๐ KEY TO ENJOY LEARNING MATHS
Class 8 Ganita Prakash · Part 2 · Chapter 5
Class 8 Ganita Prakash · Part 2 · Chapter 5
Tales by Dots and Lines
Page-wise textbook solutions (pages 103–134, Part 2) · Every step · Figures · Common mistakes · Tips
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✔ Understood 0/46
๐️ Key ideas of this chapter
- The mean is the balance point: total distances below = total distances above.
- Adding a value above (below) the mean raises (lowers) it; adding a value equal to the mean keeps it.
- Add k to every value → mean + k. Multiply every value by k → mean × k.
- Missing value = (mean × count) − (sum of known values). With frequencies: mean = \(\dfrac{\sum f x}{\sum f}\).
- Line graphs show change over time; read the scale, then infer carefully (don't claim what the data doesn't show).
๐ Textbook Page 103
๐ Page 103Think
Take two numbers and find their mean, e.g. 3 and 7; 8 and 9. What do you notice?
1\(\tfrac{3+7}{2} = 5\); \(\tfrac{8+9}{2} = 8.5\)
2On a dot plot the mean is exactly halfway between the two numbers.
✅ Answer: For two numbers the mean is the midpoint.
๐ Textbook Page 104
๐ Page 104Math Talk
How is the mean the 'centre' of a collection? Is it the midpoint of the two extremes?
1Not always. For 10, 10, 11, 17 the midpoint of 10 and 17 is 13.5, but the mean is 12.
2Distances below 12: 2 + 2 + 1 = 5. Distances above 12: 5. They are equal.
3The mean is the point where the total distance on the left equals the total distance on the right — like a see-saw balance point.
✅ Answer: The mean balances the distances on both sides.
๐ Page 104Think
Can there be more than one such centre?
1Move the centre to the right of 12: left distances grow, right distances shrink → no longer balanced.
2Move it to the left: the opposite happens.
✅ Answer: No — there is only one balance point, the mean.
๐ Textbook Page 105
๐ Page 105Math Talk
What happens to the mean when a value is included or removed? What if the value equals the mean?
1Include a value above the mean → mean increases; below → decreases.
2Remove a value above the mean → mean decreases; below → increases.
3Include or remove a value EQUAL to the mean → mean stays the same (fair share: that value already has exactly its share).
✅ Answer: Above/below the mean pulls the mean up/down; equal to the mean leaves it unchanged.
๐ Textbook Page 106
๐ Page 106Unchanging mean
Include 2 or 3 values so the mean (9) stays the same.
1Two values: equal distances on opposite sides, e.g. 7 and 11 (2 below, 2 above).
2Three values: total distance below = total distance above.
32 below, 1 above: 7 and 7 (2 + 2 = 4 below) with 13 (4 above).
42 above, 1 below: 10 and 12 (1 + 3 = 4 above) with 5 (4 below).
✅ Answer: Any values whose distances below and above the mean balance.
๐ Page 106Relatively unchanged
Mean of 8, 3, 10, 13, 4, 6, 7, 7, 8, 8, 5? Mean after adding 10 to each? After subtracting 1? After doubling?
1Sum = 79, n = 11 → mean = 79 ÷ 11 ≈ 7.18
2Add 10 to every value → mean + 10 ≈ 17.18
3Subtract 1 → ≈ 6.18 (the book shows the +10 data shifted down: 16.18)
4Double every value → mean doubles ≈ 14.36
✅ Answer: ≈ 7.18; ≈ 17.18; 1 less; double
๐ก Tip: Algebra: \(\tfrac{(x_1-2)+\cdots+(x_n-2)}{n} = a - 2\) and \(\tfrac{5x_1+\cdots+5x_n}{n} = 5a\).
๐ Textbook Page 108
๐ Page 108Median
The median of the data is 8. A value 11 is included. What happens to the median?
111 is greater than the median, so now there are more values above 8 than below it.
2With an even number of values, the median is the average of the two middle values: 8 and 11 → 9.5.
3Similarly, a value below the median makes the median decrease (or stay the same).
✅ Answer: The median increases to 9.5.
๐ Textbook Page 109
๐ Page 109Finding the unknown
Wrestlers' weights 42, 40, 39, 33, 48, 38, 42, 35, 32 and one smudged value; mean 39.2 for 10 players. Find it.
1Total = 39.2 × 10 = 392
2Known sum = 349
3w = 392 − 349
✅ Answer: 43 kg
๐ Page 109Coconuts
Average harvest of 15 trees is 25.6; one tree was counted 3 too many. Correct average?
1Total = 25.6 × 15 = 384
2Correct total = 381
3381 ÷ 15
✅ Answer: 25.4
๐ Textbook Page 110
๐ Page 110Frequencies
Family sizes 3–10 with frequencies 3, 11, 9, 7, 3, 1, 1, 1. Find the mean and median.
1Mean = \(\dfrac{9 + 44 + 45 + 42 + 21 + 8 + 9 + 10}{36} = \dfrac{188}{36} \approx 5.22\)
236 values → median = average of the 18th and 19th.
3Cumulative: 3, 14, 23 … → positions 15–23 are 5.
✅ Answer: Mean ≈ 5.22, median 5
⚠️ Common mistake: Averaging only 3, 4, …, 10 (6.5) and ignoring the frequencies.
๐ Textbook Page 112
๐ Page 112Spreadsheets
Which cell has Farooq's Maths marks? What is in B7? In which subjects did Ashwin score more than 30? Formula for the class average in Science?
1Farooq is row 5, Maths is column E → E5 (42).
2B7 = Gowri's Odia marks = 27.
3Ashwin: Telugu 31, English 33, Maths 34 (Odia 29, Social 30, Science 28 are not above 30).
4Science is column G, rows 2–23: =AVERAGE(G2:G23).
✅ Answer: E5; 27; Telugu, English, Maths; =AVERAGE(G2:G23)
๐ Textbook Page 113
๐ Page 113Spreadsheets
Is the class average in Odia greater than in Telugu? Totals for each student?
1Odia total = 686 → average ≈ 31.18. Telugu total = 739 → average ≈ 33.59.
2Total for a student: =SUM(B2:G2) in H2, then copy down.
✅ Answer: No — Telugu's average (≈ 33.6) is higher than Odia's (≈ 31.2).
๐ Page 113Figure it Out · Q1
Mean of (i) first 50 natural numbers (ii) first 50 odd numbers (iii) first 50 multiples of 4.
1(i) Sum = 50 × 51 ÷ 2 = 1275 → 25.5
2(ii) Sum = 50² = 2500 → 50
3(iii) 4 × (1 + … + 50) = 5100 → 102
✅ Answer: 25.5, 50, 102 — for equally spaced numbers, mean = (first + last) ÷ 2.
๐ Page 113Figure it Out · Q2
One dot is missing so that the mean is 9. Mark it.
1Shown: 4, 7, 8, 8, 9, 9, 9, 9, 9, 11 → sum 83, 10 values.
2With 11 values and mean 9, total = 99.
3Missing value = 99 − 83 = 16
✅ Answer: 16
๐ Textbook Page 114
๐ Page 114Figure it Out · Q3
The average height of 24 students (with 1 cm shoes) is 150.2 cm. Measure again? Correct average?
1No need to remeasure: every value is 1 cm too high.
2Subtracting 1 from every value subtracts 1 from the mean.
✅ Answer: (i) Simpler: subtract 1 (ii) (d) 149.2 cm
๐ Page 114Figure it Out · Q4
Which album (A, B, C) has a mean song length of 5.57 min?
1Estimate: B's values are mostly below 4, C's between 3.5 and 4.5 — both means are below 5.
2A: 5, 5, 5.25, 5.5, 5.75, 6, 6.5 → sum 39 → 39 ÷ 7 ≈ 5.57 ✔
✅ Answer: Album A
๐ Page 114Figure it Out · Q5
Median of 8, 10, 19, 23, 26, 34, 40, 41, 41, 48, 51, 55, 70, 84, 91, 92. Add one value / two values / remove one value without changing it.
116 values → median = (8th + 9th) ÷ 2 = (41 + 41) ÷ 2 = 41
2(i) Add 41 (any other value moves the middle).
3(ii) Add one value ≤ 41 and one ≥ 41 (e.g. 5 and 100), or two 41s.
4(iii) 15 values → median is the 8th. Removing ANY value leaves 41 in 8th place (because there are two 41s in the middle).
✅ Answer: Median 41. (i) 41 (ii) e.g. 5 and 100 (iii) any value
๐ Textbook Page 115
๐ Page 115Figure it Out · Q6
Always, sometimes or never true? (i) Removing a value below the median decreases it. (ii) Including a value below the mean decreases it. (iii) Including any 4 values doesn't affect the median. (iv) Including 4 values below the median increases it.
1(i) Removing a low value shifts the middle UP (or keeps it if values repeat) → never true.
2(ii) Always true (balance tilts to the left).
3(iii) Sometimes true (e.g. 2 below and 2 above the median keeps it).
4(iv) Adding low values pulls the middle DOWN (or keeps it) → never true.
✅ Answer: (i) never (ii) always (iii) sometimes (iv) never
๐ Page 115Figure it Out · Q7
Mean of 8, 13, 10, 4, 5, 20, y, 10 is 10.375. Find y.
1Total = 10.375 × 8 = 83
2Known values add to 70
✅ Answer: y = 13
๐ Page 115Figure it Out · Q8
15 values have mean 134. Sum?
1Sum = mean × number
✅ Answer: 2010
๐ Page 115Figure it Out · Q9
12, 47, 8, 73, 18, 35, 39, 8, 29, 25, p has median 29. Which p are possible: 10, 25, 40, 100, 29, 47, 30?
1Without p sorted: 8, 8, 12, 18, 25, 29, 35, 39, 47, 73.
2With p there are 11 values → median is the 6th.
3If p ≥ 29, the 6th value is 29 ✔. If p < 29, the 6th value is 25 or p ✘.
✅ Answer: 40, 100, 29, 47, 30
๐ Page 115Figure it Out · Q10
Cycle rides in a week (dot plot). (i) Mean (ii) median (iii) which statements are valid? (e) mean and median if everyone rides once more?
1Counts: 0→3, 1→1, 2→4, 3→7, 4→7, 5→5, 6→4, 7→6, 8→3, 10→2 (42 students).
2(i) Sum = 193 → mean = 193 ÷ 42 ≈ 4.6
3(ii) 21st and 22nd values are both 4 → median 4
4(a) Not valid — 3 students never rode. (b) Valid — almost everyone rode a few times. (c) Valid — 8 or 10 rides in 7 days means riding more than once on some day.
5(d) Not valid as stated — at least 5 did (those with 8 or 10), but others could also have ridden twice on a day.
6(e) Every value + 1 → mean ≈ 5.6, median 5.
✅ Answer: Mean ≈ 4.6, median 4; valid: (b), (c); next week mean ≈ 5.6, median 5
๐ Textbook Page 116
๐ Page 116Figure it Out · Q11
Dart throws: trials 1–10 with frequencies 1, 0, 0, 1, 4, 9, 12, 15, 10, 10. Describe the data.
1n = 62; minimum 1, maximum 10.
2Sum = 1 + 4 + 20 + 54 + 84 + 120 + 90 + 100 = 473 → mean ≈ 7.63
3Median = average of 31st and 32nd; cumulative 2, 6, 15, 27, 42 → both are 8.
✅ Answer: Min 1, max 10, mean ≈ 7.6, median 8 — most students needed 7 to 10 throws.
๐ Textbook Page 117
๐ Page 117Temperature
Do the column graph and the line graph of Kerala and Punjab maximum temperatures show the same information? How is the monthly maximum found?
1Yes — same numbers; the line graph shows the change over time more clearly.
2Weather stations record temperatures; the highest value recorded in the state that month is the monthly maximum.
3Punjab: 19 °C (Jan) up to 38 °C (Jun), down to 23 °C (Dec). Kerala: flat, 29–33 °C.
✅ Answer: Same data; Punjab varies much more than Kerala.
๐ Textbook Page 120
๐ Page 120Space jam
Which inferences are valid? Find consecutive years where the world count at least doubled.
1'Increased every year' ✘ — 2024 (~2800) is less than 2023 (~2900).
2'USA about ¾ of the world count in 2022–24' ✔ (about 2300 of 2900).
3'Nepal launched nothing' ✘ — Nepal is not shown; we cannot tell.
4'China + Russia ≈ 400 in 2024' ✔ approximately.
5World count: about 600 in 2019 and about 1250 in 2020 → more than doubled.
✅ Answer: Valid: the USA and China + Russia statements. Doubling: 2019 → 2020.
๐ Textbook Page 121
๐ Page 121Rain
How is monthly average rainfall compiled? What is common to the city groups? Peak months?
1Total rain each month is recorded for several years and averaged for each month.
2West coast (Kovalam, Udupi, Mumbai): peak June–August (south-west monsoon).
3East coast (Rameswaram, Chennai): peak October–December (north-east monsoon); Puri peaks July–September.
✅ Answer: West coast gets more rain, in the south-west monsoon.
๐ Textbook Page 122
๐ Page 122Figure it Out · Q1
Draw a line graph of customers visiting and purchasing (Mon–Sun).
1Days on the horizontal axis, number of customers on the vertical axis (scale of 5).
2Plot both rows and join the points; use two colours and a key.
✅ Answer: Both lines rise sharply at the weekend (Sunday: 35 visit, 26 buy).
๐ Page 122Figure it Out · Q2
Average rainy days per month: (i) method (ii) plot Mangaluru, Port Blair, Rameswaram (iii) read New Delhi (iv) most and least rainy days per year (v) rainy seasons of Delhi and Rameswaram.
1(i) Count rainy days each month for many years and take the average.
2(iii) New Delhi (from the graph): Jan 1, Feb 1, Mar 1, Apr 1, May 1, Jun 4, Jul 10, Aug 10, Sep 4, Oct 1, Nov 0, Dec 1 (≈ 35 days).
3(iv) Totals: Mangaluru ≈ 112, Port Blair ≈ 126, Rameswaram ≈ 42, New Delhi ≈ 35.
4(v) New Delhi: July–August. Rameswaram: October–December.
✅ Answer: Most: Port Blair; least: New Delhi. Delhi rains in Jul–Aug, Rameswaram in Oct–Dec.
๐ Textbook Page 123
๐ Page 123Figure it Out · Q2 (births)
Monthly births in India: observations, July 2017, time period, Januaries, total for 2019.
1(i) Births rise to a peak around Aug–Oct every year and dip around Feb–May — a seasonal pattern.
2(ii) July 2017 ≈ 1.9 million.
3(iii) About April 2017 to March 2020 (3 years).
4(iv) Jan 2018 ≈ 1.65 M, Jan 2019 ≈ 1.55 M, Jan 2020 ≈ 1.75 M → 2020 highest, 2019 lowest.
5(v) Monthly values in 2019 average about 1.75 M → 12 × 1.75 ≈ 21 million.
✅ Answer: About 21 million births in 2019 (estimates may differ slightly).
๐ Textbook Page 124
๐ Page 124Infographic
Wheat vs Rice map: (i) guess Karnataka (ii) top 5 rice (iii) top 5 wheat (iv) balanced states.
1(i) Karnataka is south of the red line, between Goa (+57) and Andhra (+92) → about +60 to +70 (prefers rice).
2(ii) Manipur +100, Nagaland +99, Mizoram +97, Tripura +96, Meghalaya +95
3(iii) Rajasthan −93, Haryana −81, Punjab −78, Madhya Pradesh −60, Delhi −45
4(iv) Bihar +3, Maharashtra −15, Uttarakhand −18, Himachal −19
✅ Answer: See steps (any reasonable Karnataka guess with a reason is fine).
๐ Textbook Page 125
๐ Page 125Strips
Manoj's activity strips: what does each colour show, which day is which, the movie, the lunch break?
1Light blue at night = sleeping (longest blocks).
2The strip with a long block of classes in the daytime = Friday (school day).
3A long block of 'friends/hobbies' in the afternoon/evening on one weekend strip = the movie.
4A short eating block in the middle of the school block = lunch break (around 1:00 pm).
✅ Answer: Read the strips by matching block lengths to typical routines; answers vary slightly with interpretation.
๐ Textbook Page 126
๐ Page 126Sleepy-Deepy
Observations about sleep across ages.
16-year-olds sleep about 9.5 h; sleep falls through the teens to about 8 h at ages 30–50, then rises to about 8.5 h.
280 close points make the line look smooth.
✅ Answer: Sleep is least in middle age and higher for children and older people.
๐ Textbook Page 127
๐ Page 127Figure it Out · Q1
Mean grid: fill with 9 different numbers so every row, column and diagonal averages 10. Can you change numbers and still get 10?
1Average 10 → each line adds to 30, and the centre must be 10.
2Example: 13, 6, 11 / 8, 10, 12 / 9, 14, 7 (all lines = 30).
3(ii) Yes — e.g. 7, 12, 11 / 14, 10, 6 / 9, 8, 13. Any magic square with total 30 works.
✅ Answer: Yes, many answers.
๐ Page 127Figure it Out · Q2
Two examples each: (i) 3 numbers mean 8 (ii) 4 numbers median 15.5 (iii) 5 numbers mean 13.6 (iv) 6 numbers mean = median (v) 6 numbers mean > median.
1(i) 7, 8, 9; 2, 10, 12 (sum 24)
2(ii) 10, 15, 16, 20; 1, 14, 17, 30 (middle two average 15.5)
3(iii) 10, 12, 14, 16, 16; 13, 13, 13, 14, 15 (sum 68)
4(iv) 1, 2, 3, 4, 5, 6; 5, 5, 5, 5, 5, 5
5(v) 1, 2, 3, 4, 5, 100; 2, 2, 2, 2, 2, 20
✅ Answer: See steps (many answers).
๐ Page 127Figure it Out · Q3
Median 13: 5, 21, 14, __, __, __. How many possibilities with counting numbers?
16 values → median = (3rd + 4th) ÷ 2 = 13 → they add to 26.
2Example: 1, 12, 30 → sorted 1, 5, 12, 14, 21, 30 → (12 + 14) ÷ 2 = 13 ✔
3The third blank can be any number ≥ 14, so there are endlessly many.
✅ Answer: e.g. 1, 12, 30; infinitely many possibilities.
๐ Page 127Figure it Out · Q4
Mean 6.5: 3, 11, __, __, 15, 6. How many possibilities?
1Total = 6.5 × 6 = 39; known = 35 → blanks add to 4.
2Counting numbers: (1, 3), (2, 2), (3, 1).
✅ Answer: 3 possibilities
๐ Page 127Figure it Out · Q5
True? (i) average of two even numbers is even (ii) average of two multiples of 5 is a multiple of 5 (iii) average of five multiples of 5 is a multiple of 5.
1(i) 2 and 4 → 3 ✘. (2a + 2b) ÷ 2 = a + b, which can be odd.
2(ii) 5 and 10 → 7.5 ✘.
3(iii) 5, 5, 5, 5, 10 → 6 ✘. (5a + … + 5e) ÷ 5 = a + … + e, any whole number.
✅ Answer: All three are NOT always true.
๐ Textbook Page 128
๐ Page 128Figure it Out · Q6
Two students join a class with average 150.2 cm. (i) which statements? (ii) heights 149 and 152 (iii) median?
1(i) (c) — only the two new heights are needed (the old total = 150.2 × 24 is known).
2(ii) New pair averages 150.5 > 150.2 → (b) the average increases. (New average = (3604.8 + 301) ÷ 26 ≈ 150.22.)
3(iii) (d) — we don't know the median or where 149 and 152 lie compared with it.
✅ Answer: (i) c (ii) b (iii) d
๐ Page 128Figure it Out · Q7
Is 17 the average of the dot plot?
1Balance check about 17: below: 3×2 + 2×2 + 1×3 = 13; above: 1×4 + 2×4 + 3×3 + 4 + 6 = 31.
2Not balanced (more on the right) → the mean is above 17.
3Exact: 443 ÷ 25 = 17.72
✅ Answer: No — the mean is about 17.7.
๐ Textbook Page 129
๐ Page 129Figure it Out · Q8
Mean 65.3 kg, median 67 kg. One person lost 2 kg, two gained 1 kg. Changes?
1Total change = −2 + 1 + 1 = 0 → mean stays 65.3 kg.
2The median may or may not change — it depends on whose weights changed.
✅ Answer: Mean unchanged; median cannot be decided.
๐ Page 129Figure it Out · Q9
Salt prices 2016–2025: line graph of 3 states; compare Gujarat and UP; biggest increase?
1Increase 2016 → 2025: A&N 5, Assam 6, Gujarat 3, Mizoram 10, UP 9, West Bengal 15 (rounded).
2Gujarat stays almost flat (13–17) until a jump in 2025; UP rises steadily from 16 to about 27 and dips in 2025.
3Biggest increase: West Bengal (₹9.47 → ₹23.99, more than double).
✅ Answer: West Bengal
๐ Textbook Page 130
๐ Page 130Figure it Out · Q10
Lighting source graph: which statements are valid?
1(i) 1983: rural kerosene ≈ 84%, urban electricity ≈ 64% → valid.
2(ii) Kerosene falls in both → valid.
3(iii) Urban electricity in 2000 was about 91%, not 10% → not valid.
4(iv) The graph says nothing about power cuts → not valid.
✅ Answer: Valid: (i) and (ii)
๐ Page 130Figure it Out · Q11
Hobbies graph: (i) urban 10-year-olds (ii) age when rural kids spend 1.5 h (iii) statements.
1(i) About 2 hours.
2(ii) The rural line is at 1.5 h at about age 14 → (d).
3(iii)(a) At 15 it is about 1.1 h — roughly HALF of age 10, not twice → false.
4(iii)(b) The graph shows averages, so we can't say ALL rural 15-year-olds → false.
✅ Answer: (i) ≈ 2 h (ii) 14 years (iii) both incorrect
๐ Textbook Page 131
๐ Page 131Figure it Out · Q14
Sunrise and sunset graphs: (i) earliest sunrise in January and day length (ii) longest day length.
1Lower lines (04:00–08:00) are sunrises, upper lines (16:00–20:00) are sunsets.
2(i) Kibithu: sunrise ≈ 5:50, sunset ≈ 16:30 → day ≈ 10 h 40 min.
3(ii) Srinagar in June: ≈ 5:20 to 19:45 → about 14.5 h, the longest.
✅ Answer: (i) Kibithu, about 10½ h (ii) Srinagar (in summer)
๐ก Tip: Kanyakumari (near the equator) has almost equal days all year.
๐ Textbook Page 132
๐ Page 132Figure it Out · Q15
Moonrise and moonset over a month: find amavasya and purnima.
1Moonrise is about 50 minutes later each day.
2Full moon rises at sunset (≈ 18:00): about day 13–14.
3New moon rises with the sun (≈ 06:00): about day 28.
✅ Answer: Purnima ≈ day 13–14; amavasya ≈ day 28 (about 15 days apart).
๐ Textbook Page 134
๐ Page 134Puzzle: Game of Hex
Play Hex. Can a game end in a draw?
1A full Hex board always contains a winning chain for one player.
2So there are never draws, and the first player has an advantage (with good play).
✅ Answer: No draws are possible in Hex.
๐ Where do we use this?
- Marks analysis in spreadsheets, weather and rainfall reports, prices, population and births data, news infographics.
๐ Link to higher classes
- Class 9–10: mean, median, mode of grouped data, frequency polygons. Class 11: spread (variance), statistics projects.
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