Inverse Variation- Khan Academy Math Guide with Examples

What Is Inverse Variation?

Inverse variation describes a relationship where one variable increases while the other decreases. The product of the two variables stays constant. That's the core idea — nothing more complicated than that.

Mathematically, if x and y vary inversely, then:

xy = k or equivalently y = k/x

The constant k is called the constant of variation. Find it once, and you can solve for any missing value.

Real-World Examples

Inverse variation shows up more than you think:

• Speed and time: Driving the same distance, going faster means less time. Distance = speed × time, so speed and time are inversely related when distance is fixed.
• Workers and job completion time: More workers means the job gets done faster. The number of workers and time to finish vary inversely (assuming equal productivity).
• Pressure and volume: Boyle's Law from physics — compress a gas and the pressure increases. Volume and pressure have an inverse relationship.

How to Solve Inverse Variation Problems

Here's the straightforward process:

1. Identify the two variables that vary inversely
2. Find the constant k by multiplying the given values together
3. Set up the equation xy = k or y = k/x
4. Solve for the missing variable

Example Problem

If y varies inversely with x, and y = 12 when x = 4, find y when x = 6.

Step 1: Find k

12 × 4 = 48. So k = 48.

Step 2: Set up the equation

48 = y × 6

Step 3: Solve

y = 48/6 = 8

When x increases from 4 to 6, y decreases from 12 to 8. That makes sense for inverse variation.

Inverse vs. Direct Variation

Don't confuse these two. Direct variation is simpler — as one variable goes up, the other goes up too. The ratio stays constant.

Khan Academy Resources for Inverse Variation

Khan Academy covers inverse variation in their Algebra 1 and Algebra 2 courses. You'll find it under direct and inverse variation units.

The platform walks you through:

• Recognizing inverse variation from word problems
• Writing inverse variation equations from data
• Graphing inverse variation functions
• Practice problems with immediate feedback

The videos break down each step. Watch one, then try the practice problems. That's the fastest way to get this down.

Common Mistakes to Avoid

• Mixing up k with the variables. k is the constant product, not one of the variable values.
• Forgetting that k must be positive for most real-world scenarios, but it can be negative in math problems.
• Confusing "inversely proportional" with "indirectly proportional." They mean the same thing — use whichever sounds natural.

Practice Problem Set

1. If y varies inversely with x, and y = 15 when x = 3, what is k?

Answer: k = 45

2. Using the same relationship, find y when x = 9.

Answer: y = 5

3. The time to complete a job varies inversely with the number of workers. With 4 workers, the job takes 12 hours. How long with 6 workers?

Answer: 8 hours (k = 48, so 48/6 = 8)

Graphing Inverse Variation

Inverse variation graphs are hyperbolas. They never touch the x-axis or y-axis because you'd be dividing by zero.

The graph approaches both axes but never crosses them. This is called asymptotic behavior — the curve gets closer and closer to the axes but stays away from them.

For y = k/x where k > 0, the graph sits in Quadrants I and III. For k < 0, it sits in Quadrants II and IV.

When Inverse Variation Breaks Down

Inverse variation only applies within a reasonable domain. You can't have negative workers or zero speed. Real-world constraints limit where the math actually works.

Always check if your answer makes sense in context. Mathematically correct doesn't always mean practically possible.

Bottom Line

Inverse variation is straightforward: xy = k. Find the constant, plug in your known values, solve for the unknown. Khan Academy's practice problems give you enough reps to internalize the pattern.

Master the basics first. Then move to word problems. Then graphing. That's the order that actually works.