GCSE · Maths · AQA · Spec 8300

Ratios linked to fractions and linear functions

Mix 2 litres of red paint with 3 of blue, or 20 with 30: the same colour. What stays the same is a fraction, and the slope of a line.

Mix more paint. Do the fractions change?

Batches mixed: 1/5. Red : Blue (litres) 2 : 3. Total paint 5 L. Red ÷ total 2/5. Blue ÷ total 3/5. Red ÷ blue 2/3

Red : Blue (litres)2 : 3Total paint5 LRed ÷ total2/5Blue ÷ total3/5Red ÷ blue2/3

Each square of the bar is one batch: 2 litres of red and 3 litres of blue, so red : blue = 2 : 3. Drag to mix more batches. The litres grow, but every fraction cancels back to the same value.

Lift the point: a ratio is a line through the origin

Blue with 2 L of red 3 L

Blue with 2 L of red: 3 L. Gradient = blue ÷ red: 1.50

Your point sits at 2 litres of red, and its height is the blue that goes with it: at height 3, red : blue = 2 : 3. The line joins your point to (0, 0), and every point on it apart from (0, 0) is in that same ratio. It starts on the paint from the bar: the line passes through (4, 6) and (8, 12), with gradient 3/2 = 1.5. Lift the point to try other ratios: the gradient is always blue ÷ red, the second number of the ratio over the first.

Watch out: The gradient is blue ÷ red, the second number over the first. For 2 : 3 that is 3/2, not 2/3: the line y = 2/3 x holds pairs such as (3, 2), which are in the ratio 3 : 2.

Which of these do you think is true?

A drink is made from orange juice and lemonade in the ratio orange : lemonade = 3 : 4. You make jugs of different sizes, always in this ratio.

Choose the statement closest to what you think, then say how sure you are.

Orange juice is 3/7 of the drink.Orange juice is 3/4 of the drink.Plotting lemonade (y) against orange (x) gives the line y = 3/4 x.The line y = x + 1 also keeps the drink in the ratio 3 : 4, because it goes through (3, 4).

How sure are you?

I'm sureI think soI'm guessing

Orange juice is 3/7 of the drink.

This matches the evidence

Yes. The ratio makes 3 + 4 = 7 equal parts and 3 of them are orange, so orange is 3/7 of every jug, big or small. The 3/4 in the ratio is a different fraction: there is 3/4 as much orange as lemonade.

Orange juice is 3/4 of the drink.

This is a common misconception

3/4 compares orange with lemonade, not with the whole drink. If orange really were 3/4 of the drink, lemonade would be the other 1/4, and the ratio would be 3 : 1. Add the parts first: 3 + 4 = 7, so orange is 3/7 of the drink.

Plotting lemonade (y) against orange (x) gives the line y = 3/4 x.

This is a common misconception

Test a pair: 3 litres of orange go with 4 of lemonade, but y = 3/4 x gives y = 2.25 when x = 3. The gradient is y ÷ x = lemonade ÷ orange = 4/3, so the line is y = 4/3 x. 3/4 is x ÷ y, the fraction upside down.

The line y = x + 1 also keeps the drink in the ratio 3 : 4, because it goes through (3, 4).

This is a common misconception

It does go through (3, 4), but it also goes through (6, 7), and 6 : 7 is not 3 : 4. Adding 1 to every x is not the same as multiplying every x by 4/3, so y ÷ x keeps changing along the line. Only a straight line through (0, 0), here y = 4/3 x, keeps every pair in the same ratio.

One ratio holds two different fractions: 3 : 4 makes orange 3/7 of the drink but 3/4 of the lemonade. Its graph, lemonade against orange, is the straight line through the origin y = 4/3 x.

From a ratio to an equation, and back

x and y are always in the ratio x : y = 2 : 5. Write y in terms of x. Then find y when x = 14, and find x when y = 40.

x : y = 2 : 5 means that for every 2 of x there are 5 of y.
missing step

Which line is step 2?

So y ÷ x = 2/5 for every pair.So y ÷ x = 5/2 for every pair.So y ÷ x = 5/7 for every pair.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

A ratio is a fraction of the whole, a comparison of two parts, and a straight line through the origin, all at once

What you need to know

A ratio a : b makes a + b equal parts, so the first part is a/(a + b) of the whole and the second part is b/(a + b).
Comparing the parts, the first quantity is a/b of the second, and the second is b/a of the first.
Multiplying or dividing both numbers of a ratio by the same amount does not change any of these fractions.
If x : y = a : b for every pair, then y = (b/a)x: a straight line through the origin with gradient b/a.

The big picture

A ratio a : b splits a whole into a + b equal parts, so the first part is a/(a + b) of the whole and the second is b/(a + b). Comparing the parts with each other, the first is a/b of the second. Scaling a ratio up or down never changes these fractions. Every pair of values in the ratio x : y = a : b lies on the straight line y = (b/a)x through the origin, and its gradient b/a is y ÷ x at every point except the origin.

Key points

1Part to whole: add the parts first. In 2 : 3, the parts are 2/5 and 3/5 of the whole.

2Part to part: in 2 : 3, the first quantity is 2/3 of the second.

3y ÷ x is the gradient: the second number of the ratio over the first.

4A straight line that misses the origin does not keep its pairs in one ratio.

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Worked example

Problem

Cement and sand are mixed in the ratio cement : sand = 1 : 3 by mass. (a) What fraction of the mix is cement? (b) Write the mass of sand, s kg, in terms of the mass of cement, c kg. (c) How much cement is in 60 kg of mix?

⚠ Watch out

Reading a : b as 'the first part is a/b of the whole', or writing y = (a/b)x from x : y = a : b. In 2 : 3 the first part is 2/5 of the whole, and the line is y = 3/2 x.

🧠

Memory hook

Whole? Add the parts. Slope? Second over first.

✓

Check yourself

Flour : sugar = 5 : 2. What fraction of the mixture is sugar? If you plot sugar (y) against flour (x), what is the equation of the line?

Flashcards

(6)

In a ratio a : b, what fraction of the whole is the first part?

a/(a + b). Add the parts to get the whole: in 4 : 1 the first part is 4/5 of the whole.

In a ratio a : b, what fraction of the second quantity is the first?

a/b. In 4 : 5 the first quantity is 4/5 of the second. This compares two parts, not a part with the whole.

Does scaling 2 : 3 up to 8 : 12 change its fractions?

No. 8/20 = 2/5 and 8/12 = 2/3. Multiplying both numbers by the same amount keeps every fraction of the ratio the same.

x : y = a : b for every pair. What is y in terms of x?

y = (b/a)x: the second number of the ratio over the first. For x : y = 3 : 7, y = 7/3 x.

What does a graph of pairs in a fixed ratio look like?

A straight line through the origin. Its gradient equals y ÷ x at every point except (0, 0).

Why does y = 2x + 1 not keep x and y in a fixed ratio?

It misses the origin. (1, 3) gives 1 : 3 but (2, 5) gives 2 : 5, so y ÷ x keeps changing.

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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Lift the point: a ratio is a line through the origin

Which of these do you think is true?

From a ratio to an equation, and back

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