Proportional Reasoning - 2 | FIO

Question 8

Prepare a pie chart that shows the favourite subjects of the students in your class. You can collect the data of the number of students for each subject shown in the table (each student should choose only one subject). Then write these numbers in the table and construct a pie chart:

Solution

Understand the Question

A pie chart (circle graph) represents data divided into sectors, where the size of each sector is proportional to the fraction of the total it represents.
The total central angle of a full circle is $360^\circ$ .
To find the central angle for each category: $\text{Central Angle} = \left(\dfrac{\text{Frequency of Category}}{\text{Total Frequency}}\right) \times 360^\circ$
We first assume/collect the data for each subject, find the total number of students, calculate each central angle, and then draw the pie chart.

Step 1 · Tabulate Data and Find Total Students

Let the assumed number of students choosing each subject be:

Language: $4$
Arts Education: $6$
Vocational Education: $9$
Social Science: $3$
Physical Education: $10$
Maths: $12$
Science: $16$

Calculate total number of students:

\begin{aligned} \text{Total students} &= 4 + 6 + 9 + 3 + 10 + 12 + 16 \\ &= 60 \end{aligned}

Step 2 · Calculate Central Angles

Calculate the central angle for each subject using $\text{Central Angle} = \left(\dfrac{\text{Students in Subject}}{\text{Total Students}}\right) \times 360^\circ$ :

Language:

\begin{aligned} \text{Angle for Language} &= \left(\dfrac{4}{60}\right) \times 360^\circ \\[0.6em] &= \left(\dfrac{1}{15}\right) \times 360^\circ \\[0.6em] &= 24^\circ \end{aligned}

Arts Education:

\begin{aligned} \text{Angle for Arts Education} &= \left(\dfrac{6}{60}\right) \times 360^\circ \\[0.6em] &= \left(\dfrac{1}{10}\right) \times 360^\circ \\[0.6em] &= 36^\circ \end{aligned}

Vocational Education:

\begin{aligned} \text{Angle for Vocational Education} &= \left(\dfrac{9}{60}\right) \times 360^\circ \\[0.6em] &= \left(\dfrac{3}{20}\right) \times 360^\circ \\[0.6em] &= 54^\circ \end{aligned}

Social Science:

\begin{aligned} \text{Angle for Social Science} &= \left(\dfrac{3}{60}\right) \times 360^\circ \\[0.6em] &= \left(\dfrac{1}{20}\right) \times 360^\circ \\[0.6em] &= 18^\circ \end{aligned}

Physical Education:

\begin{aligned} \text{Angle for Physical Education} &= \left(\dfrac{10}{60}\right) \times 360^\circ \\[0.6em] &= \left(\dfrac{1}{6}\right) \times 360^\circ \\[0.6em] &= 60^\circ \end{aligned}

Maths:

\begin{aligned} \text{Angle for Maths} &= \left(\dfrac{12}{60}\right) \times 360^\circ \\[0.6em] &= \left(\dfrac{1}{5}\right) \times 360^\circ \\[0.6em] &= 72^\circ \end{aligned}

Science:

\begin{aligned} \text{Angle for Science} &= \left(\dfrac{16}{60}\right) \times 360^\circ \\[0.6em] &= \left(\dfrac{4}{15}\right) \times 360^\circ \\[0.6em] &= 96^\circ \end{aligned}

Step 3 · Construct the Pie Chart

Draw a circle of any convenient radius using a compass and mark its center.
Draw a starting radius.
Using a protractor, draw adjacent sectors with angles: $24^\circ$ , $36^\circ$ , $54^\circ$ , $18^\circ$ , $60^\circ$ , $72^\circ$ , and $96^\circ$ .
Label each sector with its corresponding subject name.

Answer

The central angles for the pie chart are:

Language: $24^\circ$
Arts Education: $36^\circ$
Vocational Education: $54^\circ$
Social Science: $18^\circ$
Physical Education: $60^\circ$
Maths: $72^\circ$
Science: $96^\circ$

Common Mistakes

Sum of Angles Check: The sum of all central angles must always equal $360^\circ$ ( $24^\circ + 36^\circ + 54^\circ + 18^\circ + 60^\circ + 72^\circ + 96^\circ = 360^\circ$ ). If it does not, recheck your addition or arithmetic.
Incorrect Total: Dividing each frequency by $100$ instead of the total number of students ( $60$ ). The divisor must always be the total frequency.

More questions in FIO

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Construct a triangle with sidelengths in the ratio $3 : 4 : 5$ . Will all the triangles drawn with this ratio of sidelengths be congruent to each other? Why or why not?

Can you construct a triangle with sidelengths in the ratio $1 : 3 : 5$ ? Why or why not?

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Which of these are in inverse proportion?

Q10

Fill in the empty cells if $x$ and $y$ are in inverse proportion.

Q11

Which of the following pairs of quantities are in inverse proportion?

(i) The number of taps filling a water tank and the time taken to fill it.

(ii) The number of painters hired and the days needed to paint a wall of fixed size.

(iii) The distance a car can travel and the amount of petrol in the tank.

(iv) The speed of a cyclist and the time taken to cover a fixed route.

(v) The length of cloth bought and the price paid at a fixed rate per metre.

(vi) The number of pages in a book and the time required to read it at a fixed reading speed.

Q12

If 24 pencils cost ₹120, how much will 20 such pencils cost?

Q13

A tank on a building has enough water to supply 20 families living there for 6 days. If 10 more families move in there, how long will the water last? What assumptions do you need to make to work out this problem?

Q14

Fill in the average number of hours each living being sleeps in a day by looking at the charts. Select the appropriate hours from this list: 15, 2.5, 20, 8, 3.5, 13, 10.5, 18.

Q15

The pie chart on the right shows the result of a survey carried out to find the modes of transport used by children to go to school. Study the pie chart and answer the following questions.

(i) What is the most common mode of transport?

(ii) What fraction of children travel by car?

(iii) If 18 children travel by car, how many children took part in the survey? How many children use taxis to travel to school?

(iv) By which two modes of transport are equal numbers of children travelling?

Q16

Three workers can paint a fence in 4 days. If one more worker joins the team, how many days will it take them to finish the work? What are the assumptions you need to make?

Q17

It takes 6 hours to fill 2 tanks of the same size with a pump. How long will it take to fill 5 such tanks with the same pump?

Q18

A given set of chairs are arranged in 25 rows, with 12 chairs in each row. If the chairs are rearranged with 20 chairs in each row, how many rows does this new arrangement have?

Q19

A school has 8 periods a day, each of 45 minutes duration. How long is each period, if the school has 9 periods a day, assuming that the number of school hours per day stays the same?

Q20

A small pump can fill a tank in 3 hours, while a large pump can fill the same tank in 2 hours. If both pumps are used together, how long will the tank take to fill?

Q21

A factory requires 42 machines to produce a given number of toys in 63 days. How many machines are required to produce the same number of toys in 54 days?

Q22

A car takes 2 hours to reach a destination, travelling at a speed of $60 \text{ km/h}$ . How long will the car take if it travels at a speed of $80 \text{ km/h}$ ?