Flour. How Many Cups for 7 People?**
| Step | Cross Multiplication | Unit Rate Method |
| --- | --- | --- |
| 1 | Set up the proportion: 3/1.5 = 7/x | Find the rate per person: 1.5 ÷ 3 = 0.5 cups/person |
| 2 | Cross multiply: 3 × x = 1.5 × 7 | Scale up: 0.5 × 7 people |
| 3 | 3x = 10.5 | = 3.5 cups |
| Answer | x = 3.5 cups | 3.5 cups |

_[Figure: When to Reach for Each Method — Both always give the same answer — pick based on what the problem hands you.]_

## How to Translate Any Word Problem Into a Proportion

_[Figure: From Word Problem to Answer — The same four steps work regardless of the domain — recipes, maps, chemistry, or sports.]_

## 12 Fully Worked Word Problems

### 1. Recipes & Cooking

**A recipe for 3 people uses 1.5 cups of flour. How much flour for 7 people?**
| Step | Work | Result |
| --- | --- | --- |
| 1 | Set up: 3/1.5 = 7/x | — |
| 2 | Cross multiply: 3 × x = 1.5 × 7 | 3x = 10.5 |
| Answer | x = 10.5 ÷ 3 | 3.5 cups |

### 2. Maps & Scale Models

**A map has a scale of 1:24,000. Two towns are 5.5 cm apart on the map. What is the real distance?**
| Step | Work | Result |
| --- | --- | --- |
| 1 | Set up: 1/24,000 = 5.5/x | — |
| 2 | Cross multiply: 1 × x = 24,000 × 5.5 | x = 132,000 cm |
| Answer | Convert: 132,000 cm = 1,320 m | 1.32 km |

### 3. Discounts & Pricing

**A jacket is on sale for 30% off. The sale price is $63. What was the original price?**
| Step | Work | Result |
| --- | --- | --- |
| 1 | Sale price = 70% of original: 63/x = 70/100 | — |
| 2 | Cross multiply: 63 × 100 = 70 × x | 6,300 = 70x |
| Answer | x = 6,300 ÷ 70 | $90 |

### 4. Speed, Distance & Time

**A car travels 180 miles in 3 hours. At the same rate, how long to travel 300 miles?**
| Step | Work | Result |
| --- | --- | --- |
| 1 | Set up: 180/3 = 300/x | — |
| 2 | Cross multiply: 180 × x = 3 × 300 | 180x = 900 |
| Answer | x = 900 ÷ 180 | 5 hours |

### 5. Currency Exchange

**The exchange rate is 1 USD = 0.92 EUR. How many euros is $250?**
| Step | Work | Result |
| --- | --- | --- |
| 1 | Set up: 1/0.92 = 250/x | — |
| 2 | Cross multiply: 1 × x = 0.92 × 250 | x = 230 |
| Answer | — | €230 |

### 6. Dosage Charts

> **Illustrative Math Only** — This is a textbook-style proportion example, not medical guidance. Real dosing always follows a licensed provider's or pharmacist's instructions, not a general ratio.

**A reference chart lists 15 mg per 10 kg of body weight. What does the chart show for 42 kg?**
| Step | Work | Result |
| --- | --- | --- |
| 1 | Set up: 15/10 = x/42 | — |
| 2 | Cross multiply: 15 × 42 = 10 × x | 630 = 10x |
| Answer | x = 630 ÷ 10 | 63 mg |

_[Figure: How the Reference Chart Scales With Body Weight — Same 15 mg per 10 kg rate, plotted across four example weights — direct proportionality is a straight line.]_

### 7. Geometry & Similar Triangles

**Triangle A has sides of 6 and 8. Similar Triangle B's corresponding first side is 15. What is B's second side?**
| Step | Work | Result |
| --- | --- | --- |
| 1 | Similar triangles keep proportional sides: 6/8 = 15/x | — |
| 2 | Cross multiply: 6 × x = 8 × 15 | 6x = 120 |
| Answer | x = 120 ÷ 6 | 20 |

For the full geometry background behind this, see our [Triangle Geometry Formulas Guide](/blog/math/triangle-geometry-formulas).

### 8. Chemistry & Mixing Ratios

**A cleaning solution is mixed 2 parts concentrate to 5 parts water. For 9 parts concentrate, how much water?**
| Step | Work | Result |
| --- | --- | --- |
| 1 | Set up: 2/5 = 9/x | — |
| 2 | Cross multiply: 2 × x = 5 × 9 | 2x = 45 |
| Answer | x = 45 ÷ 2 | 22.5 parts water |

### 9. Fuel Efficiency

**A car uses 8 liters of fuel per 100 km. How many liters for a 350 km trip?**
| Step | Work | Result |
| --- | --- | --- |
| 1 | Set up: 8/100 = x/350 | — |
| 2 | Cross multiply: 8 × 350 = 100 × x | 2,800 = 100x |
| Answer | x = 2,800 ÷ 100 | 28 liters |

For trip-cost budgeting beyond just liters needed, see our [Fuel Cost Calculator](/calculators/fuel-cost-calculator).

### 10. Sports Statistics

**A player made 18 of 25 free throws. At the same rate, how many would they make out of 40 attempts?**
| Step | Work | Result |
| --- | --- | --- |
| 1 | Set up: 18/25 = x/40 | — |
| 2 | Cross multiply: 18 × 40 = 25 × x | 720 = 25x |
| Answer | x = 720 ÷ 25 | 28.8 → about 29 makes |

### 11. Photo & Image Scaling

**An image is 1,600 × 900 px. Resized proportionally to 960 px wide, what is the new height?**
| Step | Work | Result |
| --- | --- | --- |
| 1 | Set up: 1,600/900 = 960/x | — |
| 2 | Cross multiply: 1,600 × x = 900 × 960 | 1,600x = 864,000 |
| Answer | x = 864,000 ÷ 1,600 | 540 px |

### 12. Unit Price Comparison

Brand A: 12 oz for $3.60. Brand B: 18 oz for $5.22. Which is the better deal? Cross multiply the two per-ounce fractions directly, without dividing first: compare 3.60/12 vs 5.22/18 by checking whether 3.60 × 18 is greater or less than 5.22 × 12.

**Comparing 3.60/12 vs 5.22/18 by Cross Multiplication**
| Cross Product | Value |
| --- | --- |
| 3.60 × 18 | $64.80 |
| 5.22 × 12 | $62.64 |
| Result | 62.64 < 64.80, so Brand B is cheaper per ounce |

_[Figure: Price Per Ounce: Brand A vs Brand B — The cross-multiplication comparison above confirms it without ever dividing a single number.]_

## Practice Problems (With Verified Answers)

**More Word Problems to Try**
| # | Problem | Answer |
| --- | --- | --- |
| 1 | A recipe for 5 people uses 2 cups of rice. How much for 8 people? | 2/5 = x/8 → x = 3.2 cups |
| 2 | A model car is built at 1:18 scale. If the model is 25 cm long, how long is the real car in meters? | 1/18 = 25/x → x = 450 cm = 4.5 m |
| 3 | A shirt's sale price is $34 after a 15% discount. What was the original price? | 34/x = 85/100 → x = $40 |
| 4 | A train covers 240 km in 2 hours. How far does it travel in 5 hours at the same speed? | 240/2 = x/5 → x = 600 km |
| 5 | A paint mix uses 3 parts blue to 7 parts white. For 21 parts white, how many parts blue? | 3/7 = x/21 → x = 9 parts blue |

**How do I know which method to use for a word problem — cross multiplication or unit rate?**

Both always give the same answer. Use cross multiplication when the problem is naturally two fractions or ratios; use the unit rate method when you want a reusable "per one unit" number for repeated scaling.

**What is the biggest mistake in translating a word problem into a proportion?**

Mismatching units across the two sides — for example writing people/cups on one side and cups/people on the other. Keep the same unit in the same position (both numerators, both denominators) on every side.

**Can cross multiplication be used to compare two rates directly, like unit prices?**

Yes — to compare a/b vs c/d without dividing either one, cross multiply and compare a×d against b×c. The larger cross product corresponds to the larger original fraction.

**Does cross multiplication work for similar triangles and scale models?**

Yes. Similar triangles and scaled models both preserve ratios between corresponding measurements, so any two corresponding pairs form a valid proportion you can cross multiply.

**How do I check if my word-problem answer makes sense?**

Check the direction, not just the arithmetic — if you scaled a quantity up (more people, more distance, more parts), your answer should be bigger than the original matching value, and vice versa.

**Is a proportion word problem always solvable with cross multiplication?**

Only if it reduces to a single fraction = fraction equation. If the problem involves addition or subtraction of separate fractions, rearrange first — see our full Cross Multiplication Guide for that case.

**What is a real-life example of cross multiplication outside of school?**

Comparing unit prices while grocery shopping, converting a map distance to real distance, and scaling a recipe or a photo are all everyday cross-multiplication problems, as shown in the worked examples above.

**Why do some proportion answers come out as decimals or fractions instead of whole numbers?**

Real-world ratios rarely divide evenly. A fractional or decimal answer (like 3.5 cups or 28.8 free throws) is normal — round sensibly for the context, such as rounding a free-throw count down to a whole make.

## Related Math Calculators & Guides

- [Cross Multiplication Calculator](/calculators/cross-multiplication-calculator) — solve any proportion instantly with full step-by-step work.
- [Cross Multiplication: A Beginner's Guide for Students & Teachers](/blog/math/how-to-cross-multiply-step-by-step-guide) — the full rule, the butterfly method, and why cross multiplication works.
- [Ratio and Proportion Calculator](/calculators/ratio-proportion-calculator) — simplify ratios and solve proportions directly.
- [Scale Converter Calculator](/calculators/scale-converter) — convert map, model, and drawing scales.
- [Discount Calculator](/calculators/discount-calculator) — find original and sale prices using the same proportion logic.
- [Fraction Calculator](/calculators/fraction-calculator) — add, subtract, multiply, divide, and simplify fractions.
- [Triangle Geometry Formulas Guide](/blog/math/triangle-geometry-formulas) — the full background behind similar-triangle proportions.

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_Source: [Do The Calculation](https://dothecalculation.com/blog/math/cross-multiplication-word-problems). Quote freely with attribution and a link to this page._