What a Ratio Represents
A ratio compares two quantities that share the same units. If a paint formula calls for 4 parts red and 6 parts white, the ratio of red to white is 4:6. Here's the thing - it doesn't tell you anything about the actual volume. Just the relationship between the two amounts.
There are two distinct ratio types, and mixing them up is a common source of confusion:
| Type | Description | Example (4 cats, 6 dogs) |
|---|---|---|
| Part-to-Part | Compares one category to another category | Cats to dogs: 4:6 |
| Part-to-Whole | Compares one category to the combined total | Cats to all pets: 4:10 |
Ratios can be written in three equivalent forms:
The fraction form is especially handy because you can apply all the fraction rules you already know - simplification, multiplication, cross-multiplication - directly to ratios.
Simplifying a Ratio by GCF
A ratio is in its simplest form when the two numbers share no common factor other than 1. To simplify, find the Greatest Common Factor (GCF) of both parts and divide each by it.
Factors of 20: 1, 2, 4, 5, 10, 20
Factors of 28: 1, 2, 4, 7, 14, 28
GCF = 4
20 ÷ 4 = 5 | 28 ÷ 4 = 7
Simplified ratio: 5:7
Always divide both parts by the same number. Dividing only one side changes the ratio entirely. Quick check: if the two resulting numbers still share a common factor, you're not done - divide again.
GCF(36, 48) = 12
36 ÷ 12 : 48 ÷ 12 = 3:4
Equivalent Ratios: Scaling Up and Down
Multiply or divide both parts of a ratio by the same non-zero number and you get an equivalent ratio. Same relationship, different scale. That's how you scale a recipe, resize a blueprint, or convert a map distance.
| Scale Factor | Ratio | Equivalent? |
|---|---|---|
| Original | 3:5 | Yes |
| ×2 | 6:10 | Yes |
| ×5 | 15:25 | Yes |
| ×10 | 30:50 | Yes |
All four ratios above simplify back to 3:5. That's the core idea behind scaling: the ratio stays constant even when the actual quantities change. When a builder doubles every measurement in a blueprint, the proportions of the building stay identical.
Three-term ratios work the same way. If a concrete mix uses sand, gravel, and cement in the ratio 3:4:2, scaling to 18 parts total means multiplying each term by 2 to get 6:8:4.
Setting Up a Proportion Equation
A proportion is just a statement that two ratios are equal:
The key step is making sure the right quantities line up on each side. Both numerators should measure the same type of thing, and both denominators should too. For example, if your numerators are distances, your denominators should both be times.
Example - Map Scale: A map uses a scale of 3 cm to represent 75 km. Two towns are 11 cm apart on the map. How far apart are they in reality?
3/75 = 11/x
(Set up so the same types of units occupy the same position on each side)
Once you've set it up correctly, solving is the easy part.
Solving with Cross-Multiplication
When one value in a proportion is unknown, multiply the values that are diagonal from each other and solve the resulting equation.
Map example continued: 3/75 = 11/x
3 × x = 75 × 11
3x = 825
x = 825 ÷ 3 = 275 km
Recipe Example
A soup recipe designed for 3 servings requires 450 ml of broth. How much broth is needed for 8 servings?
3x = 450 × 8 = 3600
x = 3600 ÷ 3 = 1200 ml
Speed Example
A cyclist covers 28 km in 40 minutes. At the same pace, how far will they travel in one hour (60 minutes)?
40x = 28 × 60 = 1680
x = 1680 ÷ 40 = 42 km
In every case the approach is the same: write the proportion with matching units in matching positions, then cross-multiply and divide. The CalcSolver Pro ratio calculator handles this automatically when the numbers get messy.