NCERT Solutions for Class 9 Maths Chapter 10 How Quantities Combine: Understanding Data (PDF) – 2027

Method 1 gives the two group averages equal importance, as if there were as many juniors as seniors. But there are 8 seniors and only 3 juniors. Each senior’s height should count once, so the seniors’ average must count 8 times and the juniors’ average only 3 times.

Method 1 pulls the answer down towards the juniors’ average, so it is too small (157.4 cm).

The combined average is about 73.82. It is closer to 72 than to 76 because Section A has more students.

The quantities are equal, so the fraction in the mixture is the simple average of the three fractions.

Purity of the mixture is the weighted mean of the purities, with the amounts as weights.

May and July have 31 days and June has 30 days, so each month’s average is weighted by its number of days.

The same three strengths give different results because the amounts are swapped: in (i) most of the drink is 15%, in (ii) most of it is 5%.

(iii) and (iv) are correct.

First change each mark to a percentage of its own maximum: 35/50 = 70%, 44/60 ≈ 73.33%, 80/100 = 80%. Then take the weighted mean with weights 3, 4, 5.

(i) and (ii) are wrong: they treat 35 and 44 as if they were out of 100, but they are out of 50 and 60. (ii) also gives a fraction, not a percentage.

It is close to 30% because most of the sales (₹8000 of ₹9000) came from books.

Guess: a little more than 44.5 km, because one day out of 21 is above the average.

The correct option is (v).

Each substance is now spread through a larger volume, so both concentrations fall.

(i) Keerthi’s. Both have 65 in flexibility. Keerthi has 5 less in strength (weight 4) but 5 more in agility (weight 6). Agility has the larger weight, so her gain outweighs her loss.

The restaurant’s average rating is about 3.57 out of 5.

More females. The overall average 14.925 kg is closer to the female average (13.8) than to the male average (16.5), so the females carry more weight in the mean.

(i) (a) is correct (x males, y females). Since x + y = 60, (b) is also correct if read as “= 14.925”. (c) and (d) are wrong.

So there are 25 male langurs.

(ii) Yes, because 0.01% lies between 0.001% and 34%. Let w litres be added:

(iii) No. A mixture’s salinity always lies between the salinities of the parts, here between 0.01% and 34%. It can never fall to 0.001%, which is below both. So no quantity of groundwater works.

Family C (₹950). In Choice 1 the three housing bars stand side by side and Family C’s bar is clearly the shortest (Family A ₹1940, Family B ₹1750).

Family A (₹1770, against ₹1210 for Family B and ₹1300 for Family C). Choice 1 shows this at a glance: Family A’s healthcare bar is the tallest in the healthcare cluster.

About ₹1300 (the table gives ₹1280). On either chart the education bar for Family A ends a little above the 1000 line, roughly halfway to 1500.

Housing (₹1750). In Choice 2 all six of Family B’s bars are together, and the housing bar is the tallest; food (₹1546) and education (₹1500) come next.

On healthcare, but only slightly: ₹1770 on healthcare against ₹1700 on food. Choice 2 makes this easier, since both bars are in Family A’s cluster.

Choice 1 was easier for Questions 1 and 2, because they compare families within one category, and Choice 1 puts those bars side by side.

Choice 2 was easier for Questions 4 and 5, because they compare categories within one family, and Choice 2 keeps each family’s bars together.

Question 3 asks for one value, so both charts work equally well.

Family C spent the least overall (₹7040).

About ₹7000 (exactly ₹950 + ₹1700 + ₹1540 + ₹1400 + ₹1300 + ₹150 = ₹7040).

About 13.5% (roughly one-seventh) of Family C’s expenses were on housing.

No, the cluster charts made us add six values per family. The stacked bar chart (Fig. 10.2) makes it simpler: each family’s bars are joined end to end, so the length of the whole bar is the family’s total.

6. Family C’s bar is the shortest, so Family C spent the least.
7. Family C’s bar ends at about ₹7000.
8. The housing (red) piece is about 950 of the 7000-long bar, roughly one-seventh, about 13–14%.

It is possible but harder.

Only the first piece (housing) starts at 0. Every other piece starts where the previous one ends, so to read, say, Family A’s education you must subtract the start from the end (about 4920 − 3640 = 1280). Also, the same category starts at different points in different bars, so comparing, for example, healthcare across families means comparing pieces that are not lined up.

Q1 (housing) is still easy, since housing starts at 0 in every bar. The others take more effort. Stacked bars make totals easy but individual parts harder.

(i) 14,234 is the total number of animal species (all classes together) on the endangered list in 2019, that is, the full height of the 2019 column.

(ii) About 1,000 reptile species. The dark red reptile piece in 2016 is about one-third of the gap between 0 and 3,000.

(iii) Mammals, birds and amphibians. Their pieces are only a little taller in 2019 than in 2007 (a rise of roughly a tenth to a quarter), while reptiles, insects, molluscs, fish and others roughly doubled or more.

The bowler took more wickets overseas (292) than at home (194), and the most in Test matches (234).

Family B has the smallest share for healthcare. It was not easy from Fig. 10.2: a “share” compares a piece with its own total, and the bars have different totals. Pie charts, or better a 100% stacked bar chart, show each family’s spending as parts of the same whole, so shares can be compared directly.

Some observations:

• Total use doubled, from 1000 units to 2000 units.
• Lighting fell from 360 to 220 units, and its share from 36% to 11%. More efficient bulbs (such as LEDs) could explain this.
• Cooling rose from 300 to 800 units and is now the biggest share (40%), perhaps from more air conditioners and hotter summers.
• Kitchen appliances rose almost four times (120 → 460) and their share nearly doubled (12% → 23%).

Interesting: cooling’s share rose only from 30% to 40%, but its amount rose 2.7 times. Shares hide how much the total grew; amounts hide how the mix changed. We need both charts.

A stacked bar shows actual amounts. From it we can compare totals, and parts within and across bars (for example, Fatima’s 105 summer blooms against Naveen’s 48).

A 100% stacked bar shows only proportions within each bar. It tells us that 50% of Fatima’s flowers bloomed in summer, but not how many. The same 100% chart fits Case 1 (Fatima has more summer blooms) and Case 2 (Naveen has more). So it cannot be used to compare actual numbers across bars.

Yes. A stacked bar chart gives each part’s actual value, so we can find each bar’s total and work out every part as a percentage of it. For example, Fatima in Case 1: 105 ÷ 210 = 50%, 63 ÷ 210 = 30%, 42 ÷ 210 = 20%.

No, not unless we are also told the total for each bar. The 100% chart gives only the percentages. Case 1 (totals 210 and 120) and Case 2 (totals 180 and 240) both give the same percentages, so we cannot tell which stacked chart it came from.

For example, from Example 8’s 100% chart we can say “in Fatima’s garden more blooms came in summer than in monsoon”, but not “Fatima had more summer blooms than Naveen”.

I notice:

Sleep is the biggest part (about one-third of the day), then leisure. Paid and unpaid work take about the same time. Learning is small.

I wonder: why is learning only about 1 hour when students spend 6–7 hours in school? Why is paid work only about 2.4 hours when working adults work about 8 hours? The reason is that this is an average over all Indians — children, workers, homemakers and the elderly — so it may not describe any real person well.

Interesting points and inferences:

• Children spend 22% (about 5.3 h) on learning and almost nothing on paid work; they also sleep the most (38%, about 9 h).
• Youth spend only 11% on learning but 7% on paid work and 9% on unpaid work: many have started working or helping at home.
• Adults spend about 29% (about 7 h) on paid plus unpaid work and almost no time learning.
• The elderly spend the most on leisure, social and travel (33%) and less on work.
• Personal care (13–14%) is almost the same for every age group.

So the “average Indian” of Fig. 10.5 hides big differences between age groups.

The usual trend is that sleep falls as people grow from childhood to adulthood: young children need the most sleep, teenagers a little less, and adults less still.

The chart mostly agrees: children 38% (about 9.1 h), youth 34% (about 8.2 h), adults 32% (about 7.7 h). The elderly (37%, about 8.9 h) do not follow the falling trend. This may be because many have retired and spend more time resting, and not because older people need more sleep.

Learning drops from 22% (about 5.3 h) to 11% (about 2.6 h) because:

• almost all 6–14-year-olds are in school, but in the 15–24 group many have finished or left studies, especially those aged 19–24;
• many youth have started paid work (7%) or take on household and care work (9%), which children hardly do;
• the 11% is an average: full-time college students may learn as much as children, but those who are not studying bring the average down.

On average, yes: adults spend 14% (about 3.4 h) on paid work and 15% (about 3.6 h) on unpaid work and care.

But this does not mean each adult splits the time equally. The average combines people who work long hours for pay with people (often homemakers) who do mainly unpaid household and care work. Surveys show that in India women do much more of the unpaid work and men more of the paid work. A chart split by gender would show this.

Yes. A strip for one day has a fixed length (24 hours), with small pieces for each activity in the order they happened. If we join all the pieces of the same activity (for example, all the sleep pieces) into one block, and keep the blocks in the same order in every strip, each strip becomes one stacked bar of length 24 h = 100%.

Several such strips side by side form a 100% stacked bar chart like Fig. 10.6. We lose only the time order of the activities, not how much time each takes.

(i) Sample guess: lying down 9 h (sleep), sitting 10 h (about 6 h in class, plus homework, meals and travel), standing/moving 5 h (games, walking, chores). Total = 24 h.

(ii) Julie sits for 10 hours and moves for 6, so she probably has a desk job, such as an office worker, bank clerk or computer operator. She could also be someone who sits while working, such as a tailor or a driver.

(iii) Each value ÷ 24 × 100 gives the percentage (Sahana: 9 h = 37.5%, 10 h ≈ 41.7%, 5 h ≈ 20.8%).

Roughly similar, but not the same. Most teachers sleep about 7–8 hours, stand or walk while teaching, and sit while checking work, so the bar will have a similar shape. But the details depend on the person: a teacher who teaches mostly from a chair, travels a long way by bus, or plays sport after school will have a different mix.

Probably yes. Weekends are different from weekdays: people may sleep longer, students and teachers do not sit in class, a nurse may have a day off, and some people move around more doing chores or outings. The 7-day average would mix these days in, so the shares of lying down, sitting and moving would change, for example a little more lying down for most people.

Yes. For example:

• Raghu (nurse), mostly standing/moving: a shopkeeper who stands at the counter, a farmer, a delivery person, a traffic police officer, a waiter, a construction worker.
• Zakir (teacher), a mix: a salesperson, a doctor in a clinic, a chef.
• Julie, mostly sitting: an office worker, a driver, a tailor, a cashier, a software engineer.
• Pavani (patient), mostly lying down: a person recovering from an illness or injury, or a very elderly person who needs rest.

In (i) one big over changes the run rate a lot; in (ii) the same 12 runs raise it only a little, because it is shared over 20 overs.

Way: each friend tells the others the total of their prices and how many prices they have (or their average and count). Add all the totals and divide by the total count, 6 + 4 + 5 + 4 + 4 = 23.

Yes, the two are equal. Both numerators are the sum of all 23 prices (₹4063), so both give 4063 ÷ 23 ≈ ₹176.65.

Use Brahmagupta’s weighted mean with the section lengths as weights.

It does not change.

Experiment: values 10, 20 with weights 1, 3 give (10 + 60) ÷ 4 = 17.5; with weights 2, 6 they give (20 + 120) ÷ 8 = 17.5.

(i) About 10,000 objects in 2003. The total reached about 20,000 (double) in about 2021; it passes 20,000 between 2021 and 2022.

(ii) In 2025 the total is about 33,000: payload ≈ 17,000 (about 51%) and other objects ≈ 16,000 (about 49%), roughly half each.

(iii) Both numbers only increase, since the chart is cumulative. Payload objects grew slowly until about 2019 (about 4,000) and then shot up to about 18,000 by 2026. Other objects grew steadily, with jumps around 2007 and 2009, and kept rising after that.
For decades payload objects were only about one-fifth to one-quarter of the total; since 2019 their share has risen sharply to about half, so the share of other objects has fallen from about three-quarters to about half.

(i) Correct: (b) and (d).

(a) Not correct: Haryana has a higher percentage (90% vs 78%), but we do not know how many schools each state has, so we cannot compare numbers.
(b) Correct: 68% ≈ 66.7% = 2/3.
(c) Not correct: the map gives percentages, not numbers. Punjab’s 98% is the highest among the states, but Dadra & Nagar Haveli and Daman & Diu shows 99.8%.
(d) Correct: Maharashtra has both a higher percentage (93% vs 74%) and more schools, so 93% of a larger number is more than 74% of a smaller number.

(ii) No. We need the number of schools in each State/UT. The national percentage is the weighted mean of the State/UT percentages, weighted by these numbers.

(i) Play A or Play B. Each has about 71% of ratings at 4★ or 5★ and only about 14% at 1★ or 2★. Play A has a slightly bigger 5★ share, so Play A looks a touch better. Play C has the largest 5★ share (about 40%) but also about one-third 1★ ratings, so it is risky: people either love it or hate it.

(ii) Yes, I would choose Play B. The stacked chart shows that Play B has about 925 ratings, Play A only about 210 and Play C about 380. Play B’s good shares come from many more people, so they are more reliable; it is also clearly the most popular.

(i) A stacked bar chart. The race is decided by total time, and a stacked bar shows each athlete’s total as the bar length, while still showing the three segments.

(ii) Only (b): 300 ÷ 563 ≈ 0.53, a little more than half. (a) and (c) need actual times, which a 100% chart does not show.

I notice: seeing and movement disabilities form a bigger share as age increases (seeing about 18% → 25%, movement about 13% → 25%). Multiple disability is largest for 60 & above (about 12%). The shares of mental retardation (about 8% → 2%) and speech (about 9% → 4%) fall with age. Hearing stays about the same (18–20%).

Inferences: problems of eyesight, joints and walking become more common in old age, while conditions present from birth or childhood make up a larger part among the young.

I wonder: how many disabled persons are in each age group? These are only shares, so a falling share does not mean fewer people. What is included in “any other”, which is large for young people?

Sample for (i): I went through the day in time blocks (sleep, school, travel, play, homework, meals) and put each block into one body-state. Each row must add to 24 h.

Observation: I sit the most; my mother, who does most of the housework, moves the most; my father’s office job keeps him sitting.

Sample: (b) travel in a city bus.

(iv) Person 1 rated 3, 4, 2, 3, 4: rating = (3×3 + 4×3 + 2×2 + 3×1 + 4×1) ÷ 10 = 32 ÷ 10 = 3.2. The ten individual ratings were 3.2, 3.8, 2.4, 3.8, 3.2, 4.0, 2.4, 3.4, 4.5, 3.2, so the overall average = 33.9 ÷ 10 = 3.39.

(v) For each aspect, the shares of 5★, 4★, 3★, 2★ and 1★ ratings make one 100% bar (for example, safety: 20% gave 5★, 60% gave 4★, 20% gave 3★).

(vi) The bus scores well on staff behaviour and safety but poorly on cleanliness. The scheme stresses what matters most, but it misses things like fare, frequency and crowding at peak hours.

Method: for each family, multiply its average age by its number of members to get the family’s total age. Add these totals for all families and divide by the total number of family members.

Sample (5 families): (38.5 yr, 4), (31 yr, 3), (42 yr, 6), (29.4 yr, 5), (35 yr, 4).

In general it changes: it moves towards the ordinary (unweighted) average.

Experiment: values 10, 20 with weights 1, 3 give 70 ÷ 4 = 17.5. With weights 2, 4 they give 100 ÷ 6 ≈ 16.67, closer to the simple average 15.

So the new value is a weighted mean of x̄ and m. It lies between them and is unchanged only when x̄ = m (for example, when all weights were already equal).

Possible: (iii), (iv) and (v). Not possible: (i) and (ii).

The three shares must always add to 100%, so they cannot all fall or all rise.

Examples, with last year 120 cows, 50 sheep, 30 chickens (60%, 25%, 15%):