Ratio & Proportion Calculator
Simplify ratios, solve proportions, and divide amounts with step-by-step solutions and interactive visualisations. Perfect for GCSE and KS3 maths.
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What is a Ratio?
A ratio is a way of comparing two or more quantities. It shows the relative size of different amounts. Written as 2:3 or “2 to 3”, it means “for every 2 of the first quantity, there are 3 of the second.”
Ratios are dimensionless — they have no units. Whether comparing grams, litres, or people, a ratio of 2:3 always means the same proportion. This is what makes them so useful across mathematics, science, and everyday life.
Maps
1:50,000 scale — 1 cm on map = 500 m real
Recipes
2:1 flour to butter in shortcrust pastry
Finance
£1 : $1.27 exchange rate
Ratio, Fraction & Percentage
These three representations all describe proportions but in different ways. For a group split in ratio 2:3:
| Form | Part A | Part B | Note |
|---|---|---|---|
| Ratio | 2 | 3 | Compares parts to each other |
| Fraction | 2/5 | 3/5 | Each part as share of whole |
| Percentage | 40% | 60% | Per hundred of the whole |
To convert ratio A:B to fractions: Part A fraction = A/(A+B), Part B fraction = B/(A+B). Multiply by 100 for percentages.
Simplifying Ratios
A simplified ratio uses the smallest possible whole numbers while keeping the same proportion. The method relies on finding the Greatest Common Factor (GCF) of all ratio parts.
The GCF Method
- 1. List factors of each ratio part.
- 2. Find the largest factor common to all parts — this is the GCF.
- 3. Divide every part by the GCF.
- 4. Verify: no common factor greater than 1 should remain.
12 : 18
GCF = 6
2 : 3
15 : 25
GCF = 5
3 : 5
24 : 36 : 48
GCF = 12
2 : 3 : 4
Equivalent Ratios
Equivalent ratios represent the same proportion with different numbers — just like equivalent fractions. You create a family of equivalent ratios by multiplying (or dividing) every part by the same non-zero number.
Family of equivalent ratios for 2:3:
Note: Adding the same number to all parts does NOT give an equivalent ratio. 2:3 + 2 = 4:5, but 4:5 ≠ 2:3. You must multiply.
Solving Proportions (Cross Multiplication)
A proportion states that two ratios are equal: a:b = c:x. Cross multiplication gives us: a × x = b × c, so x = (b × c) ÷ a.
Example 1
Solve: 2:5 = 6:x
Cross multiply: 2 × x = 5 × 6
2x = 30
x = 15
Verify: 2:5 = 6:15 ✓ (both simplify to 2:5)
Example 2
Solve: 3:7 = 9:x
Cross multiply: 3 × x = 7 × 9
3x = 63
x = 21
Verify: 3:7 = 9:21 ✓
Dividing Amounts in a Ratio
One of the most common GCSE topics. The key is to first find how many “parts” there are, calculate the value of one part, then scale up for each share.
Add all ratio parts (total parts)
Divide amount by total parts (one part)
Multiply one part by each ratio number
Divide £100 in ratio 2:3
Total parts = 2+3 = 5
One part = £100 ÷ 5 = £20
Answer: £40 and £60
Divide £240 in ratio 1:2:3
Total parts = 1+2+3 = 6
One part = £240 ÷ 6 = £40
Answer: £40, £80 and £120
Split 500 g in ratio 3:7
Total parts = 3+7 = 10
One part = 500 ÷ 10 = 50 g
Answer: 150 g and 350 g
Combining Ratios using LCM
When you have A:B and B:C and need a single three-part ratio A:B:C, the shared “B” value must be made equal in both ratios. Use the LCM of both B values.
Worked Example
A:B = 2:3 and B:C = 4:5
B values are 3 and 4. LCM(3, 4) = 12
Scale A:B by ×4 → 8:12
Scale B:C by ×3 → 12:15
A:B:C = 8:12:15
When B values are already equal, no scaling is needed:
A:B = 2:3 and B:C = 3:5
B is already 3 in both → combine directly
A:B:C = 2:3:5
Ratios in Real Life
Ratios appear everywhere — from engineering to cooking. Here are five applications you might encounter in GCSE problems:
Map Scales
1:25,000 means 4 cm on the map = 1 km in reality. Ordnance Survey maps use this scale.
Recipe Scaling
A recipe for 4 people uses flour:butter = 3:1. For 12 people, scale up by ×3: still 3:1 proportion.
Currency Exchange
1 GBP : 1.27 USD is a proportion. £350 × 1.27 = $444.50 — the same ratio applied at scale.
Concrete Mixing
Standard concrete is cement:sand:gravel = 1:2:3 by volume. Change any ratio and the concrete weakens.
Screen Aspect Ratios
16:9 (widescreen HD), 4:3 (old TV), 21:9 (cinema ultrawide) — these are all simplified ratios.
Science Concentrations
A 1:5 dilution of solution means 1 part solution to 5 parts water — the same ratio concept.
GCSE Exam Tips
Always Simplify
Exam answers should always be in simplest form. Check if you can divide further.
Order Matters!
“Boys to girls” means boys:girls. Read the question carefully!
Add for Total Parts
When sharing, add ALL ratio parts first: 2:3:5 = 10 total parts.
Cross Multiply
For proportions a:b = c:d, use a×d = b×c to find unknowns.
Check Your Answer
Verify shares add up to the total amount. £40 + £60 = £100 ✓
LCM for Combining
When combining A:B and B:C, use LCM to equalise the B values — not GCF.
Frequently Asked Questions
What is the difference between ratio and proportion?
A ratio compares two or more quantities (e.g. 2:3). A proportion is a statement that two ratios are equal (e.g. 2:3 = 6:9). Proportions are used to find unknown values using cross multiplication.
How do you convert a ratio to a fraction?
For ratio A:B, Part A is A/(A+B) of the whole. So 2:3 means Part A = 2/5 and Part B = 3/5. The denominator is the total number of parts.
Can ratios contain decimals?
Yes, ratios can have decimal parts (e.g. 2.5:1), but they are usually converted to whole numbers first. Multiply all parts by the smallest power of 10 to clear decimals, then simplify.
When do I use GCF vs LCM?
Use GCF to simplify ratios (divide all parts by the GCF). Use LCM when combining two ratios A:B and B:C — you need to find the LCM of both B values to equalise them.
Why does my simplified ratio not match the expected answer?
Make sure you have divided ALL parts by the GCF. Also check whether the original ratio contains decimals that need to be converted to integers first.
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