Ratios and proportions, explained
How to read a ratio, simplify it, and scale it without breaking the proportion, with worked examples for mixing, recipes, and pricing.
A ratio is nothing more than a comparison between two amounts. Recipes use them, concrete mixes live or die by them, maps are drawn with them, and every shop price-per-unit label is a ratio in disguise. Proportions are the reason ratios are so powerful: once two amounts are locked in a ratio, the whole relationship can be scaled up or down without changing its character. This guide covers what ratios mean, how to simplify and scale them, why cross-multiplication works, and where the arithmetic quietly goes wrong.
What a ratio actually compares
A ratio written a : b says that for every a of the first thing, there are b of the second. A paint mix of 3 : 1 means three parts paint to one part thinner. The order matters: 3 : 1 is not the same as 1 : 3.
There are two distinct readings, and mixing them up is the source of most ratio mistakes:
- Part-to-part. 3 : 1 compares paint to thinner directly. It does not tell you what fraction of the total is paint.
- Part-to-whole. In the same mix there are 3 + 1 = 4 parts total, so paint is 3/4 of the mixture. That is a different number from the ratio 3:1.
A ratio can also be written as a fraction, a decimal, or a percentage. The notation 3 : 4, the fraction 3/4, and the decimal 0.75 all carry the same relationship when the context is part-to-part comparison of the first quantity to the second. A rate is a special ratio that compares quantities with different units, such as 80 kilometres per 1 hour.
Simplifying a ratio
Ratios are easiest to work with in lowest terms. To simplify, divide every term by their greatest common divisor. Take 6 : 8. The GCD of 6 and 8 is 2, so 6 : 8 simplifies to 3 : 4. Nothing about the relationship changed — you are looking at the same comparison through a smaller lens.
The ratio calculator performs this directly. Enter 6 and 8 in simplify mode and it returns the simplest form 3 : 4, the decimal value a/b = 0.75, the value of a as a percentage of b (75%), and the GCD it used (2). Showing the GCD matters because it is the number you would need to check the simplification by hand.
| Ratio | Simplest form | Decimal (a/b) | a as % of b | a as % of total |
|---|---|---|---|---|
| 3 : 4 | 3 : 4 | 0.75 | 75.0% | 42.9% |
| 6 : 8 | 3 : 4 | 0.75 | 75.0% | 42.9% |
| 9 : 12 | 3 : 4 | 0.75 | 75.0% | 42.9% |
| 2 : 3 | 2 : 3 | 0.667 | 66.7% | 40.0% |
| 1 : 4 | 1 : 4 | 0.25 | 25.0% | 20.0% |
| 5 : 1 | 5 : 1 | 5.0 | 500.0% | 83.3% |
The first three rows are the same ratio at different scales. The last two columns also show why “a as a percentage of b” and “a as a percentage of the total” must never be confused: for 3 : 4 they are 75% and 42.9%, and both are useful depending on the question.
Scaling without breaking the proportion
A proportion is a statement that two ratios are equal, such as 3 : 4 = 9 : 12. To move from one ratio to its equal partner, multiply or divide every term by the same non-zero number. The 3 : 4 ratio becomes 9 : 12 because both terms were multiplied by 3. This is the whole trick behind scaling a recipe, a mix, or a drawing.
When one part is known and the other is not, the scaling factor does the work. If 3 : 4 = 9 : ?, the first term grew from 3 to 9, a factor of 3, so the second term must also grow by 3: 4 × 3 = 12. The ratio calculator’s solve mode uses the equivalent formula directly — the missing value is c × (b ÷ a) — so entering a = 3, b = 4, c = 9 returns ? = 12. It is the same proportion expressed as arithmetic.
The critical rule is that scaling means multiplying, never adding. If a 2 : 3 mix is doubled, it becomes 4 : 6. If you instead add 2 to each side you get 4 : 5, which is a different mixture altogether. Adding the same amount to both terms changes the ratio; multiplying both terms by the same amount preserves it.
Cross-multiplication, and why it works
When a proportion is written as fractions, a : b = c : d becomes a/b = c/d. Cross-multiplication gives the shortcut a × d = b × c, which rearranges to whatever unknown you need.
It works because multiplying both sides of a/b = c/d by the product bd clears the denominators: a·d = b·c. The cross pattern is just a memory aid for that single step, not a separate rule.
Two worked examples:
- 6/8 = x/12. Cross-multiply: 6 × 12 = 8 × x, so 72 = 8x and x = 9.
- A recipe uses flour to sugar in a 3 : 2 ratio. You have 450 g of flour. Then 3/2 = 450/x, so 3x = 900 and x = 300 g of sugar.
Always keep matching quantities in matching positions. In the second example, flour must sit over sugar on both sides; putting flour over sugar on one side and sugar over flour on the other inverts the answer.
Ratios in the real world
Mixing and construction
Concrete is commonly specified as a 1 : 2 : 3 mix of cement : sand : aggregate by volume. A three-term ratio scales exactly like a two-term one: to make 12 parts total of concrete you would need 2 parts cement, 4 parts sand, and 6 parts aggregate. The concrete calculator converts a slab’s dimensions into the volume you need, which you then divide according to the mix ratio.
Recipes and batch cooking
Scaling a recipe is a proportion problem. If a sauce is 5 : 2 tomato to cream and you want to use 750 g of tomato, the cream is 750 × (2 ÷ 5) = 300 g. Doubling every quantity preserves the taste; adding a fixed amount to every quantity does not.
Map scale
A map scale of 1 : 50,000 means one unit on the map represents 50,000 of the same unit on the ground. One centimetre on the map is 50,000 cm, or 500 m, or 0.5 km. So 8 cm on the map is 8 × 0.5 = 4 km of real distance. The ratio has no units of its own; it only works because both sides are expressed in the same unit before you multiply.
Pricing and unit rates
Unit prices are ratios that make different package sizes comparable. A 500 ml bottle at $3.20 costs 320 ÷ 500 = 0.64 cents per ml. A 750 ml bottle at $4.50 costs 450 ÷ 750 = 0.60 cents per ml. The second is the better buy even though its sticker price is higher. This is the same comparison a unit-rate calculation makes every time.
Speed, fuel, and two-stroke mixes
Speed is a rate: 200 km in 2.5 hours is 200 ÷ 2.5 = 80 km/h. Two-stroke fuel is often mixed at 1 : 50, meaning one part oil to fifty parts petrol. For 5 litres of petrol, the oil is 5000 ml ÷ 50 = 100 ml. Notice the units had to be matched first — litres to millilitres — before the ratio could be applied. The unit converter is useful precisely for getting both sides into the same unit before you scale.
Connecting ratios to percentages
Ratios and percentages describe the same underlying relationships in different clothes. For a ratio a : b, the first quantity as a percentage of the second is (a ÷ b) × 100. For 3 : 4 that is 75%. The first quantity as a percentage of the whole is a ÷ (a + b) × 100, which for 3 : 4 is 3 ÷ 7 = 42.9%. Both answers are correct; they answer different questions. Getting comfortable switching between the two is largely what the percentage calculator is for, and it is a common place for a “3 is 75% of 4” figure to be mistaken for “3 is 75% of the total”.
Common errors
- Treating a part-to-part ratio as part-to-whole. 3 : 5 does not mean the first item is 3/5 of everything. The total is 8, so it is 3/8, or 37.5%.
- Scaling by addition. Multiplying both terms preserves a ratio; adding the same amount to both does not.
- Forgetting to convert units. A ratio of 2 m to 50 cm is not 2 : 50. Convert first: 200 cm : 50 cm = 4 : 1.
- Cross-multiplying mismatched pairs. The unknown must sit in the same position on both sides of the proportion.
- Assuming a ratio fixes the total. A 2 : 3 ratio could describe 2 and 3, or 200 and 300, or any multiple. Ratios constrain the relationship, not the size.
Frequently asked questions
What is the difference between a ratio and a fraction?
A fraction usually describes part of a whole, while a ratio compares two quantities to each other. The notation often looks the same — 3/4 can be read either way — so the deciding factor is the question being asked. If the denominator is the total, it is a fraction. If it is the second quantity being compared, it is a ratio.
Can a ratio contain zero?
A ratio can have zero as one of its parts, such as 0 : 5 or 5 : 0, but the second term cannot be zero when you convert it to a fraction, because division by zero is undefined. A 5 : 0 ratio is a valid comparison but has no decimal or percentage form.
How do I scale a recipe up or down?
Find the factor by which the ingredient you can measure changed, then apply that same factor to every other ingredient. If the original calls for 2 cups and you are using 5 cups, the factor is 2.5, so every other quantity is multiplied by 2.5.
What does a map scale of 1 : 100,000 mean?
One unit on the map equals 100,000 of the same unit on the ground. One centimetre becomes 100,000 cm, which is 1,000 m, or 1 km. To convert any map measurement, multiply by 100,000 and then convert the resulting centimetres into kilometres.