Ratios & Proportions
Ratios and proportions help us compare quantities and understand relationships between values. They are used when we want to know how much of one thing exists compared to another.
From cooking recipes and exam problems to map reading, finance, and real-life decision-making, ratios and proportions appear everywhere.
What Is a Ratio?
A ratio compares two quantities of the same kind by division. It tells us how many times one quantity contains another.
Ratios are usually written using a colon (:).
Example: 2 : 3
Understanding Ratios Visually
Visual representation makes ratios easy to understand, especially for beginners and school students.
Consider the ratio 2 : 3.
Here, the purple blocks represent the first quantity, and the green blocks represent the second quantity.
Writing Ratios in Different Forms
The same ratio can be written in multiple equivalent ways.
| Form | Example |
| Colon form | 2 : 3 |
| Fraction form | 2/3 |
| Word form | 2 to 3 |
All these forms represent the same relationship.
Simplifying Ratios
Ratios should usually be written in their simplest form to make comparison easier.
Simplification is done by dividing both terms by their highest common factor (HCF).
Example: 10 : 15
Divide both terms by 5 → 2 : 3
Ratio Between More Than Two Quantities
Sometimes ratios compare three or more quantities. This is common in mixture and sharing problems.
Example: 2 : 3 : 5
This means the first quantity has 2 parts, the second has 3 parts, and the third has 5 parts.
What Is a Proportion?
A proportion states that two ratios are equal. It shows that two comparisons describe the same relationship.
Proportions are written using the equals sign (=).
Example: 2 : 3 = 4 : 6
Understanding Proportion Using Balance
A proportion works like a balance — both ratios must represent the same value.
Even though the numbers are different, the ratios are equal in value.
Checking Whether Two Ratios Form a Proportion
To check a proportion, we use cross multiplication.
Example: 2 : 3 and 4 : 6
Cross multiply:
2 × 6 = 12 3 × 4 = 12
Since both results are equal, the ratios form a proportion.
Direct Proportion
In direct proportion, when one quantity increases, the other quantity also increases in the same ratio.
Example: More workers → more work done
If 2 workers take 4 days, then 4 workers take 2 days.
Inverse Proportion
In inverse proportion, when one quantity increases, the other quantity decreases.
Example: More speed → less time taken
If speed doubles, time becomes half.
Ratios & Proportions in Real Life
They are used naturally in many daily activities.
- Cooking recipes (ingredient ratios)
- Maps and scale drawings
- Sharing money or property
- Mixing chemicals or colors
Understanding ratios avoids waste and ensures accuracy.
Ratios & Proportions in Competitive Exams
Competitive exams often test:
- Simplification of ratios
- Direct and inverse proportion problems
- Time, work, and speed-based questions
Clear concepts lead to faster and more confident answers.
Common Mistakes to Avoid
Errors usually occur due to lack of clarity.
- Not simplifying ratios
- Mixing different units
- Applying direct proportion instead of inverse
Always understand the situation before calculating.
Practice Questions
Q1. Simplify the ratio 18 : 24
Q2. Check whether 3 : 5 and 6 : 10 form a proportion.
Q3. If 5 pens cost ₹50, what is the cost of 8 pens?
Quick Quiz
Q1. What does a ratio compare?
Q2. In which proportion does one value increase while the other decreases?
Quick Recap
- Ratios compare quantities
- Proportions show equality of ratios
- Simplification makes ratios clearer
- Direct and inverse proportions describe real situations
- Used in exams, life, and decision-making