How this proportion calculator works
Pick a mode from the row of tabs and the answer updates as you type. Solve for x finds the missing term of a/b = c/d. Is it proportional? tests two ratios, or a table of x and y values, and reports the constant k. Direct & inverse covers y ∝ x, x², x³ and √x and their inverses. Scale & resize handles scale factors, recipes, map scales and aspect ratios. Split in a ratio shares out a total, and Part of a whole turns part ÷ whole into a fraction, decimal and percentage.
Boxes accept decimals, fractions and mixed numbers, the working is shown step by step, and answers stay exact, so 3/4 = x/15 gives 45/4. Choose the decimal places shown, or round further with the rounding calculator.
The proportion formulas
Example: for 5/8 = 15/d, d = (8 × 15) ÷ 5 = 24. The cross products are then 5 × 24 = 120 and 8 × 15 = 120, which confirms the answer.
Worked examples
Example 1 — an answer that is not a whole number
- Proportion:
- 3/4 = x/15
- Unknown:
- C (third term)
- Cross-multiply: 3 × 15 = 4 × x
- Multiply: 45 = 4x
- Divide by 4: x = 45/4
x: 45/4 = 11 1/4 = 11.25
Check: 3 × 15 = 45 and 4 × 11.25 = 45, so the cross products match.
Example 2 — inverse proportion (workers and days)
- Known pair:
- 6 workers finish in 10 days
- Question:
- How long for 4 workers?
- Inverse, so y = k ÷ x: k = 6 × 10 = 60 worker-days
- Substitute x = 4: y = 60 ÷ 4
Time: 15 days
Fewer workers means more days, so this is inverse. Cross-multiplying 6/10 = 4/y here would wrongly give 6.67 days.
Direct vs inverse proportion: what happens when x doubles
| Relationship | Equation | If x doubles, y is… |
|---|---|---|
| y ∝ x | y = kx | × 2 |
| y ∝ x² | y = kx² | × 4 |
| y ∝ x³ | y = kx³ | × 8 |
| y ∝ √x | y = k√x | × 1.414 (√2) |
| y ∝ 1/x | y = k / x | ÷ 2 |
| y ∝ 1/x² | y = k / x² | ÷ 4 |
| y ∝ 1/√x | y = k / √x | ÷ 1.414 (√2) |
Where proportion is taught, and which mode covers it
| Standard | What students do | Calculator mode |
|---|---|---|
| 7.RP.A.2a | Decide whether two quantities are proportional, e.g. by testing equivalent ratios in a table | Is it proportional? |
| 7.RP.A.2b | Identify the constant of proportionality (unit rate) | Is it proportional? (table) |
| 7.RP.A.2c | Represent proportional relationships by equations such as t = pn | Direct & inverse (y ∝ x) |
| 7.RP.A.2d | Explain what a point (x, y) on the graph means | Table graph |
| 7.RP.A.3 | Use proportional relationships to solve multistep ratio and percent problems | Solve for x, Part of a whole |
Proportion in the UK GCSE
England's GCSE content treats proportion as an equality of ratios and covers direct and inverse proportion with equations and graphs, scale factors and maps, and dividing a quantity in a ratio, each matched by a mode above (source: DfE GCSE mathematics subject content).
Key facts
- In any true proportion a/b = c/d, the cross products are equal: a × d = b × c, so 3/4 = 9/12 because 3 × 12 = 36 and 4 × 9 = 36.
- A table shows a proportional relationship only if y ÷ x is the same for every row; that shared value is the constant of proportionality k, and the graph is a straight line through the origin.
- In inverse proportion the product x × y stays constant, so 6 workers taking 10 days means 4 workers take 60 ÷ 4 = 15 days.
- To split an amount in a ratio, divide it by the sum of the parts: 1,000 in the ratio 3:2 is 5 parts of 200, giving shares of 600 and 400.
Everyday uses of proportion
Scaling recipes
Every amount in a pasted ingredient list is multiplied by new servings ÷ original servings and kept as a kitchen fraction such as 2 1/4 tsp.
Maps and models
At 1:25,000, 4 cm on the map is 100,000 cm, or 1 km, converted with the exact factors in NIST SP 811.
Screens and images
Resizing a 1920 × 1080 image to 1280 wide keeps its 16:9 shape at 720 high: new height = 1280 × 1080 ÷ 1920.
Sharing costs
Ratio shares are rounded to the cent and still add up to the total. For a restaurant bill, use the bill split calculator.
Common proportion mistakes
- Mixing units. Both ratios must use the same units in the same positions: convert minutes to hours, or centimetres to metres, before you solve.
- Swapping the order. If the first ratio is price over quantity, the second must be price over quantity too.
- Cross-multiplying an inverse problem. When one quantity goes down as the other goes up, multiply the known pair to find k instead.
- Rounding too early. Keep the exact fraction until the last step; rounding k first can shift the final answer.
- A zero denominator. b and d can never be 0, so a ratio such as 5/0 has no proportion.
Frequently Asked Questions
How do you solve a proportion?
Cross-multiply, then divide. In a/b = c/d the cross products a × d and b × c are always equal, so write that equation and divide both sides by the number multiplying the unknown. For 3/4 = 9/x, 3x = 36, so x = 12. Check by confirming both cross products match.
What is the difference between a ratio and a proportion?
A ratio compares two quantities, such as 3:4 or 3/4. A proportion is an equation stating that two ratios are equal, such as 3/4 = 9/12. Every proportion contains two ratios, but a single ratio on its own is not a proportion.
How do you know if two ratios are proportional?
Two ratios a/b and c/d are proportional when their cross products are equal, a × d = b × c. For 5/8 and 15/24, 5 × 24 = 120 and 8 × 15 = 120, so they are proportional. For a table of values, every y ÷ x must give the same number.
What is the constant of proportionality?
The constant of proportionality, k, is the fixed ratio y ÷ x in a proportional relationship, so y = kx. If 4 notebooks cost $12, k = 12 ÷ 4 = 3 dollars per notebook. It is the unit rate, and on a graph it is the slope of the straight line through the origin.
What is the difference between direct and inverse proportion?
In direct proportion both quantities change in the same direction at a fixed ratio: y = kx, so doubling x doubles y. In inverse proportion one rises as the other falls and their product stays fixed: y = k/x, so doubling x halves y. Workers against time to finish a job is the classic inverse case.
How do you scale a recipe using proportions?
Divide the servings you want by the servings the recipe makes to get a scale factor, then multiply every ingredient by it. Going from 4 to 6 servings is a factor of 6 ÷ 4 = 1.5, so 2 cups of flour becomes 3 cups. Baking times and pan sizes do not always scale the same way.
How do you divide an amount in a given ratio?
Add the parts of the ratio, divide the total by that sum to get the value of one part, then multiply each part by it. To split 1,000 in the ratio 3:2 there are 5 parts, one part is 200, and the shares are 600 and 400.
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Notes & Assumptions
- Solve, check, scaling, split and part-of-whole results are worked in exact fractions; decimals are rounded half away from zero only for display.
- Direct and inverse proportion with powers and roots is worked in floating point and rounded to the places you choose; for y ∝ x², the negative root is noted alongside the positive one.
- Map-scale conversions use exact factors: 1 in = 0.0254 m, 1 ft = 0.3048 m, 1 yd = 0.9144 m and 1 mi = 1,609.344 m (NIST SP 811).
This tool is for education and general calculation. Check that a real situation really is proportional before relying on the answer.