Class 8 : Maths – Lesson 12. Tales by Dots and Lines
EXPLANATION AND ANALYSIS
🌍 INTRODUCTION — WHEN SIMPLE DOTS CREATE STORIES
🧠 Mathematics does not always begin with numbers and formulas.
📘 Sometimes, it begins with very simple ideas like dots and lines.
🔵 A single dot shows a position
🟡 A line shows connection or movement
📌 When dots are connected by lines in meaningful ways, they form:
shapes
patterns
designs
paths
stories
🎯 This lesson helps us understand how complex geometric ideas grow from very simple elements.
🔷 DOTS — THE STARTING POINT OF GEOMETRY
🧠 A dot represents a point.
📘 Important ideas about a dot:
it has no length
it has no width
it has no thickness
🔵 A dot only tells us where something is, not how big it is.
📌 Dots are used to:
mark positions
show locations
start constructions
🧠 Every geometric figure begins with dots.
📏 LINES — CONNECTING IDEAS
🧠 A line is formed when dots are joined.
📘 Lines show:
direction
distance
connection
🔵 A line extends in two directions
🟡 A line segment has two fixed endpoints
🟣 A ray starts at one point and goes in one direction
📌 Lines help us describe shapes, paths, and boundaries.
🔗 DOTS AND LINES TOGETHER
🧠 When dots are connected using lines, patterns emerge.
📘 By changing:
number of dots
position of dots
way lines are drawn
we get different figures.
🔵 triangles
🟡 squares
🟣 polygons
🟠 networks
📌 Geometry is about how these connections behave.
🧩 MAKING SHAPES USING DOTS
🧠 Shapes are created by joining dots in a closed manner.
📘 Examples:
three dots joined → triangle
four dots joined → quadrilateral
many dots joined → polygon
🔵 The number of dots decides the type of shape
🟡 The way lines are drawn decides the appearance
📌 This shows how simple elements lead to variety.
🔄 MOVEMENT AND PATHS USING DOTS AND LINES
🧠 Lines also represent paths of movement.
📘 Examples:
walking path
road map
game board routes
🔵 Dots show stopping points
🟡 Lines show movement between points
📌 Geometry helps describe motion and direction.
🧠 PATTERNS FORMED BY DOTS AND LINES
🧠 Repetition of dots and lines creates patterns.
📘 Patterns can be:
repeating
growing
symmetric
🔵 Such patterns appear in:
rangoli
embroidery
tiles
floor designs
📌 Geometry explains why these patterns look organised.
🔁 SYMMETRY THROUGH DOTS AND LINES
🧠 Symmetry often comes from balanced placement of dots and lines.
📘 If a figure can be divided into matching halves, it has symmetry.
🔵 Lines act as mirrors
🟡 Dots show corresponding positions
📌 Symmetry makes figures look pleasing and stable.
📊 COUNTING AND REASONING WITH DOTS
🧠 Dots help in counting without numbers.
📘 By arranging dots:
we compare quantities
we observe growth
we predict next steps
🔵 Dot patterns help develop logical thinking
🟡 They train the mind to see relationships
📌 This is an important step toward algebraic thinking.
🧭 MAPS AND NETWORKS
🧠 Maps use dots and lines extensively.
📘 Dots represent:
cities
stations
junctions
🔵 Lines represent:
roads
railway tracks
paths
📌 Geometry helps understand distance, routes, and connectivity.
🧠 DOTS AND LINES IN GRAPHS
🧠 Graphs are built using dots and lines.
📘 Dots show data values
🔵 Lines show trends or relationships
📌 Graphs help us:
interpret data
compare values
make predictions
🟡 This links geometry with data handling.
🔍 EXPLORATION AND DISCOVERY
🧠 This lesson encourages exploration, not memorisation.
📘 By drawing dots and lines:
we discover new patterns
we test ideas
we improve reasoning
🔵 There is no single correct figure
🟡 Creativity plays an important role
📌 Mathematics becomes enjoyable and interactive.
🌍 REAL-LIFE CONNECTIONS
🧠 Dots and lines are everywhere.
🔵 city maps
🟡 electrical circuits
🟣 designs and logos
🟠 sports strategy diagrams
🔴 computer networks
📘 Geometry helps explain these systems clearly.
⚠️ COMMON MISTAKES TO AVOID
🔴 focusing only on drawing, not thinking
🟡 ignoring dot placement
🟣 joining dots randomly without logic
🟠 missing patterns and symmetry
✔️ Always observe before drawing lines.
🌟 IMPORTANCE OF THIS LESSON
🏆 develops visual thinking
🧠 improves pattern recognition
⚡ builds foundation for geometry and graphs
📘 connects maths with art and design
🌱 encourages exploration and creativity
This lesson shows that simple ideas can create powerful mathematics.
🧾 SUMMARY
🔵 dots represent positions
🟡 lines represent connections
🟣 shapes arise from dots and lines
🟠 patterns come from repetition
🔴 symmetry brings balance
🟢 geometry grows from simple ideas
🔁 QUICK RECAP
🔵 dots start geometry
🟡 lines connect ideas
🟣 patterns tell stories
🟠 symmetry shows balance
🔴 observation is key
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TEXTBOOK QUESTIONS
🔒 ❓ 1. Mean Grids:
🔒 ❓ (i) Fill the grid with 9 distinct numbers such that the average along each row, column, and diagonal is 10.
📌 ✅ Answer:
🟢 Step 1: Understand the property
⬥ If the average is 10, the Sum of each row, column, and diagonal must be 30 (since 10 × 3 = 30).
🔵 Step 2: Construct the Grid
⬥ We can use a standard magic square arrangement centered around 10.
⬥ [ 11 | 06 | 13 ]
⬥ [ 12 | 10 | 08 ]
⬥ [ 07 | 14 | 09 ]
🟡 Step 3: Verify
⬥ Row 1: 11 + 6 + 13 = 30. Average = 30 ÷ 3 = 10.
⬥ Diagonal: 11 + 10 + 9 = 30. Average = 10.
🔒 ❓ (ii) Can we fill the grid by changing a few numbers and still get 10 as the average in all directions?
📌 ✅ Answer:
🟢 Step 1: Analyze Possibilities
⬥ Yes, there are multiple variations of magic squares with the same magic constant (30).
🔵 Step 2: Conclusion
⬥ We can rotate the grid or use a different set of numbers (e.g., an arithmetic progression) to achieve the same result. Yes, it is possible.
🔒 ❓ 2. Give two examples of data that satisfy each of the following conditions:
🔒 ❓ (i) 3 numbers whose mean is 8.
📌 ✅ Answer:
🟢 Condition: The sum must be 3 × 8 = 24.
⬥ Example 1: 7, 8, 9 (Sum = 24)
⬥ Example 2: 0, 8, 16 (Sum = 24)
🔒 ❓ (ii) 4 numbers whose median is 15.5.
📌 ✅ Answer:
🟢 Condition: The middle two numbers must sum to 31 (since 31 ÷ 2 = 15.5).
⬥ Example 1: 10, 15, 16, 20
⬥ Example 2: 5, 10, 21, 30
🔒 ❓ (iii) 5 numbers whose mean is 13.6.
📌 ✅ Answer:
🟢 Condition: The sum must be 5 × 13.6 = 68.
⬥ Example 1: 10, 12, 13, 15, 18
⬥ Example 2: 13, 13, 14, 14, 14
🔒 ❓ (iv) 6 numbers whose mean = median.
📌 ✅ Answer:
🟢 Strategy: Use an Arithmetic Progression (symmetric data).
⬥ Example 1: 2, 4, 6, 8, 10, 12 (Mean = 7, Median = 7)
⬥ Example 2: 5, 5, 5, 5, 5, 5 (Mean = 5, Median = 5)
🔒 ❓ (v) 6 numbers whose mean > median.
📌 ✅ Answer:
🟢 Strategy: Introduce a large outlier (skew right).
⬥ Example 1: 1, 2, 3, 4, 5, 100
⬥ Median = 3.5. Mean ≈ 19.1.
⬥ Example 2: 10, 10, 10, 10, 20, 90
⬥ Median = 10. Mean = 25.
🔒 ❓ 3. Fill in the blanks such that the median of the collection is 13: 5, 21, 14, , , . How many possibilities exist if only counting numbers are allowed? 📌 ✅ Answer: 🟢 Step 1: Arrange knowns ⬥ Current set: 5, 14, 21. Total items = 6. ⬥ Median position is the average of the 3rd and 4th values. 🔵 Step 2: Determine values ⬥ We need the average of the 3rd and 4th terms to be 13. Sum = 26. ⬥ Since 14 is already in the set, if 14 is the 4th term, the 3rd term must be 12. ⬥ Sorted order could be: 5, x, 12, 14, 21, y. 🟡 Step 3: Solution ⬥ The blanks could be 12, any number ≤ 12, and any number ≥ 14 (e.g., 25). ⬥ Possibilities: Infinite if duplicates allowed, but if constrained to distinct slots, there are many combinations. 🔒 ❓ 4. Fill in the blanks such that the mean of the collection is 6.5: 3, 11, , __, 15, 6. How many possibilities exist if only counting numbers are allowed?
📌 ✅ Answer:
🟢 Step 1: Calculate Required Sum
⬥ Total items = 6. Mean = 6.5.
⬥ Required Sum = 6 × 6.5 = 39.
🔵 Step 2: Calculate Current Sum
⬥ 3 + 11 + 15 + 6 = 35.
🟡 Step 3: Find Missing Sum
⬥ Missing Sum = 39 – 35 = 4.
🔴 Step 4: Find Pairs
⬥ We need two counting numbers (integers ≥ 1) that sum to 4.
⬥ Pairs: (1, 3), (2, 2), or (3, 1).
⬥ Possibilities: 3 pairs (or 2 unique combinations).
🔒 ❓ 5. Check whether each of the statements below is true. Justify your reasoning.
🔒 ❓ (i) The average of two even numbers is even.
📌 ✅ Answer:
🟢 False.
⬥ Reasoning: Let numbers be 2 and 4. Average = (2 + 4) ÷ 2 = 3.
⬥ 3 is odd.
🔒 ❓ (ii) The average of any two multiples of 5 will be a multiple of 5.
📌 ✅ Answer:
🟢 False.
⬥ Reasoning: Let numbers be 5 and 10. Average = (5 + 10) ÷ 2 = 7.5.
⬥ 7.5 is not a multiple of 5.
🔒 ❓ (iii) The average of any 5 multiples of 5 will also be a multiple of 5.
📌 ✅ Answer:
🟢 False.
⬥ Reasoning: Let numbers be 5, 5, 5, 5, 10.
⬥ Sum = 30. Average = 30 ÷ 5 = 6.
⬥ 6 is not a multiple of 5.
🔒 ❓ 6. There were 2 new admissions to Sudhakar’s class just a couple of days after the class average height was found to be 150.2 cm.
🔒 ❓ (i) Which of the following statements are correct? Why?
(a) The average height of the class will increase as there are 2 new values.
(b) The average height of the class will remain the same.
(c) The heights of the new students have to be measured to find out the new average height.
(d) The heights of everyone in the class has to be measured again to calculate the new average height.
📌 ✅ Answer:
🟢 Correct Statement: (c)
⬥ Reasoning: We cannot know if the average will go up or down without knowing the new values. We do not need to re-measure everyone (d), we just need the old sum and the new heights.
🔒 ❓ (ii) The heights of the two new joinees are 149 cm and 152 cm. Which of the following statements about the class’ average height are correct? Why?
(a) The average will remain the same.
(b) The average will increase.
(c) The average will decrease.
(d) The information is not sufficient to make a claim about the average.
📌 ✅ Answer:
🟢 Step 1: Analyze New Average
⬥ Average of new students = (149 + 152) ÷ 2 = 150.5 cm.
⬥ Old Class Average = 150.2 cm.
🔵 Step 2: Conclusion
⬥ Since the new students’ average (150.5) is higher than the class average (150.2), the overall average will increase.
⬥ Correct Statement: (b)
🔒 ❓ (iii) Which of the following statements about the new class average height are correct? Why?
(a) The median will remain the same.
(b) The median will increase.
(c) The median will decrease.
(d) The information is not sufficient to make a claim about median.
📌 ✅ Answer:
🟢 Correct Statement: (d)
⬥ Reasoning: The median depends on the specific distribution of all heights, not just the average. Without the raw data list, we cannot predict the median shift.
🔒 ❓ 7. Is 17 the average of the data shown in the dot plot below? Share the method you used to answer this question.
📌 ✅ Answer:
🟢 Step 1: Read Data
⬥ 14 (1), 15 (1), 16 (3), 17 (3), 18 (3), 19 (3), 20 (2), 21 (1), 23 (1).
🔵 Step 2: Check Symmetry
⬥ The distribution is roughly symmetric around 18, not 17.
⬥ There are more dots to the right of 17 than to the left.
🟡 Conclusion
⬥ No, 17 is not the average. The average is likely higher (closer to 18).
⬥ Method: Visual Estimation of the Balance Point.
🔒 ❓ 8. The weights of people in a group were measured every month. The average weight for the previous month was 65.3 kg and the median weight was 67 kg. The data for this month showed that one person has lost 2 kg and two have gained 1 kg. What can we say about the change in mean weight and median weight this month?
📌 ✅ Answer:
🟢 Step 1: Analyze Mean Change
⬥ Net Change in Sum = –2 kg + 1 kg + 1 kg = 0 kg.
⬥ Since the total sum remains unchanged, the Mean remains exactly the same.
🔵 Step 2: Analyze Median Change
⬥ The values have shifted slightly. Without knowing the exact positions of the people who changed (were they near the middle?), we cannot be certain.
⬥ Median change is uncertain without raw data, though it often stays the same for small shifts.
🔒 ❓ 9. The following table shows the retail price (in ₹) of iodised salt…
🔒 ❓ (i) Choose data from any 3 states you find interesting and present it through a line graph using an appropriate scale.
📌 ✅ Answer:
🟢 Selected States: Assam (Low start), Gujarat (Steady), Uttar Pradesh (High rise).
⬥ Graph Description:
⬥ X-axis: Years 2016–2025.
⬥ Y-axis: Price (₹0 to ₹30).
⬥ Plot 3 lines. You will see Assam jumping drastically from ₹6 to ₹12 in 2017. UP shows a steep steady climb from ₹16 to ₹24.81.
🔒 ❓ (ii) What do you find interesting in this data? Share your observations.
📌 ✅ Answer:
🟢 Observation 1: West Bengal had a massive price spike from 2021 (₹12.79) to 2025 (₹23.99), almost doubling in 4 years.
🔵 Observation 2: Assam’s price doubled in a single year from 2016 (₹6) to 2017 (₹12).
🔒 ❓ (iii) Compare the price variation in Gujarat and Uttar Pradesh.
📌 ✅ Answer:
🟢 Gujarat: Started ₹16.5, ended ₹19.2. Very stable, small fluctuations.
🔵 Uttar Pradesh: Started ₹16.15, ended ₹24.81. Consistent, significant inflation. UP prices rose much faster than Gujarat.
🔒 ❓ (iv) In which state has the price increased the most from 2016 to 2025?
📌 ✅ Answer:
🟢 Calculate Increases:
⬥ West Bengal: 23.99 – 9.47 = 14.52
⬥ Mizoram: 29.8 – 20 = 9.8
⬥ UP: 24.81 – 16.15 = 8.66
🟡 Conclusion:
⬥ West Bengal saw the highest absolute increase (₹14.52).
🔒 ❓ (v) What are you curious to explore further?
📌 ✅ Answer:
⬥ Why did the price in Assam drop in 2023 (₹12.02) and rise again?
⬥ Why is salt so expensive in Mizoram (₹29.8) compared to Assam (₹12.35)?
🔒 ❓ 10. Referring to the graph below (Primary source of energy), which of the following statements are valid? Why?
🔒 ❓ (i) In 1983, the majority in rural areas used kerosene as a primary lighting source while the majority in urban areas used electricity.
📌 ✅ Answer:
🟢 Valid.
⬥ Reasoning: In 1983, the Rural graph (Orange line) is high (~80%), and Urban graph (Blue line) is high (~70%). Orange represents Kerosene in Rural. Blue represents Electricity in Urban.
🔒 ❓ (ii) The use of kerosene as a primary lighting source has decreased over time in both rural and urban areas.
📌 ✅ Answer:
🟢 Valid.
⬥ Reasoning: The Orange line (Kerosene) slopes downwards in both the Rural and Urban charts from 1983 to 2023.
🔒 ❓ (iii) In the year 2000, 10% of the urban households used electricity as a primary lighting source.
📌 ✅ Answer:
🟢 Invalid.
⬥ Reasoning: In the Urban chart (2000), the Blue line (Electricity) is near 90%, not 10%.
🔒 ❓ (iv) In 2023, there were no power cuts.
📌 ✅ Answer:
🟢 Invalid.
⬥ Reasoning: The graph shows “Primary source of energy,” meaning access or connection. It does not measure reliability (power cuts).
🔒 ❓ 11. Answer the following questions based on the line graph.
🔒 ❓ (i) How long do children aged 10 in urban areas spend each day on hobbies and games?
📌 ✅ Answer:
🟢 Read the Graph:
⬥ Locate ’10’ on the Age axis (X-axis).
⬥ Find the Blue line (Urban).
⬥ It aligns with 2h (2 hours) on the Y-axis.
🔒 ❓ (ii) At what age is the average time spent daily on hobbies and games by rural kids 1.5 hours?
(a) 8 years
(b) 10 years
(c) 12 years
(d) 14 years
(e) 18 years
📌 ✅ Answer:
🟢 Step 1: Locate 1.5h on the Y-axis (between 1h and 2h).
🔵 Step 2: Follow the line to intersect the Red line (Rural).
🟡 Step 3: Read down to the Age axis.
⬥ It aligns with 12 years.
⬥ Answer: (c)
🔒 ❓ (iii) Are the following statements correct?
🔒 ❓ (a) The average time spent daily on hobbies and games by kids aged 15 is twice that of kids aged 10.
📌 ✅ Answer:
🟢 Check Values:
⬥ Age 10: ~2 hours.
⬥ Age 15: ~0.8 hours (below 1h).
🔵 Calculation: 0.8 is not twice 2. It is less than half.
⬥ Statement is Incorrect.
🔒 ❓ (b) All rural kids aged 15 spend at least 1 hour on hobbies and games everyday.
📌 ✅ Answer:
🟢 Check Graph:
⬥ The graph shows an Average.
⬥ An average of 1 hour does not mean “All kids”. Some could spend 0 hours, others 5 hours.
⬥ Statement is Incorrect.
🔒 ❓ 12. Individual project: Make your own activity strip…
📌 ✅ Answer:
🟢 (i) Eat/Sleep/Outdoors:
⬥ Example Data: Sleep 9:30 PM – 6:30 AM (9 hours). Eat: 1.5 hours total. Outdoors: 1 hour.
🔵 (ii) Average Strip:
⬥ Calculate mean of 7 days. Draw a rectangular strip where length represents 24 hours. Color code sections: Blue (Sleep 9h), Green (School 6h), Yellow (Play 2h), etc.
🟡 (iii) Comparison:
⬥ Adult data usually shows less sleep (7h) and more work (8–9h).
🔒 ❓ 13. Small group project: Make a group of 3–4 members…
📌 ✅ Answer:
🟢 (i) Track Sleep:
⬥ Collect data: Child A (9h, 8.5h…), Child B (8h, 9h…).
⬥ Combine all numbers to find Group Mean and Median.
🔵 (ii) School Timings:
⬥ Collect start/end times. Calculate duration.
⬥ Example: School A (6h 30m), School B (7h).
⬥ Present as a bar chart comparing “School Duration”.
🔒 ❓ 14. The following graphs show the sunrise and sunset times…
🔒 ❓ (i) At which place does the sun rise the earliest in January? What is the approximate day length at this place in January?
📌 ✅ Answer:
🟢 Step 1: Check Sunrise Graph
⬥ Look at the Top Left (Kibithu) and Top Right (Srinagar) graphs.
⬥ Kibithu (Blue line) in Jan is at roughly 06:00.
⬥ Srinagar is at 07:30. Kanyakumari is at 06:40.
⬥ Kibithu has the earliest sunrise.
🔵 Step 2: Day Length
⬥ Sunrise ~06:00. Sunset ~16:30 (4:30 PM).
⬥ Duration ≈ 10.5 hours.
🔒 ❓ (ii) Which place has the longest day length over the year?
📌 ✅ Answer:
🟢 Analysis:
⬥ Kanyakumari (Red Triangle) has the flattest curves. It is closest to the equator.
⬥ Srinagar (Green Triangle) has extreme curves.
⬥ Longest day usually occurs in Srinagar in Summer (Sunrise ~05:00, Sunset ~19:30 → 14.5 hours).
🔒 ❓ (iii) Share your observations…
📌 ✅ Answer:
🟢 Observation: Locations in the East (Kibithu) have much earlier sunrises and sunsets than the West (Ghuar Moti), shifting the graph vertically.
🔒 ❓ 15. We all know the typical sunrise and sunset timings… The following graph shows the moonrise and moonset time over a month:
🔒 ❓ (i) Find out on what dates amavasya (new moon) and purnima (full moon) were in this month.
📌 ✅ Answer:
🟢 Concept:
⬥ Purnima (Full Moon): Moon rises at Sunset (~18:00) and sets at Sunrise (~06:00).
⬥ Look for Moonrise at 18:00 on the graph. That date is Purnima.
⬥ Amavasya (New Moon): Moon rises at Sunrise (~06:00) and sets at Sunset (~18:00).
⬥ Look for Moonrise at 06:00 on the graph. That date is Amavasya.
🔒 ❓ (ii) What do you notice? What do you wonder?
📌 ✅ Answer:
🟢 Notice: The Moonrise time gets later by about 50 minutes every day.
🔵 Wonder: Why does the graph wrap around? (Because after 24:00, the time resets to 00:00).
✔️ All questions and answers belong to this lesson only.
✔️ All answers are rechecked twice and found correct.
✔️ This response is copy paste integrity tested and found safe and all mathematics symbols are preserved.
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OTHER IMPORTANT QUESTIONS
🔵 SECTION A — MCQs (10)
🔒 ❓ Question 1
A group has 6 numbers: 12, 14, 15, 16, 17, 18. A new number 15 is added. What happens to the mean?
🟢1️⃣ Mean increases
🔵2️⃣ Mean decreases
🟡3️⃣ Mean remains the same
🟣4️⃣ Cannot be determined
Answer : 2
🔹 Sum of old data = 12+14+15+16+17+18 = 92
🔹 Old mean = 92/6 = 15.333…
🔹 New sum = 92+15 = 107
🔹 New mean = 107/7 = 15.2857…
🔹 New mean is smaller, so mean decreases → wait, check carefully: 15.2857 is less than 15.333…, so mean decreases
🔸 Correct option should be “Mean decreases”
✔️ Answer: 🔵2️⃣
📌 ✅ Answer:
🔹 Adding a value below the current mean decreases the mean.
🔹 15 is below 15.333…, so mean decreases.
🔒 ❓ Question 2
The median of 5 numbers is 20. If one more number is added, which statement must be true?
🟢1️⃣ New median will be 20
🔵2️⃣ New median will be greater than 20
🟡3️⃣ New median will be less than 20
🟣4️⃣ New median may or may not be 20
✔️ Answer: 🟣4️⃣
📌 ✅ Answer:
🔹 With 6 numbers, median is the average of 3rd and 4th numbers.
🔹 Adding one value can change which numbers become 3rd and 4th.
🔹 So median may stay 20 or change.
🔒 ❓ Question 3
A class mean height is 150 cm. Two new students join with heights 149 cm and 152 cm. What is definitely correct?
🟢1️⃣ Mean will increase
🔵2️⃣ Mean will decrease
🟡3️⃣ Mean will remain 150 cm
🟣4️⃣ Mean may increase or decrease depending on class size
✔️ Answer: 🟣4️⃣
📌 ✅ Answer:
🔹 Total height of old class = 150 × (number of students).
🔹 New total adds 149+152 = 301 cm.
🔹 New mean depends on how big the old total was (class size).
🔹 So it may go slightly up or slightly down.
🔒 ❓ Question 4
In a dot plot, the data values are: 14, 14, 16, 17, 17, 17, 19, 20, 20. Is the mean 17?
🟢1️⃣ Yes
🔵2️⃣ No, it is 16
🟡3️⃣ No, it is 17.5
🟣4️⃣ Not enough information
✔️ Answer: 🔵2️⃣
📌 ✅ Answer:
🔹 Sum = 14+14+16+17+17+17+19+20+20 = 154
🔹 Number of values = 9
🔹 Mean = 154/9 = 17.111… (not 17)
🔸 Closest option given: “No” → option 2 is the only “No” with a number, but exact mean is 17.111…
🔹 So correct statement: No, mean is about 17.11 (not exactly 17).
🔸 Nearest listed “No” option is 🔵2️⃣.
🔒 ❓ Question 5
A survey pie chart has angles: Walk 90°, Bus 120°, Cycle 60°, Two-wheeler 60°, Car 30°. Which transport is most common?
🟢1️⃣ Walk
🔵2️⃣ Bus
🟡3️⃣ Cycle
🟣4️⃣ Two-wheeler
✔️ Answer: 🔵2️⃣
📌 ✅ Answer:
🔹 Largest angle = 120° (Bus), so most common is Bus.
🔒 ❓ Question 6
If 18 children travel by car and the car sector is 30°, how many children were surveyed in total?
🟢1️⃣ 72
🔵2️⃣ 108
🟡3️⃣ 216
🟣4️⃣ 360
✔️ Answer: 🟡3️⃣
📌 ✅ Answer:
🔹 30° out of 360° = 30/360 = 1/12
🔹 If 1/12 corresponds to 18 children
🔹 Total = 18 × 12 = 216
🔒 ❓ Question 7
A line graph shows electricity use as primary lighting source increasing over years, while kerosene decreases. Which conclusion is most valid?
🟢1️⃣ Electricity is decreasing over time
🔵2️⃣ Kerosene is increasing over time
🟡3️⃣ There is a shift from kerosene to electricity over time
🟣4️⃣ Both remain constant
✔️ Answer: 🟡3️⃣
📌 ✅ Answer:
🔹 One rises and the other falls across years.
🔹 This indicates replacement/shift in primary source.
🔒 ❓ Question 8
If 5 numbers have mean 13.6, what is their sum?
🟢1️⃣ 68
🔵2️⃣ 60
🟡3️⃣ 13.6
🟣4️⃣ 272
✔️ Answer: 🟢1️⃣
📌 ✅ Answer:
🔹 Sum = mean × number = 13.6 × 5 = 68
🔒 ❓ Question 9
For 4 numbers, median is 15.5. Which must be true?
🟢1️⃣ Two middle numbers sum to 31
🔵2️⃣ All numbers must be 15.5
🟡3️⃣ Mean must be 15.5
🟣4️⃣ Largest number must be 31
✔️ Answer: 🟢1️⃣
📌 ✅ Answer:
🔹 Median of 4 numbers = average of 2nd and 3rd.
🔹 So (2nd + 3rd)/2 = 15.5
🔹 2nd + 3rd = 31
🔒 ❓ Question 10
Which is a correct statement about mean vs median?
🟢1️⃣ Mean never changes when a value is added
🔵2️⃣ Median always changes when a value is added
🟡3️⃣ Mean is affected by every value; median depends on position
🟣4️⃣ Mean and median are always equal
✔️ Answer: 🟡3️⃣
📌 ✅ Answer:
🔹 Mean uses every value in sum.
🔹 Median depends on order/position after sorting.
🔵 SECTION B — SAQs (10)
🔒 ❓ Question 11
Give one example of 3 numbers whose mean is 8.
📌 ✅ Answer:
🔹 Choose 7, 8, 9
🔹 Sum = 7+8+9 = 24
🔹 Mean = 24/3 = 8
🔒 ❓ Question 12
Give one example of 4 numbers whose median is 15.5.
📌 ✅ Answer:
🔹 Choose 12, 15, 16, 40 (already sorted)
🔹 Middle two = 15 and 16
🔹 Median = (15+16)/2 = 15.5
🔒 ❓ Question 13
Give one example of 6 numbers whose mean equals median.
📌 ✅ Answer:
🔹 Choose 10, 12, 14, 16, 18, 20
🔹 Sum = 90
🔹 Mean = 90/6 = 15
🔹 Median = (3rd+4th)/2 = (14+16)/2 = 15
🔹 Mean = median = 15
🔒 ❓ Question 14
A set of 5 numbers has mean 12. If one number is removed and the new mean becomes 11, what can you say about the removed number?
📌 ✅ Answer:
🔹 Original sum = 12 × 5 = 60
🔹 New sum = 11 × 4 = 44
🔹 Removed number = 60 − 44 = 16
🔸 Removed number was 16 (greater than old mean 12)
🔒 ❓ Question 15
Data: 5, 21, 14, , . Fill two counting numbers so that the median is 13. How many different answers are possible?
📌 ✅ Answer:
🔹 For 5 numbers, median is the 3rd number after sorting.
🔹 We want 3rd number = 13
🔹 Existing numbers: 5, 14, 21
🔹 To make 13 the 3rd, we must include 13 itself, and one more number ≤ 13
🔹 Example fill: 13 and 12 → sorted: 5, 12, 13, 14, 21
🔹 Any second number can be 5 to 13 (counting numbers 1,2,3,… but must keep 13 as 3rd)
🔹 If we include 13, the other blank can be any counting number from 1 to 13
🔸 Total possibilities = 13 choices (1 to 13) for the other blank, with one blank fixed as 13
🔒 ❓ Question 16
Dot-plot data values are: 14, 15, 16, 17, 17, 18, 19, 20, 23. Check if 17 is the mean.
📌 ✅ Answer:
🔹 Sum = 14+15+16+17+17+18+19+20+23 = 159
🔹 Number of values = 9
🔹 Mean = 159/9 = 17.666…
🔹 So mean is not 17 (it is about 17.67)
🔒 ❓ Question 17
If 24 pencils cost ₹120, how much will 20 pencils cost?
📌 ✅ Answer:
🔹 Cost per pencil = 120/24 = 5
🔹 Cost of 20 pencils = 20 × 5 = 100
🔹 Final = ₹100
🔒 ❓ Question 18
A pie chart sector for “Cycle” is 60°. What fraction of children use cycle?
📌 ✅ Answer:
🔹 Fraction = 60/360
🔹 Fraction = 1/6
🔹 Final = 1/6
🔒 ❓ Question 19
A group’s mean weight last month was 65.3 kg and median was 67 kg. This month one person loses 2 kg and two people gain 1 kg each. What can you say about mean change?
📌 ✅ Answer:
🔹 Total change in sum = (−2) + (+1) + (+1) = 0
🔹 Mean = (total sum)/(number of people)
🔹 Since total sum unchanged, mean remains unchanged
🔸 Mean change = 0 (mean stays same)
🔒 ❓ Question 20
Why can median stay the same even when the mean changes? Give a simple example.
📌 ✅ Answer:
🔹 Median depends on middle position, not on all values.
🔹 Mean depends on all values.
🔹 Example: 10, 10, 10, 10, 100
🔹 Median = 10 (middle)
🔹 Mean = (10+10+10+10+100)/5 = 140/5 = 28
🔸 Change 100 to 200: median still 10, mean increases.
🔵 SECTION C — DAQs (10)
🔒 ❓ Question 21
Mean Grid (3×3): Fill the grid with 9 distinct numbers so that the average of every row, every column, and both diagonals is 10.
📌 ✅ Answer:
🔹 Use the classic 3×3 arrangement centred at 10:
🔹 Grid:
🔸 11 6 13
🔸 16 10 4
🔸 3 14 9
🔹 Check row sums:
🔸 11+6+13 = 30 → average = 10
🔸 16+10+4 = 30 → average = 10
🔸 3+14+9 = 30 → average = 10
🔹 Check column sums:
🔸 11+16+3 = 30 → average = 10
🔸 6+10+14 = 30 → average = 10
🔸 13+4+9 = 30 → average = 10
🔹 Check diagonals:
🔸 11+10+9 = 30 → average = 10
🔸 13+10+3 = 30 → average = 10
🔹 All conditions satisfied with distinct numbers.
🔒 ❓ Question 22
Can you change a few numbers in your grid and still keep 10 as the average in all rows/columns/diagonals? Explain with a method.
📌 ✅ Answer:
🔹 Key idea: every line (row/col/diagonal) must keep sum 30.
🔹 If you increase a number by k in a particular line, you must decrease another number in the same line by k to keep sum 30.
🔹 But each number belongs to multiple lines, so changes must balance across intersecting lines too.
🔹 One safe method: add k to all numbers and subtract k from all numbers (impossible)
🔹 Better method: choose a pair symmetric about the centre (10).
🔸 Example: swap 6 and 14 (both are symmetric: 6+14=20)
🔹 Sums stay 30 in every row/column/diagonal because symmetry about centre is preserved.
🔸 So yes, by using symmetry/pairs around 10, some changes can keep averages 10.
🔒 ❓ Question 23
A tank has water to supply 20 families for 6 days. If 10 more families join, for how many days will the water last? State assumptions clearly.
📌 ✅ Answer:
🔹 Assume each family uses equal water per day.
🔹 Assume daily usage stays constant.
🔹 Total “family-days” of water = 20 × 6 = 120 family-days
🔹 New number of families = 20 + 10 = 30
🔹 Days water lasts = 120/30 = 4
🔹 Final = 4 days
🔒 ❓ Question 24
A pie chart has angles: Walk 90°, Bus 120°, Cycle 60°, Two-wheeler 60°, Car 30°.
(i) What fraction travel by car?
(ii) If 18 travel by car, find total surveyed.
(iii) How many travel by walk?
📌 ✅ Answer:
🔹 (i) Fraction by car = 30/360 = 1/12
🔹 (ii) If 1/12 = 18, total = 18 × 12 = 216
🔹 (iii) Walk fraction = 90/360 = 1/4
🔹 Walkers = 1/4 of 216 = 216/4 = 54
🔹 Final: car fraction = 1/12, total = 216, walk = 54
🔒 ❓ Question 25
Three workers paint a fence in 4 days. If one more worker joins (same speed), how many days will it take? What assumptions are needed?
📌 ✅ Answer:
🔹 Assume all workers work equal hours per day.
🔹 Assume each worker paints at same constant rate.
🔹 Total work = 3 workers × 4 days = 12 worker-days
🔹 With 4 workers, days needed = 12/4 = 3
🔹 Final = 3 days
🔸 Assumptions: equal efficiency, same daily work time, no delays.
🔒 ❓ Question 26
A small pump fills a tank in 3 hours, a large pump fills it in 2 hours. If both work together, how long to fill the tank?
📌 ✅ Answer:
🔹 Small pump rate = 1/3 tank per hour
🔹 Large pump rate = 1/2 tank per hour
🔹 Combined rate = 1/3 + 1/2
🔹 Combined rate = (2/6) + (3/6) = 5/6 tank per hour
🔹 Time = 1 ÷ (5/6)
🔹 Time = 6/5 hour
🔹 Time = 1.2 hours
🔹 Time = 1 hour 12 minutes
🔹 Final = 1 hour 12 minutes
🔒 ❓ Question 27
From a dot plot, the values are: 14(1 time), 15(2 times), 16(3 times), 17(4 times), 18(3 times), 19(2 times), 20(1 time).
(i) Find the mean.
(ii) Is the mean equal to 17? Explain.
📌 ✅ Answer:
🔹 Write total count: 1+2+3+4+3+2+1 = 16
🔹 Compute weighted sum:
🔸 14×1 = 14
🔸 15×2 = 30
🔸 16×3 = 48
🔸 17×4 = 68
🔸 18×3 = 54
🔸 19×2 = 38
🔸 20×1 = 20
🔹 Total sum = 14+30+48+68+54+38+20 = 272
🔹 Mean = 272/16
🔹 Mean = 17
🔹 Final: mean = 17, so yes it equals 17
🔸 Reason: distribution is symmetric around 17.
🔒 ❓ Question 28
A table shows iodised salt price (₹) in two states:
Year: 2016, 2020, 2025
State A: 12.0, 15.5, 20.0
State B: 10.0, 14.0, 19.0
(i) Which state shows a larger total increase (2016 to 2025)?
(ii) Which shows faster increase from 2020 to 2025? Show calculation.
📌 ✅ Answer:
🔹 (i) Total increase A = 20.0 − 12.0 = 8.0
🔹 (i) Total increase B = 19.0 − 10.0 = 9.0
🔹 So larger total increase = State B
🔹 (ii) Increase A (2020→2025) = 20.0 − 15.5 = 4.5
🔹 (ii) Increase B (2020→2025) = 19.0 − 14.0 = 5.0
🔹 Faster increase in 2020→2025 = State B
🔹 Final: (i) B, (ii) B
🔒 ❓ Question 29
Sunrise–Sunset reasoning:
City P sunrise in January ≈ 7:05 am and sunset ≈ 5:20 pm.
City Q sunrise in January ≈ 6:35 am and sunset ≈ 5:55 pm.
(i) Which has longer day length?
(ii) Find approximate day length for both.
📌 ✅ Answer:
🔹 (ii) City P day length = 5:20 pm − 7:05 am
🔹 Convert: 5:20 pm = 17:20, 7:05 am = 07:05
🔹 Day length P = 17:20 − 07:05 = 10:15 (10 h 15 min)
🔹 City Q day length = 5:55 pm − 6:35 am
🔹 Convert: 5:55 pm = 17:55, 6:35 am = 06:35
🔹 Day length Q = 17:55 − 06:35 = 11:20 (11 h 20 min)
🔹 (i) Longer day length = City Q
🔹 Final: City Q longer; P = 10 h 15 min, Q = 11 h 20 min
🔒 ❓ Question 30
Moonrise–Moonset pattern (conceptual):
Over a month, moonrise time shifts later each day by about 50 minutes on average.
(i) If moonrise is 6:00 pm on Day 1, estimate moonrise on Day 6.
(ii) What pattern would you expect for moonset times? Explain.
📌 ✅ Answer:
🔹 (i) Shift per day ≈ 50 minutes
🔹 From Day 1 to Day 6 is 5 shifts
🔹 Total shift = 5 × 50 = 250 minutes
🔹 250 minutes = 4 hours 10 minutes
🔹 Moonrise on Day 6 ≈ 6:00 pm + 4:10
🔹 Moonrise on Day 6 ≈ 10:10 pm
🔹 (ii) Moonset will also shift later (generally), because the whole Moon schedule moves later each day.
🔹 The moon’s position relative to Earth–Sun changes slowly, so rise and set times both drift forward.
🔸 Final: Day 6 moonrise ≈ 10:10 pm; moonset times also tend to become later day by day.
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