Understanding and Simplifying Ratios
A ratio compares quantities multiplicatively: it tells you how many parts of one quantity there are for each part of another. In this lesson you will write, simplify and interpret ratios and connect them to fractions.
Foundation & Higher. You should already be comfortable with fractions and basic arithmetic. Revisit Fractions, Decimals & Percentages if needed.
Ratio notation and order
A ratio such as 3:5 is ordered. If there are 3 red counters for every 5 blue counters, red:blue = 3:5 but blue:red = 5:3.
Ratios compare parts. They do not tell you the actual total unless another piece of information is given.
Equivalent ratios and simplest form
Multiply or divide every part by the same non-zero number. For example, 12:18 = 2:3 because both parts divide by 6.
For three-part ratios the same rule applies: 8:12:20 = 2:3:5.
Worked example — ratio tables and scale factors
A ratio table makes the multiplicative structure visible. Suppose red:blue = 2:5.
| Scale factor | Red | Blue |
|---|---|---|
| ×1 | 2 | 5 |
| ×3 | 6 | 15 |
| ×7 | 14 | 35 |
Each row describes the same ratio because both quantities are multiplied by the same scale factor. This is why 2:5, 6:15 and 14:35 are equivalent.
Exam check: adding the same number to both parts does not usually preserve a ratio. 2:5 becomes 5:8 after adding 3, and 5/8 is not equal to 2/5.
Make the units match first
You cannot simplify 2 m : 50 cm directly. Convert 2 m to 200 cm, giving 200:50 = 4:1.
The same idea applies to mass, capacity, time and money.
Decimals and fractional entries
Clear decimals or fractions by multiplying every part by a convenient common factor.
0.6:1.5 → multiply both by 10 → 6:15 → 2:5.
1/2 : 3/4 → multiply both by 4 → 2:3.
Part-to-part and part-to-whole
If boys:girls = 2:3, there are 5 equal parts altogether. Boys are 2/5 of the group and girls are 3/5.
If A:B = 7:4, then A as a fraction of B is 7/4. A fraction comparing one quantity with another can be greater than 1.
Worked example — start from a fraction
If 3/8 of the counters are red, the remaining 5/8 are blue. Therefore red:blue = 3:5, while red:all = 3:8. A fraction of the whole does not directly give the ratio of the two parts.
A multiplicative comparison can also be written as a ratio: if A is 2.5 times B, then A:B = 2.5:1 = 5:2. Equivalently A = (5/2)B; this links ratio to a linear relationship.
Worked example — move between ratio and fraction
Fruit:water = 3:7. There are 10 equal parts altogether.
Fruit is 3/10 of the whole mixture, while water is 7/10. But fruit as a fraction of water is 3/7.
These answers are different because the comparison is different. Always decide whether the question asks for a part of the whole or one quantity as a fraction of another quantity.
Using 1:n
Ratios of the form 1:n are useful for comparison. 4:10 becomes 1:2.5 by dividing both parts by 4.
Do not assume n must be a whole number.
You can also write 4:10 as 0.4:1 by dividing both parts by 10. Proportion means equality of ratios: for example, 2:5 = 6:15 is equivalent to 2/5 = 6/15.
Think Like a Mathematician
Why is ratio multiplicative rather than additive?
Because equivalent ratios preserve a scale factor. 2:3 and 4:6 describe the same comparison even though both entries increased by different absolute amounts.
Common Mistakes
Reversing the order: read the quantities in the order asked.
Forgetting units: convert to the same unit before simplifying.
Adding parts incorrectly: 2:3 has 5 parts in total, not 6.
Cancelling only one part: equivalent ratios require the same operation on every part.
Practice — ratios
1. Simplify 18:30.
Divide both parts by 6: 3:5.
2. Simplify 24:36:60.
Divide all three parts by 12: 2:3:5.
3. Write 1.2:3 in simplest whole-number form.
Multiply by 10: 12:30. Divide by 6: 2:5.
4. Simplify 3/4 : 5/8.
Multiply both parts by 8: 6:5. 6:5.
5. Simplify 2 m : 80 cm.
2 m = 200 cm, so 200:80 = 5:2.
6. Red:blue = 3:7. What fraction of all counters are red?
There are 10 parts altogether. Red is 3/10.
7. A:B = 5:2. Express A as a fraction of B.
5/2.
8. Write 6:15 in the form 1:n.
Divide by 6: 1:2.5. n = 2.5.
9. A mixture has water:juice = 4:1. Is juice one quarter of the whole mixture?
No. There are 5 parts altogether, so juice is 1/5 of the mixture.
10. Three eighths of a group are adults. Write adults:children and adults:whole group.
Children are 5/8 of the group. Adults:children = 3:5; adults:whole = 3:8.
Frequently Asked Questions
Can a ratio include decimals or fractions?
Yes. Multiply every entry by a common factor to clear them if simplest whole-number form is requested.
Does a ratio tell me the total?
No. A ratio fixes the relative amounts; a total, one amount or another condition is needed to determine actual quantities.