What Is a Proportion?
A ratio compares two quantities, for example 3 cups of flour to 2 cups of sugar. A proportion is the statement that two ratios are equal: 3/2 = 6/4. Proportions are the tool you reach for whenever you need to scale one relationship to match another — doubling a recipe, reading a map, or converting currencies.
The four numbers in a proportion (A, B, C, D) are called the terms. A and D are the extremes; B and C are the means. The fundamental property is that the cross products are equal: A × D = B × C.
The Proportion Formula and Cross-Multiplication
Given A / B = C / D, multiply both sides by B and by D:
A × D = B × C
To find any one unknown, isolate it:
A = (B × C) / D | B = (A × D) / C | C = (A × D) / B | D = (B × C) / A
Example: 7 / 12 = x / 96
Cross-multiply: 7 × 96 = 12 × x
672 = 12x
x = 672 / 12 = 56
How to Use This Calculator
1
Choose a mode — Direct for A/B = C/D, Inverse for A×B = C×D, or Scale ratio to expand or shrink a ratio to a target value.
2
Enter any three of the four values and leave the unknown blank. The calculator detects which field is empty and solves for it automatically.
3
If you fill all four fields, the calculator checks whether the proportion is true or false and shows the cross products.
4
Read the step-by-step working, the decimal and fraction forms of the answer, and the constant of proportionality. Use Share to copy a link that pre-fills all values.
Direct vs Inverse Proportion
Direct proportion
Two quantities are in direct proportion when they increase or decrease together at the same rate. Doubling one doubles the other. The ratio between them stays constant: k = y / x. Example: if a car travels 60 miles in 1 hour it travels 120 miles in 2 hours — distance and time are directly proportional (k = 60 mph).
Inverse proportion
Two quantities are in inverse proportion when one increases as the other decreases, so their product is constant: k = x × y. Example: 4 workers take 6 days to finish a job; 8 workers (double) take only 3 days (half). The constant is 4 × 6 = 8 × 3 = 24 worker-days.
Inverse example: 4 × 6 = x × 3 → x = 24 / 3 = 8
Real-Life Uses of Proportions
- Recipes and cooking: A recipe serves 4 people using 300 g of pasta. How much pasta for 10 people? 300/4 = x/10 → x = 750 g.
- Map scales: A map has a scale of 1:50 000. A distance of 3 cm on the map represents 3 × 50 000 = 150 000 cm = 1.5 km in reality.
- Currency conversion: If $1 = £0.79, how many pounds is $250? 1/0.79 = 250/x → x = £197.50.
- Speed, distance, time: A car travels 240 km in 3 hours. How long for 400 km at the same speed? 240/3 = 400/x → x = 5 hours.
- Discounts and percentages: An item costs $45 after a 25% discount. What was the original price? 75/100 = 45/x → x = $60.
- Similar triangles: Two triangles are similar with corresponding sides in ratio 2:3. If the short side of the larger triangle is 9 cm, the short side of the smaller is 6 cm.
Common Mistakes to Avoid
- Swapping numerator and denominator: 3/4 = x/20 gives x = 15, but 4/3 = x/20 gives x ≈ 26.67. Always make sure corresponding quantities are in the same position on both sides.
- Unit mismatch: If one ratio uses metres and the other uses centimetres, convert both to the same unit before writing the proportion.
- Forgetting to simplify: An answer of 24/16 should be reduced to 3/2 before reporting it as a ratio.
- Confusing direct and inverse: More workers finishing faster is inverse proportion, not direct. Always ask whether quantities move in the same direction or opposite directions.