Class 9 Mathematics

Chapter 10 Solutions — How Quantities Combine: Understanding Data

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Step-by-step NCERT solutions for How Quantities Combine: Understanding Data (Chapter 10, NCERT Class 9 Mathematics) — every question and answer worked out in full, not just the final result. You can also read the How Quantities Combine: Understanding Data textbook chapter.

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All 33 questions in How Quantities Combine: Understanding Data are solved in the PDF. Here's what's inside, exercise by exercise:

Exercise Set 10.1

The average score of students on a test in Section A is 72 and that of students in Section B is 76. What is the combined average of both the sections given that Section A has 30 students and Section B has 25 students?
A farmer mixes three equal quantities of fertilisers. The first one contains 1/10 nitrogen, the second contains 9/50 nitrogen, and the third contains 3/60 nitrogen. What is the fraction of nitrogen in the mixture?
(Śrīdharācārya, Pāṭīgaṇita, c. 750 CE) A goldsmith melts together three pieces of gold: 9 units at 12 varṇa, 5 units at 10 varṇa, and 17 units at 11 varṇa. Find the purity in varṇa of the combined gold.
The average rainfall per day in the months of May, June, and July in a certain location are 3.5 mm, 10 mm and 8.7 mm respectively. Write an expression that gives their combined average.
5.
Calculate the concentration of spice mix in these two scenarios.
- (i) A 100 mL kashayam/kadha with 5% spice mix, a 200 mL one with 10% spice mix, and a 300 mL one with 15% spice mix are combined.
- (ii) A 300 mL kashayam/kadha with 5% spice mix, a 200 mL one with 10% spice mix, and a 100 mL one with 15% spice mix are mixed.

Exercise Set 10.2

Savitri's marks in Kashmiri are as follows: 35 out of 50 in internal tests, 44 out of 60 in the project, and 80 out of 100 in the final exam. The annual percentage score is calculated by combining the internals, project, and final exam in the ratio 3 : 4 : 5. Which of the following expression(s) gives her annual score (as a percentage) in Kashmiri? (i) (35×3 + 44×4 + 80×5)/(3+4+5) (ii)…

Exercise Set 10.3

A stationery shop owner made ₹8000 selling books, of which 30% is the profit amount, and ₹1000 selling book covers, of which 50% is the profit amount. What is the percentage of profit on the total sales?
A white stork's migration is tracked. The average daily distance travelled, calculated over 20 days, is 44.5 km. On the 21st day, it flew 55 km. What is the average daily distance travelled over these 21 days? Make a guess before you calculate.
A 600 mL solution with 5% salt is mixed with a 300 mL solution with 8% sugar. What are the concentrations of salt and sugar in the mixture?
At a panipuri (golgappa) stall, the concentration of spice in the pani (spiced water) was 8%. Many customers complained that it was too spicy. What quantity of regular water should be mixed into the 10 litres of pani so that the spice level is reduced to (3/4)th of the original concentration?
5.
A physical fitness evaluation is being undertaken. The final marks are calculated by combining the marks for strength, flexibility, and agility in the ratio 4 : 5 : 6. Rashi has scored 60, 65, and 70. Keerthi has scored 55, 65, and 75 respectively.
- (i) Find out whose total is more without calculating.
- (ii) What are their final marks?
A restaurant collected ratings from 10 customers on a scale of 1 to 5. Food ratings: 5★ from 5 customers, 4★ from 3, 3★ from 2, 2★ from 0, 1★ from 0. Ambience ratings: 5★ from 0, 4★ from 4, 3★ from 5, 2★ from 1, 1★ from 0. Service ratings: 5★ from 1, 4★ from 2, 3★ from 2, 2★ from 4, 1★ from 1. What is the average rating if the metrics are combined with the weights food : ambience : service = 6…
7.
The following table shows the weight data of langurs in an animal facility. Male langurs: average weight 16.5 kg. Female langurs: average weight 13.8 kg. All langurs: average weight 14.925 kg, total number 60. Without doing any computations, can you tell whether there are more male langurs or more female langurs?
- (i) Which expression describes the scenario?
- (ii) Find how many male langurs are…
8.
Dorjee has collected 1 litre of water from the Dead Sea. Using the given salinity table (Dead Sea ≈ 34%, Other seas/oceans ≈ 3.5%, Ground water ≈ 0.01%, Purified drinking water ≈ 0.001%), answer the following.
- (i) What is the salinity of the mixture if he mixes 1 litre of Dead Sea water with 2 litres of purified drinking water?
- (ii) Is it possible to mix Dead Sea water and purified drinking…

Exercise Set 10.4

The following stacked column chart shows the number of animal species in the IUCN Red List, by class, over time. Total counts at the top of each column: 2007 = 7,851; 2010 = 9,618; 2013 = 11,212; 2016 = 12,630; 2019 = 14,234. Classes shown: Mammals, Birds, Reptiles, Fish, Amphibians, Molluscs, Insects, Others. (i) What does the number 14,234 on the top of the 2019 column represent? (ii)…
2.
The wickets taken by a bowler in International Cricket matches till 2025 are shown in the table (Home — Test: 62, ODI: 100, T20: 32; Overseas — Test: 172, ODI: 49, T20: 71). Complete the given stacked bar charts:
- (i) comparing the total wickets taken at home vs. overseas (bars show Test, ODI, T20 stacked),
- (ii) the total wickets taken in each format (bars show Home, Overseas stacked).

Exercise Set 10.5

The smart watch data of 5 people was tracked from Monday to Friday. Table 10.3 shows average hours per day in each body-state (Lying down / Sitting / Standing or Moving). Pavani (Patient): 20, 3, 1. Raghu (Nurse): 7, 4, 13. Zakir (Teacher): 8, 6, 10. Sahana (Student): ?, ?, ?. Julie (___): 8, 10, 6. (i) How much time does Sahana spend per day in each body-state? Make a reasonable guess. (ii)…

End-of-Chapter Exercises

1.
In cricket, the run rate is the average number of runs scored per over. In a T20 match, a team scored 6 runs in the first over making the run rate 6.
- (i) In the second over they scored 12 runs. What is the run rate now?
- (ii) After Over 19, their run rate was 6. What is the run rate after 20 overs given the team made 12 runs in the last over?
Five friends collected prices of shuttlecocks (N = Nylon, F = Feather, in ₹): Yusuf: 40(N), 105(N), 383(F), 108(N), 165(F), 116(F); Srikanth: 194(N), 85(N), 93(N), 121(N); Kashvi: 49(N), 297(F), 105(N), 275(F), 40(N); Prasanna: 124(N), 333(F), 182(N), 258(F); Gracy: 220(F), 458(F), 129(F), 183(N). (i) Each calculated the average of prices they gathered (a_y, a_s, a_k, a_p, a_g respectively)…
3.
Shreyas holds 25 shares of a company at an average price of ₹150 and Vaishnavi holds 5 shares of the same company at an average price of ₹150.
- (i) Shreyas buys 10 shares of this company at a price of ₹30 each. What is the average price per share for him after the purchase?
- (ii) Vaishnavi buys some shares of this company at a price of ₹30 each and the average price per share for her after the…
(Pṛthūdakasvāmī, commentary on Brahmagupta's Brāhmasphuṭasiddhānta, c. 864 CE) A pool 30 hastas long is divided into 5 sections having lengths 4, 5, 6, 7, and 8 hastas, and is dug to depths of 9, 7, 7, 3, and 2 hastas, respectively. Find the mean depth of the pool.
5.
Suvarna had purchased 1g gold at ₹15k. This is shown by point O on a number line (0 to 30). Six scenarios for her next transaction are given. For each, estimate and mark the new average price of gold she will have after the transaction.
- (i) Purchase 1g at ₹30k
- (ii) Purchase 2g at ₹30k
- (iii) Purchase 1g at ₹10k
- (iv) Purchase 10g at ₹10k
- (v) Purchase 0.5g at ₹30k
- (vi) Purchase 0.5g at ₹15k
Given some data with corresponding weights, how would the weighted average change if all the weights are doubled? If required, experiment with some data. What do you observe? Justify your answer using algebra.
7.
Answer the following questions based on the stacked column chart: 'Cumulative number of objects currently orbiting Earth, by year of launch/separation (as of Apr. 24, 2026)' — showing Payload objects and Other objects from 1960 to 2025+.
- (i) In 2003, approximately how many total objects were found orbiting Earth? In what year did this number double?
- (ii) Find the approximate number and share of…
8.
Observe the infographic showing 'What Percentage of Schools have a Playground?' — a map of India with state-wise percentages (e.g., Haryana: 90%, Uttarakhand: 78%, Arunachal Pradesh: 68%, Punjab: 98%, Maharashtra: 93%, Telangana: 74%).
- (i) Identify the correct inference(s):
- (a) More schools have a playground in Haryana than Uttarakhand.
- (b) Approximately every 2 out of 3 schools in Arunachal…
9.
Decision Dilemma:
- (i) Which of the plays would you choose to watch based on the 100% stacked bar chart showing share of ratings (★★★★★, ★★★★, ★★★, ★★, ★) for Play A, Play B, Play C?
- (ii) The corresponding stacked bar chart shows absolute number of ratings (bars extend to about 100 for A, 750 for B, and 100 for C). Would you change your decision after seeing this? Why/Why not?
10.
A triathlon consists of swimming 3.8 km, cycling 180 km, and running 42.2 km. Table 10.4 shows finish times: Athlete 1 — Swimming 01:08, Cycling 05:00, Running 03:15; Athlete 2 — 01:05, 05:10, 03:35; Athlete 3 — 01:22, 05:55, 03:50.
- (i) Is a stacked bar chart or a 100% stacked bar chart more suitable?
- (ii) If a 100% stacked bar chart is drawn, which questions can be answered:
- (a) Who finished…
Look at the following graph: 'Distribution of disabled persons by age group and type of disability in India (Census 2011)' — a 100% stacked bar chart with age groups 0–19, 20–39, 40–59, 60 & above on the y-axis and percentage (0–100) on the x-axis. Disability types: seeing, hearing, speech, movement, mental retardation, mental illness, any other, multiple disability. What do you notice? What do…
12.
Individual project: Do at least one of the following.
- (i) Recall the previous day and fill in the approximate time spent lying down, sitting, and standing/moving around, etc. Ask at least 2 family members about their day. Visualise using a stacked bar chart.
- (ii) Visualise your family's monthly expenditure using a 100% stacked bar chart for at least 3 months.
13.
Small-group project: Make a group of 3–4 students. Choose one scenario to design a custom rating scheme:
- (a) shopping experience at a cloth store,
- (b) travel experience in a bus,
- (c) tourism experience of a nearby tourist spot,
- (d) clinic/hospital experience.
- (i) Discuss and arrive at 4–6 aspects to rate.
- (ii) Decide relative weights and justify.
- (iii) Collect/imagine ratings from at least 10…
Whole class project: Each student shares the average age of their family and the number of family members. Discuss among the class and come up with a way to find out the average age of all the families of the class.
*(Challenge) Given some data with corresponding weights, what would happen to the weighted average if all the weights are increased by a constant value, say 1? If required, experiment with some data. What do you observe? Justify your answer using algebra.
16.
*(Challenge) A farm has some cows, sheep, and chickens. Last year the cows made up 60%, the sheep 25%, and the chickens 15%. There was a decrease in the number of all three animals' population over the year. Choose the possibilities for the change in their respective shares of the population:
- (i) % cows ↓, % sheep ↓, % chickens ↓
- (ii) % cows ↑, % sheep ↑, % chickens ↑
- (iii) % cows same, % sheep…

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