Ratios & Proportion

Mixing orange juice takes 3 scoops of concentrate to 2 scoops of water. Whether it tastes right depends entirely on that pair. A ratio records the recipe: $3 : 2$ , read "3 to 2".

A scoreboard's 3:2 is not a ratio — it only counts points, and one more point wins. A mathematical ratio means two quantities mixed in a fixed recipe.

Ratios: two quantities, one recipe

$3 : 2$ says: split everything into $3 + 2 = 5$ parts; concentrate takes 3 of them, water takes 2. The concentrate is $\frac{3}{2}$ times the water — ratios and fractions are close relatives.

InteractiveRatio Mixer

3/4

Numerator

Denominator

3/4 > 5/8

A greater than B

5/8

Numerator

Denominator

Compare with: 3/4 = 6/8 —— Equivalent multiplier ×2

Equivalent ratios: scale both sides

If 3-to-2 tastes right, a double batch is 6-to-4 and a triple is 9-to-6:

$3 : 2 = 6 : 4 = 9 : 6$

Multiplying or dividing both terms by the same non-zero number leaves the flavour unchanged — such pairs are equivalent ratios. Run it in reverse to simplify: divide both terms of $12 : 8$ by 4 to reach $3 : 2$ , the same move as reducing a fraction.

How do you split 20 candies between two people in the ratio 3:2, fairly and without breaking any?

Example: 20 candies in a 3:2 split

Total parts: $3 + 2 = 5$ . Each part: $20 \div 5 = 4$ candies. One person takes 3 parts: $3 \times 4 = 12$ ; the other takes 2 parts: $2 \times 4 = 8$ . Check: $12 + 8 = 20$ , and $12 : 8$ simplifies to exactly $3 : 2$ .

The recipe never changes: count the total parts, size one part, then deal out the shares.

Proportion: two ratios in balance

When two ratios are equal you can write a proportion: $3 : 2 = 15 : 10$ . A quick equality test is cross-multiplying: $3 \times 10 = 30$ and $2 \times 15 = 30$ — equal, so the proportion holds.

Proportion's favourite job is scaling: a recipe uses 2 cups of rice to 3 cups of water. You now cook 6 cups of rice — how much water? Call it $x$ : $2 : 3 = 6 : x$ , cross-multiply to get $2x = 18$ , so $x = 9$ cups. Scale the recipe up or down and the taste holds.

Same idea, other outfits

In the $3:2$ mix, concentrate owns $\frac{3}{5}$ of the parts — that is 60%. Ratios, fractions and percentages are three spellings of one idea; fluency means translating between them.

Check yourself

Quick quiz

1. Simplify 18 : 12 as far as it goes.
2. 30 candies are shared in the ratio 2:3 — the larger share is…
3. If 2 : 3 = 8 : x, what is x?

Ratios: two quantities, one recipe

Equivalent ratios: scale both sides

Sharing in a ratio

Proportion: two ratios in balance

Check yourself

Quick quiz