Class 8 Mathematics

Chapter 12 Solutions — Tales by Dots and Lines

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Solutions

Overview

Step-by-step NCERT solutions for Tales by Dots and Lines (Chapter 12, NCERT Class 8 Mathematics) — every question and answer worked out in full, not just the final result. You can also read the Tales by Dots and Lines textbook chapter.

Solved

What these solutions cover

All 28 questions in Tales by Dots and Lines are solved in the PDF. Here's what's inside, exercise by exercise:

The Balancing Act (Mean and Median)

1.
Find the mean of the following data and share your observations:
- (i) The first 50 natural numbers.
- (ii) The first 50 odd numbers.
- (iii) The first 50 multiples of 4.
The dot plot below shows a collection of data and its average; but one dot is missing. Mark the missing value so that the mean is 9.
Sudhakar, the class teacher, asks Shreyas to measure the heights of all 24 students in his class and calculate the average height. Shreyas informs the teacher that the average height is 150.2 cm. Sudhakar discovers that the students were wearing uniform shoes when the measurements were taken and the shoes add 1 cm to the height. (i) Should the teacher get all the heights measured again without…
The three dot plots below show the lengths, in minutes, of songs of different albums. Which of these has a mean of 5.57 minutes? Explain how you arrived at the answer.
5.
Find the median of 8, 10, 19, 23, 26, 34, 40, 41, 41, 48, 51, 55, 70, 84, 91, 92.
- (i) If we include one value to the data (in the given list) without affecting the median, what could that value be?
- (ii) If we include two values to the data without affecting the median what could the two values be?
- (iii) If we remove one value from the data without affecting the median what could the value be?
6.
Examine the statements below and justify if the statement is always true, sometimes true, or never true.
- (i) Removing a value less than the median will decrease the median.
- (ii) Including a value less than the mean will decrease the mean.
- (iii) Including any 4 values will not affect the median.
- (iv) Including 4 values less than the median will increase the median.
The mean of the numbers 8, 13, 10, 4, 5, 20, y, 10 is 10.375. Find the value of y.
The mean of a set of data with 15 values is 134. Find the sum of the data.
9.
Consider the data: 12, 47, 8, 73, 18, 35, 39, 8, 29, 25, p. Which of the following number(s) could be p if the median of this data is 29?
- (i) 10
- (ii) 25
- (iii) 40
- (iv) 100
- (v) 29
- (vi) 47
- (vii) 30
10.
The number of times students rode their cycles in a week is shown in the dot plot below. Four students rode their cycles twice in that week.
- (i) Find the average number of times students rode their cycles.
- (ii) Find the median number of times students rode their cycles.
- (iii) Which of the following statements are valid? Why?
- (a) Everyone used their cycle at least once.
- (b) Almost everyone used…
A dart-throwing competition was organised in a school. The number of throws participants took to hit the bull's eye (the centre circle) is given in the table below. Describe the data using its minimum, maximum, mean and median. No. of trials: 1 2 3 4 5 6 7 8 9 10 No. of students: 1 0 0 1 4 9 12 15 10 10

Visualising and Interpreting Data (Line Graphs)

The average number of customers visiting a shop and the average number of customers actually purchasing items over different days of the week is shown in the table below. Visualise this data on a line graph. Day: Mon Tue Wed Thu Fri Sat Sun Visiting: 16 19 10 14 20 22 35 Purchasing: 10 8 7 11 12 16 26
2.
The average number of days of rainfall in each month for a few cities is shown in the table below.
- (i) What could be the possible method to compile this data?
- (ii) Mark the data for Mangaluru, Port Blair, and Rameswaram in the line graph shown. You can round off values to the nearest integer.
- (iii) Based on the line for New Delhi in the graph fill the data in the table.
- (iv) Which city among…
3.
The following line graph shows the number of births in every month in India over a time period.
- (i) What are your observations?
- (ii) What was the approximate number of births in July 2017?
- (iii) What time period does the graph capture?
- (iv) Compare the number of births in the month of January in the years 2018, 2019, and 2020.
- (v) Estimate the number of births in the year 2019.
4.
Share your observations on the infographic (Wheat vs Rice preference in Indian states). Based on this infographic, answer the following:
- (i) The value of Karnataka is hidden. Can you guess what it could be?
- (ii) Which are the top 5 states where rice is the most popular?
- (iii) Which are the top 5 states where wheat is the most popular?
- (iv) List a few states where the preference between rice and…

Data Story and Further Practice (Mean, Median, Line Graphs)

1.
Mean Grids:
- (i) Fill the grid with 9 distinct numbers such that the average along each row, column, and diagonal is 10.
- (ii) Can we fill the grid by changing a few numbers and still get 10 as the average in all directions?
2.
Give two examples of data that satisfy each of the following conditions:
- (i) 3 numbers whose mean is 8.
- (ii) 4 numbers whose median is 15.5.
- (iii) 5 numbers whose mean is 13.6.
- (iv) 6 numbers whose mean = median.
- (v) 6 numbers whose mean > median.
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?
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?
5.
Check whether each of the statements below is true. Justify your reasoning. Use algebra, if necessary, to justify.
- (i) The average of two even numbers is even.
- (ii) The average of any two multiples of 5 will be a multiple of 5.
- (iii) The average of any 5 multiples of 5 will also be 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?
- (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…
Is 17 the average of the data shown in the dot plot below? Share the method you used to answer this question.
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?
9.
The following table shows the retail price (in ₹) of iodised salt in the month of January in a few states over 10 years (2016–2025).
- (i) Choose data from any 3 states you find interesting and present it through a line graph using an appropriate scale.
- (ii) What do you find interesting in this data? Share your observations.
- (iii) Compare the price variation in Gujarat and Uttar Pradesh.
- (iv) In…
10.
Referring to the graph showing primary source of energy for household lighting over time, which of the following statements are valid?
- (i) In 1983, the majority in rural areas used kerosene while the majority in urban areas used electricity.
- (ii) The use of kerosene as a primary lighting source has decreased over time in both rural and urban areas.
- (iii) In the year 2000, 10% of the urban…
11.
Answer the following questions based on the line graph showing average daily time spent on hobbies and games by urban and rural kids.
- (i) How long do children aged 10 in urban areas spend each day on hobbies and games?
- (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.
- (iii) Are the…
12.
The following graphs show the sunrise and sunset times across the year at 4 locations in India (Kibithu, Ghuar Moti, Srinagar, Kanyakumari).
- (i) At which place does the sun rise the earliest in January? What is the approximate day length at this place in January?
- (ii) Which place has the longest day length over the year?
- (iii) Share your observations.
13.
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.
- (ii) What do you notice? What do you wonder?

Sample

Worked examples

3 questions from Tales by Dots and Lines, solved step by step.

The Balancing Act (Mean and Median)

Find the mean of the following data and share your observations: (i) The first 50 natural numbers. (ii) The first 50 odd numbers. (iii) The first 50 multiples of 4.

(i) First 50 natural numbers: 1, 2, 3, …, 50.

Sum = 50 × 51 / 2 = 1275.

Mean = 1275 / 50 = 25.5.

(ii) First 50 odd numbers: 1, 3, 5, …, 99.

Sum = 50² = 2500.

Mean = 2500 / 50 = 50.

Observation: the mean equals the 25th and 26th odd numbers (49 and 51), i.e., the middle of the sequence. It also equals n = 50, the count of terms.

(iii) First 50 multiples of 4: 4, 8, 12, …, 200.

Sum = 4 × (1 + 2 + … + 50) = 4 × 1275 = 5100.

Mean = 5100 / 50 = 102.

Observation across all three: in each case the mean equals the average of the first and last term, confirming that for evenly-spaced (arithmetic) sequences the mean lies exactly at the midpoint of the sequence.

Answer: (i) 25.5 (ii) 50 (iii) 102

Data Story and Further Practice (Mean, Median, Line Graphs)

Is 17 the average of the data shown in the dot plot below? Share the method you used to answer this question.

Read the dot plot. Values and their frequencies (from the dot plot image):

14: 2, 15: 2, 16: 3, 17: 5, 18: 4, 19: 4, 20: 3, 21: 1, 23: 1.

Total n = 2+2+3+5+4+4+3+1+1 = 25.

Method 1 — Balance / distance method:

Compute total distance from 17 on each side.

LHS (values < 17): 14 is 3 below (2 dots, total 6), 15 is 2 below (2 dots, total 4), 16 is 1 below (3 dots, total 3). LHS total = 13.

RHS (values > 17): 18 is 1 above (4 dots, total 4), 19 is 2 above (4 dots, total 8), 20 is 3 above (3 dots, total 9), 21 is 4 above (1 dot, total 4), 23 is 6 above (1 dot, total 6). RHS total = 31.

LHS (13) ≠ RHS (31). Since RHS > LHS, the true mean is above 17.

Method 2 — Direct calculation:

Sum = 14×2 + 15×2 + 16×3 + 17×5 + 18×4 + 19×4 + 20×3 + 21×1 + 23×1

= 28 + 30 + 48 + 85 + 72 + 76 + 60 + 21 + 23 = 443.

Mean = 443 / 25 = 17.72.

17.72 ≠ 17. Therefore, 17 is NOT the average of this data.

Answer: No, 17 is not the average. The actual mean = 443/25 = 17.72. The right-side distances from 17 (total 31) exceed the left-side distances (total 13), confirming the mean is above 17.

The Balancing Act (Mean and Median)

The dot plot below shows a collection of data and its average; but one dot is missing. Mark the missing value so that the mean is 9.

From the dot plot the existing 10 values are: 4, 7, 8, 8, 9, 9, 9, 9, 9, 11 (one dot at 4, one at 7, two at 8, five at 9, one at 11).

Sum of existing values = 4 + 7 + 8 + 8 + 9 + 9 + 9 + 9 + 9 + 11 = 83.

For the mean of all 11 values (10 existing + 1 missing) to be 9:

Required total sum = 9 × 11 = 99.

Missing value = 99 − 83 = 16.

Verify: (4 + 7 + 8 + 8 + 9 + 9 + 9 + 9 + 9 + 11 + 16) / 11 = 99 / 11 = 9. ✓

Place the missing dot at 16 on the dot plot.

Answer: The missing value is 16.

All 28 questions in Tales by Dots and Lines are solved step by step in the Solutions PDF above.

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Chapter 12 Solutions — Tales by Dots and Lines

Overview

What these solutions cover

The Balancing Act (Mean and Median)

Visualising and Interpreting Data (Line Graphs)

Data Story and Further Practice (Mean, Median, Line Graphs)

Worked examples

Find the mean of the following data and share your observations: (i) The first 50 natural numbers. (ii) The first 50 odd numbers. (iii) The first 50 multiples of 4.

Is 17 the average of the data shown in the dot plot below? Share the method you used to answer this question.

The dot plot below shows a collection of data and its average; but one dot is missing. Mark the missing value so that the mean is 9.

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