Ganita Prakash · Grade 6 · Maths

Chapter 4 — Data Handling and Presentation

Complete, step-by-step solutions to every question with figures, charts & clear explanations.

Collecting Data · Tally Marks & Frequency · Pictographs · Bar Graphs · Infographics

💡In-text QuestionsExplore · Math Talk · Examples 📝Exercise QuestionsAll "Figure it Out" solutions

Jump to a section

Section 4.1 Section 4.2 Section 4.3 Section 4.4 Section 4.5

💡 In-text Questions

Answers to the in-text prompts, "Math Talk" boxes and worked examples within the chapter.

(4.1 Collecting & Organising Data) To figure out the most popular game in their class, what should Navya and Naresh do?

Solution

A single opinion is not enough — they need to look at the whole class. So they should collect data from everyone and then organise it.

Answer

Go to each student and ask their favourite game, record every answer, then organise the data (for example with tally marks in a table). The game with the highest count (frequency) is the most popular.

(4.2 Pictographs) In the travel pictograph (☺ = 1 student), which mode of travel is used by the most students, and which by the least?

Solution

Count the symbols in each row — each ☺ stands for one student.

Answer

Most: School bus (11 students, the longest row). Least: Cycle (3 students, the shortest row).

(4.2 Worked Example) Nand Kishor's pictograph (▲ = 10 children) shows how often children slept at least 9 hours. Find how many Always, Sometimes, and Never.

Solution

Always: 5 full symbols → \(5\times10=50\).

Sometimes: 2 full symbols and 1 half → \(2\times10 + \tfrac12\times10 = 20+5 = 25\).

Never: 4 full symbols → \(4\times10 = 40\). These 40 children never sleep 9 hours, i.e. they always sleep less than 9 hours.

Answer

Always \(=50\), Sometimes \(=25\), Never \(=40\) children.

(4.2 Drawing a Pictograph) Lakhanpal showed 1 student as one ☺. His friends had large numbers (present students), so Jarina used ☺ = 5 and Sangita used ☺ = 10 (with a half symbol for 5). What is the advantage of a larger key?

Lakhanpal's pictograph of students absent (☺ = 1 student)

Solution

When numbers are large, one-symbol-per-item needs too many symbols and too much space. Choosing a bigger scale/key (one symbol = 5 or 10) makes the pictograph compact, and a half symbol handles the leftover 5.

Answer

A larger key saves time and space. Each symbol then stands for many items, and half symbols show the in-between values (like 25 = two-and-a-half tens).

(4.2 Math Talk) What problems arise in preparing such a pictograph if the total number of students present in a class is 33 or 27?

Solution

With a key of ☺ = 10 and a half symbol = 5, we can only show multiples of 5 neatly.

Answer

☺ represents 10 and a half symbol represents 5. But 33 or 27 are not multiples of 5, so the extra 3 or 7 students cannot be shown exactly by dividing a symbol — the pictograph becomes imprecise.

(4.3 Bar Graphs, Example) The bar graph shows India's population (in crores) each decade. How much did the population increase over the 50 years, and in each decade?

Population of India in crores (1951–2001)

Solution

Read the bar tops: 36, 44, 54, 68, 84, 102 (crores).

Over 50 years: \(102-36 = 66\) crores.

Each decade: \(44-36=8\), \(54-44=10\), \(68-54=14\), \(84-68=16\), \(102-84=18\) crores.

Answer

The population rose by 66 crores in 50 years. Decade increases: 8, 10, 14, 16 and 18 crores — the growth got larger each decade.

(4.3 Bar Graphs in real life) Newspapers and TV often show bar graphs. What does this one tell us at a glance?

Number of animal species on the IUCN Red List, by year

Solution

The bars grow taller from left to right, so the total keeps rising each year.

Answer

It shows the number of endangered animal species on the IUCN Red List rising steadily — from about 7,851 in 2007 to around 16,900 by 2022. A bar graph makes this upward trend obvious instantly, which a long table of numbers would not.

📝 Exercise Questions

Detailed solutions to all the "Figure it Out" exercises, section by section.

Section 4.1 — Collecting & Organising Data

What would you do to find the most popular game among Naresh's and Navya's classmates?

Solution

The raw name-and-game list is hard to read. Organising it groups equal answers together.

Answer

Arrange and organise the data in a table (for example using tally marks), then count how many chose each game. Think of other ways too, such as making a pictograph or bar graph.

What is the most popular game in their class?

Solution

Counting each game from the list: Hockey 8, Kabaddi 6, Cricket 6, Satoliya 5, Football 4, Badminton 2.

Answer

Hockey is the most popular (frequency \(=8\)).

Try to find out the most popular game among your classmates.

Solution

This is an activity to do in your own class.

Answer

Ask each classmate their favourite game, record every answer with a tally mark, then the game with the highest tally is the most popular in your class.

Pari wants to answer the questions below. Tick (✓) those needing data collection and cross (✗) those that don't:

a. Most popular TV show among her classmates? b. When did India get independence? c. How much water is wasted in her locality? d. What is the capital of India?

Solution

A question needs data collection when the answer is not already known and varies from place to place; general facts do not.

Answer

a. ✓ (must survey classmates) b. ✗ (known fact: 1947) c. ✓ (must measure/observe locally) d. ✗ (known fact: New Delhi)

Sweets 1

Complete the sweets table (using the tally marks) to help Shri Nilesh: a. jalebi? b. barfi? c. gujiya? d. rasgulla? e. gulab jamun?

Sweet	Tally	No. of Students
Jalebi	\|\|\|\|̅ \|	6
Gulab jamun	\|\|\|\|̅ \|\|\|\|	9
Gujiya	\|\|\|\|̅ \|\|\|\|̅ \|\|\|	13
Barfi	\|\|\|	3
Rasgulla	\|\|\|\|̅ \|\|	7

Solution

Each completed group of five (̶||||) is 5; count the leftover single marks.

Answer

a. Jalebi \(=6\) b. Barfi \(=3\) c. Gujiya \(=13\) d. Rasgulla \(=7\) e. Gulab jamun \(=9\). These counts are the frequencies of each sweet.

Sweets 2

Is the above table sufficient to distribute each type of sweet to the correct student? Explain. If not, what is the alternative?

Solution

The table gives only how many chose each sweet, not who chose what.

Answer

No. It shows only the number of students for each sweet, not the sweet chosen by each particular student. Alternative: list the students' names under their chosen sweet (categorise students by their choice).

Shoes 1

Sushri Sandhya's shoe-size data, arranged in ascending order:

3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7

Find: a. largest size b. smallest size c. students with size 5 d. students with size > 4.

Solution

Read directly from the ordered list and count.

Answer

a. Largest \(=7\) b. Smallest \(=3\) c. Size 5: \(10\) students d. Larger than 4 (sizes 5, 6, 7): \(10+4+1 = 15\) students.

Shoes 2

How did arranging the data in ascending order help to answer these questions?

Solution

Answer

In ordered form, equal values sit together, so the frequency of any size is easy to count and the smallest/largest values are simply the first and last entries.

Shoes 3

Are there other ways to arrange the data?

Solution

Answer

Yes — the data can be organised in a frequency (tally) table, listing each size against the number of students.

4 (Trees)

Record the trees you see on the way to school and fill a table (Peepal, Neem, …). a. Which tree was most? b. Which least? c. Any two equal?

Solution

This is a data-collection activity.

Answer

Observe and tally each tree along your route, then compare the counts to find the greatest, the smallest, and any ties. Answers depend on your own locality.

5 (Letters)

Count the letters c, e, i, r, x in a news clipping. a. most frequent letter? b. least frequent? c. list c, e, i, r, x in ascending order of frequency (most people get x, c, r, i, e). Why?

Solution

This is an activity, but the order is predictable.

Answer

Almost everyone gets the ascending order x, c, r, i, e because letter frequencies are fairly constant in English: 'e' and 'i' are very common, 'r' fairly common, 'c' less so, and 'x' is rare. Different articles still follow this same pattern.

Section 4.2 — Pictographs

The pictograph shows books borrowed in a week from Middle School, Ginnori. a. On which day were the minimum books borrowed? b. Total books during the week? c. On which day were the maximum borrowed, and a possible reason?

Solution

Each book symbol = 1 book. Count each row; the empty row is the minimum, the longest row the maximum, and the sum of all rows is the weekly total.

Answer

a. Thursday (no books borrowed). b. Total \(=24\) books. c. Saturday (maximum). Likely reason: Sunday is a holiday, so students borrow more on Saturday to read at home.

Magan Bhai sold kites to six shopkeepers: Chaman 250, Rani 300, Rukhsana 100, Jasmeet 450, Jetha Lal 250, Poonam Ben 700. Draw a pictograph (🪁 = 100 kites) and answer: a. symbols for Rani? b. who bought the most? c. Jasmeet or Chaman? d. Is Poonam Ben's more than double Rani's?

Solution

With 🪁 = 100 kites, each 100 is one symbol and 50 is a half symbol:

🪁 = 100

Chaman🪁🪁🪁250

Rani🪁🪁🪁300

Rukhsana🪁100

Jasmeet🪁🪁🪁🪁🪁450

Jetha Lal🪁🪁🪁250

Poonam Ben🪁🪁🪁🪁🪁🪁🪁700

🪁 = 100 kites (half kite = 50)

d. Double of Rani \(= 2\times300 = 600\); Poonam Ben \(=700 = 600+100\), which is more than 600.

Answer

a. Rani \(=300 \Rightarrow 3\) symbols. b. Maximum: Poonam Ben (700). c. Jasmeet (450) bought more than Chaman (250). d. Yes — \(700 > 2\times300\).

Section 4.3 — Bar Graphs

Bar 1

Using the bar graph of students absent in each class: 1. In Class 2, ___ students were absent. 2. Which class had the maximum absent? 3. Which class had full attendance?

Solution

Read each bar's height (1 unit = 1 student). The tallest bar is the maximum; the zero-height bar means nobody was absent (full attendance).

Answer

1. Class 2 \(=5\) students. 2. Maximum: Class 8 (7 students). 3. Full attendance: Class 5 (0 absent).

Traffic 1

The bar graph shows vehicles at a Delhi crossing each hour, 6 a.m.–noon (1 unit = 100 vehicles). 1. Total cars 6 a.m.–noon? 2. Why so little traffic 6–7 a.m.? 3. Why heaviest 7–8 a.m.? 4. Why less each hour after 8 a.m.?

Solution

Add the bar values: \(150+1200+1000+800+700+600\).

\[150+1200+1000+800+700+600 = 4450\]

Answer

1. Total \(=4450\) vehicles. 2. 6–7 a.m. is very early, so few people are travelling. 3. 7–8 a.m. is the peak office/school rush hour. 4. After 8 a.m. the rush is over, so fewer commuters travel each following hour.

Section 4.4 — Drawing a Bar Graph

Expenditure

Using the bar graph of Imran's family monthly expenditure: 1. Most and second-most spending? 2. Is electricity about half of education? 3. Is education less than one-fourth of food?

Solution

Values: House rent ₹3000, Food ₹3400, Education ₹800, Electricity ₹400, Transport ₹600, Miscellaneous ₹1200.

2. \(400 = \tfrac12\times 800\). 3. One-fourth of food \(= \tfrac{3400}{4} = 850\), and \(800 < 850\).

Answer

1. Most on Food (₹3400), second-most on House rent (₹3000). 2. Yes, electricity (₹400) is half of education (₹800). 3. Yes, education (₹800) is less than one-fourth of food (₹850).

1 (Insects)

Samantha's tea-garden data — Mites 6, Caterpillars 10, Beetles 5, Butterflies 3, Grasshoppers 2. Help her draw a bar graph.

Solution

Take 1 unit length = 1 insect and draw a bar for each, as high as its count:

Answer

Caterpillars are the most common (10) and grasshoppers the least (2). The bar graph above represents the data.

2 (Tickets)

Pooja's bar graph of tickets sold was partly erased. a. Tickets for Vidisha? b. for Jabalpur? c. the scale? d. draw Sagar's bar. e. add the vertical scale. f. are Seoni and Indore correct?

Solution

Vidisha's bar is 6 units and equals 24 tickets, so \(\text{scale}= \dfrac{24}{6}=4\) tickets per unit. Using this scale on the true data (Vidisha 24, Jabalpur 20, Seoni 16, Indore 28, Sagar 16):

Answer

a. Vidisha \(=24\). b. Jabalpur \(=20\). c. Scale: 1 unit \(=4\) tickets. d. Sagar \(=16\) (a 4-unit bar). e. Mark the axis 0, 4, 8, 12, …, 28. f. Seoni is correct (16), but Indore is wrong — it should be 28, not the drawn height.

3 (Transport)

Chinu listed transport passing his house (9–10 a.m.). a. Make a frequency table. b. Which was used most? c. How would you collect this data?

Solution

Tallying Chinu's list gives:

Means of Transport	Number
Bike	13
Car	6
Bicycle	8
Auto rickshaw	8
Scooter	9
Bus	4
Bullock cart	2

Answer

b. Bike was used the most (13). c. To collect it: make a two-column table (transport vs frequency), watch the road and put a tally mark for each vehicle as it passes, then count the tallies.

4 (Die)

Roll a die 30 times, record each result, and make a tally frequency table. Find the number appearing: a. minimum times, b. maximum times, c. equal number of times.

Solution

This is a hands-on activity (results differ each time).

Answer

Tally each of the 30 rolls under 1–6. Then the value with the smallest tally appeared the fewest times, the largest tally the most times, and any values with the same tally appeared an equal number of times.

5 (Wickets)

Faiz's table of wickets Bumrah took in his last 30 matches. a. What information does it give? b. A suitable title? c. What caught your attention? d. In how many matches did he take 4 wickets? e. Can Mayank total the wickets by adding 0+1+…+7? f. How to correctly find the total?

Wickets Taken	No. of Matches
0	2
1	4
2	6
3	8
4	3
5	5
6	1
7	1

Solution

To get the total wickets, multiply each wicket value by its number of matches, then add:

\[0(2)+1(4)+2(6)+3(8)+4(3)+5(5)+6(1)+7(1)\]\[=0+4+12+24+12+25+6+7 = 90\]

Answer

a. It shows how many matches he took each number of wickets in. b. e.g. “Wickets taken by Jaspreet Bumrah in his last 30 matches.” c. e.g. he took 7 wickets in just 1 match. d. 3 matches. e. No — adding \(0+1+\dots+7\) ignores how many matches each happened in. f. Multiply wickets × matches for each row and add → total \(=90\) wickets.

6 (Tractors)

The pictograph shows tractors in five villages (🚜 = 1 tractor). a. Smallest number? b. Most tractors? c. How many more does C have than B? d. Komal says “D has half of E” — is she right?

Solution

Count the tractor symbols in each row (Village A 6, B 5, C 8, D 3, E 6).

Answer

a. Smallest: Village D (3). b. Most: Village C (8). c. \(C-B = 8-5 = 3\) more. d. Yes — D \(=3\) is half of E \(=6\).

7 (Girls)

The pictograph shows girl students per class (👧 = 4 girls). a. Class with the least girls? b. Difference between Classes 5 and 6? c. If 2 more girls join Class 2, how does the graph change? d. Girls in Class 7?

Number of girl students per class (👧 = 4 girls)

Solution

Each full symbol = 4 girls and a half symbol = 2 girls; read each row accordingly.

Answer

a. Class 8 has the least. b. Difference between Classes 5 and 6 \(=6\). c. Class 2's last half-symbol becomes a full symbol (2 more girls = one half → full). d. Class 7 \(=12\) girls.

8 (Mudhol)

Mudhol dogs — A 18, B 36, C 12, D 48, E 18, F 24. Prepare a pictograph and answer: a. a useful scale? b. symbols for Village B? c. Kamini says B + D together exceed the other four villages — right?

Solution

All the numbers are multiples of 6, so a key of 🐕 = 6 dogs works neatly:

🐕 = 6

Village A🐕🐕🐕18

Village B🐕🐕🐕🐕🐕🐕36

Village C🐕🐕12

Village D🐕🐕🐕🐕🐕🐕🐕🐕48

Village E🐕🐕🐕18

Village F🐕🐕🐕🐕24

🐕 = 6 dogs

c. \(B+D = 36+48 = 84\); other four \(= 18+12+18+24 = 72\); and \(84 > 72\).

Answer

a. Scale: 🐕 = 6 dogs (all counts are multiples of 6). b. Village B \(=36\Rightarrow 6\) symbols. c. Yes, Kamini is right — \(84 > 72\).

9 (Activities)

Survey of 120 students' free-time activity: Playing 45, Reading story books 30, Watching TV 20, Listening to music 10, Painting 15. Draw a bar graph (1 unit = 5 students). Which is most preferred other than playing?

Solution

With 1 unit = 5 students, bar heights are \(45/5=9,\ 30/5=6,\ 20/5=4,\ 10/5=2,\ 15/5=3\) units:

Answer

Other than playing, Reading story books (30) is preferred by the most students.

10 (Saplings)

Bar graph of saplings planted in one week. a. Total on Wednesday and Thursday? b. Total for the whole week? c. Greatest and least days, and possible reasons?

Solution

Read the bars: Mon 50, Tue 40, Wed 30, Thu 40, Fri 50, Sat 60, Sun 40.

a. \(30+40 = 70\). b. \(50+40+30+40+50+60+40 = 310\).

Answer

a. Wednesday + Thursday \(=70\). b. Whole week \(=310\) saplings. c. Greatest on Saturday (60), least on Wednesday (30). Possible reasons: more volunteers on weekends, or differences in attendance/weather (e.g. rain) on some days. You could check by asking how many people came each day.

11 (Tigers)

Shagufta and Divya's bar graph of tigers in India (2006–2022) has mistakes compared with their table. Find and fix them.

The tiger table and the graph with mistakes

Solution

The table says: 2006 → 1400, 2010 → 1700, 2014 → 2200, 2018 → 3000, 2022 → 3700. Comparing each bar with the table, only 2022 matches. The correct graph is:

Answer

The bars for 2006, 2010, 2014 and 2018 are wrong — they don't match the table values (only 2022 is correct). Redrawing them to 1400, 1700, 2200 and 3000 respectively fixes the graph.

Section 4.5 — Artistic & Aesthetic Considerations

Mountains

(Artistic & Aesthetic Considerations) The table lists the tallest mountain on each continent. How much taller is Everest than Koscuiszko? Why is a bar graph clearer than the table?

Continent	Tallest Mountain	Height
Asia	Everest	8848 m
S. America	Aconcagua	6962 m
N. America	Denali	6194 m
Africa	Kilimanjaro	5895 m
Europe	Elbrus	5642 m
Antarctica	Vinson Massif	4892 m
Australia	Koscuiszko	2228 m

Solution

\(8848 - 2228 = 6620\) m. A bar (or column) graph turns these numbers into lengths, so heights can be compared at a glance instead of reading many figures.

Answer

Everest is \(6620\) m taller than Koscuiszko. A column graph (vertical bars) shows the comparison instantly — far easier than scanning a table of numbers.

Infographic caution

In the mountain infographic, taller triangles were also drawn wider, and Everest looks about twice as tall as Elbrus. Is that accurate? What is \(5642\times 2\)?

Solution

\(5642\times 2 = 11284\) m, but Everest is only \(8848\) m — not double Elbrus. Also, making taller mountains wider adds information (width) that isn't real.

Answer

No, it is misleading. Everest (8848 m) is not twice Elbrus, since \(5642\times2 = 11284 \neq 8848\). ‘Fancy’ pictures can accidentally mislead, so we must read and make infographics carefully.

To show the heights of the tallest persons in each class, would you use vertical or horizontal bars? Why?

Solution

Answer

A vertical bar (column) graph is more natural, because height is measured upwards — taller bars directly suggest taller persons. (Either can be used, but vertical is more intuitive.)

For a table of the longest rivers on each continent, would you use vertical or horizontal bars? Why?

Solution

Answer

A horizontal bar graph suits lengths, since a river's length runs along the ground — longer bars naturally represent longer rivers. (Either works, but horizontal fits length better.)

Educational solutions compiled from NCERT Ganita Prakash, Grade 6 · for practice & revision · @edugrown