⚖️ Ratio Province · Proportion

Inverse proportion

Model inverse proportion with a constant relationship and use it to find missing values. In this lesson, focus on inverse proportion uses a constant relationship.

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Understand Inverse proportion

Model inverse proportion with a constant relationship and use it to find missing values. In this lesson, focus on inverse proportion uses a constant relationship.

Inverse proportion uses a constant relationship. The product xy stays constant as one quantity rises and the other falls. For inverse proportion, the final written answer should make that exact relationship visible rather than hiding it inside an unexplained result.

Start here

Inverse proportion: Inverse proportion uses a constant relationship. Recalculate the constant product xy from the final pair. Keep the inverse proportion representation visible until the final line.

Picture the idea

Move one coordinate on a proportion graph and inspect the constant ratio or product at every point. Use the model to explain one change you notice while working on inverse proportion.

Check as you go

Recalculate the constant product xy from the final pair. Write that check beside the final inverse proportion answer.

Key vocabulary

proportionconstantscale factordirect proportioninverse proportioninverse

Rules and key facts

Given information: Inverse proportion — y is inversely proportional to x. When x = 3, y = 4. Find y when x = 2. Method choice: Write each labelled part against its own total before comparing or scaling. Keep ratio parts in the stated order and scale every part by the same factor. Calculation or reasoning: For inverse proportion, k = xy = 3 × 4 = 12. Then y = 12 ÷ 2 = 6. Final answer: 6. Check: Keep ratio parts in the stated order and scale every part by the same factor.

  • Identify whether the relationship is direct or inverse.
  • Use a known pair to calculate the constant.
  • Write the matching relationship.
  • Substitute the new value and check the constant remains unchanged. Record the check explicitly for inverse proportion.

Step-by-step method

  1. Identify whether the relationship is direct or inverse.
  2. Use a known pair to calculate the constant.
  3. Write the matching relationship.
  4. Substitute the new value and check the constant remains unchanged. Record the check explicitly for inverse proportion.

What you need first

  • Recognise the vocabulary: proportion, constant, scale factor.
  • Be able to explain the purpose of inverse proportion before calculating.
  • Keep the relevant values, units and representation visible while you work.

Real-world use

  • Unit pricing
  • Journey time and speed

Visual / interactive

See the idea, then move it around

Skip to Practice

Move one coordinate on a proportion graph and inspect the constant ratio or product at every point. Use the model to explain one change you notice while working on inverse proportion.

Interactive maths model Connected to this topic; move controls, check outputs, then earn XP only from verified actions.
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Worked examples

Examples, methods and exam thinking

Level 1 · Foundation

Understand the idea with small numbers, one representation and one clear step.

Level 2 · Secure

Use the standard Year 8 method with mixed examples and normal wording.

Level 3 · Challenge

Handle multi-step or less familiar questions and explain choices.

Level 4 · Exam-style

Solve a worded question, show reasoning, check accuracy and write a final sentence.

Foundation example

Build confidence

Given information: Inverse proportion — y is inversely proportional to x. When x = 3, y = 4. Find y when x = 2. Method choice: use the inverse proportion method and show each step with the stated values. Calculation or reasoning: For inverse proportion, k = xy = 3 × 4 = 12. Then y = 12 ÷ 2 = 6. Final answer: 6. Check: substitute or compare with the original information to confirm the result fits the question.

  1. Identify whether the relationship is direct or inverse.
  2. Use a known pair to calculate the constant.
  3. Write the matching relationship.
Secure example

Use the normal method

Given information: Inverse proportion — y is inversely proportional to x. When x = 3, y = 140. Find y when x = 14. Method choice: use the inverse proportion method and show each step with the stated values. Calculation or reasoning: For inverse proportion, k = xy = 3 × 140 = 420. Then y = 420 ÷ 14 = 30. Final answer: 30. Check: substitute or compare with the original information to confirm the result fits the question.

Check: Check the inverse proportion result against the original information.

Challenge example

Stretch the idea

Given information: Inverse proportion — y is inversely proportional to x. When x = 3, y = 91. Find y when x = 13. Method choice: use the inverse proportion method and show each step with the stated values. Calculation or reasoning: For inverse proportion, k = xy = 3 × 91 = 273. Then y = 273 ÷ 13 = 21. Final answer: 21. Check: substitute or compare with the original information to confirm the result fits the question.

Try explaining why each step works before checking the answer.

Exam-style example

Show your reasoning

Given information: Inverse proportion — y is inversely proportional to x. When x = 3, y = 48. Find y when x = 12. Method choice: use the inverse proportion method and show each step with the stated values. Calculation or reasoning: For inverse proportion, k = xy = 3 × 48 = 144. Then y = 144 ÷ 12 = 12. Final answer: 12. Check: substitute or compare with the original information to confirm the result fits the question.

Exam tip: Underline the units and command word. Show the key calculation and write the answer in context.

Common mistakes

  • Assuming every increasing relationship is direct proportion. This is a key trap when answering inverse proportion questions.
  • Using y ÷ x for an inverse relationship.

How to check your answer

Recalculate the constant product xy from the final pair. Write that check beside the final inverse proportion answer.

Extension challenge

Create a inverse proportion problem with a tempting incorrect answer. Solve it, apply the check, and explain exactly where the incorrect method breaks down.

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Inverse proportion challenge

Use proportion quest controls to solve three checked inverse proportion rounds. Solve at least two of three marked rounds and use feedback to correct any error.

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Boss challenge

Challenge Inverse proportion Guardian

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Inverse proportion Guardian

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Study cards

Core idea

Inverse proportion: Inverse proportion uses a constant relationship. Recalculate the constant product xy from the final pair. Keep the inverse proportion representation visible until the final line.

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Key vocabulary

proportion · constant · scale factor · direct proportion · inverse proportion · inverse

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Rules

Identify whether the relationship is direct or inverse. Use a known pair to calculate the constant. Write the matching relationship. Substitute the new value and check the constant remains unchanged. Record the check explicitly for inverse proportion.

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Formula / fact

Given information: Inverse proportion — y is inversely proportional to x. When x = 3, y = 4. Find y when x = 2. Method choice: Write each labelled part against its own total before comparing or scaling. Keep ratio parts in the stated order and scale every part by the same factor. Calculation or reasoning: For inverse proportion, k = xy = 3 × 4 = 12. Then y = 12 ÷ 2 = 6. Final answer: 6. Check: Keep ratio parts in the stated order and scale every part by the same factor.

Tap to mark reviewed
Foundation example

Given information: Inverse proportion — y is inversely proportional to x. When x = 3, y = 4. Find y when x = 2. Method choice: use the inverse proportion method and show each step with the stated values. Calculation or reasoning: For inverse proportion, k = xy = 3 × 4 = 12. Then y = 12 ÷ 2 = 6. Final answer: 6. Check: substitute or compare with the original information to confirm the result fits the question.

Tap to mark reviewed
Secure example

Given information: Inverse proportion — y is inversely proportional to x. When x = 3, y = 140. Find y when x = 14. Method choice: use the inverse proportion method and show each step with the stated values. Calculation or reasoning: For inverse proportion, k = xy = 3 × 140 = 420. Then y = 420 ÷ 14 = 30. Final answer: 30. Check: substitute or compare with the original information to confirm the result fits the question.

Tap to mark reviewed
Challenge example

Given information: Inverse proportion — y is inversely proportional to x. When x = 3, y = 91. Find y when x = 13. Method choice: use the inverse proportion method and show each step with the stated values. Calculation or reasoning: For inverse proportion, k = xy = 3 × 91 = 273. Then y = 273 ÷ 13 = 21. Final answer: 21. Check: substitute or compare with the original information to confirm the result fits the question.

Tap to mark reviewed
Exam-style example

Given information: Inverse proportion — y is inversely proportional to x. When x = 3, y = 48. Find y when x = 12. Method choice: use the inverse proportion method and show each step with the stated values. Calculation or reasoning: For inverse proportion, k = xy = 3 × 48 = 144. Then y = 144 ÷ 12 = 12. Final answer: 12. Check: substitute or compare with the original information to confirm the result fits the question.

Tap to mark reviewed
Common mistake

Assuming every increasing relationship is direct proportion. This is a key trap when answering inverse proportion questions.

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Exam tip

For inverse proportion, show the key representation before the final calculation. Use this final check: Recalculate the constant product xy from the final pair.

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Real-world use

Unit pricing, Journey time and speed

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Checklist

I can explain inverse proportion, use the method, check for mistakes, and answer an exam-style question.

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Flashcards

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Help for Inverse proportion

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Simple explanation

Inverse proportion: Inverse proportion uses a constant relationship. Recalculate the constant product xy from the final pair. Keep the inverse proportion representation visible until the final line.

Think of inverse proportion as a careful model: make the important values visible, change one thing at a time, and use the check to prove the answer fits.

Step-by-step breakdown

  1. Identify whether the relationship is direct or inverse.
  2. Use a known pair to calculate the constant.
  3. Write the matching relationship.
  4. Substitute the new value and check the constant remains unchanged. Record the check explicitly for inverse proportion.

Hint 1

Start by naming the given information and the exact result required for inverse proportion.

Hint 2

Identify whether the relationship is direct or inverse.

Full worked solution

Given information: Inverse proportion — y is inversely proportional to x. When x = 3, y = 4. Find y when x = 2. Method choice: use the inverse proportion method and show each step with the stated values. Calculation or reasoning: For inverse proportion, k = xy = 3 × 4 = 12. Then y = 12 ÷ 2 = 6. Final answer: 6. Check: substitute or compare with the original information to confirm the result fits the question.

Method: Identify whether the relationship is direct or inverse. → Use a known pair to calculate the constant. → Write the matching relationship. → Substitute the new value and check the constant remains unchanged. Record the check explicitly for inverse proportion.

Common mistake warning

Assuming every increasing relationship is direct proportion. This is a key trap when answering inverse proportion questions.

Choose a support button above when you need a nudge.

Mastery milestones

Badges reward learning, not locked clicking

  • I can explain inverse proportion in my own words.
  • I can use these words accurately: proportion, constant, scale factor.
  • I can follow the 4-step method without guessing.
  • I can avoid this mistake: Assuming every increasing relationship is direct proportion.
  • I can apply this check: Recalculate the constant product xy from the final pair.
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