ICSE Class 6 Mathematics · Solutions for Class 6 RS Aggarwal ICSE Foundation Mathematics · 33 solved questions
The number of cars sold at a showroom during a week are given below:
| Day | Number of cars sold |
|---|---|
| Monday | 15 |
| Tuesday | 6 |
| Wednesday | 9 |
| Thursday | 3 |
| Friday | 18 |
| Saturday | 12 |
Choose an appropriate scale and draw a pictograph to represent the given data.
The number of symbols required for each day is :
| Day | Number of cars sold | Number of symbols |
|---|---|---|
| Monday | 15 | 5 |
| Tuesday | 6 | 2 |
| Wednesday | 9 | 3 |
| Thursday | 3 | 1 |
| Friday | 18 | 6 |
| Saturday | 12 | 4 |
The required pictograph is given below:
In a class test, the number of students passed in various subjects are given below:
| Subject | Number of students passed |
|---|---|
| English | 15 |
| Hindi | 10 |
| Mathematics | 5 |
| Social Science | 25 |
| Science | 20 |
Draw a pictograph to represent the above data.
The number of symbols required for each subject is :
| Subject | Number of students passed | Number of symbols |
|---|---|---|
| English | 15 | 3 |
| Hindi | 10 | 2 |
| Mathematics | 5 | 1 |
| Social Science | 25 | 5 |
| Science | 20 | 4 |
The required pictograph is given below:
The number of trees of various kinds in a fruit orchard are given below:
| Tree | Number of trees |
|---|---|
| Apple | 56 |
| Banana | 40 |
| Peach | 16 |
| Apricot | 32 |
| Plum | 24 |
Represent the above data by a pictograph.
The number of symbols required for each kind of tree is :
| Tree | Number of trees | Number of symbols |
|---|---|---|
| Apple | 56 | 7 |
| Banana | 40 | 5 |
| Peach | 16 | 2 |
| Apricot | 32 | 4 |
| Plum | 24 | 3 |
The required pictograph is given below:
The number of chairs in five classrooms of a school are given below:
| Classroom | Number of chairs |
|---|---|
| I | 30 |
| II | 40 |
| III | 60 |
| IV | 50 |
| V | 20 |
Represent the above data by a pictograph.
The number of symbols required for each classroom is :
| Classroom | Number of chairs | Number of symbols |
|---|---|---|
| I | 30 | 3 |
| II | 40 | 4 |
| III | 60 | 6 |
| IV | 50 | 5 |
| V | 20 | 2 |
The required pictograph is given below:
The following pictograph shows the number of computers sold by a showroom during a week.
Look at the above pictograph and answer the questions given below:
(i) On which day the minimum number of computers were sold?
(ii) How many computers were sold on Thursday?
(iii) On which day the number of computers sold was one-third of that sold on Tuesday?
(iv) How many computers were sold in all during the week?
(v) What is the difference between the maximum and minimum number of computers sold in a day?
From the pictograph, the number of computer symbols against each day is :
Monday = 3, Tuesday = 6, Wednesday = 4, Thursday = 5, Friday = 2 and Saturday = 4.
Since one symbol represents 6 computers,
(i) The least number of symbols is against Friday, i.e. 2 symbols.
∴ Computers sold on Friday = 2 × 6 = 12, which is the minimum.
Hence, the minimum number of computers were sold on Friday.
(ii) Number of computers sold on Thursday = 5 × 6
= 30
Hence, 30 computers were sold on Thursday.
(iii) Number of computers sold on Tuesday = 6 × 6
= 36
One-third of computers sold on Tuesday = $\frac{1}{3} \times 36$
= 12
Number of computers sold on Friday = 2 × 6
= 12
Hence, on Friday the number of computers sold was one-third of that sold on Tuesday.
(iv) Total number of computers sold during the week = (3 + 6 + 4 + 5 + 2 + 4) × 6
= 24 × 6
= 144
Hence, 144 computers were sold in all during the week.
(v) Maximum number of computers sold in a day (i.e. on Tuesday) = 6 × 6
= 36
Minimum number of computers sold in a day (i.e. on Friday) = 2 × 6
= 12
∴ Required difference = 36 − 12
= 24
Hence, the difference between the maximum and minimum number of computers sold in a day is 24.
The following table shows the favourite sports of 250 students in a school.
| Sports | Number of students |
|---|---|
| Cricket | 75 |
| Football | 35 |
| Tennis | 50 |
| Badminton | 25 |
| Swimming | 65 |
Represent the above data by a column graph.
Steps:
The line OX is taken horizontal and is called the x-axis, whereas the line OY is taken vertical and is called the y-axis.
On the x-axis, starting from O, mark points at equal distances. At these points write the different sports given. In this question, the sports are Cricket, Football, Tennis, Badminton and Swimming.
Along y-axis, mark the heights of the bars (rectangles) in proportion to the given data (number of students). Along the y-axis, take 1 cm to represent 10 students.
The required column graph is as given below:
The birth rate per thousand of five countries over a period of time is shown below:
| Country | Birth rate per thousand |
|---|---|
| China | 42 |
| India | 35 |
| Germany | 14 |
| U.K. | 28 |
| Sweden | 21 |
Represent the above data by a bar graph.
Steps:
The line OX is taken horizontal and is called the x-axis, whereas the line OY is taken vertical and is called the y-axis.
On the x-axis, starting from O, mark points at equal distances. At these points write the different countries given. In this question, the countries are China, India, Germany, U.K. and Sweden.
Along y-axis, mark the heights of the bars (rectangles) in proportion to the given data (birth rate per thousand). Along the y-axis, take 1 cm to represent 7 infants (i.e. a birth rate of 7 per thousand).
The required bar graph is as given below:
The marks obtained by a student in his annual examinations in various subjects are given below:
| Subject | Marks obtained |
|---|---|
| Hindi | 75 |
| Maths | 40 |
| English | 50 |
| Science | 85 |
| S.Studies | 35 |
Represent the above data by a bar graph.
Steps:
The line OX is taken horizontal and is called the x-axis, whereas the line OY is taken vertical and is called the y-axis.
On the x-axis, starting from O, mark points at equal distances. At these points write the different subjects given. In this question, the subjects are Hindi, Maths, English, Science and S.Studies.
Along y-axis, mark the heights of the bars (rectangles) in proportion to the given data (marks obtained). Along the y-axis, take 1 cm to represent 10 marks.
The required bar graph is as given below:
The table given below depicts the monthly salary (in ₹) of six employees of a company:
| Employee | Monthly salary (₹) |
|---|---|
| A | 2000 |
| B | 3500 |
| C | 4000 |
| D | 3000 |
| E | 2500 |
| F | 4500 |
Represent the above data by a column graph.
Steps:
The line OX is taken horizontal and is called the x-axis, whereas the line OY is taken vertical and is called the y-axis.
On the x-axis, starting from O, mark points at equal distances. At these points write the different employees given. In this question, the employees are A, B, C, D, E and F.
Along y-axis, mark the heights of the bars (rectangles) in proportion to the given data (monthly salary in ₹). Along the y-axis, take 1 cm to represent a salary of ₹ 500.
The required column graph is as given below:
The table given below shows the monthly expenditure (in percentage) on various items, in a family.
| Item | Expenditure (in percentage) |
|---|---|
| Rent | 24% |
| Food | 36% |
| Education | 16% |
| Clothing | 20% |
| Misc. | 4% |
Represent the above data by a column graph.
Steps:
The line OX is taken horizontal and is called the x-axis, whereas the line OY is taken vertical and is called the y-axis.
On the x-axis, starting from O, mark points at equal distances. At these points write the different items given. In this question, the items are Rent, Food, Education, Clothing and Misc.
Along y-axis, mark the heights of the bars (rectangles) in proportion to the given data (expenditure in percentage). Along the y-axis, take 1 cm to represent 4%.
The required column graph is as given below:
Read the given column graph and answer the questions that follow:
(i) What information is represented by the column graph?
(ii) In which month was the rainfall maximum?
(iii) Which month was the driest?
(iv) What was the rainfall in March?
(v) Find the total amount of rainfall in all the six months taken together.
From the column graph, the rainfall in the six months is :
Jan. = 5 cm, Feb. = 2 cm, Mar. = 7 cm, Apr. = 10 cm, May = 8 cm and June = 15 cm.
(i) The column graph shows the rainfall (in cm) month-wise from January to June.
Hence, the column graph represents the rainfall (in cm) at a place during the first six months of a year.
(ii) The maximum rainfall is 15 cm, which is in June.
Hence, the rainfall was maximum in June.
(iii) The least rainfall is 2 cm, which is in February.
Hence, February was the driest month.
(iv) The height of the column against March is 7 cm.
Hence, the rainfall in March was 7 cm.
(v) Total amount of rainfall in the six months = (5 + 2 + 7 + 10 + 8 + 15) cm
= 47 cm
Hence, the total amount of rainfall in all the six months taken together is 47 cm.
Read the column graph given below:
Now, answer the following questions:
(i) What information is represented by the column graph?
(ii) What was the production of food grains in 2013?
(iii) In which year the production was maximum?
(iv) After which year there was a sudden fall in the production?
(v) Find the percentage increase in the production of food grains in the year 2015.
From the column graph, the production of food grains (in million tons) is :
2011 = 65, 2012 = 80, 2013 = 90, 2014 = 55 and 2015 = 110.
(i) The column graph shows the production of food grains (in million tons) year-wise from 2011 to 2015.
Hence, the column graph represents the production of food grains in an Indian state during the years 2011 to 2015.
(ii) The height of the column against 2013 is 90.
Hence, the production of food grains in 2013 was 90 million tons.
(iii) The maximum production is 110 million tons, which is in 2015.
Hence, the production was maximum in the year 2015.
(iv) The production fell from 90 million tons in 2013 to 55 million tons in 2014.
Hence, there was a sudden fall in the production after the year 2013.
(v) Production in 2014 = 55 million tons
Production in 2015 = 110 million tons
Increase in production = (110 − 55) million tons
= 55 million tons
∴ Percentage increase = $\frac{55}{55} \times 100$%
= 100%
Hence, the percentage increase in the production of food grains in the year 2015 is 100%.
In a survey of 130 families of a colony, the number of children in each family was recorded and the data has been represented by the bar graph, given below:
Read the bar graph carefully and answer the questions given below:
(i) What information does the bar graph give?
(ii) How many families have 2 children?
(iii) How many families have no child?
(iv) What percentage of families have 4 children?
From the bar graph, the number of families having 1, 2, 3 and 4 children are 20, 35, 60 and 15 respectively.
(i) The bar graph shows the number of families against the number of children in each family.
Hence, the bar graph gives the number of children in each of the 130 families of a colony.
(ii) The height of the bar against 2 children is 35.
Hence, 35 families have 2 children.
(iii) Number of families having one or more children = 20 + 35 + 60 + 15
= 130
∴ Number of families having no child = 130 − 130
= 0
Hence, there is no family with no child, i.e. 0 families.
(iv) Number of families having 4 children = 15
∴ Required percentage = $\frac{15}{130} \times 100$%
= $\frac{150}{13}$%
= $11\frac{7}{13}$%
Hence, $\bm{11\frac{7}{13}}$% of the families have 4 children.
The data represented through the pictures of objects is called
The representation of data through pictures of objects is called a pictograph.
Hence, option 3 is the correct option.
A graph drawn using vertical bars is called
A graph drawn using bars (rectangles) of equal width with equal spacing between them is called a bar graph.
Hence, option 2 is the correct option.
One symbol represents 65 pens.
The second figure has 6 such symbols.
∴ Number of pens represented = 6 × 65
= 390
Hence, option 3 is the correct option.
One symbol represents 18 balloons.
∴ Number of symbols required for 90 balloons = $\frac{90}{18}$
= 5
Hence, option 1 is the correct option.
6 symbols stand for 42 articles.
∴ 1 symbol stands for $\frac{42}{6}$ = 7 articles.
∴ 2 symbols stand for 2 × 7 = 14 articles.
Hence, option 2 is the correct option.
The sizes of shoes of 8 students of a class are given as 4, 6, 4, 5, 4, 5, 6, 5. If the shoe of size less than 5 is observed for girls, then how many boys are there in the list?
Total number of students = 8
Shoe sizes less than 5 are 4, 4 and 4, i.e. 3 students, which are observed for girls.
∴ Number of girls = 3
∴ Number of boys = 8 − 3
= 5
Hence, option 3 is the correct option.
Number of children in 10 families are given as 2, 1, 2, 1, 3, 3, 2, 3, 2, 1. The number of families having less than 3 children is
Families having less than 3 children are those having 1 or 2 children.
In the given data, 1 occurs 3 times and 2 occurs 4 times.
∴ Number of families having less than 3 children = 3 + 4
= 7
Hence, option 3 is the correct option.
A graph drawn using vertical bars is called
A graph drawn using vertical bars (rectangles) of equal width with equal spacing between them is called a bar graph.
Hence, option 1 is the correct option.
The given bar graph shows the savings of 5 friends. A, B, C, D and E. The ratio of the amounts saved by B and D is
From the bar graph,
Amount saved by B = ₹ 300
Amount saved by D = ₹ 500
∴ Required ratio = $\frac{300}{500}$
= $\frac{3}{5}$
= 3 : 5
Hence, option 3 is the correct option.
The following graph shows the marks obtained by a student in different subjects. If 35 marks are needed to pass in a subject, then the student failed in
From the bar graph, the marks obtained are :
Hindi = 30, English = 40, Maths = 50, S. Studies = 20 and Science = 40.
Since 35 marks are needed to pass in a subject, the student failed in the subjects in which he scored below 35 marks.
The marks in Hindi (30) and Social Studies (20) are below 35.
∴ The student failed in Hindi and Social Studies.
Hence, option 3 is the correct option.
Fill in the blanks:
(i) A representation using pictures is called a ............... .
(ii) A bar graph consists of bars of equal ............... .
(iii) A pictograph represents data through ............... .
(iv) A collection of some numbers to give some information is called ............... .
(v) ............... is the simplest way to represent data graphically.
(i) The representation of data through pictures is called a pictograph.
Hence, a representation using pictures is called a pictograph.
(ii) A bar graph is a representation of data by bars (rectangles) of uniform width.
Hence, a bar graph consists of bars of equal width.
(iii) In a pictograph, each observation is shown by drawing a proportional number of pictures.
Hence, a pictograph represents data through pictures.
(iv) A collection of numerical figures gathered to give some information is called data.
Hence, a collection of some numbers to give some information is called data.
(v) A bar graph is simple to draw, convenient to read and provides for easy comparison of the given data.
Hence, a bar graph is the simplest way to represent data graphically.
Write (T) for true and (F) for false for each of the following statements:
(i) A data is a collection of numbers gathered to give some information.
(ii) In a bar graph, the bars can be horizontal or vertical.
(iii) If $\bigoplus$ represents 10 doctors, then $\bigoplus$ $\bigoplus$ $\bigoplus$ represent 15 doctors.
(iv) If $\square$ $\square$ $\square$ represents 75 books, then $\square$ represents 25 books.
(i) Data is a collection of numbers (observations) gathered to give some information.
Hence, the given statement is true (T).
(ii) In a bar graph, the bars can be drawn either horizontally or vertically.
Hence, the given statement is true (T).
(iii) One symbol represents 10 doctors.
∴ Three symbols represent 3 × 10 = 30 doctors, not 15 doctors.
Hence, the given statement is false (F).
(iv) Three symbols represent 75 books.
∴ One symbol represents $\frac{75}{3}$ = 25 books.
Hence, the given statement is true (T).
The following pictograph depicts the number of TVs sold by an electronics shop during a certain week:
Look at the pictograph and answer the questions given below:
1. From the pictograph, the number of TV symbols against each day is :
Monday = $4\frac{1}{2}$, Tuesday = 4, Wednesday = $2\frac{1}{2}$, Thursday = 1, Friday = $5\frac{1}{2}$ and Saturday = 3.
The maximum number of symbols is against Friday, i.e. $5\frac{1}{2}$ symbols.
∴ Number of TVs sold on Friday = $5\frac{1}{2} \times 6$
= 33, which is the maximum.
Hence, option (c) is the correct option.
2. Maximum number of TVs sold in a day (i.e. on Friday) = $5\frac{1}{2} \times 6$
= 33
Minimum number of TVs sold in a day (i.e. on Thursday) = 1 × 6
= 6
∴ Required percentage = $\frac{33 - 6}{6} \times 100$%
= $\frac{27}{6} \times 100$%
= 450%
Hence, option (a) is the correct option.
3. Number of TVs sold on Monday = $4\frac{1}{2} \times 6$
= 27
Let the number of TVs sold on the required day be $y$.
Given,
$$\begin{aligned}\frac{27}{y} = \frac{3}{2} \\[1em] \Rightarrow 3y = 27 \times 2 \\[1em] \Rightarrow 3y = 54 \\[1em] \Rightarrow y = \frac{54}{3} \\[1em] \Rightarrow y = 18\end{aligned}$$
Number of TVs sold on Saturday = 3 × 6
= 18
∴ The required day is Saturday.
Hence, option (d) is the correct option.
4. Number of TVs sold on Tuesday = 4 × 6
= 24
Number of TVs sold on Wednesday = $2\frac{1}{2} \times 6$
= 15
Decline in sale = 24 − 15
= 9
∴ Percentage decline = $\frac{9}{24} \times 100$%
= 37.5%
Hence, option (b) is the correct option.
The following graph represents the production of fertilisers (in 10000 tonnes) by company X from 2014 to 2019.
Study the above information and answer the questions given below:
1. From the graph, the production of fertilisers (in 10000 tonnes) is :
2014 = 360, 2015 = 620, 2016 = 540, 2017 = 800, 2018 = 740 and 2019 = 900.
Highest production (in 2019) = 900
Lowest production (in 2014) = 360
∴ Required ratio = $\frac{900}{360}$
= $\frac{5}{2}$
= 5 : 2
Hence, option (a) is the correct option.
2. Production in 2014 = 360
Production in 2016 = 540
Increase in production = 540 − 360
= 180
∴ Percentage increase = $\frac{180}{360} \times 100$%
= 50%
Hence, option (b) is the correct option.
3. Production in 2017 = 800
Production in 2018 = 740
Decrease in production = 800 − 740
= 60
∴ Percentage decrease = $\frac{60}{800} \times 100$%
= 7.5%
Hence, option (c) is the correct option.
4. Sum of the productions of all the years = 360 + 620 + 540 + 800 + 740 + 900
= 3960
and, number of years = 6
∴ Average production = $\frac{3960}{6}$
= 660
The production was more than 660 in the years 2017 (800), 2018 (740) and 2019 (900).
∴ Number of such years = 3
Hence, option (a) is the correct option.
Assertion (A): In a bar graph, the gap between two consecutive bars may not be the same.
Reason (R): In a bar graph, each bar represents only one value of the numerical data.
In a bar graph, the gap (distance) between each pair of consecutive bars must be the same.
So, Assertion (A) is false.
In a bar graph, each bar represents only one value of the numerical data.
So, Reason (R) is true.
Hence, option 4 is the correct option.
Assertion (A): Pictograph and bar graphs are graphical representation of the numerical data.
Reason (R): In a bar graph, bars of uniform width are drawn vertically only.
Pictographs and bar graphs are both graphical representations of numerical data.
So, Assertion (A) is true.
In a bar graph, bars of uniform width can be drawn either horizontally or vertically.
So, Reason (R) is false.
Hence, option 3 is the correct option.
The students of a school washed cars every Sunday for a month to raise money for the flood affected people.
The students charged ₹100 for each car they washed. The pictograph below shows the total number of cars washed by the students.
What is the total amount of money raised by the students?
From the pictograph, the number of car symbols against the four Sundays are 3, 2, 2 and 5 respectively.
Since each symbol represents 15 cars,
Total number of cars washed = (3 + 2 + 2 + 5) × 15
= 12 × 15
= 180
Amount charged for each car = ₹ 100
∴ Total amount of money raised = 180 × ₹ 100
= ₹ 18000
Hence, option 2 is the correct option.
The sale of the number of notebooks on different days of a week is shown below. Study the pictograph carefully and answer the following question.
How many more notebooks were sold on Friday than on Wednesday?
From the pictograph, the number of notebook symbols against Friday and Wednesday are 7 and 4 respectively.
Since each symbol represents 4 notebooks,
More notebooks sold on Friday than on Wednesday = (7 − 4) × 4
= 3 × 4
= 12
Hence, option 2 is the correct option.
The sale of the number of notebooks on different days of a week is shown below. Study the pictograph carefully and answer the following question.
If one notebook was sold for ₹70, then what was the total earnings on Tuesday?
From the pictograph, the number of notebook symbols against Tuesday is 8.
∴ Number of notebooks sold on Tuesday = 8 × 4
= 32
Selling price of one notebook = ₹ 70
∴ Total earnings on Tuesday = 32 × ₹ 70
= ₹ 2,240
Hence, option 4 is the correct option.
One symbol represents 2 persons.
The given figure has $4\frac{1}{2}$ symbols.
∴ Number of persons represented = $4\frac{1}{2} \times 2$
= 9
Hence, option 1 is the correct option.
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