TACHS: Three-part ratios: find the value of one part
Use a three-part ratio to find any group's share, the total, or a new ratio, starting from whatever number the question gives.
How do you split an amount in a ratio like 3:5:7, or work back from one group's share?
Short answer
Match the number you are given to the number of parts it covers, divide to find one part, then multiply by the parts you need.
Know first: Multiplying and dividing whole numbers, Simplifying a ratio
Step 1 of 6
A ratio counts equal parts
A ratio of 3:5:7 means the three groups get 3, 5 and 7 equal parts. Together that is 15 parts. Once you know the size of one part, every share, total and difference follows.
The question always gives one number to unlock the part size. The skill is deciding how many parts that number covers.
| You are given | It covers | Example with 3:5:7 |
|---|---|---|
| The total | All the parts | 15 parts |
| One group's amount | That group's parts | The middle group is 5 parts |
| A difference between two groups | The difference in parts | Largest minus smallest is 7 − 3 = 4 parts |
| Two groups together | The sum of their parts | First two groups are 3 + 5 = 8 parts |
Divide the given number by the parts it covers to get one part.
Step 2 of 6
One part unlocks every share
Worked example: Marbles in the ratio 3:5:7
A teacher divides 120 marbles among three tables in the ratio 3:5:7. How many marbles does the middle table get?
- A8
- B24
- C40
- D56
- 1
Count the parts
3 + 5 + 7 = 15 parts.
- 2
Find one part
The total covers all 15 parts: 120 ÷ 15 = 8 marbles.
- 3
Multiply for the middle table
5 parts × 8 = 40 marbles.
- 4
Check
3 × 8 = 24, 5 × 8 = 40, 7 × 8 = 56, and 24 + 40 + 56 = 120.
- 5
See why the other choices are there
8 is one part, not a table's share. 24 is the first table's share, or 120 ÷ 5. 56 is the third table's share.
Answer
The middle table gets 40 marbles.
Pens, pencils and markers
Pens, pencils and markers are in the ratio 4:5:9. There are 35 pencils. How many items are there in all?
- 1
Total parts: 4 + 5 + 9 = 18.
- 2
One part: 35 ÷ 18.
What went wrong
The 35 pencils cover only the pencils' 5 parts, not all 18 parts.
- 3
Total items: 18 × one part.
The fix
Pencils are 5 parts, so one part is 35 ÷ 5 = 7. The total is 18 × 7 = 126. Check: 28 pens + 35 pencils + 63 markers = 126.
Step 3 of 6
Differences and changes use the same part size
When you are given a difference, the number covers the difference in parts. In a 3:4:8 ratio, if the largest group has 25 more than the smallest, then 8 − 3 = 5 parts equal 25. One part is 5, and the total is 15 × 5 = 75.
When items are added to one group, find the real amounts first, change them, then simplify.
A ratio after a change
Ratio 1:3:5, with 6 in the first group
One part: 6 ÷ 1 = 6, so the groups are 6, 18 and 30
Add 12 to the first group: 18, 18 and 30
Divide each by 6: the new ratio is 3:3:5
Step 4 of 6
Match the number to its parts
Check yourself · Question 1
Lemonade, iced tea and water bottles are in the ratio 5:2:3. There are 140 bottles in all. How many are iced tea?
Answer: B
Total parts: 5 + 2 + 3 = 10. One part: 140 ÷ 10 = 14. Iced tea is 2 parts: 2 × 14 = 28.
Trap answer: 14
Stopping at one part
- Why it looks right
- Finding one part takes real work, so it feels like the answer when it appears in the choices.
- Why it's wrong
- 14 is the size of one part. The question asks for iced tea, which gets 2 parts.
- The right answer
- 28, because 2 × 14 = 28.
Check yourself · Question 2
Red, white and blue flags are in the ratio 2:5:6. There are 24 more blue flags than red flags. How many white flags are there?
Answer: C
The difference covers 6 − 2 = 4 parts, so one part is 24 ÷ 4 = 6. White is 5 parts: 5 × 6 = 30. Check: blue 36 − red 12 = 24.
Common mistake
"There are three groups, so I divide the total by 3."
- Why it's tempting
- Three names in the question make three equal shares feel natural.
- Do this instead
- Divide by the number of parts, not the number of groups. A 3:5:7 split has 15 parts, and the groups get different numbers of them.
Step 5 of 6
What to remember
Remember
- Add the ratio numbers to count the parts.
- Decide how many parts the given number covers: all, one group, a difference or a sum. Divide to find one part.
- Multiply one part by the parts you need, and check that the shares add up.
Step 6 of 6
Practice on a real question
Use what you just learned on this question, then check the explanation.
Try another one
In your own words
In the practice question, how big is one part, and which wrong choices are the shares of the other two groups?