Write a ratio from a description and use ratio notation correctly
Understand that order matters in a ratio and that ratios can have more than two parts
Simplify a ratio by dividing by the largest common factor
Understand the difference between a ratio and a fraction
Find equivalent ratios and share a quantity in a given ratio
Apply ratios to real-life problems including map scales and recipes
Real-World Applications
Cooking: Scaling recipes up or down uses ratios — if a recipe for 4 people uses 200 g of flour, a ratio of 1:2 scales it to 400 g for 8 people.
Construction: Mixing materials like concrete or mortar requires exact ratios — cement, sand and gravel are combined in a 1:2:3 ratio to achieve the correct strength.
Finance: Splitting costs or profits between partners uses ratios — if two partners invest in a 3:2 ratio, they share returns in the same proportion.
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A ratio compares two or more quantities of the same type. It shows how much of one there is relative to another.
Ratio notation: \(a : b\) means "for every \(a\) of the first, there are \(b\) of the second"
Key ideas to remember:
• A ratio is a comparison, not a fraction of a total — "for every 3 red counters, there are 2 blue counters" is the ratio 3:2.
• Order matters: 3:2 is not the same as 2:3.
• Ratios can have more than two parts: e.g. cement:sand:gravel = 2:3:5.
Example 1: In a bag there are 5 red and 3 blue counters. Write the ratio of red to blue.
Step 1
Red : Blue = \(5 : 3\) This means: for every 5 red counters, there are 3 blue counters.
Step 2
The total number of parts is \(5 + 3 = 8\).
Note: 5:3 and 3:5 are different ratios. The order must match the order in which items are named.
Equivalent ratios are made by multiplying (or dividing) both parts by the same number — just like equivalent fractions.
Equivalent Ratios — Pattern Table
×1
×2
×3
×4
×5
1:2
2:4
3:6
4:8
5:10
2:3
4:6
6:9
8:12
10:15
Multiplying both parts of a ratio by the same number always gives an equivalent ratio.
✓ Quick Check 1 — What is a Ratio?
Question 1 of 10
Simplifying Ratios
Simplify a ratio by dividing all parts by their largest common factor — exactly like simplifying a fraction. (Full method and worked examples: Lesson 9.2 — Simplifying Ratios.)
⚠ Units must match. To write 50 cm : 2 m as a ratio, first convert to the same unit:
50 cm : 200 cm = \(50:200\) — divide both by 50 — = 1:4
Example 3b: Simplify the ratio 12:18
Step 1
Find the largest number that divides into both 12 and 18 exactly — list their factors: 12 → 1,2,3,4,6,12. 18 → 1,2,3,6,9,18. The largest one they share is 6.
Step 2
Divide both parts by 6: \(12 \div 6 : 18 \div 6 = \mathbf{2:3}\)
For the full method on trickier cases — three-part ratios, decimals, and unit-conversion ratios — see Lesson 9.2 — Simplifying Ratios.
Ratios, Fractions and Percentages
Each part of a ratio can be written as a fraction of the total, and then converted to a percentage.
Example 4: The ratio of boys to girls is \(3:2\). What fraction and percentage are boys?
Step 1
Total parts: \(3 + 2 = 5\)
Step 2
Boys = \(\dfrac{3}{5}\) of the class → \(\dfrac{3}{5} \times 100 = \mathbf{60\%}\)
Step 3
Girls = \(\dfrac{2}{5}\) of the class → \(\dfrac{2}{5} \times 100 = \mathbf{40\%}\)
✓ Quick Check 2 — Simplifying & Fractions
Question 1 of 10
Sharing a Quantity in a Given Ratio
To share an amount in a ratio, use the unitary method: find the value of one part, then multiply. (Full method and worked examples: Lesson 9.3 — Dividing in a Ratio.)
1. Add the ratio parts to find the total number of parts.
2. Divide the total amount by the number of parts (= value of 1 part).
3. Multiply by each ratio number to find each share.
Example 4b: Share £60 between Amir and Bo in the ratio 2:3
Step 1
Total parts = \(2 + 3 = 5\)
Step 2
Value of 1 part = \(£60 \div 5 = £12\)
Step 3
Amir gets \(2 \times £12 = £24\). Bo gets \(3 \times £12 = £36\).
Check
\(£24 + £36 = £60\) ✓
For sharing between three or more parts, and for problems where you're only given one share and need to find the total, see Lesson 9.3 — Dividing in a Ratio.
Scaling Ratios — Finding a Missing Value
If you know one part of a ratio, you can find the other by multiplying or dividing by the same scale factor.
Example 7: Blue:Yellow = 2:5. If blue = 8, how many yellow?
Step 1
Scale factor: \(8 \div 2 = 4\)
Step 2
Yellow = \(5 \times 4 = \mathbf{20}\)
Example 8: Map scale 1:50 000. A road is 3 cm on the map. How far is this in real life?
Step 1
Real distance = \(3 \times 50\,000 = 150\,000\) cm
Tip: For map scales, multiply the map measurement by the scale number to get the real distance in the same units, then convert.
✓ Quick Check 3 — Sharing & Scaling Ratios
Question 1 of 10
Practice
Work out each answer, then type it in and click Check to see if you're right.
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Question
Your answer
Result
1
Share £24 in the ratio 1:3. What is the larger share?
2
Write 15:25 in its simplest form.
3
In a class the ratio of boys to girls is 2:3. There are 30 students. How many girls are there?
4
Share £40 in the ratio 3:5. What is the smaller share?
5
A recipe uses flour and sugar in the ratio 5:2. If you use 350 g of flour, how much sugar do you need?
6
Write 12:18 in its simplest form.
7
Share £63 in the ratio 2:7. What is the larger share?
⚠ Things to Watch Out For
Writing ratios in the wrong order: "Boys to girls in ratio 2:3" means 2 parts boys, 3 parts girls. If there are 4 boys and 6 girls, write 4:6 not 6:4.
Confusing ratio with fraction: Ratio 2:3 does NOT mean ⅔. It means 2 parts out of 5 total (2+3), so boys = 2/5 of the group.
Not simplifying: 6:9 simplifies to 2:3. Divide all parts by the largest common factor. A ratio is fully simplified when all parts share no common factors.
Three-part ratios: A:B:C = 2:3:5 means 10 parts total. Each part = total ÷ 10. Scale all three parts equally.
Units in ratios: Both quantities in a ratio must be in the same units before writing the ratio. 50p : £2 = 50p : 200p = 1:4.