How to Solve a Proportion: Direct and Inverse

Formula

• Direct proportion: a / b = c / x, so x = b × c / a
• Inverse proportion: a × b = c × x, so x = a × b / c

What the letters mean:

• a: a known value of the first quantity, for example a time
• b: the value of the second quantity that goes with a, for example a distance
• c: the new value of the first quantity
• x: the unknown value of the second quantity that goes with c
• k: the constant of proportionality; k = b / a for a direct proportion, k = a × b for an inverse one

In a direct proportion, cross multiplication gives the same result: if a / b = c / x, then a × x = b × c.

Step-by-step example

A car covers 300 km in 4 hours. How far does it go in 7 hours at the same speed?

1. Separate the quantities: time is the first quantity (a = 4, c = 7) and distance is the second (b = 300).
2. Choose the type: more time means more distance, so this is a direct proportion.
3. Find the constant: k = 300 / 4 = 75. The car covers 75 km per hour.
4. Calculate x: x = 300 × 7 / 4 = 525. The car covers 525 km in 7 hours.

An inverse proportion example

A car at 60 km/h finishes a trip in 5 hours. How long does the same trip take at 75 km/h?

1. A higher speed means a shorter time, so this is an inverse proportion. a = 60, b = 5, c = 75.
2. Find the constant: k = 60 × 5 = 300. This product is the length of the trip: 300 km.
3. Calculate x: x = 60 × 5 / 75 = 4. The trip takes 4 hours.

Direct or inverse?

A common mistake

Choosing the wrong type changes the answer completely. If 6 workers finish a job in 12 days, 8 workers finish it in 9 days as an inverse proportion. Solve the same question as a direct proportion and you get 16 days, yet more workers never make a job take longer. Before you solve, ask: when one grows, does the other grow or shrink?

The proportion calculator solves a simple proportion of two quantities. A compound proportion, where three or more quantities change, can be solved one pair of quantities at a time.