Chapter 10 Class 9 - How Quantities Combine: Understanding Data

Study Chapter 10: How Quantities Combine: Understanding Data from NCERT Ganita Manjari, Class 9 Maths, Part II, with Teachoo. Learn how to combine averages correctly, calculate the concentration of mixtures, use different weights in assessments and ratings, and interpret stacked bar charts.

Can you find the average marks of two classes by simply averaging their class averages? Does mixing two solutions always give a concentration halfway between the original concentrations? Can a higher percentage represent a smaller actual quantity?

This chapter helps you answer these questions. Teachoo makes the reasoning easier to follow, connecting formulas with practical examples so that you understand which calculation or chart to use—and why.

Whether you are learning the concepts for the first time, practising Exercise Sets 10.1–10.5, or working through the End-of-Chapter Exercises, make Teachoo your study companion for understanding data with confidence.

What will you learn in Chapter 10?

The chapter has two main sections: Combining Things and Visualising and Interpreting Data.

Combining things: averages, mixtures and weights

Learn how to:

• Find the combined average of groups with different numbers of observations.

• Recognise when taking a simple average of averages works.

• Calculate the weighted mean, also called the weighted average.

• Find the concentration of a mixture using the quantities and concentrations of its ingredients.

• Apply custom weights to marks, assessments and rating systems.

• Calculate how an average changes when observations are added or removed.

• Use weighted averages in contexts such as rainfall, alloy purity, prices and family ages.

Visualising and interpreting data

Learn how to:

• Read clustered bar or column charts.

• Use stacked bar charts to compare components and totals.

• Interpret 100% stacked bar charts to compare percentage shares.

• Understand the relationship between pie charts and 100% stacked bar charts.

• Distinguish absolute quantities from proportions.

• Decide which conclusions a chart supports and what additional information you need.

With Teachoo, you can connect the calculations to the charts and understand the story behind the data.

Understand weighted averages with a simple example

Suppose one group has 10 students with an average score of 60, and another has 30 students with an average score of 80.

Simply averaging 60 and 80 gives 70. But this treats the two groups equally, even though the second group has three times as many students.

The correct combined average is:

Combined average = [(10 × 60) + (30 × 80)] ÷ (10 + 30)
= 3,000 ÷ 40
= 75

The larger group has a greater influence on the result. Here, the numbers of students are the weights.

This is the central idea of a weighted average: each value contributes according to its weight. Teachoo helps you identify those weights before applying the formula.

Weighted average formula

For values x₁, x₂, …, xₙ with corresponding weights w₁, w₂, …, wₙ:

Weighted average = (w₁x₁ + w₂x₂ + … + wₙxₙ) ÷ (w₁ + w₂ + … + wₙ)

The weights might represent numbers of students, quantities of solutions, numbers of days, or the importance assigned to different assessment components.

When all the weights are equal, the weighted average becomes the ordinary arithmetic mean.

Exercise Sets 10.1–10.5 and End-of-Chapter Exercises

Ganita Manjari Class 9 Chapter 10 contains five exercise sets—10.1, 10.2, 10.3, 10.4 and 10.5—followed by End-of-Chapter Exercises.

The questions develop your understanding through calculations, graphs, explanations and practical activities. You will encounter contexts such as:

• Combining average test scores and daily rainfall.

• Mixing fertilisers, solutions and alloys.

• Calculating assessment scores with different weightages.

• Interpreting customer ratings and animal-weight data.

• Reading charts about family expenditure and endangered species.

• Comparing how people spend their time.

• Calculating average purchase prices.

• Interpreting percentages, totals and changes in shares.

• Collecting data and creating charts through individual and group projects.

Teachoo helps you approach these questions systematically: identify the quantities, choose the correct weights or representation, calculate carefully, and explain what the result means.

Why study Chapter 10 with Teachoo?

This chapter requires more than substituting numbers into a formula. You need to decide whether groups should contribute equally, whether a chart shows totals or percentages, and whether the available information is enough to support a conclusion.

Teachoo teaches these distinctions clearly, helping you avoid common mistakes such as averaging unequal groups incorrectly or assuming that a larger percentage always means a larger number.

Start with averages, move to mixtures and custom weights, and then practise interpreting charts. Choose Teachoo for clear, connected learning that helps you understand the chapter and write answers with confidence.

Frequently asked questions

1. What is Chapter 10 in Class 9 Ganita Manjari?

Chapter 10 in Ganita Manjari, Class 9 Maths, Part II, is “How Quantities Combine: Understanding Data.” It covers weighted averages, combined averages, mixtures, custom weights, stacked bar charts and 100% stacked bar charts. Teachoo helps you understand the calculations and interpret what the data shows.

2. What is a weighted average?

A weighted average is an average in which each value contributes according to its assigned weight. Multiply each value by its weight, add the products, and divide by the sum of the weights.

For example, when combining two class averages, the number of students in each class is the weight.

3. What is the difference between an ordinary average and a weighted average?

An ordinary arithmetic mean gives every observation equal weight. A weighted average allows different values or group averages to have different weights.

If two classes have different numbers of students, their class averages generally need different weights when calculating the average for all students together.

4. Can you calculate a combined average by averaging the group averages?

This method works when the groups contain equal numbers of observations. When group sizes differ, use the group sizes as weights to calculate the combined average correctly.

For two groups with averages a and b and sizes n and m:

Combined average = (na + mb) ÷ (n + m)

Teachoo helps you check the group sizes before choosing the method.

5. How do you calculate the concentration of a mixture?

First calculate the amount of the ingredient contributed by each component. Add these amounts and divide by the total quantity of the mixture.

For example, mixing 100 mL of a 10% solution with 300 mL of a 20% solution, using the textbook’s concentration convention, gives:

Final concentration = [(100 × 10) + (300 × 20)] ÷ 400
= 17.5%

The result is closer to 20% because more of the mixture comes from the 20% solution.

6. What are custom weights in Chapter 10?

Custom weights represent the importance assigned to different components. For example, an assessment may give 20% weight to internal work and 80% to a final examination.

Convert the component marks to a common scale, such as percentages, before applying the weights. Otherwise, marks from assessments with different maximum scores cannot be compared directly.

7. What happens to a weighted average if all weights are doubled?

The weighted average stays the same. Doubling every weight doubles both the numerator and denominator of the formula, so the factor of 2 cancels.

The same applies when all weights are multiplied by any positive constant.

8. What is a stacked bar chart?

A stacked bar chart represents a total using a bar divided into segments. Each segment shows one component, and the entire bar shows the total.

For example, a family’s expenditure bar may contain segments for food, housing, education and healthcare. You can compare both the spending categories and the families’ total expenditure.

9. What is a 100% stacked bar chart?

A 100% stacked bar chart represents every total as 100%. Each segment shows the percentage share of one component.

All bars have the same overall length or height, making it easier to compare how different totals are divided. However, the chart alone does not show the actual size of those totals.

10. What is the difference between a stacked bar chart and a 100% stacked bar chart?

A stacked bar chart shows actual quantities, allowing you to compare components and totals.

A 100% stacked bar chart shows proportions, allowing you to compare percentage shares even when the totals differ.

Teachoo helps you choose the representation that answers the question you are investigating.

11. Does a higher percentage always mean a larger actual number?

No. The actual number depends on both the percentage and the total.

For example, 80% of 100 schools is 80 schools, while 60% of 200 schools is 120 schools. The second group has a lower percentage but a larger number of schools.

This distinction is important when interpreting the chapter’s charts and infographics.

12. Can you convert a 100% stacked bar chart into a stacked bar chart?

You need the total quantity represented by each bar. Percentage shares alone are not enough to calculate actual quantities.

However, you can convert a stacked bar chart into a 100% stacked bar chart by dividing each component by its bar’s total and multiplying by 100.

13. Which exercises are included in Ganita Manjari Class 9 Chapter 10?

The chapter includes Exercise Sets 10.1, 10.2, 10.3, 10.4 and 10.5, followed by End-of-Chapter Exercises. It also includes practical projects involving data collection, weighted ratings and chart construction.

14. How should I study Chapter 10 effectively?

Begin with ordinary averages and then learn why unequal groups require weights. Practise mixtures and assessment weightages before moving to charts.

For every graph, ask: Does it show actual quantities or percentages? What comparisons are valid? What information is missing?

Use Teachoo to work through the reasoning, then attempt questions independently to check your understanding.

15. Why choose Teachoo for Class 9 Chapter 10?

Teachoo helps you connect weighted averages, mixtures and data interpretation, so you understand both how to calculate an answer and what it means. Its focus on clear reasoning makes it a strong choice for studying Ganita Manjari Class 9 Chapter 10, practising textbook questions and revising confidently.

Learn Chapter 10 with Teachoo—understand the weights, read the charts and make sense of the data.