Ratio and Proportion
1. Introduction to Ratio
A ratio is a comparison of two quantities of the same kind by division.
The ratio of a to b is written as a : b or a/b.
Key Rules:
- Both quantities must be in the same unit.
- Ratio has no unit (it is a pure number).
- The order matters: a : b is different from b : a.
Example: In a class of 30 students, 18 are boys and 12 are girls.
Ratio of boys to girls = 18 : 12 = 3 : 2 (in simplest form).
Ratio of girls to boys = 12 : 18 = 2 : 3.
Common Mistake: Writing ratio of 2 kg to 500 g as 2 : 500. Wrong! Convert to same unit: 2000 g : 500 g = 4 : 1.
2. Ratio as a Fraction
A ratio a : b can be written as the fraction a/b.
Worked Example: Express 25 : 40 in simplest form.
25 : 40 = 25/40 = 5/8 (dividing numerator and denominator by 5).
Simplest form = 5 : 8.
Worked Example: Three numbers are in the ratio 2 : 3 : 5. If the smallest is 14, find the others.
The smallest part corresponds to 2 in the ratio.
2 units = 14, so 1 unit = 7.
Second number = 3 x 7 = 21. Third number = 5 x 7 = 35.
Exam Focus (3 marks): 'Divide 360 between A and B in the ratio 4 : 5.'
Total parts = 4 + 5 = 9.
A's share = (4/9) x 360 = 160. B's share = (5/9) x 360 = 200.
3. Proportion
A proportion states that two ratios are equal.
a : b = c : d is written as a : b :: c : d (read as 'a is to b as c is to d').
- a and d are the extremes.
- b and c are the means.
- Product of extremes = Product of means (cross-multiplication): a x d = b x c.
Worked Example: Check if 4 : 6 and 8 : 12 are in proportion.
4 x 12 = 48, 6 x 8 = 48. Since 4 x 12 = 6 x 8, they are in proportion.
Worked Example: Find x if 5 : 8 :: x : 24.
Product of extremes = 5 x 24 = 120.
Product of means = 8 x X.
8X = 120, so X = 15.
Common Mistake: Writing the proportion in the wrong order. If a : b = c : d, then a : b :: c : d is correct. Make sure the order matches.
4. Unitary Method
Find the value of one unit first, then find the value of the required number of units.
Worked Example: 8 pens cost 96. Find the cost of 15 pens.
Cost of 1 pen = 96 / 8 = 12.
Cost of 15 pens = 15 x 12 = 180.
Worked Example: A car travels 210 km in 3 hours. How far will it travel in 5 hours at the same speed?
Distance in 1 hour = 210 / 3 = 70 km.
Distance in 5 hours = 5 x 70 = 350 km.
Exam Focus (4 marks): 'If 15 workers can build a wall in 8 days, how many days will 12 workers take?'
This is inverse proportion. 15 workers take 8 days.
1 worker takes 15 x 8 = 120 days.
12 workers take 120 / 12 = 10 days.
Common Mistake: Applying direct proportion where inverse is needed. More workers means fewer days (inverse), not more days (direct).
5. Comparison Table: Ratio vs Proportion
| Aspect | Ratio | Proportion |
|---|---|---|
| Definition | Comparison of two quantities | Equality of two ratios |
| Notation | a : b or a/b | a : b :: c : d |
| Number of terms | 2 terms | 4 terms |
| Property | Can be simplified | Product of extremes = Product of means |
6. Self-Test
- Write the ratio 750 g to 2 kg in simplest form.
- A sum of 480 is divided among A, B, and C in the ratio 3 : 4 : 5. Find each share.
- Check if 12 : 18 and 14 : 21 are in proportion.
- Find the missing term: 8 : 3 :: 40 : x.
- 5 kg of rice costs 220. Find the cost of 12 kg.
- If 6 pipes fill a tank in 4 hours, how long will 8 pipes take?
- The ratio of boys to girls is 4 : 3. If there are 28 boys, find the number of girls.
7. Answers to Self-Test
- 750 g : 2000 g = 750/2000 = 3/8 = 3 : 8.
- Total parts = 3 + 4 + 5 = 12. A = (3/12) x 480 = 120, B = (4/12) x 480 = 160, C = (5/12) x 480 = 200.
- 12 x 21 = 252, 18 x 14 = 252. Yes, they are in proportion.
- 8 x x = 3 x 40 => 8x = 120 => x = 15.
- Cost of 1 kg = 220/5 = 44. Cost of 12 kg = 12 x 44 = 528.
- 1 pipe takes 6 x 4 = 24 hours. 8 pipes take 24/8 = 3 hours.
- If 3:x = x:27 (continued proportion), find x. (Answer: x² = 3×27 = 81 → x = 9.)
Ratio and Proportion in Daily Life
Ratios are used everywhere: cooking recipes (2 cups flour : 1 cup sugar), map scales (1:50000 means 1cm = 500m), sports (win-loss ratio), and finance (profit ratio). 'The unitary method — finding the value of ONE unit first, then multiplying — is the most powerful technique for solving proportion problems. It works for BOTH direct and inverse variation. Always ask: if I know the value for some quantity, what is the value for ONE unit?'
ICSE Exam Tips
'In ICSE Class 6 exams, ratio and proportion problems appear in both short-answer (2 marks) and long-answer (4 marks) formats. Always write ratios in SIMPLEST form. For proportion, check that the product of extremes equals the product of means. Show your working clearly — marks are awarded for the method, not just the answer.' 7. 4 units = 28, so 1 unit = 7. Girls = 3 x 7 = 21.
