What Is a Ratio?
A ratio compares two (or more) quantities. If a bag contains 3 red marbles and 5 blue ones, the ratio of red to blue is 3:5. That's it. It tells you the relative size of one thing compared to another — not the actual amounts, just the relationship between them.
Ratios can be written three ways:
All three mean the same thing. The fraction form is especially useful because it lets you use fraction operations on ratios.
Simplifying a Ratio
Like fractions, ratios should be reduced to their simplest form by dividing both parts by their GCF.
GCF(12, 18) = 6
12 ÷ 6 : 18 ÷ 6 = 2:3
In cooking, if a recipe uses 12 tablespoons of butter and 18 tablespoons of sugar, the ratio 12:18 simplifies to 2:3. You could scale it down to 2 tablespoons of butter and 3 of sugar and get the same result proportionally — though good luck baking anything useful with that.
Part-to-Part vs Part-to-Whole Ratios
There are two different ways to frame a ratio:
| Type | Example | Meaning |
|---|---|---|
| Part-to-Part | 3:5 (red:blue) | Comparing parts to each other |
| Part-to-Whole | 3:8 (red:total) | Comparing one part to the total |
If 3 out of every 8 marbles are red, 3:5 is the part-to-part ratio and 3:8 is the part-to-whole ratio. Percentages are always part-to-whole ratios expressed out of 100.
What Is a Proportion?
A proportion says two ratios are equal. It's the equation form of a ratio comparison.
(both simplify to 3:5, so they're equivalent)
Think of a proportion as saying "these two ratios describe the same relationship, just at different scales." A map might say 1 cm = 10 km — that's a proportion. Double the distance on the map and you double the real distance too.
Solving Proportions with Cross-Multiplication
The most useful thing about proportions is you can solve for an unknown value using cross-multiplication. If three values are known and one is missing, set up the proportion and cross-multiply.
Example: 3/5 = x/20
Cross-multiply: 3 × 20 = 5 × x
60 = 5x
x = 60 ÷ 5 = 12
Check: 3/5 = 12/20 → both simplify to 3:5. ✓
Real-World Examples
Scaling a Recipe
A pancake recipe serves 4 and uses 200g of flour. How much flour for 10 people?
4x = 200 × 10 = 2000
x = 2000 ÷ 4 = 500g
Map Distance
On a map, 2 cm = 50 km. Two cities are 7 cm apart. What's the actual distance?
2x = 350
x = 175 km
Mixing Paint
To make a certain shade of green, you mix blue and yellow in a 3:2 ratio. You want 25 litres of green. How much of each?
Blue = (3/5) × 25 = 15 litres
Yellow = (2/5) × 25 = 10 litres
Unit Rate
A car travels 240 miles on 8 gallons of fuel. How far per gallon?
Unit rates are a proportion where one side is per 1 unit — miles per hour, price per kg, calories per serving. They're everywhere.
Ratio Tables
A ratio table is just a way to list equivalent ratios. If the base ratio is 2:3, you can extend it:
| Part A | Part B | Ratio |
|---|---|---|
| 2 | 3 | 2:3 |
| 4 | 6 | 2:3 |
| 10 | 15 | 2:3 |
| 100 | 150 | 2:3 |
All equivalent. Multiplying both parts by the same number doesn't change the ratio — same as how equivalent fractions work.
Ratios with Three Terms
Ratios don't have to be just two numbers. Three-way ratios (a:b:c) describe a split between three groups.
Total parts = 2 + 3 + 4 = 9
Each part = £360 ÷ 9 = £40
Share A = 2 × £40 = £80
Share B = 3 × £40 = £120
Share C = 4 × £40 = £160
Check: 80 + 120 + 160 = 360 ✓
Using a Calculator
The SolveCalc ratio calculator lets you simplify ratios, find equivalent ratios, and solve proportions with one unknown. It's handy when the numbers get messy.
Conclusion
The core idea behind ratios and proportions is simple: ratios compare quantities, and proportions say two ratios are equal. From there, cross-multiplication lets you solve for unknowns in almost any practical scenario — recipes, maps, fuel efficiency, sharing, mixing. The key steps are: write the ratio clearly, simplify if needed, and set up the proportion as two equal fractions before cross-multiplying. That covers 90% of ratio problems you'll ever see.