Direct & Inverse Variation: When One Quantity Drives Another

Some quantities move together: drive twice as long at a fixed speed and you cover twice the distance. Others trade off: at a fixed distance, double your speed and you halve your time. The first is direct variation, the second inverse. Both are governed by a single number — the constant of variation — and the entire skill is finding that constant and using it.

This guide builds both relationships from a clear graph, shows how to find the constant, and finishes with worked and practice problems matched to real test difficulty at Northside Tutoring.

Why Variation Matters

Variation is proportional reasoning in algebraic clothing, and it appears on the SAT and ACT in science and real-world contexts: speed and time, pressure and volume, intensity and distance. It's not tested on the SSAT, but it's central to Algebra II and to interpreting formulas, so it pays off well beyond a single exam.

Direct Variation: y = kx

In direct variation, y is a constant multiple of x: y = kx. Their ratio y/x always equals the constant k. The graph is a straight line through the origin with slope k.

Inverse Variation: y = k/x

In inverse variation, y is the constant divided by x: y = k/x. Now the product xy stays constant. As x increases, y decreases. The graph is a curved hyperbola, never touching the axes.

Seeing the Difference

Finding the Constant

You only need one matching pair of values to find k. For direct variation, divide: k = y/x. For inverse, multiply: k = xy. Once you have k, you have the complete equation.

Solving for a New Value

With k in hand, plug in the new x (or y) and solve. The cleanest method for these problems: set up the constant from the first pair, then apply it to the second — for direct, y₁/x₁ = y₂/x₂; for inverse, x₁y₁ = x₂y₂.

Where You'll See This — Test by Test

Variation needs no reference sheet — just the two equations. It's a recurring SAT and ACT topic in real-world and science contexts. It is not on the SSAT, which stops before this Algebra II material.

Watch the Lesson

Sometimes a diagram needs a voice. In the short video below, one of our Northside tutors walks through the core idea and works through test-style problems in real time.

Worked Example Problems

These problems are calibrated to the difficulty you'll actually see on test day. Try each one before opening the solution.

Common Mistakes to Avoid

Practice Problems — You Try

Three problems below. Work each before checking the solution.

The Northside Method — How We Teach This 1-on-1

Reading a blog is a great starting point. But there's a meaningful gap between understanding a concept and reflexively applying it under timed conditions. That gap is exactly what our tutors close.

Every Northside student works through a four-step framework:

1. Assessment. We diagnose which specific skills are slowing your student down — not just whether they "get it" in the abstract.
2. Perfect-match coach. We pair them with an elite tutor (we accept only the top 1% of applicants) whose teaching style fits how your student actually learns.
3. Bespoke plan. A roadmap built around your student's target score, target timeline, and current pacing data.
4. Data-driven adjustment. Every session ends with a check on whether the student's accuracy and speed are moving in the right direction.

And if a student meets all eligibility requirements but doesn't hit the defined score improvement? We provide 5 additional hours of cohort learning at no cost. That's the Northside guarantee — built on 25 years of measured outcomes.