Mixing cement for a wall, adjusting a recipe for guests, reading the scale on a map, comparing the price of two packs of rice — all of these everyday tasks rely on the same idea: comparing quantities. That idea is called a ratio, and when two such comparisons are equal, we call it a proportion.
Ratio and proportion are among the most useful topics in basic maths because they connect to fractions, percentages, speed, money and science. In this guide you will learn what ratios are, how to simplify them, how to share a quantity in a given ratio, how to solve proportion problems, and how direct and inverse proportion differ. You will also see real-life examples and practice questions. For a quick check on your own numbers, try our free ratio calculator.
Quick Answer: Ratio and Proportion in One Minute
- A ratio compares two quantities of the same kind, written a:b or a/b. Example: 24:36 simplifies to 2:3.
- A proportion states that two ratios are equal: a/b = c/d. Solve it by cross multiplication (a × d = b × c).
- To share a quantity in the ratio 2:3, add the parts (5), divide the total by 5, then multiply by 2 and 3.
What Is a Ratio?
A ratio is a way of comparing two or more quantities by showing how many times one contains the other. It can be written in three ways: with a colon (2:3), as a fraction (2/3), or in words (“2 to 3”). All three mean the same thing.
If a class has 12 boys and 18 girls, the ratio of boys to girls is 12:18. The order matters. The ratio of girls to boys is 18:12, which is a different comparison. Always read the question carefully and write the numbers in the order asked.
Ratios have two important features:
- Same units. The quantities compared must be in the same unit. To compare 50 cm and 2 m, first convert 2 m to 200 cm, giving 50:200, which simplifies to 1:4.
- No units in the final ratio. Once the units match, the ratio is just a pair of plain numbers.
Each number in a ratio is called a term. In a:b, the first term (a) is the antecedent and the second term (b) is the consequent.
How to Simplify a Ratio
A ratio is in its simplest form when its terms have no common factor other than 1. Simplifying works exactly like reducing a fraction.
Step-by-Step Method
- Find the greatest common divisor (GCD) of all terms.
- Divide each term by the GCD.
Example 1
Simplify 24:36. The GCD of 24 and 36 is 12. Divide both: 24 ÷ 12 = 2 and 36 ÷ 12 = 3. The simplified ratio is 2:3.
Example 2
Simplify 45:60. The GCD is 15, giving 3:4.
Example 3: Ratios with Decimals or Fractions
Simplify 0.5:2.5. Multiply both terms by 10 to remove decimals: 5:25. Then divide by 5 to get 1:5. For fractions, such as 1/2 : 3/4, multiply both terms by the common denominator (4): 2:3. If decimals or fractions give you trouble, see our guide on converting fractions to decimals.
Example 4: Different Units
Find the ratio of 750 g to 3 kg. Convert 3 kg to 3,000 g. The ratio is 750:3,000, which simplifies by dividing by 750 to 1:4.
Types of Ratios
Part-to-Part Ratio
This compares one part of a group with another part. In a class of 12 boys and 18 girls, the ratio of boys to girls (12:18, or 2:3) is a part-to-part ratio.
Part-to-Whole Ratio
This compares one part with the entire group. Boys to all students is 12:30, which simplifies to 2:5. Part-to-whole ratios are directly related to fractions and percentages: 2/5 of the class is boys, which is 40%.
Rate and Unit Rate
A rate is a ratio of two quantities in different units, such as 180 km in 3 hours. A unit rate has 1 as its second term: 180 ÷ 3 = 60 km per hour. Unit rates are the key to comparing prices fairly. If 12 apples cost 96, the unit rate is 96 ÷ 12 = 8 per apple.
How to Divide a Quantity in a Given Ratio
One of the most common exam and real-life tasks is sharing an amount in a given ratio.
Step-by-Step Method
- Add the terms of the ratio to get the total number of parts.
- Divide the total quantity by the total number of parts to find the value of one part.
- Multiply the value of one part by each term of the ratio.
Example 1: Sharing Money
Share 600 between two people in the ratio 2:3. Total parts = 2 + 3 = 5. One part = 600 ÷ 5 = 120. The shares are 2 × 120 = 240 and 3 × 120 = 360. Check: 240 + 360 = 600.
Example 2: Three Parts
Divide 900 in the ratio 2:3:4. Total parts = 9. One part = 900 ÷ 9 = 100. The shares are 200, 300 and 400.
Example 3: Mixtures
Milk and water are mixed in the ratio 5:3 to make 40 litres. Total parts = 8. One part = 40 ÷ 8 = 5 litres. Milk = 25 litres and water = 15 litres.
Example 4: Given Total Population
The ratio of boys to girls in a school is 5:7 and there are 240 students. One part = 240 ÷ 12 = 20. Boys = 100, girls = 140.
Business partners use exactly this method to split profit in proportion to their investments. If you run a business, our guide on profit margin vs markup will help you understand the profit you are sharing.
Equivalent Ratios and Comparing Ratios
Equivalent ratios are ratios that represent the same comparison. You create them by multiplying or dividing every term by the same non-zero number. For example, 2:3 is equivalent to 4:6, 6:9 and 20:30.
Comparing Ratios
To check which of two ratios is larger, convert them to fractions and compare, or convert to decimals. For instance, which is greater, 3:5 or 4:7? Write them as 3/5 = 0.6 and 4/7 ≈ 0.571. Since 0.6 is larger, 3:5 is the greater ratio.
| Multiplier | Ratio |
|---|---|
| × 1 | 2:3 |
| × 2 | 4:6 |
| × 5 | 10:15 |
| × 10 | 20:30 |
| × 50 | 100:150 |
What Is a Proportion?
A proportion is an equation that says two ratios are equal. It is written as a:b = c:d, or a/b = c/d. The first and last terms (a and d) are called the extremes, while the middle terms (b and c) are the means.
The fundamental property of a proportion is that the product of the extremes equals the product of the means:
a × d = b × c
This is the cross multiplication rule, and it is the main tool for solving proportions.
Example 1: Find the Missing Term
Solve 3/5 = x/20. Cross multiply: 3 × 20 = 5 × x, so 60 = 5x and x = 12.
Example 2
Solve 7/x = 21/45. Cross multiply: 7 × 45 = 21 × x, so 315 = 21x and x = 15.
Example 3: Checking a Proportion
Is 4:9 proportional to 12:27? Cross multiply: 4 × 27 = 108 and 9 × 12 = 108. They are equal, so yes, the ratios are in proportion.
Direct Proportion
Two quantities are in direct proportion when they increase or decrease together at the same rate. Double one, and the other doubles as well. The ratio between them stays constant.
Example 1: Cost of Items
If 5 notebooks cost 150, how much do 8 notebooks cost? One notebook costs 150 ÷ 5 = 30, so 8 notebooks cost 8 × 30 = 240. This is called the unitary method. Using proportion: 5/150 = 8/x, so x = (150 × 8) ÷ 5 = 240.
Example 2: Distance and Time
A car covers 180 km in 3 hours at a steady speed. In 5 hours it will cover 180 ÷ 3 × 5 = 300 km.
Example 3: Recipes
A recipe for 4 people needs 300 g of flour. For 10 people, you need 300 ÷ 4 × 10 = 750 g.
Inverse Proportion
Two quantities are in inverse proportion when one increases while the other decreases, so that their product stays the same. More workers means fewer days; higher speed means less time.
Example 1: Workers and Days
6 workers can finish a job in 10 days. How long will 4 workers take? The total work is 6 × 10 = 60 worker-days. With 4 workers: 60 ÷ 4 = 15 days.
Example 2: Pipes Filling a Tank
8 identical pipes fill a tank in 6 hours. With 12 pipes, the time is (8 × 6) ÷ 12 = 4 hours.
Example 3: Speed and Time
A trip takes 4 hours at 60 km/h. At 80 km/h it takes (60 × 4) ÷ 80 = 3 hours.
| Feature | Direct proportion | Inverse proportion |
|---|---|---|
| Relationship | Both increase or both decrease | One increases, the other decreases |
| Constant | Ratio (x ÷ y) | Product (x × y) |
| Example | Items and total cost | Workers and days |
| Formula | x₁/y₁ = x₂/y₂ | x₁ × y₁ = x₂ × y₂ |
Combining Two Ratios
Sometimes you are given a:b and b:c and asked for a:b:c. Make the shared term equal in both ratios first.
Example: a:b = 2:3 and b:c = 4:5. The shared term is b, which is 3 in the first ratio and 4 in the second. The lowest common multiple of 3 and 4 is 12. Multiply the first ratio by 4 to get 8:12, and the second by 3 to get 12:15. Now b is 12 in both, so a:b:c = 8:12:15.
Real-Life Applications
Cooking and Recipes
Recipes are proportions. If the ratio of flour to sugar is 3:1 and you need 600 g of dry mix in total, one part is 150 g, so use 450 g flour and 150 g sugar. Scaling a dish for more people is direct proportion in action.
Maps and Scale Drawings
A map scale of 1:50,000 means 1 cm on the map represents 50,000 cm in real life. A distance of 4 cm on the map is 200,000 cm, which is 2,000 m, or 2 km. Architects and model makers rely on the same principle.
Shopping and Unit Prices
To find the better deal, compare unit rates. A 500 g pack for 60 costs 0.12 per gram, while a 750 g pack for 84 costs 0.112 per gram, so the larger pack is cheaper per gram. This pairs well with the skills in our guide on how to calculate discount.
Finance and Business
Partners split profits in the ratio of their investments. Analysts use financial ratios such as debt-to-equity to judge a company’s health, and return on investment is a ratio too, as explained in our article on how to calculate ROI.
Health and Fitness
Body mass index compares weight with height squared, and diet plans specify macronutrient ratios. To see how this works, read our BMI calculator guide.
Science and Engineering
Chemists mix solutions in fixed ratios, engineers use gear ratios, and mechanics specify fuel-to-oil ratios for two-stroke engines. Even unit conversion is a proportion; see our introduction to unit conversion basics.
Art and Design
Screen aspect ratios such as 16:9, photo sizes and the golden ratio (about 1.618:1) all use ratios to create balanced designs.
Ratio, Fraction and Percentage
These three ideas are closely linked. A part-to-whole ratio can be written as a fraction and then as a percentage.
- The ratio 3:2 (part to part) means 3 parts out of 5 in total, so the shares are 3/5 = 60% and 2/5 = 40%.
- The ratio 1:3 (part to part) gives 1/4 = 25% and 3/4 = 75%.
- A part-to-whole ratio of 7:20 is the fraction 7/20 = 0.35 = 35%.
If you need to refresh the conversions, see our guides on how to calculate percentage and converting fractions to decimals.
Common Mistakes
- Mixing units. Always convert to the same unit before writing a ratio.
- Writing the terms in the wrong order. The ratio of A to B is not the same as B to A.
- Forgetting to add the parts when sharing a quantity. In the ratio 2:3, divide by 5, not by 2 or 3.
- Treating a part-to-part ratio as a part-to-whole ratio. A 2:3 ratio of boys to girls means boys are 2/5 of the class, not 2/3.
- Using direct proportion for an inverse situation. If more workers should reduce the time, you need inverse proportion.
- Not simplifying. Give the final answer in simplest form unless asked otherwise.
- Cross-multiplying incorrectly. Multiply diagonally, and keep the unknown on one side.
Practice Questions
- Simplify 45:60.
- Divide 900 in the ratio 4:5.
- Solve 7/x = 21/45.
- A car travels 180 km in 3 hours. How far will it go in 5 hours at the same speed?
- 8 pipes fill a tank in 6 hours. How long will 12 pipes take?
- The ratio of boys to girls is 5:7 and there are 240 students. How many boys and girls are there?
Answers
- GCD = 15, so 3:4.
- Total parts = 9, one part = 100, shares are 400 and 500.
- 7 × 45 = 315, and 315 ÷ 21 = 15.
- 180 ÷ 3 × 5 = 300 km.
- (8 × 6) ÷ 12 = 4 hours.
- One part = 240 ÷ 12 = 20, so 100 boys and 140 girls.
Frequently Asked Questions
What is a ratio?
A ratio compares two or more quantities of the same kind, written as a:b or a/b. Two boys to three girls is the ratio 2:3.
What is a proportion?
A proportion is an equation stating that two ratios are equal, such as 3/5 = 12/20.
How do I simplify a ratio?
Divide all terms by their greatest common divisor. For 24:36, dividing by 12 gives 2:3.
How do I divide a quantity in a ratio?
Add the ratio terms, divide the total by that sum to find one part, then multiply one part by each term.
What is the difference between direct and inverse proportion?
In direct proportion both quantities rise or fall together and their ratio is constant. In inverse proportion one rises as the other falls and their product is constant.
How do I solve a proportion?
Use cross multiplication: for a/b = c/d, set a × d equal to b × c and solve for the unknown.
Can a ratio have more than two terms?
Yes. A ratio such as 2:3:4 compares three quantities, and the same methods for simplifying and sharing still work.
Conclusion
Ratio and proportion give you a reliable way to compare quantities and scale them up or down. Keep the units matching, simplify with the greatest common divisor, add the parts before sharing, and use cross multiplication for proportions. Decide first whether the situation is direct or inverse, because that choice determines the method.
When you want a fast answer, use our free ratio calculator, and continue learning with these related guides: