KS3 · Maths

Inverse proportion in context

A rectangle 2 cm wide and 12 cm long, and another 3 cm wide and 8 cm long: different shapes, but what have they got in common?

Every rectangle with an area of 24 cm²

1611162106121824Width (cm)Length (cm)(11, 2.18)Width × length: 24 cm²

Width (cm): 11. Length (cm): 2.18. Width × length: 24 cm²

Drag the point along the curve and read each rectangle's width and length. Try widths 2, 3, 4 and 10. (The width scale starts at 1 cm.)

Watch out: A point on the curve isn't always a sensible answer. The context decides. A rectangle can be 2.4 cm wide, because you can always measure more accurately. But if one person takes six days to paint a fence, finishing in five days would need 1.2 people, and you can't have one and a bit of a person.

Maths · Ratio and proportion

Direct, inverse or neither?

Is each relationship direct proportion, inverse proportion, or neither?

Still to sort

Direct proportion (0)

As one goes up, the other goes up at the same rate.

Where the line is: Its graph is a straight line that starts at the origin.

Inverse proportion (0)

As one goes up, the other comes down in proportion.

Where the line is: Its graph is a curve, and the product of each pair stays the same.

Neither (0)

A pattern, but not a proportion.

Where the line is: A straight line that doesn't start at the origin isn't direct proportion.

6 of 6 still to sort.

Pick a card, then pick the column you think it belongs in. You'll see the reason straight after.

Maths · Ratio and proportion

Which of these is true?

Three students are each trying to sum up inverse proportion in one sentence.

Which sentence is right? Pick the one you'd back, then say how sure you are.
How sure are you?

Worked example: the machines

Problem

10 machines take 5 days to make a lorry load of items. Working at the same rate, how long would it take (a) 2 machines and (b) 3 machines?

Maths · Ratio and proportion

Your turn: find the missing value

A and B are inversely proportional. When A = 15, B = 8. Find B when A = 24.

  1. Ratio table: A goes from 15 to 24. B starts at 8, and we need the new B.
  2. missing step
Which line is step 2?

Predict, then check

Commit to an answer first. Then see the ratio table.

Four machines make one million items in 7 days. A fifth machine is added. Working at the same rate, how long do the five machines take?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Make a rectangle wider and it has to get shorter to keep the same area. One goes up, the other comes down, and something stays fixed.

What you need to know

  • Two quantities are in inverse proportion when, as one increases, the other decreases in proportion. Their product stays the same.
  • More precisely, two variables are inversely proportional if there is a constant multiplicative relationship between one variable and the reciprocal of the other.
  • Everyday examples: rectangles with the same area (width and length), a car's speed and the time it takes to cover a fixed distance, and the number of people doing a job and the time it takes.
  • The graph of inverse proportion is a curve. It never meets either axis. Direct proportion is a straight line through the origin.
  • In a ratio table for inverse proportion, do the opposite operation to the other column: multiply one by a number, divide the other by the same number.
  • Check an answer by multiplying each pair. In inverse proportion the products are all the same.

The big picture

Two quantities are in inverse proportion when one goes up as the other comes down in proportion, so their product stays the same. Every rectangle with an area of 24 cm² shows it: 1 cm by 24 cm, 2 cm by 12 cm, 3 cm by 8 cm, 4 cm by 6 cm. Its graph is a curve that never touches the axes, unlike the straight line through the origin you get for direct proportion. To solve problems, use a ratio table and do the opposite operation to the other column, then check that the products match. And always ask whether the answer makes sense in context.

Key points

1One goes up, the other comes down in proportion, and the product stays fixed.
2Graph: a curve that never touches the axes, not a straight line like x + y = a.
3Ratio table: whatever you do to one column, do the opposite to the other.
4Any pair of numbers has a multiplier, so the method always works, even for awkward numbers like 3/10.
5The context decides whether an answer makes sense: 2.4 cm is fine for a length, but 1.2 people isn't possible.

Worked example

Problem

10 factory workers take 4 hours to pack a lorry. Two of them are on holiday. Working at the same rate, how long do the other workers take?

⚠ Watch out

Doing the same thing to both columns of a ratio table, as you would for direct proportion. If 4 people take 2 days to paint a house, 8 people do NOT take 4 days. The people doubled, so the days halve: 8 people take 1 day.

🧠

Memory hook

Double one, halve the other. In inverse proportion the two quantities pull in opposite directions, but their product never moves.

✓

Check yourself

Without looking back: why can an inverse proportion graph never touch the axes, and how do the products of each pair check an answer?

Flashcards

(12)
What is inverse proportion?
As one quantity increases, the other decreases in proportion.
What stays the same when two quantities are inversely proportional?
Their product. For rectangles of area 24 cm², width × length is always 24.
Give the precise definition of inversely proportional.
There is a constant multiplicative relationship between one variable and the reciprocal of the other.
Give two everyday examples of inverse proportion.
Speed and the time taken to cover a fixed distance; the number of people doing a job and the time it takes.
What shape is the graph of inverse proportion?
A curve, not a straight line.
Does an inverse proportion graph ever meet the axes?
No. On the rectangle graph, a point on an axis would mean a side of 0 cm, which isn't a rectangle.
What does a direct proportion graph look like?
A straight line with a constant positive gradient that starts at the origin.
A straight line with a positive gradient crosses the y-axis above 0. Which kind of proportion is it?
Neither. It isn't direct because it doesn't start at the origin, and a straight line isn't inverse proportion.
In an inverse proportion ratio table, you multiply one column by 5. What do you do to the other?
Divide it by 5: the opposite operation.
Is the line x + y = 24 inverse proportion?
No. It keeps the sum at 24, not the product, and it's a straight line that meets both axes.
Reading a graph gives 1.2 people. Is that a sensible answer?
No. The context decides: you can't have one and a bit of a person, although a length of 2.4 cm is fine.
Before calculating an inverse proportion problem, what quick check should you make?
Decide whether the answer should be bigger or smaller. More workers means less time.

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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