Ratio Calculator
Simplify ratios, scale recipes, divide amounts in ratio, and find equivalent ratios instantly. Enter your values below for step-by-step solutions. Ideal for GCSE maths, cooking, map reading, and everyday problem-solving.
Enter a ratio A:B or A:B:C to find its simplest form by dividing by the GCF (HCF).
Scale a ratio by a factor or find what ratio gives a target total.
Enter either a scale factor or a target total — the other will be calculated.
Split a total amount between two or three parties according to a ratio.
Find 5 equivalent ratios for any ratio A:B.
What is a Ratio?
A ratio is a mathematical way to compare two or more quantities of the same type. It tells you how much of one thing there is compared to another. Ratios are written using a colon, such as 3:1 or 2:3:5, or as fractions (3/1 or 2/5).
For example, if a bag contains 3 red sweets and 1 blue sweet, the ratio of red to blue is 3:1. This means for every 3 red sweets, there is 1 blue sweet. Ratios are everywhere in daily life: recipes, maps, currencies, mixing paint, diluting squash, and financial reports.
Ratios can be simplified, scaled, and used to divide totals
Types of Ratios: Part-to-Part vs Part-to-Whole
Understanding the two main types of ratios helps you apply them correctly:
| Type | Definition | Example |
|---|---|---|
| Part-to-Part | Compares one part of a group to another part | Girls to boys in a class: 3:2 (3 girls for every 2 boys) |
| Part-to-Whole | Compares one part to the total group | Girls to total students: 3:5 (3 out of every 5 are girls) |
| Three-part ratio | Compares three quantities | Red:blue:green paint = 2:3:5 |
Part-to-whole ratios are directly related to fractions and percentages. If girls to total is 3:5, then 3/5 (60%) of students are girls.
Simplifying Ratios: Step-by-Step Guide
To simplify a ratio, you divide all parts by their Highest Common Factor (HCF), also called the Greatest Common Factor (GCF). The Euclidean algorithm is the most efficient method for finding the HCF.
The Euclidean Algorithm for HCF
To find HCF(a, b): repeatedly replace the larger number with the remainder when the larger is divided by the smaller, until the remainder is 0. The last non-zero remainder is the HCF.
Worked Example 1: Simplify 24:36
Step 1: Find HCF(24, 36) using the Euclidean algorithm:
Step 2: Divide both parts by 12:
Result: 24:36 simplified = 2:3
Worked Example 2: Simplify 15:25:35
Step 1: Find HCF(15, 25, 35):
Step 2: Divide all three parts by 5:
Result: 15:25:35 simplified = 3:5:7
Equivalent Ratios and Proportions
Equivalent ratios are ratios that represent the same relationship. You create them by multiplying or dividing both parts of a ratio by the same number.
| Multiplier | Equivalent Ratio | In Words |
|---|---|---|
| ×2 | 4:6 | 4 to 6 |
| ×3 | 6:9 | 6 to 9 |
| ×4 | 8:12 | 8 to 12 |
| ×5 | 10:15 | 10 to 15 |
| ×10 | 20:30 | 20 to 30 |
Two ratios are in proportion if they are equivalent. This is written a:b = c:d, or as the equation a/b = c/d. You can cross-multiply to check: ad = bc.
Dividing Amounts in Ratio
One of the most practical uses of ratios is dividing a total amount between multiple parties. This appears in:
- Inheritance: An estate of £180,000 divided between two children in ratio 2:1
- Business profits: Three partners sharing profits in ratio 3:2:1
- Splitting a restaurant bill based on how much each person ate
- Recipe scaling: Adjusting ingredient quantities maintaining their ratios
Worked Example 3: Divide £240 in the ratio 3:5
Step 1: Find the total number of parts: 3 + 5 = 8 parts
Step 2: Find the value of one part: £240 ÷ 8 = £30 per part
Step 3: Multiply each ratio part by the value per part:
Check: £90 + £150 = £240 ✓
Worked Example 4: Share £3,600 profit in ratio 3:2:1
Step 1: Total parts = 3 + 2 + 1 = 6
Step 2: One part = £3,600 ÷ 6 = £600
Step 3: Each share:
Check: £1,800 + £1,200 + £600 = £3,600 ✓
Map Scales and Ratio
Maps use ratios to represent real distances. The Ordnance Survey (OS) uses several standard map scales:
| OS Map Series | Scale | 1 cm on map equals |
|---|---|---|
| OS Explorer (Orange) | 1:25,000 | 250 m on the ground |
| OS Landranger (Pink) | 1:50,000 | 500 m on the ground |
| OS Road (Blue) | 1:250,000 | 2.5 km on the ground |
| OS Travel Map | 1:1,000,000 | 10 km on the ground |
A 1:50,000 scale means 1 cm on the map = 50,000 cm = 500 m = 0.5 km in real life. If a path on the map is 4 cm long, the actual path is 4 × 500 m = 2 km.
Recipe Scaling with Ratios
Ratios are essential in cooking when scaling recipes up or down. A Victoria sponge recipe for 8 servings uses:
| Ingredient | For 8 (original) | Ratio | For 4 (÷2) | For 16 (×2) |
|---|---|---|---|---|
| Self-raising flour | 200g | 4 | 100g | 400g |
| Butter | 200g | 4 | 100g | 400g |
| Caster sugar | 200g | 4 | 100g | 400g |
| Eggs | 4 | 1 | 2 | 8 |
The classic ratio for a sponge cake is flour:butter:sugar = 1:1:1, with approximately 1 egg per 50g of flour. This is why experienced bakers can scale any sponge recipe mentally.
Finance: Currency Exchange as a Ratio
Exchange rates are ratios between currencies. If £1 = $1.27 (USD), the ratio of pounds to dollars is 1:1.27. To convert:
- £500 to USD: 500 × 1.27 = $635
- $1,000 to GBP: 1,000 ÷ 1.27 = £787.40
Similarly, interest rates express the ratio of interest earned to principal: a 5% annual interest rate is the ratio 5:100 (or 1:20) of interest to original investment.
GCSE Ratio: Exam Technique and Mark Scheme
At GCSE level (Edexcel, AQA, OCR), ratio questions typically award marks as follows:
| Step | Action | Marks |
|---|---|---|
| 1 | Find total number of parts by adding ratio values | M1 |
| 2 | Divide total amount by number of parts | M1 |
| 3 | Multiply each part by the value per unit | A1 |
| 4 | Check: sum of all shares equals original total | C1 (checking) |
A common mistake is to add the ratio parts incorrectly or to divide by just one part of the ratio instead of the sum. Always write out "total parts = A + B" explicitly in your working.
10 Worked Examples with Full Solutions
Example 5: Simplify 48:72
HCF(48,72): 72 = 1×48+24; 48 = 2×24+0 → HCF = 24
48÷24 = 2, 72÷24 = 3 → Answer: 2:3
Example 6: Scale ratio 3:7 so the first part equals 12
Scale factor = 12 ÷ 3 = 4
Scaled ratio = 3×4 : 7×4 = 12:28
Example 7: Divide 500g in ratio 2:3
Parts = 5; one part = 100g; A = 200g, B = 300g → 200g and 300g
Example 8: A map scale of 1:25,000. A path on the map is 6.4 cm. How far in km?
Real distance = 6.4 × 25,000 = 160,000 cm = 1,600 m = 1.6 km
Example 9: Express 45 minutes to 2 hours as a simplified ratio
Convert to same units: 45 min : 120 min; HCF = 15
45÷15 : 120÷15 = 3:8
Example 10: Three siblings share an inheritance of £156,000 in ratio 4:3:2. How much does each receive?
Parts = 9; one part = £17,333.33; A = £69,333.33, B = £52,000, C = £34,666.67
Or: £156,000 ÷ 9 × 4 = £69,333, × 3 = £52,000, × 2 = £34,667
Frequently Asked Questions
How do you simplify a ratio?
To simplify a ratio, find the Highest Common Factor (HCF) of all the numbers in the ratio, then divide each part by that HCF. For example, to simplify 12:18: the HCF of 12 and 18 is 6 (using the Euclidean algorithm: 18 = 1×12 + 6, then 12 = 2×6). Divide both parts: 12÷6 = 2 and 18÷6 = 3. The simplified ratio is 2:3. A ratio is fully simplified when the only common factor of all parts is 1.
What is the difference between ratio and proportion?
A ratio compares two or more quantities, written as a:b (e.g. 3:1). A proportion is a statement that two ratios are equal: a:b = c:d. In other words, a ratio is a comparison; a proportion is an equation. For example, the ratio of flour to butter in a recipe might be 3:1. If you double the recipe, the ratio stays 3:1 — the two ratios (original and doubled) are in proportion. Proportions are solved by cross-multiplication: if 3:1 = x:4, then x = 12.
How do you divide £240 in the ratio 3:5?
Step 1: Add the ratio parts: 3 + 5 = 8 total parts. Step 2: Find the value of one part: £240 ÷ 8 = £30. Step 3: Multiply each part: 3 × £30 = £90 and 5 × £30 = £150. Check: £90 + £150 = £240. This method works for any division in ratio problem — always add all ratio parts first, divide the total, then multiply.
What does 1:50,000 mean on a map?
A map scale of 1:50,000 means that 1 unit on the map represents 50,000 of the same units in real life. In practical terms: 1 cm on the map = 50,000 cm = 500 metres = 0.5 km on the ground. This is the scale of Ordnance Survey Landranger maps (the pink ones). So if a walking route measures 8 cm on the map, the actual distance is 8 × 0.5 km = 4 km.
How do you simplify 12:18 to lowest terms?
Use the Euclidean algorithm to find HCF(12, 18): 18 = 1 × 12 + 6; 12 = 2 × 6 + 0. The HCF is 6. Divide both parts by 6: 12 ÷ 6 = 2 and 18 ÷ 6 = 3. The simplest form is 2:3. You can verify this is fully simplified because 2 and 3 share no common factor greater than 1 (they are coprime).
What is the golden ratio?
The golden ratio (φ ≈ 1.6180339887) is a special irrational number where a line divided into two parts a and b (with a > b) satisfies a:b = (a+b):a. Numerically, 1:φ ≈ 1:1.618. It appears throughout nature (spiral patterns in shells and sunflowers), classical art and architecture (the Parthenon, da Vinci's Vitruvian Man), and modern design. The golden ratio is also the limiting ratio of consecutive Fibonacci numbers: 89÷55 ≈ 1.618, 144÷89 ≈ 1.618.