Inverse Proportion Calculator

For quantities that move in opposite directions — more workers, fewer days. Enter one known pair to find the constant product, then solve for whichever value you are missing. The graph shows the hyperbola and the table shows the product staying fixed.

Formula x × y = k

Use this when one quantity going up means the other goes down — more workers means fewer days, faster speed means less time. If both move the same way, you want the direct proportion calculator instead.

The known pair

The new value

Try an example
Method

Multiply to find k, divide to use it

The mirror image of direct proportion: where that one divides to find the constant, this one multiplies.

Check the relationship really is inverse

Ask what happens if you double one quantity. If the other halves, it is inverse. If it also doubles, it is direct. If it changes by some other amount, it may be neither — a job that gets less efficient with more people on it is not a clean inverse proportion.

Multiply the known pair to get k

k = x₁ × y₁. Give the result its units and it usually names something real: workers × days is person-days of work, speed × time is distance, pressure × volume is proportional to the energy in the gas.

Write x × y = k

With k = 48 the equation is xy = 48, or y = 48/x. Any pair of numbers multiplying to 48 is a valid state of this system.

Divide k by the value you know

Unlike direct proportion, the operation is the same in both directions: y = k/x and x = k/y. Whichever value you have, divide k by it. This symmetry is a useful check that you have the right kind of relationship.

Confirm the direction of the answer

If you increased x, the answer for y must be smaller than y₁. If it came out larger, you multiplied where you should have divided.

x × y = k    y = k / x

Inverse proportion. The product is fixed. Because the equation is symmetric in x and y, finding either one from the other is the same division.

x₁ · y₁ = x₂ · y₂

Pair form. Both pairs give the same product, so you can equate them directly and skip calculating k — handy for a one-off answer.

The assumption hiding in the maths

"6 workers take 8 days, so 48 workers take 1 day" is arithmetically right and practically wrong. Inverse proportion assumes the work divides perfectly and everyone works at the same rate. Real jobs have coordination costs and tasks that cannot be parallelised, so the model breaks down at the extremes.

Worked examples

Inverse proportion in practice

Workers and time

6 workers finish a job in 8 days. How long for 12 workers?

Known pair
x₁ = 6, y₁ = 8
Find k
k = 6 × 8 = 48 person-days
Equation
xy = 48
Divide by 12
y = 48 ÷ 12
Check
12 × 4 = 48 ✓

4 days — twice the workers, half the time

Speed and journey time

A trip takes 4 hours at 60 mph. How long at 80 mph?

Known pair
x₁ = 60, y₁ = 4
Find k
k = 60 × 4 = 240 miles
Equation
speed × time = 240
Divide by 80
y = 240 ÷ 80
Check
80 × 3 = 240 ✓

3 hours — and k is the distance

Boyle's law

A gas at 300 kPa occupies 2 litres. What pressure at 5 litres, same temperature?

Known pair
V₁ = 2, P₁ = 300
Find k
k = 2 × 300 = 600
Equation
PV = 600
Divide by 5
P = 600 ÷ 5
Check
5 × 120 = 600 ✓

120 kPa

Working backwards

How many workers are needed to finish that same 48 person-day job in 3 days?

Known k
k = 48 person-days
Equation
x × 3 = 48
Divide by 3
x = 48 ÷ 3
Check
16 × 3 = 48 ✓

16 workers

Comparison

Direct against inverse, side by side

Choosing the wrong one is the single biggest source of error in proportion problems.

Direct proportion

y = kx · the ratio y/x is fixed

  • Both quantities rise and fall together
  • Constant found by dividing: k = y/x
  • Double x and y doubles
  • Graph: straight line through the origin
  • Example: 4 kg costs $6, so 8 kg costs $12

Inverse proportion

xy = k · the product xy is fixed

  • One rises as the other falls
  • Constant found by multiplying: k = xy
  • Double x and y halves
  • Graph: hyperbola with both axes as asymptotes
  • Example: 6 workers take 8 days, so 12 take 4
Situation Which type Why
Litres of petrol and total cost Direct More litres means more money — the price per litre is the constant.
Speed and time for a fixed distance Inverse Going faster cuts the time — speed × time is the fixed distance.
Number of servings and flour needed Direct Twice the servings needs twice the flour.
People sharing a fixed bill Inverse More people means a smaller share each — the total is the constant.
Gear teeth and rotation speed Inverse A bigger gear turns more slowly; teeth × rpm is fixed along the chain.
Hours worked and pay earned Direct The hourly rate is the constant of proportionality.
Taps filling a tank and time taken Inverse More taps means less time; tank volume is the constant.
Distance from a lamp and brightness Inverse square Brightness falls as 1/d², not 1/d — twice as far is a quarter as bright.
Background

The constant is usually the thing you care about

In inverse proportion the constant k tends to be more meaningful than in the direct case, because it names a fixed total that is being divided up differently. Six workers over eight days is 48 person-days of labour; that 48 is the size of the job, and it does not change however you staff it. Sixty miles an hour for four hours is 240 miles; that 240 is the length of the route. Recognising what k is often answers the question faster than the algebra does.

This also gives you a quick way to check whether a problem is genuinely inverse. Multiply the two quantities and ask whether the product is a thing. Workers × days = work: yes. Kilograms × price = nothing in particular: so cost and weight are not inversely proportional, they are directly proportional. The test takes a few seconds and catches the most common misclassification.

Why the curve bends the way it does

On a hyperbola the same absolute change in x has wildly different effects depending on where you are. Going from 1 worker to 2 halves the time — an enormous saving. Going from 20 workers to 21 barely moves it. That flattening is why real projects hit a point where adding people stops helping, and it is visible in the shape of the graph long before you reach the practical limits.

Combined proportion

Many real relationships are direct in one variable and inverse in another at the same time. The time to finish a job is inversely proportional to the number of workers but directly proportional to the size of the job: t = kS/w. These are handled by finding the single constant that satisfies a fully specified case, then substituting the new values for every variable at once. The principle is the same; there are simply more quantities to keep track of.

Where to go next

For relationships where both quantities move together, use the direct proportion calculator. If you have two complete pairs and only want to know whether they are consistent, the cross multiplication calculator tests it in one step — remembering that for an inverse relationship you compare x₁y₁ with x₂y₂ rather than cross-multiplying. And to turn a constant into a per-unit figure, the unit ratio calculator does that conversion.

Questions

Inverse proportion — questions

Starting with the distinction from direct proportion, which is where most errors originate.

What does inversely proportional mean?
Two quantities are inversely proportional when their product stays constant rather than their ratio. Double one and the other halves; triple one and the other becomes a third. The equation is x × y = k, or equivalently y = k / x. The classic case is workers and time: if it takes 6 people 8 days, then 12 people take 4 days, because 6 × 8 = 12 × 4 = 48 person-days of work.
How do you find the constant of an inverse proportion?
Multiply a known pair together: k = x₁ × y₁. For 6 workers taking 8 days, k = 48. That constant usually has a concrete meaning — here it is the total amount of work measured in person-days, which is why it does not change when you redistribute the same job among more or fewer people.
What is the difference between direct and inverse proportion?
In direct proportion the ratio y/x is fixed, so both quantities rise and fall together and the graph is a straight line through the origin. In inverse proportion the product xy is fixed, so one rises as the other falls and the graph is a hyperbola that approaches but never touches either axis. Compare with the direct proportion calculator.
What are real examples of inverse proportion?
Speed and journey time for a fixed distance; the number of workers and the time to finish a fixed job; pressure and volume of a gas at constant temperature (Boyle's law); the number of people sharing a fixed cost and each share; gear teeth and rotation speed; and light intensity against distance squared, which is an inverse-square rather than a simple inverse relationship.
Can x or y be zero in an inverse proportion?
No. If either were zero the product would be zero, so k would be zero, and then every value of the other variable would satisfy the equation — the relationship would carry no information. Practically it also makes sense: zero workers never finish the job, and at zero speed the journey takes forever. This is what the asymptotes on the graph represent.
How do I tell whether a table is inversely proportional?
Multiply x by y on every row. If every product is the same, the table is inversely proportional and that product is k. If the products drift, it is not — and if instead the quotients y ÷ x are constant, you have a direct proportion.
What is inverse square proportion?
A related but steeper relationship where y = k / x², so doubling x divides y by four rather than two. Gravity, sound intensity and light brightness all fall off this way with distance. This calculator handles the simple inverse case y = k / x; for the squared version you would find k as y × x² instead.
Why does the graph never touch the axes?
Because y = k / x has no value at x = 0 — division by zero is undefined — and y can never actually reach 0 either, since k / x only shrinks towards zero as x grows without ever arriving. The two axes are asymptotes: lines the curve approaches indefinitely closely but never meets.