NCERT Solutions Ganita Prakash (Part 2) Chapter 5 –132Section 5.2 Visualising and Interpreting Data — Figure it Out

(i) Each line holds 3 numbers, so an average of 10 means every row, column and diagonal must add up to 30. Take the ordinary 3 × 3 magic square (which sums to 15 in every direction) and add 5 to each entry:

(ii) Yes, and there are endless ways. The centre is forced to be 10, but everything else is free. Fill the grid like this, choosing any two numbers a and b:

Every row, column and diagonal here adds to 30 whatever a and b are — the a’s and b’s cancel out. Taking a = 3, b = 1 gives the grid above; taking a = 4, b = 1 gives a fresh one:

There will be six numbers, so the median is the average of the 3rd and 4th when sorted.

The fixed values sorted are 5, 14, 21, and 13 lies between 5 and 14. Only two pairs of whole numbers can occupy the two middle places:

• 12 and 14. Put 12 in one blank; then one blank must be 12 or less and the other 14 or more, so 12 is 3rd and 14 is 4th.
  Example: 5, 21, 14, 12, 3, 40 → sorted 3, 5, 12, 14, 21, 40 → median (12 + 14) ÷ 2 = 13 ✓
• 13 and 13. Put 13 in two blanks; the third must be 13 or less.
  Example: 5, 21, 14, 13, 13, 2 → sorted 2, 5, 13, 13, 14, 21 → median 13 ✓

How many possibilities? Infinitely many. In the first family the largest blank may be any counting number from 14 upwards — 14, 15, 16, …, 1000, … — and each choice gives a fresh answer with the median still 13.

With counting numbers (1, 2, 3, …) there are only 3 possibilities:

If you count 1 and 3 as the same answer whichever blank they go in, there are just 2 possibilities.

(i) False.

(ii) False.

(iii) False.

(i) Only (c) is correct.

• (a) Wrong. Adding values does not automatically raise a mean. Two very short students would pull it down.
• (b) Wrong. It stays the same only in the special case where the two new heights add to 2 × 150.2 = 300.4 cm. There is no reason to expect that.
• (c) Correct. The two new heights are the only missing information — until they are measured, the new average cannot be found.
• (d) Wrong. Nobody needs re-measuring. The old average already carries the old total: 150.2 × 24 = 3604.8 cm.

(ii) Statement (b) is correct — the average will increase, very slightly.

(iii) Statement (d) is correct — there is not enough information.

No. The average is 17.72, not 17.

Reading the crosses column by column:

Method 1 — the balancing method (no long addition needed). Measure every cross from 17 and see whether the two sides balance.

Method 2 — the direct calculation, as a check.

The mean does not change at all. About the median we cannot be sure.

Median. The median depends on which people changed and where they sat in the sorted order.

• If the three people who changed are all well away from the middle, the middle person is untouched and the median stays 67 kg.
• If one of them is near the middle, the median can shift — by at most a kilogram or two, and it can move either way.

(i) Mizoram, Uttar Pradesh and West Bengal make an interesting trio — the most expensive state, a steady riser and the fastest riser. Years along the horizontal axis, price in ₹ up the vertical axis, scale 0 to 30 in steps of 5.

(ii) Observations. Salt costs very different amounts in different states in the same month — in 2016 it was ₹6 in Assam and ₹20 in Mizoram, more than three times as much. Mizoram is the dearest every single year. Prices are not always rising: Andaman and Nicobar fell from ₹16 to ₹12 between 2016 and 2017 and stayed there for three years, and Gujarat dipped to ₹13 in 2020. But from about 2021 onwards almost every state rises together, which suggests a common cause such as higher transport and packaging costs.

(iii) Gujarat compared with Uttar Pradesh.

Gujarat’s price hardly moved for nine years — it sat between ₹13 and ₹16.5 right up to 2024, and only jumped to ₹19.2 in 2025. Uttar Pradesh climbed steadily year after year, and its total rise is more than three times Gujarat’s. So Gujarat is the steadier of the two and Uttar Pradesh the more variable.

(iv) West Bengal, by a clear margin.

(v) Things to explore. Why is salt so much dearer in Mizoram than in Assam next door — is it the cost of carrying goods into the hills? Why did West Bengal’s price more than double while Gujarat’s barely moved? Do coastal states where salt is made pay less than landlocked ones? And how do these rises compare with the rise in the price of other everyday things over the same ten years?

The two graphs show, for rural and for urban households, the percentage using each source as their primary lighting source.

(i) Valid. Reading the two graphs at 1983:

In rural areas kerosene was well above half, and in urban areas electricity was well above half. Both halves of the statement hold.

(ii) Valid. The kerosene line slopes downward throughout in both graphs — from about 84% to nearly 0% in rural areas, and from about 34% to nearly 0% in urban areas.

(iii) Not valid. In 2000 about 88–90% of urban households used electricity; it was kerosene that was down to roughly 10%. The statement has the two lines swapped.

(iv) Not valid. The graph is about which source a household uses as its main one for lighting. It says nothing about how reliable the supply is. A household counted under ‘electricity’ may still face power cuts every day.

The blue line is Urban and the orange line is Rural. Both fall steadily with age.

(i) Reading the blue line above age 10: about 2 hours a day (a shade over 2 hours).

(ii) Follow the orange line down to a height of 1.5 hours:

The orange line crosses 1.5 hours a little after age 14, so the answer is (d) 14 years.

(iii)(a) Not correct — it is the other way round, and the other way about.

(iii)(b) Not correct. The graph gives an average of about 1.3 hours for rural 15-year-olds. An average of 1.3 hours is perfectly consistent with some children spending 3 hours and others spending none at all. A statement about all children can never be read off a graph of averages.

How to do it. Draw a strip of 48 boxes for each day, one box per half hour from midnight to midnight, and fix a colour for each activity before you start. Fill the strip in the same evening while you still remember the day; filling in a whole week at the weekend gives poor data.

(i) Look down the strips at the same hour on different days. If your green meal boxes and your blue sleep boxes fall in the same places all week, your routine is regular. Count the outdoor boxes and halve the count to get hours.

(ii) For each activity, add the boxes across all seven strips and divide by 7.

Do the same for every activity. Because a day has exactly 48 boxes, the seven averages must themselves add up to 48 — a useful check on your arithmetic. Round each average to the nearest box and colour a single ‘average day’ strip. Note that this average strip is not any real day of yours: it puts the same amount of sleep and study into one picture, but the order of the blocks is your choice.

(iii) Make an average strip for an adult at home the same way and set the two side by side. Compare the sleep block, the largest daytime block, and how many blocks each day is broken into. Adults usually sleep fewer boxes and have far more, shorter blocks.

Option (i): sleep across three age groups.

1. Collect. Each member records every family member’s sleep for seven days, in half hours, counting naps. Agree beforehand on the age boundaries — for example children up to 18, adults 19 to 59, elderly 60 and above — so that everyone’s data can be pooled.
2. (a) Strips. One 48-box strip per person per day, with sleep in one colour. Stacking a person’s seven strips shows at a glance whether their bedtime is regular.
3. (b) Averages and medians. Find each person’s average over the week, then pool the group. A worked example for a group of 12 children:
   Daily sleep (hours): 9, 9.5, 8, 10, 9, 8.5, 9, 10.5, 8, 9.5, 9, 7.5
   Total = 107.5, mean = 107.5 ÷ 12 = 8.96 h
   Sorted: 7.5, 8, 8, 8.5, 9, 9, 9, 9.5, 9.5, 10, 10.5
   For 12 values the median is the average of the 6th and 7th → (9 + 9) ÷ 2 = 9 h
   Repeat for adults and for the elderly.
4. (c) Observations. Expect children to sleep the most and adults the least, with the elderly somewhere between — the same shape as the graph on page 127. Compare the mean and the median in each group: if one person sleeps unusually little, the mean will sit below the median, and the median is then the better summary.

Option (ii): school timings. Collect start time, end time and total break time for at least ten Grade 8 schools. Set them out like this:

Then find the mean and median class time, note the earliest and latest start, and draw a dot plot of the total hours. Look for a pattern: schools in hotter regions often start earlier, and schools that run two shifts have shorter days.

Reading the graphs. Each place has two lines. The lower pair, between about 5:00 and 8:00, are the sunrise times; the upper pair, between about 17:00 and 20:00, are the sunset times. The gap between a place’s two lines is the length of its day.

(i) Kibithu, in the first graph, where the sun rises at about 5:50 am in January — nearly two hours before Ghuar Moti (about 7:35 am).

(ii) Srinagar. Its two lines spread furthest apart in June:

(iii) Observations.

• Kanyakumari’s two lines stay almost flat all year — its day length hardly changes, staying near 12 hours. Srinagar’s lines bow strongly apart in summer and close in winter, from about 14½ hours in June down to about 10 hours in January.
• So the further north a place is, the more its day length swings; near the equator it barely moves.
• Kibithu and Ghuar Moti have curves of almost the same shape, just shifted up or down the clock — because they are at similar latitudes but far apart in longitude. Srinagar and Kanyakumari have curves of very different shapes, because they differ in latitude.
• Longitude shifts the clock times; latitude changes the length of the day. The four graphs separate these two effects rather neatly.
• Curious to find out: should India have more than one time zone, given that sunrise differs by nearly two hours from end to end? How do people in Kibithu arrange a school day when it is dark by 4:30 pm? And what would these graphs look like for a place in the southern hemisphere?

(i) Use what each phase means in terms of rising and setting:

• Purnima (full moon) — the moon is opposite the sun, so it rises around sunset and sets around sunrise, staying up the whole night.
• Amavasya (new moon) — the moon is in the same direction as the sun, so it rises around sunrise and sets around sunset, keeping the sun’s company all day and never showing at night.

Now read the graph. Counting the tick marks from the left edge:

So purnima falls around the 13th–14th of this month and amavasya around the 28th–29th — about a fortnight apart, as it should be.

(ii) What to notice

• The moon rises about 50 minutes later each day. Over the first twenty days the moonrise slides from about 8:25 am to about 11:45 pm — 15 hours and 20 minutes later in 19 days, an average of 48 minutes a day.
• Because of that slide, both lines climb steadily and then drop right off the top of the graph and restart at the bottom. That is not an error: the moonrise has simply passed midnight and moved into the next day.
• There is one day with no moonrise at all (the 21st here) and one day with no moonset (the 6th). If moonrise falls just before midnight one day, the next rise is after midnight of the following day, so one calendar date gets skipped.
• The moon is often up in broad daylight — around amavasya it is in the sky only during the day. We simply do not notice it against a bright sky.
• The pattern is regular, but its period is about 29½ days, not 24 hours — so unlike the sun it does not repeat with the calendar day.