To share a quantity in a three-part ratio a:b:c, add the parts to find the total (a + b + c), divide the quantity by this total to find the value of one part, then multiply each ratio number by one part to get each share. This method works for any amount and any three-part ratio.
What does a three-way ratio mean?
A ratio a:b:c divides a quantity into three shares in the proportions given. For example, a paint mixture in the ratio red:white:blue = 2:5:3 means that for every 2 parts red, there are 5 parts white and 3 parts blue — a total of 10 parts.
| Ratio | Parts | Fraction of total |
|---|---|---|
| Red part | 2 | 2/10 = 1/5 |
| White part | 5 | 5/10 = 1/2 |
| Blue part | 3 | 3/10 |
The total number of parts = 2 + 5 + 3 = 10. Each "part" is the same size; you find its value and scale up.
How do you share a quantity in a three-way ratio?
Method — four steps:
- Add the ratio numbers to find the total number of parts.
- Divide the total quantity by the total number of parts to find the value of one part.
- Multiply each ratio number by one part to find each share.
- Check: add the three shares and verify they equal the original quantity.
Worked example 1:
Share £240 in the ratio 3:2:1.
Step 1: Total parts = 3 + 2 + 1 = 6
Step 2: One part = £240 ÷ 6 = £40
Step 3: Shares:
- First person: 3 × £40 = £120
- Second person: 2 × £40 = £80
- Third person: 1 × £40 = £40
Step 4: £120 + £80 + £40 = £240 ✓
Worked example 2:
Three friends Aisha, Ben and Chloe share a prize of £360 in the ratio 5:3:4.
Total parts = 5 + 3 + 4 = 12
One part = £360 ÷ 12 = £30
| Person | Ratio | Calculation | Share |
|---|---|---|---|
| Aisha | 5 | 5 × £30 | £150 |
| Ben | 3 | 3 × £30 | £90 |
| Chloe | 4 | 4 × £30 | £120 |
| Total | 12 | £360 ✓ |
How do you combine two two-way ratios into a three-way ratio?
Sometimes a question gives two separate ratios and asks for a combined three-way ratio.
Worked example 3:
The ratio of apples to oranges is 2:3. The ratio of oranges to bananas is 3:5. Write the ratio apples:oranges:bananas.
Since the orange value is already the same (3 in both), combine directly:
Apples : Oranges : Bananas = 2 : 3 : 5
Worked example 4 (matching values needed):
The ratio of cats to dogs is 3:4. The ratio of dogs to rabbits is 2:5. Find cats:dogs:rabbits.
The dog value in the first ratio is 4; in the second it is 2. Find the LCM: LCM(4, 2) = 4.
- Scale first ratio by 1: cats:dogs = 3:4
- Scale second ratio by 2: dogs:rabbits = 4:10
Cats : Dogs : Rabbits = 3 : 4 : 10
Check: 3:4 still gives cats:dogs ✓; dogs:rabbits = 4:10 = 2:5 ✓.
How do you find the original total when given one share?
Work backwards: if one share is given, use it to find the value of one part, then multiply by the total number of parts.
Worked example 5:
Three parts are divided in the ratio 4:3:2. The largest share is £80. Find the total.
Largest share corresponds to 4 parts: one part = £80 ÷ 4 = £20
Total parts = 4 + 3 + 2 = 9
Total = 9 × £20 = £180
Other shares: 3 × £20 = £60; 2 × £20 = £40. Check: £80 + £60 + £40 = £180 ✓.
Common mistakes in three-way ratio problems
| Mistake | How it goes wrong | Fix |
|---|---|---|
| Dividing by one ratio number only | £240 ÷ 3 = £80 (wrong "one part") | Always add ALL ratio numbers first |
| Not checking the sum | Shares don't add back to the original | Add the three shares as a final check |
| Wrong LCM when combining ratios | Incorrect combined three-way ratio | Find LCM of the shared value, scale both ratios to match |
Frequently asked questions
Can a three-way ratio have a ratio number of 0?
Technically yes, but a ratio part of 0 means that person or category receives nothing. In practice, GCSE problems use positive whole numbers. A 0 would make one of the fractions zero, which is fine mathematically but unusual in a real-world sharing context.
What if the quantity doesn't divide evenly into the total number of parts?
Quantities that do not divide evenly give non-integer shares, which can happen with lengths, weights and volumes. At KS3 these problems usually give amounts that divide neatly into whole numbers. If a decimal arises, it is not an error — just round appropriately or leave as a fraction.
How is a three-way ratio different from a two-way ratio?
The method is identical — add all parts, find one part, multiply. The only difference is that you add three numbers instead of two in step 1 and produce three shares in step 3. The two-way method (divide into two parts) is a special case of the same approach.
Can I use three-way ratios with different units?
Only if all three quantities are in the same units. You cannot share "£240 and 3 kg" in a ratio without converting. When the quantity is a single measurement (money, length, mass), the method above applies directly. For problems mixing units, convert everything to one unit first.
Try ratio problems at your own pace with Professor Pi at aitutors.me.