How Quantities Combine: Understanding Data Class 9 Notes PDF | Free – USP

Class 9 Maths Chapter 10 – How Quantities Combine: Understanding Data

A complete 79-page study guide for Class 9 Mathematics Chapter 10 from Ganita Manjari Part II (NCERT 2026–27). This guide covers weighted averages, mixtures and concentrations, custom weights, and stacked bar charts — with concept explanations, solved examples, topic-wise NCERT solutions, a formula sheet, extra practice questions, and an unsolved worksheet with answer key.

Combining Things — Weighted Average

When two or more groups of different sizes are combined, you cannot simply take the ordinary average of their values. A weighted average gives each value an importance (weight) based on how much of it there is.

Example: If 20 students score an average of 60 marks and 30 students score an average of 80 marks, the combined average is not 70. It is (20×60 + 30×80) ÷ 50 = (1200 + 2400) ÷ 50 = 72 marks — pulled closer to 80 because the larger group scored higher.

Mixtures and Concentrations

Concentration tells you how much of one substance is present in a mixture, usually as a percentage. When two solutions of different concentrations are mixed, the result is a weighted average of the two — weighted by the quantity of each solution.

Example: Mixing 2 litres of 30% sugar solution with 3 litres of 50% sugar solution gives (2×30 + 3×50) ÷ 5 = (60 + 150) ÷ 5 = 42% concentration.

Custom Weights — Ratings and Marks

Weights do not always come from quantity. Sometimes they are assigned to show importance — for example, when a school counts the final exam more heavily than unit tests, or when product ratings count verified reviews more than anonymous ones.

Visualising Data — Stacked Bar Charts

A stacked bar chart shows how a total is divided into parts. Each bar represents a whole, and the segments within it show the contribution of each category — useful for comparing both totals and their composition at the same time.

100% Stacked Bar Charts

In a 100% stacked bar chart, every bar is drawn to the same height and each segment shows a percentage share instead of an actual value. This makes it easy to compare proportions across groups of very different sizes — for example, comparing subject preferences in a class of 30 with a class of 80.

Common Mistakes to Avoid

• Taking a simple average instead of a weighted average when group sizes differ
• Forgetting to divide by the total of all weights, not the number of groups
• Confusing a stacked bar chart (shows totals) with a 100% stacked bar chart (shows proportions)
• Mixing up quantity-based weights with importance-based weights

Frequently Asked Questions

Q1. What is a weighted average?
A weighted average is an average where each value is multiplied by its weight, the results are added, and the total is divided by the sum of all weights. It is used when the values do not carry equal importance or come from groups of different sizes.

Q2. How is a weighted average different from a normal average?
A normal average treats every value equally. A weighted average gives more influence to values with larger weights — so the answer moves closer to the values that carry more weight.

Q3. How do you find the concentration of a mixture?
Multiply each solution's quantity by its concentration, add the results, then divide by the total quantity of the mixture.

Q4. When should I use a 100% stacked bar chart instead of a normal one?
Use a 100% stacked bar chart when you want to compare proportions across groups of different sizes. Use a normal stacked bar chart when the actual totals also matter.

Q5. Is a weighted average always between the smallest and largest value?
Yes. A weighted average can never fall outside the range of the values being combined — it always lies between the smallest and the largest.

What's Included in This Free Download

• 79-page complete study guide with concept explanations and solved examples
• Topic-wise solutions to all NCERT end-of-chapter exercises
• Chapter summary and formula sheet for quick revision
• Extra practice question bank plus an unsolved worksheet
• Full answer key

Prepared by Sumeet Sahu, Unique Study Point, Indore. Free for all Class 9 CBSE students.