Unit 4: In Proportion

How much of this, for how much of that

A squash recipe says 1 part juice to 4 parts water. We write that ratio as 1 : 4.

It does not tell you how much squash to make. It tells you the *shape* of the mix — and that shape stays the same whether you make a cup or a bucket.

2 parts juice needs 8 parts water. Same drink, more of it.

Ratios cancel down, like fractions

10 : 15 and 2 : 3 describe exactly the same mix — both sides were divided by 5.

So a ratio is really a fraction in disguise, and the same tidying-up works: divide both sides by whatever fits into them both.

Sharing in a ratio

Share £20 between two people in the ratio 2 : 3.

Count the parts first: 2 and 3 make 5 parts altogether. So one part is 20 ÷ 5 = 4.

Then the shares are 2 × 4 = 8 and 3 × 4 = 12. Check: they add back to 20.

Direct proportion

If 3 pens cost 60p, then 6 pens cost 120p. Double the pens, double the price.

That is direct proportion: one thing doubles, the other doubles too. What stays fixed is the price *per pen* — 20p — and finding that one-unit value is usually the fastest route through any question.

Inverse proportion

Not everything grows together. If 2 painters take 12 days, then 4 painters take 6 days — twice as many people, half the time.

That is inverse proportion. Here what stays fixed is the total work: 2 × 12 = 24 painter-days, and 4 × 6 = 24 as well.

So the two numbers multiply to something constant instead of dividing to something constant.

Ratios can have as many parts as you like

A cake recipe might use flour, sugar and butter in the ratio 4 : 2 : 1.

The method does not change. Count the parts — 4 + 2 + 1 = 7 — then work out what one part is worth.

With 350 g altogether, one part is 50 g. So it is 200 g of flour, 100 g of sugar and 50 g of butter.

Ratios can be scaled to match

If a : b = 2 : 3 and b : c = 6 : 7, the two bs do not agree yet.

Double the first ratio so its b is also 6: a : b = 4 : 6.

Now they join up: a : b : c = 4 : 6 : 7. Making a shared part match is the whole technique.

A rate compares two different things

A ratio compares like with like. A rate compares one thing with another kind of thing altogether.

Kilometres per hour. Pence per gram. Words per minute.

That little word "per" always means divide, and it is the fastest way to make two offers comparable.

Which is the better buy?

A 500 g jar costs £2.00. A 750 g jar costs £2.70. Which is better value?

Work out the price for the same amount of each. Per 100 g that is 40p and 36p.

The big jar wins. Comparing prices directly tells you nothing — comparing rates answers it in one line.

A scale is a ratio in disguise

A map with scale 1 : 50000 means one of anything on the map is fifty thousand of the same thing in the world.

So 1 cm on the map is 50000 cm on the ground — which is 500 m, or half a kilometre.

The units must match on both sides of a scale. That conversion at the end is where most mistakes happen.

Scaling areas, not just lengths

Careful: a scale of 1 : 10 makes lengths ten times bigger, but areas one hundred times bigger.

Both the width and the height grew by ten, and area is the two multiplied.

This is the same k^2 fact the geometry units will meet again — and the reason a doll’s house needs far less carpet than you would guess.

Keeping a mixture the same

A paint mix is 3 parts blue to 2 parts white. You have 12 litres of blue. How much white?

Blue went up by a factor of 4, so white must too: 2 × 4 = 8 litres.

The factor is the thing to find. Once you have it, everything else follows in one multiplication.

When more workers means less time

Be careful which kind of proportion you are in.

More people eating means more food — direct. More people digging means less time — inverse.

Ask yourself: if I double one, does the other double or halve? Getting that backwards is the only real danger in this whole unit.

Out of a hundred

Per cent literally means "per hundred". So 25% is the ratio 25 : 100, which cancels to 1 : 4, which is the fraction 1/4.

That is all a percentage is — a fraction that has agreed to always use 100 on the bottom, so that any two of them can be compared at a glance.

Increases and decreases in one step

A 20% rise does not mean "find 20% then add it" — not if you want to be quick.

You end up with the original 100% plus 20%, which is 120%, so multiply by 1.2 and you are done.

A 20% drop leaves 80%, so multiply by 0.8. One step, no adding on.

The trap: a rise then a fall

A price goes up 10%, then down 10%. Is it back where it started?

No. £100 rises to £110, and 10% of *110* is 11, so it drops to £99.

The percentages were taken from different starting numbers. This one catches grown-ups every day.