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.
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 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
Quick Recap: The Cross Multiplication Rule
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.
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| 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 |
When to Reach for Each Method
Both always give the same answer — pick based on what the problem hands you.
Cross Multiplication
When the problem is already stated as two fractions or ratios
- Works directly on algebraic fractions, e.g. (2x-1)/3 = (x+2)/4
- No intermediate rounding — exact fractions throughout
- The standard method for solving proportion equations
Unit Rate Method
When you want an intuitive "per one" number to reason with
- Gives a reusable rate (cups per person, dollars per ounce) for repeated scaling
- Often faster mentally for simple whole-number ratios
- Can introduce rounding error with repeating decimals
Many people mix both intuitively: eyeball a unit rate to sanity-check a cross-multiplication answer, or vice versa.
How to Translate Any Word Problem Into a Proportion
From Word Problem to Answer
The same four steps work regardless of the domain — recipes, maps, chemistry, or sports.
1. Identify the two known quantities
Find the pair of numbers that are already related — e.g. "3 people, 1.5 cups."
2. Set up matching units on each side
People/cups = People/cups, not People/cups = cups/people.
3. Cross multiply and solve
a × d = b × c, then isolate the unknown.
4. Check the answer makes sense
A bigger group should need more, not less — sanity-check the direction, not just the arithmetic.
Step 2 is where most mistakes happen — mismatched units on the same side of the proportion will give a wrong (but confident-looking) answer.
12 Fully Worked Word Problems
1. Recipes & Cooking
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| 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
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| 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
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| 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
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| 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
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| 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
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| 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 |
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.
This is what "direct proportion" looks like visually: double the input, double the output.
7. Geometry & Similar Triangles
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| 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.
8. Chemistry & Mixing Ratios
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| 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
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| 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.
10. Sports Statistics
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| 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
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| 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.
Swipe sideways to compare columns.
| 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 |
Price Per Ounce: Brand A vs Brand B
The cross-multiplication comparison above confirms it without ever dividing a single number.
Brand A ($3.60 / 12 oz)
Brand B ($5.22 / 18 oz)
Brand B costs about 1 cent less per ounce — small per unit, but it adds up on a full grocery run.
Practice Problems (With Verified Answers)
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| # | 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 — solve any proportion instantly with full step-by-step work.
- Cross Multiplication: A Beginner's Guide for Students & Teachers — the full rule, the butterfly method, and why cross multiplication works.
- Ratio and Proportion Calculator — simplify ratios and solve proportions directly.
- Scale Converter Calculator — convert map, model, and drawing scales.
- Discount Calculator — find original and sale prices using the same proportion logic.
- Fraction Calculator — add, subtract, multiply, divide, and simplify fractions.
- Triangle Geometry Formulas Guide — the full background behind similar-triangle proportions.
Written by
Do The Calculation Team
Do The Calculation Editorial Board
The Do The Calculation Editorial Board is comprised of software engineers, finance analysts, and technical contributors focused on building clean, accurate, and easy-to-use calculator tools.
Reviewed & Verified By
Dr. Elena Rostova, PhD
Mathematics Advisor
Professor of mathematics with 20+ years of teaching experience. Dr. Rostova oversees the formulas, proofs, and algorithms behind our math division tools, fractions, logarithms, and scientific equations.