How to Solve: Inverse Variation

How to Solve: Inverse Variation

Introduction

"If you’ve ever wondered why doubling your speed cuts your travel time in half—or how scientists predict how bright a star will look from Earth—you’re already thinking about inverse variation. Master this, and you’ll crush word problems on your exam and in real life!

What You Need To Know First

Before diving into inverse variation, make sure you understand: 1. Direct variation – When one quantity increases, the other increases proportionally (e.g., y = kx). 2. Solving for a constant (k) – How to isolate k in equations like y = kx. 3. Basic algebra – Rearranging equations and substituting values.

If any of these feel shaky, review them first!

Key Vocabulary

Formulas To Know

1. Basic Inverse Variation Formula

Formula: y = k / x Variables: - y = dependent variable (what you’re solving for) - x = independent variable - k = constant of variation (must be found first)

Memorise This. – It’s the foundation of all inverse variation problems.

2. Product Rule (Alternative Form)

Formula: x₁y₁ = x₂y₂ or xy = k Variables: - x₁, y₁ = first pair of values - x₂, y₂ = second pair of values - k = constant of variation

Given on exam sheet? Sometimes, but memorize it—it saves time!

3. Joint Variation (Bonus for Advanced Problems)

Formula: y = k / (xz) or y = kx / z Variables: - y = dependent variable - x, z = independent variables - k = constant of variation

Memorise This. if your exam includes joint variation.

Step-by-Step Method

Follow these steps exactly for every inverse variation problem.

Step 1: Identify the Relationship

• Read the problem carefully.
• Look for phrases like:
• "y varies inversely with x"
• "y is inversely proportional to x"
• "as x increases, y decreases"
• If you see these, write: y = k / x

Step 2: Find the Constant (k)

• Use the first set of values given in the problem.
• Plug them into y = k / x and solve for k.
• Example: If y = 4 when x = 3, then 4 = k / 3 → k = 12.

Step 3: Write the Equation

• Substitute k back into y = k / x.
• Example: If k = 12, then y = 12 / x.

Step 4: Solve for the Unknown

• Use the equation to find the missing value.
• Example: If x = 6, then y = 12 / 6 = 2.

Step 5: Check Your Answer

• Verify that the product xy is constant (equal to k).
• Example: For x = 3, y = 4 → 3 × 4 = 12. For x = 6, y = 2 → 6 × 2 = 12. ✅

Worked Example Using the Steps

Problem: The number of hours (h) it takes to paint a house varies inversely with the number of painters (p). If 4 painters take 15 hours, how long will 6 painters take?

Step 1: Identify the Relationship - "Varies inversely" → h = k / p

Step 2: Find the Constant (k) - Given: p = 4, h = 15 - 15 = k / 4 → k = 15 × 4 = 60

Step 3: Write the Equation - h = 60 / p

Step 4: Solve for the Unknown - Find h when p = 6 - h = 60 / 6 = 10

Step 5: Check Your Answer - 4 × 15 = 60 and 6 × 10 = 60 ✅

Final Answer: 6 painters will take 10 hours.

Worked Examples

Example 1 – Basic

Problem: y varies inversely with x. When x = 5, y = 12. Find y when x = 10.

Solution: 1. y = k / x 2. 12 = k / 5 → k = 60 3. y = 60 / 10 = 6

What we did and why: We used the first pair of values to find k, then plugged in the new x to find y. The product xy stayed constant (5 × 12 = 60 and 10 × 6 = 60).

Example 2 – Medium

Problem: The time (t) it takes to download a file varies inversely with the internet speed (s). If a speed of 20 Mbps takes 30 seconds, how long will it take at 50 Mbps?

Solution: 1. t = k / s 2. 30 = k / 20 → k = 600 3. t = 600 / 50 = 12 seconds

What we did and why: We treated t and s like y and x, found k, then solved for the new t. The faster speed (50 Mbps) means less time (12 seconds).

Example 3 – Exam Style

Problem: A car’s fuel efficiency (E, in miles per gallon) varies inversely with its speed (S, in mph). At 50 mph, the efficiency is 30 mpg. What speed gives an efficiency of 25 mpg?

Solution: 1. E = k / S 2. 30 = k / 50 → k = 1500 3. 25 = 1500 / S → S = 1500 / 25 = 60 mph

What we did and why: We set up the inverse variation, found k, then solved for S when E changed. The key was recognizing that E and S are inversely related.

Common Mistakes

Exam Traps

1-Minute Recap

"Alright, let’s lock this in for your exam. Inverse variation means when one thing goes up, the other goes down—like speed and travel time. The formula is y = k / x, and the product xy always equals k. Here’s how to solve any problem in 5 steps:

1. Write y = k / x (or whatever variables the problem uses).
2. Plug in the first pair of values to find k.
3. Write the equation with your k.
4. Plug in the new value to solve for the unknown.
5. Check that xy is constant.

Watch out for traps—examiners love hiding inverse variation in word problems. If you see ‘varies inversely’ or ‘as one increases, the other decreases,’ you’ve got this. Now go practice two problems tonight, and you’ll own this on exam day!