Direct and inverse proportion
Two quantities can grow together or trade off against each other. Deciding which relationship applies before calculating prevents nearly every error in this topic.
Part of your national curriculum
- Ratio, proportion and rates of change: Solve problems involving direct and inverse proportion, including graphical and algebraic representations
Lesson overview
What you'll learn in this lesson
Solve problems involving direct and inverse proportion, including graphical and algebraic representations
Key learning points
- • Direct proportion
- • Inverse proportion
- • Recognising which is which
- • Modelling assumptions
This lesson at a glance
- 30 minutes
- 21 parts to scroll through
- 4 quick checks
- Marked quiz at the end
- Gentle pace: short sittings with pauses
Words to know
Scroll down — the lesson carries on below
Watch & discover
Part 1 of 21
Visual introductionPicture this
Direct and inverse proportion
Two quantities can grow together or trade off against each other. Deciding which relationship applies before calculating prevents nearly every error in this topic.
In a nutshell
Solve problems involving direct and inverse proportion, including graphical and algebraic representations
Learning cycle
Part 2 of 21
Learning cycle 1 of 2
Part 1 · Direct proportion
A short piece of teaching, then a check to make sure it has landed.
Explore the idea
Part 3 of 21
Learn
Direct proportion
If y is directly proportional to x, then y = kx and the graph is a straight line through the origin. Doubling x doubles y. If 6 books cost £27, then k = 4.5 and 10 books cost £45. The unitary method, finding the value of one, is the same idea.
Reset break
Part 4 of 21
Pause
That's sitting 1 of 5 done
Stretch, get a drink, look out of the window. There is no timer and nothing is counting down — your place is saved, so you can come back in five minutes or tomorrow.
Explore the idea
Part 5 of 21
Learn
Inverse proportion
If y is inversely proportional to x, then y = k/x and the product xy is constant. The graph is a curve that never touches the axes. If 4 workers take 9 hours, the total work is 36 worker-hours, so 6 workers take 6 hours, assuming they work at the same rate.
Quick check
Part 6 of 21
Quick check
Part 7 of 21
Reset break
Part 8 of 21
Pause
That's sitting 2 of 5 done
Stretch, get a drink, look out of the window. There is no timer and nothing is counting down — your place is saved, so you can come back in five minutes or tomorrow.
Learning cycle
Part 9 of 21
Learning cycle 2 of 2
Part 2 · Recognising which is which
A short piece of teaching, then a check to make sure it has landed.
Explore the idea
Part 10 of 21
Learn
Recognising which is which
Ask what happens when one quantity doubles. If the other doubles, the relationship is direct; if it halves, it is inverse. Recipes, currency conversion and fuel use are direct; the number of workers against time, and speed against journey time, are inverse.
Explore the idea
Part 11 of 21
Learn
Modelling assumptions
Inverse proportion for workers assumes everyone works at the same constant rate and does not get in each other's way. Stating that assumption is part of a full answer, because in reality doubling the workforce rarely halves the time exactly.
Reset break
Part 12 of 21
Pause
That's sitting 3 of 5 done
Stretch, get a drink, look out of the window. There is no timer and nothing is counting down — your place is saved, so you can come back in five minutes or tomorrow.
Quick check
Part 13 of 21
Quick check
Part 14 of 21
Explore the idea
Part 15 of 21
Worked example
Worked answer: 4 workers build a wall in 9 hours. How long would 6 identical workers take? (3 marks)
The total work is 4 × 9 = 36 worker-hours, which stays constant (1). With 6 workers, time = 36 ÷ 6 = 6 hours (1). This assumes all workers work at the same steady rate and can work at once without obstruction (1). Answering 13.5 hours treats the relationship as direct, which is the classic error.
Reset break
Part 16 of 21
Pause
That's sitting 4 of 5 done
Stretch, get a drink, look out of the window. There is no timer and nothing is counting down — your place is saved, so you can come back in five minutes or tomorrow.
Challenge round
Part 17 of 21
Game · Sort it
Which of these are true?
Drag each card into the right column. Tap a card first if dragging is fiddly.
True
Not true
Challenge round
Part 18 of 21
Game · Fill the gaps
Finish the sentences
Choose the word that belongs in each gap.
If the other doubles, the relationship is ____; if it halves, it is inverse.
Recipes, currency conversion and fuel use are direct; the number of workers against time, and speed against journey time, are ____.
Inverse ____ for workers assumes everyone works at the same constant rate and does not get in each other's way.
Challenge round
Part 19 of 21
Game · Recall cards
5 items cost £12.50. What do 8 cost?
Card 1 of 4
Mastery quiz
Part 20 of 21
Marked quiz
End of lesson quiz: Direct and inverse proportion
4 questions, marked with the reasoning shown. No timer.
1. 5 items cost £12.50. What do 8 cost?
2. A graph of a proportional relationship must…
3. A tracker records 18 km in 3 hours. At the same rate, how far in 7 hours?
4. If 4 tickets cost £26, 6 tickets cost…
Lesson round-up
Part 21 of 21
Lesson round-up
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Solve problems involving direct and inverse proportion, including graphical and algebraic representations