Ratios & Ratio Tables
A ratio compares two amounts, and scaling both by the same factor leaves it unchanged.
Mix two tins of blue paint with three of yellow.
You get a particular green. Now mix four blue with six yellow.
It is the same green. Twice as much of it, but not a shade different.
The amounts changed and the colour did not, because both were doubled together.
That is what a ratio captures: the relationship between the amounts, not the amounts themselves. 2:3 and 4:6 are two names for one recipe.
Reading a ratio
2 : 3 means two of the first for every three of the second. Order matters as much as it does in coordinates.
Blue to yellow 2:3 is a different instruction from yellow to blue 2:3.
Always write down which is which before doing anything else.
Name the parts
A ratio with no labels is ambiguous. Write "blue : yellow" above the numbers and half the errors in this unit disappear.
Equivalent ratios
Multiply both parts by the same number and the ratio is unchanged: 2:3 = 4:6 = 10:15.
Divide both by the same number and it is still unchanged, which is how you simplify.
12:18 divides by 6 to give 2:3, the simplest form.
Both parts, same number
Adding to both parts does not keep a ratio: 2:3 plus one each gives 3:4, a different mixture entirely.
Ratio tables
Write the two quantities as rows and fill in equivalent columns by scaling.
Each new column is the first one times something, so you can jump straight to the number you need.
Tables make the arithmetic visible and are much harder to get backwards than a remembered rule.
Go through one
Finding the amount for a single unit first — the unit rate — lets you reach any other amount in one step.
Part-to-part and part-to-whole
With 2 blue to 3 yellow, the total is 5 parts.
So blue to yellow is 2:3, but blue to the whole mixture is 2:5.
A question asking "what fraction is blue?" wants 25, not 23.
Add the parts first
The total is nowhere in the ratio as written. 2:3 hides the 5, and most word problems need it.
Sharing in a ratio
To share $40 in the ratio 3:5, add the parts to get 8, then find one part: $5.
Then multiply: three parts is $15 and five parts is $25.
Check by adding — the shares must come back to the total.
One part first
Dividing the total by the sum of the parts gives the value of a single part. Everything else is one multiplication away.
Worked examples
Simplifying
Write the ratio 12 : 18 in its simplest form.
- Find the largest number dividing both: 6. The greatest common factor.
- Divide both parts: 12 ÷ 6 = 2 and 18 ÷ 6 = 3. Both parts, same number.
- So 12:18 = 2:3. The relationship is unchanged.
2:3
Scaling up
A recipe uses 2 cups of flour to 3 eggs. How much flour for 12 eggs?
- 12 ÷ 3 = 4, so the eggs have been multiplied by four. Find the scale factor.
- Multiply the flour by four as well: 2 × 4. Both parts, same number.
- = 8 cups. Check: 8:12 simplifies back to 2:3.
8 cups
Sharing
Share $40 between two people in the ratio 3:5.
- Add the parts: 3 + 5 = 8. The total is divided into eight parts.
- One part is $40 ÷ 8 = $5. Find a single part first.
- So the shares are 3 × 5 = $15 and 5 × 5 = $25. They add to $40, as they must.
$15 and $25
Words to know
- Ratio
- A comparison of two amounts, written a : b.
- Equivalent ratios
- Ratios naming the same relationship, like 2:3 and 4:6.
- Ratio table
- A table of equivalent ratios, useful for scaling up or down.
- Part-to-part
- Comparing one part with another, like blue to yellow.
- Part-to-whole
- Comparing one part with the total, like blue to all the paint.
Mistakes to watch for
Adding to both parts instead of multiplying. 2:3 is not equivalent to 3:4 — adding one to each changes the mixture. Only multiplying or dividing both by the same number keeps a ratio.
Confusing part-to-part with part-to-whole. With 2 blue to 3 yellow, blue is 25 of the mixture, not 23. The total has to be worked out; it is not written in the ratio.
Writing the parts in the wrong order. "Blue to yellow is 2:3" and "yellow to blue is 2:3" describe different paints. Label the parts before writing the numbers.
Scaling only one part. Doubling the eggs without doubling the flour changes the recipe. Whatever is done to one part must be done to the other.
Forgetting to check the total when sharing. Shares of $15 and $25 come back to $40; if they do not, the value of one part was wrong.
Ratios are how every recipe scales, how maps work, and how paint is mixed in shops — the machine adds pigment in a fixed ratio whatever the tin size.
The colon in 2:3 was first used for ratios in the 1600s, borrowed from the punctuation mark.
Screen sizes are ratios: a 16:9 television and a 32:18 one would be exactly the same shape.