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12
NCERT Solutions · Maths Ch 12
Tales by Dots and Lines
29 questions · 3 exercises
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Ex 12.1
Ex 12.2
Ex 12.3
NCERT answers with full exam-style working, step by step. Tap a question to open its solution.
1
EXERCISE 12.1 · 11 QUESTIONS
Mean and median (Balancing act)
Q1
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.
(5 marks)
Q2
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. (The plotted values are 4, 7, 8, 8, 9, 9, 9, 9, 9 and 11.)
(3 marks)
Q3
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 shoes to find the correct average height? Or is there a simpler way? (ii) What is the correct average height of the class? (a) 174.2 cm (b) 126.2 cm (c) 150.2 cm (d) 149.2 cm (e) 151.2 cm (f) None of the above (g) Insufficient information
(3 marks)
Q4
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.
(3 marks)
Q5
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 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?
(5 marks)
Q6
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.
(5 marks)
Q7
The mean of the numbers 8, 13, 10, 4, 5, 20, y, 10 is 10.375. Find the value of y.
(3 marks)
Q8
The mean of a set of data with 15 values is 134. Find the sum of the data.
(2 marks)
Q9
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
(3 marks)
Q10
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 their cycle a few times. (c) There are some students who cycled more than once on some days. (d) Exactly 5 students have used their cycles more than once on some days. (e) The following week, if all of them cycled 1 more time than they did the previous week, what would be the average and median of the next week's data?
(5 marks)
Q11
A dart-throwing competition was organised in a school. The number of throws participants took to hit the bull's eye 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
(3 marks)
2
EXERCISE 12.2 · 3 QUESTIONS
Line graphs
Q1
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
(3 marks)
Q2
The average number of days of rainfall in each month for a few cities is shown in the table below (Mangaluru, New Delhi [blank], Port Blair, Rameswaram). (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 below. You can round off the 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 these receives the most number of days of rainfall per year? Which city gets the least number of days of rainfall per year? (v) Looking at the table, when is the rainy season in New Delhi and Rameswaram?
(5 marks)
Q2b
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.
(3 marks)
3
EXERCISE 12.3 · 15 QUESTIONS
More on data
Q1
Mean Grids: (i) Fill the 3 × 3 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?
(5 marks)
Q2
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.
(5 marks)
Q3
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?
(5 marks)
Q4
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?
(3 marks)
Q5
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.
(5 marks)
Q6
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. (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. (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.
(5 marks)
Q7
Is 17 the average of the data shown in the dot plot below? Share the method you used to answer this question.
(5 marks)
Q8
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?
(3 marks)
Q9
The following table shows the retail price (in ₹) of iodised salt in the month of January in a few states over 10 years (2016 to 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 which state has the price increased the most from 2016 to 2025? (v) What are you curious to explore further?
(5 marks)
Q10
Referring to the graph below (primary source of energy for household lighting, rural and urban, 1983 to 2023), 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. (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 households used electricity as a primary lighting source. (iv) In 2023, there were no power cuts.
(3 marks)
Q11
Answer the following questions based on the line graph (average daily time spent on hobbies and games, by age, urban and rural). (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 following statements correct? (a) The average time spent daily on hobbies and games by kids aged 15 is twice that of kids aged 10. (b) All rural kids aged 15 spend at least 1 hour on hobbies and games everyday.
(3 marks)
Q12
Individual project: Make your own activity strip for different days of the week. (i) Do you eat and sleep at regular times every day? Typically how long do you spend outdoors? (ii) Calculate the average time spent per activity. Represent this average day using a strip. (iii) Similarly, track the activities of any adult at home. Compare your data with theirs.
(3 marks)
Q13
Small group project: Make a group of 3 – 4 members. Do at least one of the following: (i) Track daily sleep time of all your family members for a week. (a) Represent this on strips. (b) Put together the data of all your group members. Calculate the average and median sleep time of children, adults, elderly. (c) Share your findings and observations. (ii) When do schools start and end? Collect information on the daily timings of different schools for Grade 8, including class time and break time. Analyse and present the data collected.
(3 marks)
Q14
The following graphs show the sunrise and sunset times across the year at 4 locations in India (Kibithu, Ghuar Moti, Srinagar and Kanyakumari). Observe how the graphs are organised. Are you able to identify which lines indicate the sunrise and which indicate the sunset? Answer the following questions based on the graphs: (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 — what do you find interesting? What are you curious to find out?
(3 marks)
Q15
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?
(3 marks)