Direct and Inverse Proportions Class 8 Maths Chapter Notes

Key Terms and Definitions

Variation

The change in one quantity due to the change in another quantity. This can be a direct or inverse change.

Direct Proportion

Two quantities x and y are in direct proportion if they increase or decrease together in such a way that the ratio x/y remains constant.

Inverse Proportion

Two quantities x and y are in inverse proportion if an increase in x causes a proportional decrease in y (and vice-versa) in such a way that their product xy remains constant.

Constant of Proportionality (k)

The constant value that relates two proportional quantities. For direct proportion, k = x/y. For inverse proportion, k = xy.

Ratio

A comparison of two quantities by division, expressed as a:b or a/b.

Understanding Direct and Inverse Variation

At its core, this chapter is about understanding relationships. When two things are connected, how does a change in one affect the other? This is called variation.

Direct Proportion (or Direct Variation) is the simpler relationship. Think about buying apples. If you buy more apples, the total cost goes up. If you buy fewer apples, the cost goes down. Here, the number of apples and the total cost are in direct proportion. As one quantity (apples) increases, the other quantity (cost) also increases at a steady rate. This steady rate is defined by the constant of proportionality (k). In this case, 'k' would be the price of one apple. The formula is that the ratio of the two quantities is always constant: x/y = k. So, if x₁ and y₁ are one pair of values and x₂ and y₂ are another, then x₁/y₁ = x₂/y₂.

Inverse Proportion (or Inverse Variation) is the opposite. Imagine you need to paint a wall. If you hire more painters, the time it takes to finish the job will decrease. If you have fewer painters, it will take more time. Here, the number of painters and the time taken are in inverse proportion. As one quantity (painters) increases, the other quantity (time) decreases proportionally. The key here is that the product of the two quantities remains constant. The formula is xy = k. So, if x₁ and y₁ are one pair of values and x₂ and y₂ are another, then x₁y₁ = x₂y₂.

Direct vs. Inverse Proportion: A Quick Comparison

Must Remember for Exams

• {"point":"The core formula for Direct Proportion is x₁/y₁ = x₂/y₂. The ratio of corresponding values is constant."}
• {"point":"The core formula for Inverse Proportion is x₁y₁ = x₂y₂. The product of corresponding values is constant."}
• {"point":"To identify the type of proportion, ask yourself: 'If I increase the first quantity, does the second quantity increase or decrease?'"}
• {"point":"Increase-Increase or Decrease-Decrease means Direct Proportion."}
• {"point":"Increase-Decrease or Decrease-Increase means Inverse Proportion."}
• {"point":"The constant 'k' is your check. In direct proportion, x/y must be the same for all pairs. In inverse, x*y must be the same."}
• {"point":"Common direct proportion examples: distance-time (at constant speed), cost-quantity, wages-hours worked."}
• {"point":"Common inverse proportion examples: speed-time (for a fixed distance), number of workers-time taken, number of pipes-time to fill a tank."}

How to Solve Proportion Word Problems

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Worked Examples

• {"title":"Example 1: Direct Proportion","bodyMarkdown":"Problem: A car travels 90 km in 2 hours. How much time is required to cover 135 km with the same speed?\n\nSolution:\n1. Quantities: Distance (km) and Time (hours).\n2. Relationship: More distance means more time. This is Direct Proportion.\n3. Setup:\n x₁ = 90 km, y₁ = 2 hours\n x₂ = 135 km, y₂ = ?\n4. Formula: x₁ / y₁ = x₂ / y₂\n5. Solve: 90 / 2 = 135 / y₂ => 45 = 135 / y₂ => y₂ = 135 / 45 = 3 hours.\n\nAnswer: It will take 3 hours to cover 135 km."}
• {"title":"Example 2: Inverse Proportion","bodyMarkdown":"Problem: 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?\n\nSolution:\n1. Quantities: Number of pipes and Time (minutes).\n2. Relationship: Fewer pipes means more time. This is Inverse Proportion.\n3. Setup: (Note: 1 hr 20 min = 80 minutes)\n x₁ = 6 pipes, y₁ = 80 minutes\n x₂ = 5 pipes, y₂ = ?\n4. Formula: x₁ y₁ = x₂ y₂\n5. Solve: 6 80 = 5 y₂ => 480 = 5 * y₂ => y₂ = 480 / 5 = 96 minutes.\n\nAnswer: It will take 96 minutes (or 1 hour 36 minutes)."}

Exam Tip: Avoid Common Mistakes

The biggest trap in this chapter is using the wrong formula. Students often mix up the direct and inverse proportion equations under exam pressure.

Pro Tip: Before you write the formula, write one line explaining why you chose it. For example: "As the number of workers increases, the time taken to complete the job decreases. Therefore, this is a case of inverse proportion, and we will use x₁y₁ = x₂y₂." This not only helps you select the correct formula but also shows the examiner your clear understanding, which can fetch you partial marks even if you make a calculation error later.

Practice Questions with Solutions

• Q: If the cost of 12 notebooks is ₹180, what is the cost of 8 notebooks? A: This is direct proportion. (12/180) = (8/x) => 1/15 = 8/x => x = ₹120.
• Q: A school has 8 periods a day, each of 45 minutes duration. How long would each period be if the school has 9 periods a day, assuming the school hours remain the same? A: This is inverse proportion. (8 45) = (9 x) => 360 = 9x => x = 40 minutes.
• Q: A machine fills 600 bottles in 3 hours. How many bottles will it fill in 5 hours? A: This is direct proportion. (600/3) = (x/5) => 200 = x/5 => x = 1000 bottles.
• Q: 15 men can build a wall in 48 hours. How many men will be required to do the same work in 30 hours? A: This is inverse proportion. (15 48) = (x 30) => 720 = 30x => x = 24 men.

Frequently Asked Questions

Frequently Asked Questions