Ratio and Proportion Basics: A Simple Beginner's Guide
Ratio and proportion are among the most useful ideas in school maths. They appear in board exams, in daily life like cooking and shopping, and in almost every competitive aptitude test. This guide explains what they mean and shows you how to solve typical problems step by step.
What is a ratio?
A ratio compares two quantities of the same kind by division. If a class has 12 boys and 18 girls, the ratio of boys to girls is written as 12:18, read as "12 to 18". A ratio has no units because the units cancel out.
To simplify a ratio, divide both terms by their highest common factor (HCF). The HCF of 12 and 18 is 6, so 12:18 = 2:3. This is just like reducing a fraction, a skill you can revise in our fractions, decimals and percentages guide.
What is a proportion?
A proportion is a statement that two ratios are equal, such as 2:3 = 8:12. In any proportion a:b = c:d, the outer terms (a and d) are called extremes and the inner terms (b and c) are called means.
- Cross-multiplication rule: product of means = product of extremes, so b × c = a × d.
Check 2:3 = 8:12. Means give 3 × 8 = 24 and extremes give 2 × 12 = 24. They match, so it is a true proportion. This rule lets you find any missing term.
Dividing a quantity in a given ratio
Suppose you must split ₹500 between two friends in the ratio 2:3. Add the ratio parts: 2 + 3 = 5. So each part is 500 ÷ 5 = ₹100. The shares are 2 × 100 = ₹200 and 3 × 100 = ₹300. Always check the shares add back to the total: 200 + 300 = 500.
Direct proportion
Two quantities are in direct proportion when one increases as the other increases at the same rate, following y = kx for a constant k. More items cost more money.
Worked example: If 5 pens cost ₹75, what do 8 pens cost? First find the cost of one pen: 75 ÷ 5 = ₹15. Then 8 pens cost 8 × 15 = ₹120. This "find one, then find many" method (the unitary method) is a favourite in Class 9 Maths.
Inverse proportion
Two quantities are in inverse proportion when one increases as the other decreases, so their product stays constant: x × y = k. More workers finish a job in fewer days.
Worked example: If 4 workers build a wall in 6 days, how long will 3 workers take? The total work is 4 × 6 = 24 worker-days. With 3 workers, time = 24 ÷ 3 = 8 days. Notice fewer workers means more days, which confirms the relationship is inverse.
Tips to avoid mistakes
- Keep the order: the ratio of boys to girls is not the same as girls to boys.
- Same units: convert quantities to the same unit before forming a ratio.
- Simplify at the end: reduce your final ratio to lowest terms.
With practice these problems become quick marks. To build on ratios and move into percentages and algebra, explore our structured Class 10 Maths lessons.
Frequently asked questions
What is the difference between ratio and proportion?
A ratio compares two quantities, such as 2:3. A proportion states that two ratios are equal, such as 2:3 = 4:6. So a proportion is made from two equal ratios.
How do I know if a problem is direct or inverse proportion?
If both quantities increase or decrease together, it is direct proportion. If one increases while the other decreases, such as workers and days, it is inverse proportion.
Can a ratio have more than two terms?
Yes. A ratio can compare three or more quantities, for example 2:3:5. You simplify it by dividing every term by their common HCF, just like a two-term ratio.