A shade of orange paint uses 2 cans of red for every 3 cans of yellow (a ratio of $2 : 3$). To paint a bigger wall you make more batches. Two batches use 4 cans of red and 6 cans of yellow; three batches use 6 and 9.
Key idea
Multiply (or divide) both numbers of a ratio by the same number to get an equivalent ratio: $2 : 3 = 4 : 6 = 6 : 9 = 8 : 12$. Equivalent ratios describe the same mix.
Why it works
Multiplying both numbers by 3 is the same as making 3 batches. Each batch still has 2 cans of red for 3 cans of yellow, so the color does not change. It is the same idea as equivalent fractions: $\frac{2}{3} = \frac{4}{6}$ because both the top and the bottom were multiplied by 2.
| batches | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| red (cans) | 2 | 4 | 6 | 8 | 10 |
| yellow (cans) | 3 | 6 | 9 | 12 | 15 |
Worked example: Scale up
A lemonade uses 3 scoops of mix for every 4 cups of water. How many cups of water go with 15 scoops?
- Find the multiplier: $15 = 3 \times 5$, so this is 5 batches.
- Multiply the water by the same 5: $4 \times 5 = 20$.
- Check: $3 : 4$ and $15 : 20$ are equivalent, since both numbers were multiplied by 5.
Answer: 20 cups of water.
Worked example: Scale down, then up
A paint uses 6 cans of red for every 9 cans of yellow. How many cans of yellow go with 10 cans of red?
- 10 is not a whole-number multiple of 6, so first make the ratio smaller.
- Divide both numbers by 3: $6 : 9 = 2 : 3$.
- Now $10 = 2 \times 5$, so multiply both parts of $2 : 3$ by 5: $10 : 15$.
Answer: 15 cans of yellow.
Common mistake
Adding the same amount to both numbers does not make an equivalent ratio. $2 : 3$ and $4 : 5$ look alike, but in $2 : 3$ the red is $\frac{2}{3}$ of the yellow, and in $4 : 5$ it is $\frac{4}{5}$ of the yellow, a redder orange. Quick check: an equivalent ratio has one multiplier that works for both numbers. $2 \times 2 = 4$, but $3 \times 2 = 6$, not 5.