Ganita Manjari · Class 9 · Part II · Chapter 10

Class 9 Maths Ganita Manjari Chapter 10 Exercise 10.1 Solutions

All 5 questions, worked step by step with the weighted mean. Each answer lists the values and weights, estimates the result first, and then calculates.

Last updated 6 October 2026

5Questions
3Diagrams
Ch 10How Quantities Combine: Understanding Data

How to solve every question

weighted mean = w₁x₁ + w₂x₂ + … + w_nx_nw₁ + w₂ + … + w_n

List the groups Write each group's average (or concentration) x and its size w, the weight.
Multiply Find x × w for every group. This is the group's total.
Add Add all the totals, and add all the weights.
Divide Total of (weight × value) over total weight is the weighted mean. Check it lies between the smallest and largest averages.

Weighted mean: an average in which each value counts according to its weight.

Weight of a value is how big or how important it is, such as the number of students or the volume of a mixture.

Equal weights: when all weights are the same, the weighted mean is just the simple average.

Combining class averages

Question 1: Combined average of two sections

Answer ≈ 73.82

Question: Section A (30 students) averaged 72 on a test and Section B (25 students) averaged 76. Find the combined average of both sections.

Step 1 · Collect the values and weights

Section A: average 72, weight (students) 30
Section B: average 76, weight (students) 25

The sections have different sizes, so the two averages cannot be treated equally.

Step 2 · Estimate before calculating

Section A has more students, so the combined average should be closer to 72 than to 76. That means it should be below the simple average 74.

Q1 · the combined average sits closer to the larger section

Step 3 · Apply the weighted mean

Combined average = 72 × 30 + 76 × 2530 + 25

= 2160 + 190055 = 406055 = 81211 ≈ 73.82

Here 72 × 30 = 2160 is the total marks of Section A and 76 × 25 = 1900 is the total marks of Section B.

The combined average score is 81211 ≈ 73.82, below 74 as the estimate predicted.

Mixtures

Question 2: Nitrogen in a mixture of three equal parts

Answer: 11100

Question: A farmer mixes three equal quantities of fertilisers containing 110, 950 and 360 nitrogen. Find the fraction of nitrogen in the mixture.

Step 1 · Notice the quantities are equal

Let each quantity be y. Equal weights mean the nitrogen fraction of the mixture is the simple average of the three fractions.

Step 2 · Write the fractions with the same denominator

First simplify 360 = 120. Then use denominator 100:

110 = 10100, 950 = 18100, 120 = 5100

Step 3 · Find the nitrogen in the mixture

10y100 + 18y100 + 5y100 = 33y100 of nitrogen in 3y of mixture, so the fraction = 33y100 × 3y = 33300 = 11100

The answer lies between the smallest (5100) and the largest (18100) fractions, as it should.

The mixture contains 11100 nitrogen, that is 11%.

Question 3: Gold purity in varṇa

Answer ≈ 11.13 varṇa

Question: Gold of k varṇa is k16 pure. A goldsmith melts 9 units at 12 varṇa, 5 units at 10 varṇa and 17 units at 11 varṇa. Find the purity of the combined gold in varṇa.

Step 1 · List the values and weights

Piece	Units (weight)	Purity (varṇa)	Units × purity
1	9	12	108
2	5	10	50
3	17	11	187
Total	31		345

Q3 · the widest piece has the biggest say in the final purity

Step 2 · Apply the weighted mean

9 × 12 + 5 × 10 + 17 × 119 + 5 + 17 = 108 + 50 + 18731 = 34531

Since 34531 = 11 and 431, the value is about 11.13.

Step 3 · Check the result

11.13 lies between the smallest purity (10) and the largest (12). It is close to 11 because the largest piece, 17 units, has purity 11.

The combined gold has a purity of about 11.13 varṇa. In gold content this is 345496, about 69.6% gold (roughly 16.7 karat).

Averages over months

Question 4: Combined average rainfall per day

Answer ≈ 7.37 mm per day

Question: The average rainfall per day in May, June and July is 3.5 mm, 10 mm and 8.7 mm respectively. Write an expression for the combined average.

Step 1 · Find the weights

The weight of each month is its number of days.

Month	Average per day (mm)	Days (weight)
May	3.5	31
June	10	30
July	8.7	31

Step 2 · Apply the weighted mean

Combined average = 3.5 × 31 + 10 × 30 + 8.7 × 3131 + 30 + 31

Each term (average × days) is the total rainfall of that month, so the numerator is the total rainfall and the denominator is the total number of days.

= 108.5 + 300 + 269.792 = 678.292 ≈ 7.37 mm

Step 3 · Why not 3.5 + 10 + 8.73?

That treats the three months as equal. It is correct only when every month has the same number of days. June has 30 days and May and July have 31, so the days-weighted expression above is the right one. (The simple average, 7.4 mm, is close but not exact.)

The combined average is 3.5 × 31 + 10 × 30 + 8.7 × 3192 ≈ 7.37 mm per day.

Concentration of mixtures

Question 5: Spice mix in two kashayam mixtures

Answers: 11.67% and 8.33%

Question: (i) 100 mL of kashayam with 5% spice mix, 200 mL with 10% and 300 mL with 15% are combined. (ii) 300 mL with 5%, 200 mL with 10% and 100 mL with 15% are mixed. Find the concentration of spice mix in each case.

Q5 · the same three kashayams in different amounts

Step 1 · Scenario (i)11.67%

100 × 5 + 200 × 10 + 300 × 15100 + 200 + 300

= 500 + 2000 + 4500600 = 7000600 ≈ 11.67%

That is 70 mL of spice mix in 600 mL.

Step 2 · Scenario (ii)8.33%

300 × 5 + 200 × 10 + 100 × 15300 + 200 + 100

= 1500 + 2000 + 1500600 = 5000600 ≈ 8.33%

That is 50 mL of spice mix in 600 mL.

Step 3 · Compare the two

Both mixtures use the same three concentrations and the same total volume, yet the results differ. In (i) the strongest kashayam (15%) has the largest volume, so the mixture is stronger. In (ii) the weakest kashayam (5%) has the largest volume, so the mixture is weaker.

(i) 11.67% spice mix, (ii) 8.33% spice mix.

Answers at a glance

Every question uses the weighted mean: multiply each value by its weight, add, and divide by the total weight.

Question	What is asked	Weights used	Answer
Q1	Combined average score	Students: 30 and 25	81211 ≈ 73.82
Q2	Fraction of nitrogen	Equal quantities, so equal weights	11100 = 11%
Q3	Purity of combined gold	Units: 9, 5 and 17	34531 ≈ 11.13 varṇa
Q4	Combined average rainfall per day	Days: 31, 30 and 31	3.5 × 31 + 10 × 30 + 8.7 × 3192 ≈ 7.37 mm
Q5	Spice mix concentration	Volumes in mL	(i) 11.67%, (ii) 8.33%

Source of the questions: NCERT, Ganita Manjari, Grade 9, Part II, Chapter 10, Exercise 10.1. The solutions, explanations and diagrams on this page are our own working.