Proportional Reasoning - 2 | FIO

Question 15

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?

Solution

Understand the Question

A pie chart represents a whole divided into sectors where the sum of all central angles is $360^\circ$ .
The fraction of children using any transport mode is given by: $\text{Fraction} = \dfrac{\text{Central Angle}}{360^\circ}$
Modes with larger angles represent more children, and equal angles represent equal numbers of children.

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

Step 1 · Find angle for Car and identify largest sector

Diagram 1

The sum of all central angles in a pie chart is $360^\circ$ .

\begin{aligned} A_{\text{Walk}} + A_{\text{Bus}} + A_{\text{Cycle}} + A_{\text{Two-wheeler}} + A_{\text{Car}} &= 360^\circ \\[0.6em] 90^\circ + 120^\circ + 60^\circ + 60^\circ + A_{\text{Car}} &= 360^\circ \\[0.6em] 330^\circ + A_{\text{Car}} &= 360^\circ \\[0.6em] A_{\text{Car}} &= 360^\circ - 330^\circ = 30^\circ \end{aligned}

Comparing the central angles:

Bus: $120^\circ$
Walk: $90^\circ$
Cycle: $60^\circ$
Two-wheeler: $60^\circ$
Car: $30^\circ$

The largest central angle is $120^\circ$ , which corresponds to the Bus.

Answer

(i) Bus

(ii) What fraction of children travel by car?

Step 1 · Calculate fraction of children travelling by car

\begin{aligned} \text{Fraction by car} &= \dfrac{\text{Angle for Car}}{\text{Total angle}} \\[0.6em] &= \dfrac{30^\circ}{360^\circ} \\[0.6em] &= \dfrac{1}{12} \end{aligned}

Answer

(ii) $\dfrac{1}{12}$

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

Step 1 · Calculate total number of children

Let $x$ be the total number of children in the survey.

\begin{aligned} \dfrac{1}{12} \times x &= 18 \\[0.6em] x &= 18 \times 12 \\[0.6em] x &= 216 \end{aligned}

Step 2 · Find number of children using taxis

Since there is no sector for 'Taxi' in the pie chart, the number of children using taxis is $0$ .

Answer

(iii) Total children $= 216$ ; Children using taxis $= 0$

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

Step 1 · Identify modes with equal central angles

Both Cycle and Two-wheeler have the same central angle of $60^\circ$ , so an equal number of children travel by these two modes.

Answer

(iv) Cycle and Two-wheeler

Common Mistakes

Angle Base Error: Calculating the fraction out of $100$ instead of the total central angle of $360^\circ$ .
Missing Sector: Forgetting to calculate the central angle for 'Car' before evaluating its fraction or total count.
Unlisted Category: Assuming a non-zero count for 'Taxi', even though it is not present in the survey data.

More questions in FIO

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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

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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 $80 \text{ km/h}$ ?