GCSE · Maths · AQA · Spec 8300 · Foundation

Inverse proportion equivalence and equations

Make a rectangle of 12 tiles twice as wide and it has to get exactly half as tall. Not just shorter: exactly half. That is inverse proportion.

Double the width, halve the height

width, w (cm): 3. height, h (cm): 4. w × h: 12 cm²

Every point is a rectangle with an area of 12 cm², so h = 12/w. Drag the point to w = 1, then 2, then 4. Each time you double the width, what happens to the height?

What does 'inversely' really mean?

Going down isn't enough

Your friend says: "y = 10 − x is inverse proportion. Look: when x goes up, y goes down."

Which is closest to what you think 'y is inversely proportional to x' means right now?

When x goes up, y goes down. Any relationship that does that counts.If x goes up by some amount, y goes down by the same amount.y is proportional to 1/x: if x is multiplied by any number, y is divided by that same number.

How sure are you?

I'm sureI think soI'm guessing

When x goes up, y goes down. Any relationship that does that counts.

This is a common misconception

Every inverse proportion does go down as x goes up, but lots of other relationships do too, so 'going down' can't be the definition. Test y = 10 − x: at x = 2, y = 8, and at x = 4, y = 6. Doubling x should have halved y to 4, and it didn't. So y = 10 − x goes down, but it is not inverse proportion.

If x goes up by some amount, y goes down by the same amount.

This is a common misconception

That's adding and taking away, and it's exactly how y = 10 − x behaves: x and y always add up to 10. Inverse proportion is about multiplying and dividing. From x = 2 to x = 4, y has to be divided by 2 (8 becomes 4), not reduced by 2 (8 becomes 6).

y is proportional to 1/x: if x is multiplied by any number, y is divided by that same number.

This matches the evidence

Yes, that's it. Treble x and y becomes a third of what it was. Multiply x by 10 and y becomes a tenth. It isn't just 'down', it's down by the matching factor. That's why y = 10 − x fails the test: doubling x took y from 8 to 6, not from 8 to 4.

Inverse proportion is a multiplying pattern, not just a going-down pattern: multiply x by a number and y is divided by that same number.

Reading equations

Direct, inverse or neither?

Put each equation in the right group.

Still to sort

Direct proportion (0)

Can be written as y = kx for a fixed number k.

Where the line is: A division sign doesn't make it inverse. y = x/4 is y = ¼x, which is still y = kx.

Inverse proportion (0)

Can be written as y = k/x for a fixed number k.

Where the line is: Nothing can be added to or taken away from k/x. y = 6/x + 1 has an extra '+ 1', so it is not y = k/x.

Neither (0)

Can't be written as y = kx or as y = k/x.

Where the line is: Going down as x goes up doesn't make an equation inverse. It must fit y = k/x exactly.

7 of 7 still to sort.

Read each equation and decide what kind of proportion it describes before you check.

Higher

Building an inverse proportion equation

Problem

y is inversely proportional to x. When x = 4, y = 15. (a) Find an equation for y in terms of x. (b) Find y when x = 10.

Higher

Spot the slip

Where does this answer go wrong?

p is inversely proportional to q. When q = 2, p = 18. Find p when q = 6.

A student's answer — which line goes wrong?

p ∝ 1/q
p = k/q
18 = k/2
k = 18 ÷ 2 = 9
p = 9/q
When q = 6, p = 9 ÷ 6 = 1.5

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

y ∝ 1/x

Read ∝ as 'is proportional to'. An inverse proportion is a direct proportion in disguise: y keeps pace with 1/x.

What you need to know

'y is inversely proportional to x' means exactly the same as 'y is proportional to 1/x'.
In an inverse proportion, multiplying x by any number divides y by that same number.
Direct proportion has the equation y = kx, and inverse proportion has y = k/x, where k is a fixed number called the constant of proportionality.
An equation that can't be written as y = kx or y = k/x, such as y = 12 − 2x, is neither.
To build the equation, substitute one pair of values to find k: for y = kx, k = y ÷ x, and for y = k/x, k = x × y.

The big picture

'y is inversely proportional to x' means exactly the same as 'y is proportional to 1/x': multiply x by any number and y is divided by that same number. Its equation is y = k/x, next to y = kx for direct proportion, and anything else, like y = 12 − 2x, is neither. To build either equation, use one pair of values to find the constant k.

Key points

1y ∝ 1/x is how you write 'y is inversely proportional to x'. Both statements mean the same thing.

2In an inverse proportion the product x × y always equals the same constant, k.

3Going down as x goes up is not enough for inverse proportion. The change has to be by the matching factor.

4y = x/4 is direct proportion, not inverse, because it is y = ¼x.

5When you build an equation, check your answer with the factor test.

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Worked example

Problem

An equation is y = 45/x. (a) Write the relationship between y and x in two equivalent ways using the word 'proportional'. (b) Find y when x = 3 and when x = 9, and use your answers to describe what trebling x does to y.

⚠ Watch out

Calling any 'one goes up, the other goes down' relationship inverse proportion. y = 12 − 2x goes down, but doubling x doesn't halve y, so it is neither. Only y = k/x is inverse proportion.

🧠

Memory hook

Flip x and inverse becomes direct: y ∝ 1/x. Double one, halve the other.

✓

Check yourself

y = 24/x. Work out y when x = 6, then again when x = 12. Is y proportional to x, to 1/x, or to neither? How do your two answers show it?

Flashcards

(8)

'y is inversely proportional to x' means the same as what?

y is proportional to 1/x. Written with symbols: y ∝ 1/x.

Write the equation for direct proportion and the equation for inverse proportion.

Direct: y = kx. Inverse: y = k/x. In both, k is a fixed number called the constant of proportionality.

y is inversely proportional to x, and x is multiplied by 5. What happens to y?

y is divided by 5.

In an inverse proportion, what stays the same for every pair of values?

The product x × y. It always equals k.

How do you find k for inverse proportion from one pair of values?

Multiply them: k = x × y.

How do you find k for direct proportion from one pair of values?

Divide: k = y ÷ x.

Is y = 10 − x inverse proportion?

No. y goes down as x goes up, but doubling x doesn't halve y (x = 2 gives 8, x = 4 gives 6). It's neither.

Direct, inverse or neither: y = x/3?

Direct. y = x/3 is y = ⅓x, so k = ⅓.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Double the width, halve the height

Going down isn't enough

Direct, inverse or neither?

Where does this answer go wrong?

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