A ratio such as 3:7 compares part with part. A fraction of the total such as 3/10 compares part with the whole. You move between them by adding the ratio numbers, and this lesson shows how in ratio and proportional reasoning.
What is the difference?
Take a bag of 40 beads with red and blue in the ratio 3:7. The ratio says that for every 3 red beads there are 7 blue ones.
The fraction of red beads compares red with all the beads.
The whole is 3 + 7 = 10 parts, so red is 3/10 of the beads and blue is 7/10. The fractions add to 1. The ratio numbers do not add to 1, and that is the quickest way to tell them apart.
How to convert, step by step
Ratio to fraction:
- Add the ratio numbers to get the whole.
- Write each number over that whole.
Fraction to ratio:
- Find the other part by subtracting from 1.
- Write the two parts as a ratio and simplify.
Worked example
A bag of 40 beads has red and blue in the ratio 3:7. Find the fraction that is red and the number of each colour.
Step 1, whole: 3 + 7 = 10 parts.
Step 2, fractions: red = 3/10 and blue = 7/10.
Step 3, one part: 40 ÷ 10 = 4, so red = 3 × 4 = 12 and blue = 7 × 4 = 28.
Check: 12 + 28 = 40, and 12/40 = 3/10. The fraction matches the ratio.
Now the reverse: 2/7 of a team are juniors. The other part is 1 − 2/7 = 5/7, so the ratio of juniors to seniors is 2:5.
The mistake to watch for
A common slip is to read the second ratio number as the whole.
Mistaken working: red to blue is 3:7, so 3/7 of the beads are red. 3/7 of 40 = 17.14…
The answer is not a whole number of beads, which is a warning sign.
Red is compared with blue here, not with all the beads. The whole is 10 parts, so red is 3/10, and 3/10 of 40 = 12. Whenever a count of objects comes out as a decimal, re-check what the denominator is.
Check yourself
1. The ratio of cats to dogs at a shelter is 4:5. What fraction of the animals are dogs?
Show answer
Whole: 4 + 5 = 9 parts. Dogs are 5/9 of the animals.
2. 3/8 of a club’s members are under 16. Write the ratio of members under 16 to members 16 or over.
Show answer
The rest are 1 − 3/8 = 5/8. The ratio is 3/8 : 5/8, which simplifies to 3 : 5.
3. A drink is made from orange and water in the ratio 1:4. How much orange is in a 2 litre jug?
Show answer
Whole: 1 + 4 = 5 parts, so orange is 1/5 of the drink.
2 litres = 2000 ml, and 2000 ÷ 5 = 400 ml of orange. Check: water is 4 × 400 = 1600 ml, and 400 + 1600 = 2000.
Where this leads next
Once you can switch freely, share a quantity in a stated ratio will feel faster, and modelling direct proportion from a table is the next step. The non-calculator working trainer is useful for checking fraction work.
Some students know both definitions but choose the wrong one when a question is worded in a new way. A teacher in online one-to-one Mathematics tuition can test that with fresh wording until the choice becomes automatic.