Inverse Variation

Drive miles at mph and the trip takes hours. Speed up to mph and it takes . The faster you go, the less time it takes, and the speed times the time always comes back to the same . That’s inverse variation: as one quantity grows, the other shrinks in exact proportion. It’s the mirror image of direct variation, and it turns up everywhere from physics to splitting a bill.

What Is Inverse Variation?

varies inversely with when

The number is the constant of variation. You’ll also see ” is inversely proportional to ”. The key fact is that the product never changes:

A nice way to picture it: every pair is the width and height of a rectangle with the same area.

The Graph of Inverse Variation

Plot those pairs and you get a curve that drops steeply, then flattens out, getting closer to the -axis without ever reaching it. This curve is a hyperbola, one of the rational functions.

Two features to remember:

• It never passes through the origin. would mean dividing by zero.
• For positive the curve lies in quadrants I and III; for negative , in II and IV. In real-world problems you usually only see the first-quadrant branch.

How to Solve Inverse Variation Problems

1. Find by multiplying a known pair: .
2. Write the equation .
3. Substitute the new value and solve.

Example. varies inversely with , and when . Find when .

Step 1 — find :

Step 2 — write the equation and use it:

You can also use directly: , so .

Direct vs Inverse Variation

To decide which one a table shows, test both: if is constant it’s direct; if is constant it’s inverse.

Inverse Variation Word Problems

Speed and Time

A trip takes hours at mph. How long does it take at mph?

Step 1 — the constant is the distance:

Step 2 — the new time:

Workers and Days

Six painters can paint a building in days. Working at the same rate, how long would painters take?

Pressure and Volume (Boyle’s Law)

At a constant temperature, the pressure of a gas varies inversely with its volume. A gas at atmospheres fills liters. What’s the pressure if it’s squeezed into liters?

Inverse Square Variation

” varies inversely with the square of ” means . Light, sound and gravity all weaken this way with distance. If you move twice as far from a light, it looks a quarter as bright.

Example. The brightness of a lamp is units at meters. How bright is it at meters?

Joint and Combined Variation

Joint variation means one quantity varies directly with the product of two others: . The volume of a cylinder varies jointly with and , and .

Combined variation mixes direct and inverse: . For example, the time to finish a job varies directly with its size and inversely with the number of workers.

Common Mistakes to Avoid

• Dividing to find . For inverse variation, . Dividing gives the direct-variation constant.
• Expecting a straight line. The graph of is a curve.
• Plugging in . It’s never allowed; the equation is undefined there.
• Forgetting the square in inverse square problems: .
• Calling any decreasing relationship inverse variation. decreases, but isn’t constant, so it isn’t inverse variation.

Practice Problems

Work each one before opening the answer.

Problem 1. varies inversely with , and when . Write the equation.

Problem 2. Using Problem 1’s equation, find when .

Problem 3. Does this table show direct variation, inverse variation, or neither? : 1, 2, 5; : 20, 10, 4.

Problem 4. A trip takes hours at mph. How long does it take at mph?

Problem 5. Eight workers can build a fence in days. How many days would workers take?

Problem 6. varies inversely with the square of , and when . Find when .

Inverse variation equations are rational equations in disguise. Solving for is the same skill as the ones in the rational equations lesson.