Find the missing value in an inverse proportion.
The product of the two quantities stays constant — as one increases, the other decreases proportionally.
Solve: A₁ × B₁ = A₂ × B₂
An inverse proportion calculator finds the missing value when one quantity goes down exactly as the other goes up, so that their product stays the same. If 12 workers take 15 days to build a wall, it tells you how many days 18 workers would take, assuming each works at the same pace.
Inverse proportion is a staple of Class 8 NCERT and of time-and-work and time-speed-distance questions in bank, SSC and railway exams. In daily life it describes speed and travel time over a fixed route, the number of people sharing a fixed amount of food and how long it lasts, and the number of taps filling a tank. The calculator answers: if one side changes to this, what must the other become to keep the product fixed?
1. Check the relationship is inverse: doubling one quantity should halve the other.
2. Write the known pair as A₁ and B₁, and the new value of the first quantity as A₂.
3. Use the constant product rule: A₁ × B₁ = A₂ × B₂.
4. Solve B₂ = (A₁ × B₁) ÷ A₂. A₂ must not be zero.
5. Check the direction: if A₂ is larger than A₁, B₂ must be smaller than B₁.
6. Round sensibly for the context, for example up to whole days or whole workers.
Saying y is inversely proportional to x means y = k ÷ x, or x × y = k. The constant k usually has a physical meaning. For workers and days it is the total work in worker-days: 12 workers × 15 days = 180 worker-days. For speed and time it is the fixed distance. Once k is known, any new x gives y = k ÷ x. With 18 workers, 180 ÷ 18 = 10 days. The formula B₂ = A₁B₁ ÷ A₂ is this idea in one line.
The graph of y = k ÷ x is a curve called a rectangular hyperbola, not a straight line. As x grows, y shrinks towards zero but never reaches it, and as x shrinks towards zero, y grows without limit. That is why A₂ = 0 has no answer: zero workers can never finish the job. In practice there are also limits the maths ignores. A thousand workers cannot build a small wall in a few minutes, because they would get in each other's way.
Inverse proportion assumes every unit contributes equally and independently. All workers work at the same rate, every tap has the same flow, and the total work or distance is fixed. If new workers are less skilled, or if the job has a fixed setup time, the simple product rule overstates the savings. When one quantity depends on two others, such as workers, hours per day and days, combine the rule: workers × hours × days stays constant for a fixed job.
A contractor in Nagpur knows that 12 masons take 15 days to finish a boundary wall and wants to know how long 18 masons working at the same pace would take.
Answer: B₂ (where A₁ × B₁ = A₂ × B₂) 10
Using direct proportion and getting a longer time for more workers.
Forgetting that the total work, distance or quantity must stay fixed.
Ignoring a third factor such as hours per day when it also changes.
Rounding days down when the job cannot be completed in a fractional day.
Time-and-work questions in competitive exams.
Estimating how long food or supplies will last for a different number of people.
Finding travel time at a different speed over the same distance.
Planning how many machines or taps are needed to finish within a deadline.
Gear problems where the number of teeth and rotation speed are inversely related.
What's an example of an inverse proportion?
Speed and time, for a fixed distance. If you travel twice as fast, the journey takes half the time. Multiplying them together (which gives distance) always stays the same.