What is the inverse rule of three?
The inverse rule of three finds a missing value when two quantities change in opposite directions and follow an inverse proportion. For example, if several people complete exactly the same task at the same pace, a larger crew may need fewer hours. The rule organizes the known values and calculates the one that is missing.
It is called simple because it relates two quantities: workers and time, speed and duration for a fixed distance, or flow rate and the time needed to fill a fixed volume. If three or more quantities change together, this rule alone is not enough; each relationship needs to be examined separately.
Inverse does not simply mean that one value rose while another fell in a single observation. The relationship must hold under the conditions of the problem. If the task changes, the crew works at a different pace, or a pause is introduced, the proportion may no longer describe the situation accurately.
The mathematical test: a constant product
In an inverse proportion, the product of corresponding values stays the same. If A workers take B hours and C workers take X hours to finish the same task, then A × B = C × X. Solving for X gives X = (A × B) ÷ C.
Check this condition before calculating. If 4 people take 18 hours, their initial product is 72 person-hours. Six people taking 12 hours also gives 72. This check explains why time falls as the crew grows, provided the task is unchanged and each person's productivity is comparable.
How to tell whether a problem is inverse
Read the question and name both quantities clearly. Ask what each number represents and what must stay fixed: the same route, the same amount of work, or the same volume of liquid. Without that condition, a table showing opposite movement can be misleading.
Next, imagine increasing only one quantity while everything else stays fixed. If the other must decrease so that the same total amount is preserved, and the product remains constant, an inverse model may fit. If you cannot explain what stays constant, do not pick a formula from intuition; review the model or look for more information.
Fix what does not change. Write down the task, distance, or volume that is the same in both situations.
Change one quantity at a time. Ask what happens to the unknown when only the known quantity increases.
Check the product. Verify that each corresponding pair keeps the same product.
The inverse rule of three formula
Arrange the values by situation, not by the order in which they appear in the question. The first situation contains A and B; in the new situation, C is known and X is the unknown. The equality A × B = C × X expresses the constant product. To isolate X, divide both sides by C.
This gives X = (A × B) ÷ C. Multiply the two values from the initial situation, then divide by the new value of the quantity that changes in the opposite direction. Keep the units visible: if A counts people and B is measured in hours, the product is person-hours; dividing by people leaves X in hours.
A × B = C × X → X = (A × B) ÷ C
How to solve an inverse rule of three in five steps
A clear setup prevents the most common mistake: putting numbers into a formula before deciding what each one represents. Write the units beside the values and check whether the expected result makes sense before rounding.
1. Define the unknown
Identify the missing value and its unit. If you are looking for time, note whether the question uses hours, minutes, or days.
2. Arrange the two situations
Write the two values from the initial situation together, then place the new value beside X in a second column. Do not switch units halfway through.
3. Justify the inverse relationship
Explain what stays the same and why increasing one quantity should reduce the other. If you cannot justify this, the relationship may not be inverse.
4. Substitute into the formula
Multiply the initial pair A × B and divide by C. Keep the parentheses so it is clear which values belong to each situation.
5. Check units and direction
Multiply the final pair to check the product. Also confirm that more workers did not mistakenly produce more time for the same task.
Three inverse rule of three examples
Each example states what remains fixed. That sentence is not a minor detail: it is the reason the inverse model is appropriate.
The examples use simple quantities so you can follow the substitution. If the result contains decimals, keep precision during the calculation and round only at the end, according to the unit and context.
Workers and task duration
Four people complete a task in 18 hours. How long would six people take at the same pace if the work and shift are unchanged?
The total amount of work stays the same. More people sharing that task means fewer hours per person, so the product people × hours is constant.
4 × 18 = 6 × X → X = (4 × 18) ÷ 6 = 12 hResult: six people would take 12 hours under these conditions.
Check the product: 4 × 18 = 72 and 6 × 12 = 72 person-hours. If the added workers cannot work in parallel, the model needs adjustment.
Speed and time for a fixed distance
A trip over a fixed distance takes 4 hours at 60 km/h. How long would it take at 80 km/h with no stops and a steady speed?
The distance is the same and equals speed × time. As speed increases, time decreases, so the product of those two values stays constant.
60 × 4 = 80 × X → X = 240 ÷ 80 = 3 hResult: the trip would take 3 hours in the ideal model.
Both situations represent 240 km. On a real route, traffic, speed limits, breaks, and changing speeds mean that time cannot be predicted by this proportion alone.
Flow rate and time to fill a tank
A flow rate of 8 litres per minute fills a 240-litre tank in 30 minutes. How long would a steady rate of 10 litres per minute take?
The volume to be delivered does not change. A larger flow supplies the same amount in less time, so flow rate × minutes remains the total volume.
8 × 30 = 10 × X → X = 240 ÷ 10 = 24 minResult: the higher flow rate would take 24 minutes.
Both calculations give 240 litres. This answer assumes a steady flow with no losses or pressure changes.
Direct versus inverse rule of three
In a direct relationship, increasing one quantity also increases the other in the same proportion; the ratio between corresponding values stays constant. In an inverse relationship, one rises as the other falls, and the product stays constant. Direction can suggest a model, but the conditions of the question confirm it.
| Situation | Direct | Inverse |
|---|---|---|
| Price and quantity bought | More units at the same unit price cost more. | It is not inverse if only the quantity bought changes. |
| Workers and hours for the same job | This does not describe the usual fixed-work case. | More workers may reduce the hours if their pace stays comparable. |
| Speed and time for the same distance | It is not direct when distance and route are fixed. | A higher steady speed means less ideal travel time. |
What to check before trusting the result
The rule solves a proportional model; it does not prove that the model captures every detail of a real situation. Treat the conditions in the question as part of the calculation, not as fine print.
- Do not call a relationship inverse just because one value rose and another fell once. Identify what stays fixed and check the product.
- Do not mix minutes with hours or metres with kilometres. Convert first so that the compared values use compatible units.
- Do not divide by zero: the formula is undefined when C is zero. A negative answer also rarely makes sense for people, time, or volume.
- If workers, hours per shift, and days all change, examine each quantity separately. The problem may require a compound rule of three.
- In real work, adding people can create waiting or interference. Treat the answer as a mathematical estimate when productivity may vary.
Questions about the inverse rule of three
What is the inverse rule of three formula?
If A and B are the initial values and C is the new value of the first quantity, use X = (A × B) ÷ C. It follows from A × B = C × X, which states that the product remains constant.
How can I tell whether a rule of three is inverse?
Check whether one quantity increases while the other decreases to preserve the same task, distance, or amount. Then confirm that the products of the corresponding pairs match.
Can the inverse rule of three calculate speed and time?
Yes, when the distance is fixed and speed stays constant. A higher speed corresponds to less time in the model; stops or traffic break that simplification.
What does a constant product mean?
It means multiplying each related pair gives the same amount. For example, 4 people × 18 hours and 6 people × 12 hours both equal 72 person-hours.
What if more than two quantities change?
First identify how each quantity relates to the unknown while holding the others fixed. If several values change, consult the compound rule of three guide and justify each ratio separately.