Class 9 Maths Ganita Manjari Chapter 10 Solutions — How Quantities Combine: Understanding Data

The eight senior heights sum to 1324 cm and the three junior heights to 448 cm. The simple mean of the two group averages gives the small junior group as much influence as the senior group, so it fails. Use the exact junior mean $\frac{448}{3}$; 149.33 is already rounded, and multiplying it by 3 gives 447.99, not exactly 448.

Each week contains five recorded weekdays, so averaging 12.8, 15.8 and 18 gives the same answer as averaging all 15 observations. Their sum is 46.6, not the printed 46.63. Equal group sizes guarantee that the unweighted average of means works; unequal sizes do not generally do so, although special data can give accidental equality.

For equal quantities the concentration is halfway between 10% and 20%. With 500 mL at 10% and only 200 mL at 20%, the result is closer to 10%. The textbook’s approximately 13% is a rounded answer.

Check: copper totals $140+60+108=308\,\mathrm{kg}$ out of 560 kg, which is 55%.

The ratio 3:2:5 weights the final examination more strongly. The equal-weight mean would be $\frac{197}{3}\%=65\frac23\%$ (about 65.67%); the printed 65.6% is not the usual rounding to one decimal place.

For older reviews with mean a and count m, and recent reviews with mean b and count n given twice the weight per review, the combined rating is $\frac{ma+2nb}{m+2n}$. Counts still matter; do not just average the two means using weights 1 and 2 unless the review counts are equal.

The weights are 30 and 25 students. The result is slightly closer to 72 than to 76.

The nitrogen fraction is $\frac{11}{100}$, or 11%. This treats the listed fractions as shares on the same quantity basis.

The corresponding gold fraction is $\frac{345}{496}$. Purity in varṇa is a linear measure, so the amounts of alloy are the weights.

May and July each have 31 days, while June has 30. Weight the daily means by these day counts.

Reversing the amounts changes the weights. The first mixture has more of the strongest solution; the second has more of the weakest.

Correct expressions: (iii) and (iv). First convert each score to a proportion of its own maximum; then apply the weights. Raw marks with different maxima are not directly comparable.

Expressions (i) and (ii) ignore the different maximum marks. Expression (iv) is equivalent to (iii) because it converts each proportion to a percentage before averaging.

The denominator is total sales because the question states that profit is a share of the sales amount. This differs from a profit percentage based on cost price.

Guess: slightly more than 44.5 km/day, because just one higher-distance day is added to 20 earlier days.

Option (v). Track the two solutes separately; assume the salt solution contains no sugar and the sugar solution contains no salt.

The target concentration is $\frac34(8\%)=6\%$. The spice amount remains unchanged.

Add $3\frac13$ litres of plain water. The instruction reduces the concentration to three quarters, not by three quarters.

(i) Keerthi loses 5 marks in strength but gains 5 in agility, which has the larger weight. The weighted change is $-5(4)+5(6)=10$, so Keerthi scores more.

The final rating is approximately 3.57 out of 5. Each metric has ten responses, so its mean can be weighted directly.

The combined mean 14.925 kg is below the midpoint 15.15 kg, so there are more females, whose mean is lower. (i) Only (a) is correct as printed; (b), (c) and (d) have a negative right-hand side. With x males and y females, $x+y=60$.

(iii) Female mean after admission:

(iv) Male mean after the two releases:

(v) Taking “now” to mean after part (iv), the subsequent 1 kg loss gives:

If part (v) is instead treated as an independent change to the original 25 males, the answer is $\frac{412.5-1}{25}=16.46$ kg. The sequential interpretation is the natural one.

Use the approximate salinity figures supplied in the exercise and its ideal volume-weighted mixture model; these calculations do not establish whether water is safe to drink.

(ii) The target 0.01% lies between 0.001% and 34%, so a positive mixture is possible in this model. Let x litres of purified water be added:

(iii) It is impossible using only Dead Sea water and groundwater: any positive mixture has salinity at least 0.01%, above 0.001%. The equation would require a negative amount, which is not a physical mixture.

Sample observations: the total doubles. Lighting decreases both in amount (360 to 220) and share (36% to 11%); cooling increases from 300 to 800 and from 30% to 40%. The stated household “daily” kWh amounts are unusually large; use the exercise’s supplied figures for the percentage calculation rather than presenting them as typical household consumption.

Only statement (i) follows: within Fatima’s garden, 50% in summer exceeds 30% in monsoon. Statements (ii)–(iv) compare unknown totals and cannot be concluded. For example, if Fatima totals 210 and Naveen 120, their summer counts are 105 and 48. If the totals are 180 and 240, summer counts become 90 and 96 and the comparison reverses.

Page 26 reflections: (1) Ordinary stacks retain absolute counts; 100% stacks retain proportions. (2) Divide each component by its positive stack total to make a 100% chart. (3) The reverse is not unique unless each total is known. (4) Within-stack rankings can be read from either; across-stack counts need absolute totals, while across-stack shares can be compared in 100% charts.

In the overall-day chart, 34% sleeping corresponds to 8.16 hours; the figures are rounded averages, not a timetable for every person. The age-group sleep shares are about 38%, 34%, 32% and 37%: sleep decreases from childhood to adulthood and then increases for the oldest group. This broadly matches a decline through younger ages, but is not a strictly decreasing lifelong trend.

Sample interpretations: learning falls from about 22% for ages 6–14 to 11% for ages 15–24, plausibly because the latter group includes more people in work and fewer in full-time school. That is a possible explanation, not proof of causation from the graph. Adults’ paid and unpaid work shares are close (14% and 15%), approximately 3.36 and 3.60 hours. Leisure/social/travel is largest for the oldest group at about 33%.

Chart correction: the axis label “55%” aligned with 12 hours should be 50%, since 12 hours is half of 24. These bars can be read both as absolute time and as percentages because every day totals 24 hours. Grouping the separate strips of one activity together preserves duration and produces the same kind of stack.

(i) 14,234 is the sum of all the coloured animal-class counts represented in the 2019 bar. It is not the total number of animal species on Earth, nor necessarily every assessment in the full IUCN database. (ii) The reptile segment for 2016 is approximately 1000 species; a nearby visual estimate is reasonable because no exact segment value is printed. (iii) Mammals and birds show comparatively small increases. Compare segment thicknesses, not their upper boundary heights. These are readings of the textbook’s historical graph, not current conservation statistics.

In (i), join format segments within each location. In (ii), join location segments within each format. Use a common zero and the same wicket scale across bars.

(i) Sample estimate: Sahana could spend 8 hours lying down, 10 sitting and 6 standing/moving. These sum to 24 hours; other plausible estimates are valid. (ii) Julie might do desk-based office work, but posture data alone cannot identify her occupation. (iii) For each entry use $\frac{\text{hours}}{24}\times100\%$.

The mean after 19 overs represents 19 observations, so it cannot be averaged equally with the final over.

Each friend can report both the sum and count of prices; add the sums and divide by the total count. Equivalently, weight each friend’s mean by the number of observations.

There are 13 nylon prices totalling ₹1429 and 10 feather prices totalling ₹2634. Both groupings contain exactly the same 23 prices, so both give:

This is the mean of the collected list; it is not automatically a representative mean of every product sold in the city.

Vaishnavi buys 10 additional shares. These are arithmetic exercises using the stated purchase costs.

The weights are the lengths of the five sections. This is the mean depth along the stated length; a volume-per-surface-area mean would also require appropriate width information.

Each case starts independently from 1 g costing ₹15,000; the transactions are not cumulative. Values below are in thousands of rupees per gram.

The mean is unchanged because numerator and denominator scale by the same nonzero factor. This holds for any common positive multiplier.

(i) The 2003 bar is approximately 10,000 objects; a total around 20,000 appears around 2021. (ii) In 2025 a reasonable graph reading is roughly 33,000 total: about 17,000 payloads and 16,000 other objects, or approximately 52% and 48%. Values within the chart’s reading precision are acceptable; the graph does not print exact annual figures.

(iii) Both cumulative components increase, but payloads grow particularly quickly in the most recent years. Their share rises sharply in that period, while the share of other objects falls even though its count increases. Shares need not change monotonically over the entire graph.

Read the title carefully: this chart groups objects still orbiting as of 24 April 2026 by launch/separation year. Its 2003 bar is not necessarily the number that was orbiting in 2003; objects that have since re-entered are not represented. The estimates above answer the intended reading of the plotted bars.

(i) Correct inferences: (b) and (d). Arunachal Pradesh’s 68% is approximately two thirds. Maharashtra has both a higher percentage (93% versus Telangana’s 74%) and, by the added assumption, more schools, so it has more schools with playgrounds. Haryana’s 90% versus Uttarakhand’s 78% does not establish the larger count without totals. Punjab’s high percentage likewise does not prove the highest number.

(ii) We need the number of schools in each state/territory, with consistent coverage and reference year. Then:

Here pᵢ is the percentage and Nᵢ the school count. Do not simply average state percentages. The map is historical textbook data, not a current national estimate.

Sample judgement: Plays A and B have similar high shares of four- and five-star ratings; either can be a reasonable initial choice. Play C has the largest five-star share but also a noticeably larger one-star share, suggesting more divided opinions. The second graph shows that B has many more ratings than A. I might prefer B because its similar positive profile is based on a larger sample. This is a judgement, not a uniquely correct answer; audience selection and individual preferences can still matter.

(i) Use an ordinary stacked bar chart to compare completion times and their components. Athlete 1 finishes fastest. (ii) From a 100% chart alone, only (b) can be answered:

The shares alone do not determine the fastest finisher or the largest absolute running time. Convert hh:mm to minutes before calculating; 01:08 means 68 minutes, not 1.08 hours.

Sample observations: the graph shows the distribution of disability types among persons with disabilities within each age group. The hearing share is much greater in the 60+ group than in 0–19. The movement share is larger in the adult groups than in 0–19, while the speech share is smaller in the oldest group. The “any other” share appears lower in the oldest group. These are approximate comparisons of segment widths.

The chart does not show the proportion of all people in each age group who have a disability, or the numbers of people in those groups. It cannot establish causes. A useful question is how counts and shares differ by age and access to support. Use respectful current language, such as “persons with disabilities” and “intellectual disability”; one label in the historical census graphic uses older terminology.

Choose at least one option. (i) Sample invented data: weekday (8, 10, 6) hours; Saturday (9, 8, 7); Sunday (9, 7, 8), in the order lying, sitting, standing/moving. Each sums to 24. Record a real diary in mutually exclusive intervals, merge repeated activities, then make the chart. Estimates may differ from smartwatch readings and should be described honestly.

(ii) Sample invented household budget: amounts below illustrate the method and are not a real family’s finances. Agree on categories and record consistent monthly totals from receipts or a diary; avoid counting transfers twice.

Divide each expense by its own month’s total and multiply by 100. Housing is unchanged in rupees but falls from 50% to 40% in month 2. Collect your own three months before drawing conclusions about your household.

Complete invented example: bus-travel experience. Rate comfort (weight 4: long journeys need tolerable seating), cleanliness (3: a usable environment matters), punctuality (2: passengers plan around arrivals), and accessibility (1: entry and information should be usable). These weights illustrate a discussion, not a universal priority; a group could reasonably give accessibility the highest weight. Agree on a 1–5 scale before collecting responses.

For the 100% chart, show the rating distribution for each aspect: each of the ten responses represents 10%. Comfort and cleanliness have mean 3.9, punctuality 4.0 and accessibility 3.6. The weighted overall score hides differences between individuals and depends on the chosen priorities. The small imagined sample cannot represent all bus users. Replace it with your group’s observations for the actual project.

For family i, multiply its mean age aᵢ by its number of members nᵢ to recover its age total. Then divide the sum of these totals by the total number of people:

If siblings in the class report the same family, agree whether the target counts distinct people or family reports; remove duplicate families when calculating the mean for distinct people. Rounded family means give an approximate result.

Let W be the original total weight, μ the weighted mean and a the ordinary mean of the n data values. The new mean is:

Thus it moves towards the ordinary mean: it increases if a > μ, decreases if a < μ, and stays unchanged if they are equal. It is not generally unchanged, unlike multiplying every weight by 2. Example: values 0 and 10 with weights 1 and 3 have mean 7.5; increasing the weights to 2 and 4 gives $\frac{20}{3}\approx6.67$, closer to the ordinary mean 5.

Possible: (iii), (iv) and (v). (i) and (ii) are impossible because the three shares must still sum to 100%. Start with 600 cows, 250 sheep and 150 chickens, total 1000:

Each new total is 500 and every animal count has fallen. A falling count can still have a rising share if the total falls more quickly.