# NCERT Class 8 Mathematics Chapter 10 Proportional Reasoning-2: Summary, Concepts, and Revision Notes | SwaVid

This chapter builds upon your understanding of proportional reasoning by introducing inverse proportion, compound proportion, and their applications in...

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# Proportional Reasoning-2

## Concepts in this chapter

## The ideas this chapter keeps returning to

## Where this chapter usually trips people up

## Find the gaps before this chapter finds them for you.

## Related chapters

### Inverse Proportion

Chapter 10 · Class 8 Mathematics

This chapter builds upon your understanding of proportional reasoning by introducing inverse proportion, compound proportion, and their applications in real-world problems like time and work. You will learn to differentiate between direct and inverse relationships and apply appropriate methods to solve complex scenarios involving multiple quantities.

Your study route

Key topics

Two quantities are in inverse proportion if an increase in one quantity causes a decrease in the other quantity in such a way that their product remains constant. If 'x' and 'y' are inversely proportional, then xy = k (a constant).

Example

Example 1: If 4 workers can complete a task in 12 days, how many days will 6 workers take to complete the same task?

Watch out

Confusing inverse proportion with direct proportion, especially when quantities seem related but the product, not the ratio, is constant.

Two quantities are in inverse proportion if an increase in one quantity causes a decrease in the other quantity in such a way that their product remains constant. If 'x' and 'y' are inversely proportional, then xy = k (a constant).

Tap the card for an example

Example

Example 1: If 4 workers can complete a task in 12 days, how many days will 6 workers take to complete the same task?

Why it matters

Helps in understanding relationships where resources or effort affect time or output inversely, like more workers meaning less time for a job.

Watch out

Confusing inverse proportion with direct proportion, especially when quantities seem related but the product, not the ratio, is constant.

Ask at home

Ask your child to explain what happens to the time taken to complete a task if the number of workers increases, and why. Ensure they can state that the product of workers and days remains constant.

Chapter 10, "Proportional Reasoning-2", expands on the concept of proportionality. It begins by defining inverse proportion, where an increase in one quantity leads to a proportional decrease in another, keeping their product constant. The chapter then introduces compound proportion, which involves relationships between three or more quantities, requiring careful analysis to determine if each pair is directly or inversely proportional. Key methods for solving these problems, including the unitary method and formulaic approaches, are explained with examples. Finally, the chapter delves into 'Time and Work' problems, teaching how to calculate individual and combined work rates to determine the time taken to complete tasks. Mastery of these concepts is crucial for solving practical problems involving rates, resources, and efficiency.

Chapter summary

Chapter 10, "Proportional Reasoning-2", expands on the concept of proportionality. It begins by defining inverse proportion, where an increase in one quantity leads to a proportional decrease in another, keeping their product constant. The chapter then introduces compound proportion, which involves relationships between three or more quantities, requiring careful analysis to determine if each pair is directly or inversely proportional. Key methods for solving these problems, including the unitary method and formulaic approaches, are explained with examples. Finally, the chapter delves into 'Time and Work' problems, teaching how to calculate individual and combined work rates to determine the time taken to complete tasks. Mastery of these concepts is crucial for solving practical problems involving rates, resources, and efficiency.

What you should learn

Keep these close

Direct Proportion: x/y = k or x1/y1 = x2/y2.

Inverse Proportion: xy = k or x1y1 = x2y2.

In inverse proportion, as one quantity increases, the other decreases proportionally.

The unitary method can be used for both direct and inverse proportion problems.

Compound proportion involves relationships between three or more quantities.

For compound proportion, identify if each pair of quantities is directly or inversely proportional.

A common formula for compound proportion is M1D1H1/W1 = M2D2H2/W2 (Men, Days, Hours, Work).

If a person completes a work in 'n' days, their one-day work is 1/n.

To find combined work rate, add the individual one-day work fractions.

The reciprocal of the combined one-day work gives the total time taken to complete the work together.

In problems where workers leave or join, calculate the work done and remaining work carefully.

Always check if the answer makes logical sense in the context of the problem.

Common confusions

It is easy to think

If 3 people take 10 days to do a job, then 6 people will take 20 days.

The clearer idea

This assumes direct proportion. In reality, more people usually mean less time for the same job (inverse proportion). So, 6 people would take 5 days (3 * 10 = 6 * x => x = 5).

It is easy to think

To find the time taken by A and B together, I just add the days A takes and B takes.

The clearer idea

You cannot directly add the days. You must add their one-day work fractions (e.g., 1/A + 1/B) to find their combined one-day work, then take the reciprocal for the total time.

It is easy to think

All problems with multiple quantities are solved the same way.

The clearer idea

Each pair of quantities in a compound proportion problem must be individually analyzed to determine if they are directly or inversely proportional to the quantity being sought.

It is easy to think

If a car travels faster, it will take more time to cover the same distance.

The clearer idea

Speed and time to cover a fixed distance are inversely proportional. Higher speed means less time.

It is easy to think

The formula M1D1/W1 = M2D2/W2 always works for all compound proportion problems.

The clearer idea

This formula is a specific case. It's crucial to understand the underlying direct/inverse relationships. For example, if hours per day (H) are involved, it becomes M1D1H1/W1 = M2D2H2/W2. Always derive or adapt based on the specific quantities.

Before you push ahead

Most stuck chapters trace back to one earlier idea. Check the prerequisites first, or let SwaVid adapt this chapter to the way your child learns.

Keep exploring Class 8

Source

- Differentiate between direct and inverse proportion in various contexts.
- Solve problems involving inverse proportion using appropriate methods.
- Understand and apply the concept of compound proportion to solve multi-variable problems.
- Calculate individual and combined work rates to solve time and work problems.
- Apply proportional reasoning to solve real-life scenarios efficiently.
- Differentiate between direct and inverse proportion in various contexts.
- Solve problems involving inverse proportion using appropriate methods.
- Understand and apply the concept of compound proportion to solve multi-variable problems.
- Calculate individual and combined work rates to solve time and work problems.
- Apply proportional reasoning to solve real-life scenarios efficiently.
- NCERT Class 8 Mathematics textbook: Ganita Prakash-II : Chapter 10: Proportional Reasoning-2

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