Class 8 Mathematics Notes Chapter 13 (Direct and Inverse Proportions) – Mathematics Book

Alright class, let's focus on Chapter 13: Direct and Inverse Proportions. This is a fundamental chapter, not just for your Class 8 exams, but also because the concepts appear frequently in various government exam aptitude sections. Understanding how quantities relate to each other is crucial.

Chapter 13: Direct and Inverse Proportions - Detailed Notes

1. Introduction: What are Proportions?

• Many situations involve two quantities that change in relation to each other. For example, if you buy more notebooks, the total cost increases. If you increase the speed of a car, the time taken to cover a fixed distance decreases.
• Proportion deals with how a change in one quantity affects another quantity in a predictable way.
• We will study two main types of relationships: Direct Proportion and Inverse Proportion.

2. Direct Proportion

• Definition: Two quantities, say x and y, are said to be in direct proportion if they increase or decrease together in such a way that the ratio of their corresponding values remains constant.
• Meaning: If x increases, y increases proportionally. If x decreases, y decreases proportionally.
• Mathematical Representation:
  • We write this as x ∝ y (read as 'x is directly proportional to y').
  • This means x / y = k (where k is a positive constant, known as the constant of proportionality).
  • Alternatively, x = ky.
• Key Characteristic: The ratio x/y is always the same for any corresponding pair of values (x, y).
• Condition for Checking: If we have two pairs of values (x₁, y₁) and (x₂, y₂):
  • They are in direct proportion if x₁ / y₁ = x₂ / y₂.
  • This can be rearranged as x₁ / x₂ = y₁ / y₂. This form is very useful for solving problems.
• Examples:
  • Cost and Quantity: More articles purchased means more cost (Cost ∝ Quantity). If 1 pen costs ₹10, 5 pens cost ₹50. (Ratio 1/10 = 5/50).
  • Distance and Time (Constant Speed): More time travelled means more distance covered (Distance ∝ Time). If speed is 50 km/h, in 1 hour distance is 50 km, in 2 hours distance is 100 km. (Ratio 50/1 = 100/2).
  • Wages and Hours Worked: More hours worked means more wages earned (Wages ∝ Hours).
• Problem Solving Technique:
  1. Identify the two quantities (x and y).
  2. Recognize that they increase/decrease together (confirming direct proportion).
  3. Set up the relationship: x₁ / y₁ = x₂ / y₂ or x₁ / x₂ = y₁ / y₂.
  4. Substitute the known values and solve for the unknown value.

Example Problem (Direct Proportion):
If the cost of 12 meters of cloth is ₹300, find the cost of 5 meters of cloth.

• Quantities: Length of cloth (x) and Cost (y).
• Relationship: As length increases, cost increases. Direct Proportion.
• Let x₁ = 12 m, y₁ = ₹300.
• Let x₂ = 5 m, y₂ = ? (the unknown cost).
• Using x₁ / y₁ = x₂ / y₂:
  12 / 300 = 5 / y₂
12 * y₂ = 5 * 300
y₂ = (5 * 300) / 12
y₂ = 1500 / 12
y₂ = 125
• Answer: The cost of 5 meters of cloth is ₹125.

3. Inverse Proportion

• Definition: Two quantities, say x and y, are said to be in inverse proportion if an increase in one quantity causes a proportional decrease in the other quantity (and vice-versa) in such a way that the product of their corresponding values remains constant.
• Meaning: If x increases, y decreases proportionally. If x decreases, y increases proportionally.
• Mathematical Representation:
  • We write this as x ∝ 1/y (read as 'x is inversely proportional to y').
  • This means x * y = k (where k is a positive constant, the constant of proportionality).
  • Alternatively, x = k/y.
• Key Characteristic: The product x * y is always the same for any corresponding pair of values (x, y).
• Condition for Checking: If we have two pairs of values (x₁, y₁) and (x₂, y₂):
  • They are in inverse proportion if x₁ * y₁ = x₂ * y₂.
  • This can be rearranged as x₁ / x₂ = y₂ / y₁. Notice the inversion of y terms compared to direct proportion. This form is useful for solving problems.
• Examples:
  • Speed and Time (Fixed Distance): Higher speed means less time taken to cover the same distance (Speed ∝ 1/Time). If distance is 100 km, at 50 km/h time is 2 hours (502=100), at 100 km/h time is 1 hour (1001=100).
  • Number of Workers and Time (Fixed Work): More workers means less time to complete the same job (Workers ∝ 1/Time). If 2 workers take 10 days, 4 workers will take 5 days (210 = 45).
  • Number of People and Food Duration (Fixed Food Stock): More people means the food stock lasts for fewer days (People ∝ 1/Days).
• Problem Solving Technique:
  1. Identify the two quantities (x and y).
  2. Recognize that as one increases, the other decreases (confirming inverse proportion).
  3. Set up the relationship: x₁ * y₁ = x₂ * y₂ or x₁ / x₂ = y₂ / y₁.
  4. Substitute the known values and solve for the unknown value.

Example Problem (Inverse Proportion):
6 pipes are required to fill a tank in 1 hour 20 minutes. How long will it take if only 5 pipes of the same type are used?

• Quantities: Number of pipes (x) and Time taken (y).
• Relationship: More pipes means less time. Inverse Proportion.
• First, convert time to a single unit (minutes): 1 hour 20 minutes = 60 + 20 = 80 minutes.
• Let x₁ = 6 pipes, y₁ = 80 minutes.
• Let x₂ = 5 pipes, y₂ = ? (the unknown time).
• Using x₁ * y₁ = x₂ * y₂:
  6 * 80 = 5 * y₂
480 = 5 * y₂
y₂ = 480 / 5
y₂ = 96 minutes.
• Answer: It will take 96 minutes (or 1 hour 36 minutes) if 5 pipes are used.

4. Identifying the Type of Proportion

• This is the most critical step. Read the problem carefully.
• Ask yourself: "If I increase quantity A, what happens to quantity B?"
  • If B increases proportionally -> Direct Proportion.
  • If B decreases proportionally -> Inverse Proportion.
  • If there's no clear proportional relationship -> Neither.

5. Summary Table

Practice MCQs for Government Exam Preparation

Here are 10 Multiple Choice Questions based on this chapter. Remember to identify the type of proportion first!

1. If the cost of 8 toys is ₹192, what will be the cost of 14 toys?
(a) ₹336
(b) ₹320
(c) ₹256
(d) ₹350

2. A car travels 180 km in 4 hours. How far will it travel in 7 hours at the same speed?
(a) 280 km
(b) 300 km
(c) 315 km
(d) 350 km

3. If 15 workers can build a wall in 48 hours, how many workers will be required to do the same work in 30 hours?
(a) 18
(b) 20
(c) 24
(d) 28

4. A garrison of 500 men had provisions for 27 days. If 300 more men join the garrison, how many days will the provisions last?
(a) 15 days
(b) 16.875 days (approx 17 days)
(c) 18 days
(d) 20 days

5. Which of the following is an example of inverse proportion?
(a) The number of items purchased and their total cost.
(b) The distance covered and the petrol consumed by a car.
(c) The speed of a vehicle and the time taken to cover a fixed distance.
(d) The wages earned and the number of hours worked.

6. If x and y are directly proportional, and x = 6 when y = 24, what is the value of y when x = 9?
(a) 30
(b) 36
(c) 18
(d) 40

7. If p and q are inversely proportional, and p = 10 when q = 6, what is the value of p when q = 15?
(a) 4
(b) 25
(c) 9
(d) 11

8. A machine fills 420 bottles in 3 hours. How many bottles will it fill in 5 hours?
(a) 600
(b) 700
(c) 750
(d) 840

9. 6 oxen or 8 cows can graze a field in 28 days. How long would 9 oxen and 2 cows take to graze the same field? (Hint: Convert cows to equivalent oxen or vice-versa).
(a) 16 days
(b) 18 days
(c) 20 days
(d) 14 days

10. In direct proportion, if quantity x is doubled, the quantity y becomes:
(a) Halved
(b) Doubled
(c) Remains same
(d) Four times

Answer Key for MCQs:

1. (a) ₹336 (Direct: 192/8 = x/14 => x = (192 * 14) / 8 = 336)
2. (c) 315 km (Direct: 180/4 = x/7 => x = (180 * 7) / 4 = 315)
3. (c) 24 (Inverse: 15 * 48 = x * 30 => x = (15 * 48) / 30 = 720 / 30 = 24)
4. (b) 16.875 days (Inverse: Total men = 500 + 300 = 800. 500 * 27 = 800 * x => x = (500 * 27) / 800 = 13500 / 800 = 135 / 8 = 16.875)
5. (c) (Speed increases, time decreases for fixed distance)
6. (b) 36 (Direct: x/y = k => 6/24 = 1/4. So, 9/y = 1/4 => y = 9 * 4 = 36)
7. (a) 4 (Inverse: p*q = k => 10 * 6 = 60. So, p * 15 = 60 => p = 60 / 15 = 4)
8. (b) 700 (Direct: 420/3 = x/5 => x = (420 * 5) / 3 = 2100 / 3 = 700)
9. (a) 16 days (Inverse: First find equivalence: 6 oxen = 8 cows => 1 ox = 8/6 = 4/3 cows. Or 1 cow = 6/8 = 3/4 oxen. Let's use oxen. 9 oxen + 2 cows = 9 oxen + 2*(3/4) oxen = 9 + 1.5 = 10.5 oxen. Now, use inverse proportion with oxen: 6 oxen * 28 days = 10.5 oxen * x days => x = (6 * 28) / 10.5 = 168 / 10.5 = 16)
10. (b) Doubled (Direct: x/y = k. If x becomes 2x, then (2x)/y' = k = x/y => y' = (2x * y) / x = 2y)

Make sure you practice numerous problems of both types. Identifying the relationship correctly is half the battle won. Good luck!