A ratio compares two quantities; a proportion says two ratios are equal. Together they power recipes, maps, scale models and mixing — anywhere amounts have to keep the same relationship.
What a ratio is
A ratio compares two quantities side by side. Writing \( 2 : 3 \) means "for every 2 of the first, there are 3 of the second". Order matters — \( 2 : 3 \) is not the same as \( 3 : 2 \), just as two parts water to three parts juice is a different drink from three parts water to two parts juice.
Ratios simplify exactly like fractions: divide both parts by a common factor and the comparison stays the same. So \( 4 : 6 = 2 : 3 \), because both sides shrink by the factor \( 2 \) while the relationship between them is untouched.
Drag the slider below to change how many beads sit on each side and watch the ratio settle into its simplest form, just as a fraction would.
Ratios as fractions of the whole
A ratio doesn't just compare two amounts — it also tells you how to slice a whole into shares. The ratio \( 2 : 3 \) splits something into \( 2 + 3 = 5 \) equal parts. The first share is \( \frac{2}{5} \) of the whole and the second is \( \frac{3}{5} \).
Suppose you split \$20 in the ratio \( 2 : 3 \). First find the size of one part: there are \( 5 \) parts in all, so one part is \( 20 \div 5 = \$4 \). Then the two shares are \( 2 \times 4 = \$8 \) and \( 3 \times 4 = \$12 \). As a check, \( 8 + 12 = 20 \) — the whole is accounted for.
Proportion: when two ratios are equal
A proportion is a statement that two ratios are equal. Written as fractions, \( \frac{2}{3} = \frac{8}{12} \) is a proportion — both describe the very same relationship at different scales.
This is what lets you find a missing value: scale both parts by the same factor. To solve \( 2 : 3 = \square : 12 \), look at the parts you know. Since \( 3 \times 4 = 12 \), the whole ratio has been scaled up by \( 4 \), so \( \square = 2 \times 4 = 8 \).
Unit rates and best buys
A unit rate is the amount per one — the price of a single item, the distance in one hour, the cost per litre. Finding it is just dividing to bring the second quantity down to \( 1 \).
If \( 3 \) apples cost \$1.50, then one apple costs \( 1.50 \div 3 = \$0.50 \). Unit rates let you compare deals fairly: once everything is measured per one, the cheaper option is simply the smaller number.
Cross-multiplication
When two fractions are equal, their cross-products are equal too. For \( \frac{a}{b} = \frac{c}{d} \), this means \( ad = bc \). It is a fast, reliable way to solve a proportion for an unknown.
To solve \( \frac{4}{5} = \frac{x}{20} \), cross-multiply to get \( 5x = 4 \times 20 = 80 \), then divide both sides by \( 5 \):
\[ \frac{4}{5} = \frac{x}{20} \;\Rightarrow\; 5x = 80 \;\Rightarrow\; x = 16 \]In words A ratio carries no units of its own — it is a pure comparison. "2 parts water to 1 part juice" describes exactly the same recipe whether you measure in cups or in litres, because both sides are measured the same way and the units cancel out.
Tip To keep a ratio equal, scale both parts by the same factor. Multiplying just one part changes the recipe — double only the juice and the drink turns out wrong. Whatever you do to one side, do to the other.
- Find a common factor of both parts. The greatest common factor of \( 12 \) and \( 18 \) is \( 6 \).
- Divide each part by \( 6 \): \( 12 \div 6 = 2 \) and \( 18 \div 6 = 3 \).
So \( 12 : 18 = \mathbf{2 : 3} \). Both parts shrank by the same factor, so the comparison is unchanged.
- The scale says every \( 1 \) cm on the map stands for \( 50000 \) cm in reality.
- For \( 3 \) cm, scale up by \( 3 \): \( 3 \times 50000 = 150000 \) cm.
- Convert to a friendlier unit: \( 150000 \) cm \( = 1500 \) m \( = 1.5 \) km.
So \( 3 \) cm on the map is \( \mathbf{1.5 \text{ km}} \) on the ground.
Practice
Try each one yourself, then reveal the full solution.
1. Simplify \( 15 : 25 \).
Look for a common factor of both parts. The greatest common factor of \( 15 \) and \( 25 \) is \( 5 \).
Divide each part by \( 5 \): \( 15 \div 5 = 3 \) and \( 25 \div 5 = 5 \).
So \( 15 : 25 = \mathbf{3 : 5} \).
2. Solve \( 2 : 5 = x : 20 \).
Compare the parts you know. The second part goes from \( 5 \) to \( 20 \), and \( 5 \times 4 = 20 \), so the ratio is scaled up by \( 4 \).
Scale the first part by the same factor: \( x = 2 \times 4 = 8 \).
So \( x = \mathbf{8} \).
3. Paint is mixed blue to yellow in the ratio \( 3 : 2 \). For \( 15 \) litres of blue, how much yellow?
The blue part goes from \( 3 \) to \( 15 \), and \( 3 \times 5 = 15 \), so the recipe is scaled up by \( 5 \).
Scale the yellow part by the same factor: \( 2 \times 5 = 10 \). The full mix is \( 3 : 2 \) scaled by \( 5 \), which is \( 15 : 10 \).
So you need \( \mathbf{10} \) litres of yellow.