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Ratio & Proportion

Direct & Inverse Proportion Calculator

Solve proportion problems step by step. Give two known values and a third — the calculator finds the missing value, identifies the constant of proportionality, and explains the working clearly.

Direct & Inverse Proportion

If y ∝ x (y = kx), enter three values — find the fourth.

x₁ (known x)
y₁ (known y)
x₂ (new x)
y₂ (calculated)

Step-by-step working

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Enter values to solve proportion
Enter three known values on the left to calculate the constant of proportionality (k), find the missing value, and view detailed step-by-step working.

Direct & Inverse Proportion Calculator Explained

Core concepts, classroom applications, and plain-English explanations for all audiences.

When would you actually use this?

Every time you scale a recipe, you use direct proportion. If a cake serves 8 and uses 240 g of flour, how much flour for 12 people? Because servings and flour are directly proportional (double the servings = double the flour), you find k = 240 ÷ 8 = 30 g per person, then multiply: 30 × 12 = 360 g.

Worked example

"£1 = $1.27. How many dollars do you get for £45?"

Step 1: k = $1.27 ÷ £1 = 1.27

Step 2: y = k × x = 1.27 × 45

Result: $57.15

More pounds always means more dollars — classic direct proportion. The rate k stays constant.

Unit pricing in supermarkets

Which is better value — 400 g of pasta for £1.20 or 900 g for £2.52? Find the cost per 100 g for each: £1.20 ÷ 4 = 30p vs £2.52 ÷ 9 = 28p. The 900 g bag wins. Supermarkets display this as "price per 100 g" — it's direct proportion applied to every shelf in every aisle.

Direct vs inverse — the key test

Ask: "If x gets bigger, does y get bigger too?" If yes → direct (y = kx). If y gets smaller → inverse (xy = k). More fuel = more distance travelled is direct. More workers = less time to finish is inverse.

Good to know

Direct vs inverse proportion

y=kx
Direct proportion
When y is directly proportional to x (written y ∝ x), doubling x doubles y. The relationship is y = kx, where k is the constant of proportionality. The graph is a straight line through the origin. Examples: distance = speed × time, cost of items, exchange rates.
y ∝ x  →  y = kx
Find k: k = y ÷ x = 12 ÷ 3 = 4
So y = 4x

When x = 7: y = 4 × 7 = 28
When x = 10: y = 4 × 10 = 40
xy=k
Inverse proportion
When y is inversely proportional to x (written y ∝ 1/x), doubling x halves y. The relationship is xy = k, or y = k/x. The graph is a hyperbola — it never touches the axes. Examples: speed and time for a fixed distance, pressure and volume of a gas (Boyle's Law), workers and time to complete a job.
y ∝ 1/x  →  xy = k
Find k: k = x × y = 4 × 6 = 24
So y = 24/x

When x = 3: y = 24 ÷ 3 = 8
When x = 8: y = 24 ÷ 8 = 3
k
The constant of proportionality
The constant k links x and y. For direct proportion, k = y/x (always the same for every pair of values). For inverse proportion, k = xy (the product is always the same). Once you have k, you can find any value. The first step in every proportion problem is to find k using a known pair of values.
Direct: k = y/x
Check: 12/3=4, 28/7=4, 40/10=4 ✓

Inverse: k = x×y
Check: 4×6=24, 3×8=24, 8×3=24 ✓
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Common mistakes
The biggest mistake is using direct proportion when the problem is actually inverse (or vice versa). Ask yourself: "If x gets bigger, does y get bigger too?" If yes → direct. "If x gets bigger, does y get smaller?" If yes → inverse. Always find k first — never cross-multiply without understanding the relationship type.
Direct: "More workers, more walls built" ✓
Inverse: "More workers, less time to finish" ✓

If 6 workers take 4 days:
k = 6×4 = 24 worker-days
12 workers take: 24÷12 = 2 days