Class 6 Mathematics Notes Chapter 12 (Ratio and Proportion) – Mathematics Book

Alright class, let's dive straight into Chapter 12: Ratio and Proportion. This is a foundational chapter, and understanding it well is crucial, not just for your Class 6 exams, but also as building blocks for many quantitative aptitude questions you'll face in government exams later on.

Chapter 12: Ratio and Proportion - Detailed Notes for Exam Preparation

1. What is Ratio?

• Definition: A ratio is a way to compare two quantities of the same kind by division. It tells us how much of one quantity there is compared to another.
• Notation: The ratio of a quantity 'a' to a quantity 'b' is written as a : b or as a fraction a/b. Here, 'a' is called the antecedent and 'b' is called the consequent.
• Key Points:
  • Same Kind & Same Units: To find the ratio between two quantities, they must be of the same kind (e.g., length to length, weight to weight) and expressed in the same units (e.g., cm to cm, kg to kg). If units are different, convert them to the same unit first.
    • Example: Ratio of 50 cm to 2 meters. First convert 2 meters to 200 cm. The ratio is 50 : 200.
  • No Units: A ratio itself has no units. It's a pure number showing comparison.
  • Order Matters: The ratio a : b is different from b : a. For example, the ratio of apples to oranges is not the same as the ratio of oranges to apples unless the number of apples and oranges is equal.
  • Simplest Form: Ratios are usually expressed in their simplest form. To do this, divide both the antecedent and the consequent by their Highest Common Factor (HCF).
    • Example: The ratio 50 : 200. HCF of 50 and 200 is 50. Dividing both by 50, we get (50/50) : (200/50) = 1 : 4.
  • Equivalent Ratios: Ratios that represent the same comparison are called equivalent ratios. You can get equivalent ratios by multiplying or dividing both terms of the ratio by the same non-zero number.
    • Example: 1 : 4 is equivalent to 2 : 8 (multiplied by 2), 3 : 12 (multiplied by 3), 50 : 200 (multiplied by 50), etc.

2. What is Proportion?

• Definition: A proportion is an equality of two ratios. If the ratio a : b is equal to the ratio c : d, then a, b, c, and d are said to be in proportion.
• Notation: We write it as a : b :: c : d or a/b = c/d. This is read as "a is to b as c is to d".
• Terms:
  • a and d are called the extreme terms (or extremes).
  • b and c are called the middle terms (or means).
• Key Property (Very Important for Exams!): If four quantities are in proportion, then the Product of Extremes = Product of Means.
  • If a : b :: c : d, then a × d = b × c.
• Checking for Proportion: To check if four numbers (say p, q, r, s) are in proportion, check if p : q is equal to r : s. This can be done by:
  • Simplifying both ratios to their simplest form. If they are the same, the numbers are in proportion.
  • Checking if p × s = q × r. If the product of extremes equals the product of means, they are in proportion.
    • Example: Are 10, 20, 30, 60 in proportion?
      • Method 1: Ratio 1 = 10:20 = 1:2. Ratio 2 = 30:60 = 1:2. Since ratios are equal, they are in proportion.
      • Method 2: Extremes are 10 and 60. Means are 20 and 30. Product of Extremes = 10 × 60 = 600. Product of Means = 20 × 30 = 600. Since products are equal, they are in proportion.
• Continued Proportion: Three quantities a, b, c are said to be in continued proportion if a : b :: b : c (i.e., a/b = b/c or b² = ac). Here, 'b' is called the mean proportional between 'a' and 'c'.

3. Unitary Method

• Definition: The unitary method is a technique used to first find the value of one unit and then use that value to find the value of the required number of units.
• Steps:
  1. Find the value of one unit: If the value of many units is given, divide the total value by the number of units.
  2. Find the value of the required number of units: Multiply the value of one unit (found in step 1) by the required number of units.
• Application in Ratio/Proportion: This method is frequently used in word problems involving cost, distance, time, work, etc., which often relate to proportional reasoning.
  • Example: If the cost of 6 pens is ₹90, what is the cost of 10 pens?
    • Step 1: Cost of 1 pen = ₹90 / 6 = ₹15.
    • Step 2: Cost of 10 pens = ₹15 × 10 = ₹150.
  • Notice this is a proportion problem too: 6 pens : 10 pens :: ₹90 : Cost of 10 pens. Let the cost be x. 6/10 = 90/x => 6 * x = 10 * 90 => 6x = 900 => x = 900/6 = 150.

Government Exam Relevance:

• Direct Questions: Finding ratios, simplifying ratios, checking proportions, finding missing terms in proportions (a : b :: c : ?).
• Word Problems: Many quantitative aptitude problems are based on ratios (e.g., sharing amounts, mixtures, ages) and proportions (e.g., time and work, time and distance, cost variations).
• Unitary Method: Forms the basis for solving many practical problems quickly.
• Speed and Accuracy: Understanding the core concepts (especially Product of Extremes = Product of Means) allows for faster problem-solving. Practice converting units quickly and simplifying ratios efficiently.

Multiple Choice Questions (MCQs)

Here are 10 MCQs based on the concepts we've just covered. Try to solve them!

1. The ratio of 20 minutes to 1 hour is:
   a) 20 : 1
   b) 1 : 3
   c) 1 : 4
   d) 20 : 100

2. Which of the following ratios is equivalent to 3 : 5?
   a) 6 : 15
   b) 9 : 10
   c) 12 : 20
   d) 15 : 20

3. If the cost of a dozen bananas is ₹60, what is the cost of 5 bananas? (Hint: A dozen = 12)
   a) ₹20
   b) ₹25
   c) ₹30
   d) ₹35

4. Are the numbers 2, 4, 8, 16 in proportion?
   a) Yes
   b) No
   c) Cannot be determined
   d) Only the first three are in proportion

5. Find the missing term in the proportion: 4 : 8 :: 12 : ?
   a) 16
   b) 20
   c) 24
   d) 32

6. The simplest form of the ratio 75 paise to ₹3 is:
   a) 75 : 3
   b) 3 : 75
   c) 1 : 4
   d) 4 : 1

7. In a class of 45 students, the number of boys is 25. What is the ratio of the number of girls to the number of boys?
   a) 25 : 45
   b) 20 : 25
   c) 4 : 5
   d) 5 : 4

8. If a : b = 2 : 3 and b : c = 3 : 4, then a : c is:
   a) 2 : 4
   b) 1 : 2
   c) 3 : 4
   d) 2 : 3

9. A car travels 180 km in 3 hours. How far will it travel in 5 hours at the same speed?
   a) 240 km
   b) 280 km
   c) 300 km
   d) 320 km

10. In a proportion p : q :: r : s, which of the following is always true?
    a) p × q = r × s
    b) p × r = q × s
    c) p × s = q × r
    d) p + s = q + r

Answer Key for MCQs:

1. b (1 hour = 60 mins. Ratio is 20:60 = 1:3)
2. c (3x4 = 12, 5x4 = 20 => 12:20)
3. b (Cost of 1 banana = 60/12 = ₹5. Cost of 5 bananas = 5x5 = ₹25)
4. a (2:4 = 1:2, 8:16 = 1:2. OR 2x16 = 32, 4x8 = 32. Extremes = Means)
5. c (4 * ? = 8 * 12 => 4 * ? = 96 => ? = 96/4 = 24)
6. c (₹3 = 300 paise. Ratio is 75:300. HCF is 75. 75/75 : 300/75 = 1:4)
7. c (Total = 45, Boys = 25. Girls = 45 - 25 = 20. Ratio Girls:Boys = 20:25. HCF is 5. 20/5 : 25/5 = 4:5)
8. b (a:b = 2:3, b:c = 3:4. Since 'b' term is same (3), we can directly say a:c = 2:4, which simplifies to 1:2)
9. c (Speed = 180/3 = 60 km/hr. Distance in 5 hours = 60 * 5 = 300 km. OR Unitary method: In 1 hr, car travels 180/3 = 60 km. In 5 hrs, it travels 60 * 5 = 300 km)
10. c (Product of Extremes = Product of Means)

Make sure you understand the reasoning behind each answer. Practice more problems from your textbook and other resources focusing on these concepts. Good luck!