Ratio and Proportion

1. Ratio

A RATIO is a comparison of two quantities of the SAME kind by division. Ratio of a to b is written as a : b = a/b.

Key Points

• Both quantities must be in the SAME unit.
• Ratio has NO unit (it is a pure number).
• Order matters: a : b is NOT the same as b : a.
• Ratio is expressed in SIMPLEST FORM (divide by HCF).

Simplifying Ratios

Divide both terms by their HCF.

Example: 36 : 24. HCF of 36 and 24 = 12. 36 ÷ 12 = 3, 24 ÷ 12 = 2. Simplest form = 3 : 2.

Example with different units: Convert to same unit first. 'Find ratio of 2 m to 50 cm.' 2 m = 200 cm. Ratio = 200 : 50 = 4 : 1.

2. Ratio in Different Contexts

Comparing Three Quantities

If a : b = 3 : 4 and b : c = 2 : 5, find a : b : c. Make b the SAME in both ratios. a : b = 3 : 4 (multiply by 1) = 3 : 4. b : c = 2 : 5 (multiply by 2) = 4 : 10. Therefore a : b : c = 3 : 4 : 10.

Dividing a Quantity in a Given Ratio

'Divide ₹600 between A and B in ratio 3 : 2.' Total parts = 3 + 2 = 5. A's share = (3/5) × 600 = ₹360. B's share = (2/5) × 600 = ₹240.

Worked Example (ICSE 2024, 3 marks)

'Two numbers are in ratio 5 : 7. If their sum is 108, find the numbers.' Let numbers be 5x and 7x. 5x + 7x = 108. 12x = 108. x = 9. Numbers: 5 × 9 = 45 and 7 × 9 = 63.

3. Proportion

Four quantities a, b, c, d are in PROPORTION if a : b = c : d. Written as a : b :: c : d (read as 'a is to b as c is to d').

Product of Extremes = Product of Means

In a : b :: c : d:

• a and d are the EXTREMES (outer terms).
• b and c are the MEANS (middle terms).
• a × d = b × c (Product of extremes = Product of means).

Checking Proportion

'Check if 3, 5, 9, 15 are in proportion.' Product of extremes = 3 × 15 = 45. Product of means = 5 × 9 = 45. YES, they are in proportion.

Finding the Missing Term

'Find x: 4 : 7 :: 12 : x.' 4 × x = 7 × 12 = 84. x = 84/4 = 21.

4. Continued Proportion

Three quantities a, b, c are in CONTINUED PROPORTION if a : b = b : c. Then b² = a × c. b is called the MEAN PROPORTIONAL.

Finding Mean Proportional

'Find the mean proportional between 4 and 16.' b² = 4 × 16 = 64. b = √64 = 8.

Third Proportional

If a, b, c are in continued proportion, c is the THIRD PROPORTIONAL. c = b²/a.

Fourth Proportional

If a : b = c : d, then d is the FOURTH PROPORTIONAL. d = (b × c)/a.

5. Unitary Method

Find the value of ONE unit first. Then multiply.

Direct Proportion

When one quantity INCREASES, the other INCREASES at the same rate.

Example: 'If 8 pens cost ₹96, find the cost of 15 pens.' 1 pen = ₹96 ÷ 8 = ₹12. 15 pens = 15 × ₹12 = ₹180.

Inverse Proportion

When one quantity INCREASES, the other DECREASES at the same rate.

Example: 'If 6 workers finish a job in 12 days, how many workers finish it in 8 days?' Total work = 6 × 12 = 72 worker-days. Number of workers = 72 ÷ 8 = 9 workers.

Worked Example (ICSE 2023, 3 marks)

'A car travels 180 km in 4 hours. How far will it travel in 7 hours at the same speed?' Distance in 1 hour = 180 ÷ 4 = 45 km. Distance in 7 hours = 45 × 7 = 315 km.

6. ICSE Exam Focus

Common Mistakes

1. Writing ratio without same units (MUST convert both to same unit).
2. Inverting ratio order — a : b ≠ b : a.
3. Using unitary method in inverse proportion without checking.
4. Forgetting product of extremes = product of means.

Self-Test (5 Questions)

Q1. Simplify: 2.4 m : 60 cm. (2 marks)

• A) 2 : 1
• B) 4 : 1
• C) 1 : 4
• D) 3 : 1

Q2. Find x: 9 : 12 :: x : 20. (2 marks)

Q3. Find the mean proportional between 9 and 25. (2 marks)

Q4. 'If a dozen eggs cost ₹54, what is the cost of 20 eggs?' (2 marks)

Q5. '5 pipes can fill a tank in 8 hours. How long will 4 pipes take?' (3 marks)

Answers

A1. B) 4 : 1. (2.4 m = 240 cm. 240 : 60 = 4 : 1.) A2. x = 15. (9 × 20 = 12 × x. 180 = 12x. x = 15.) A3. 15. (b² = 9 × 25 = 225. b = √225 = 15.) A4. ₹90. (1 dozen = 12. 1 egg = 54/12 = ₹4.50. 20 eggs = 20 × 4.50 = ₹90.) A5. 10 hours. (Total work = 5 × 8 = 40 pipe-hours. Time = 40 ÷ 4 = 10 hours.)