Proportional Reasoning - 2 | FIO

Question 14

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.

Solution

Understand the Question

A full circle represents a complete 24-hour day.
The shaded blue sector represents the portion of the day spent sleeping, while the remaining section represents waking hours.
The fraction $F$ of the day spent sleeping for a sleep duration of $S$ hours is: $F = \dfrac{S}{24}$
By calculating the corresponding fraction or sector size for each given sleep duration and comparing it with the visual pie charts, we can match each living being to its correct sleep time.

Step 1 · Calculate the sleep fractions of a 24-hour day

Diagram 1

Let $S$ be the sleep duration in hours and $F$ be the fraction of the 24-hour day spent sleeping: $F = \dfrac{S}{24}$

Calculating the fraction for each value in the given list:

For $2.5\text{ hours}$ : $F = \dfrac{2.5}{24} \approx 0.104$
For $3.5\text{ hours}$ : $F = \dfrac{3.5}{24} \approx 0.146$
For $8\text{ hours}$ : $F = \dfrac{8}{24} = \dfrac{1}{3} \approx 0.333$
For $10.5\text{ hours}$ : $F = \dfrac{10.5}{24} = \dfrac{7}{16} = 0.4375$
For $13\text{ hours}$ : $F = \dfrac{13}{24} \approx 0.542$
For $15\text{ hours}$ : $F = \dfrac{15}{24} = \dfrac{5}{8} = 0.625$
For $18\text{ hours}$ : $F = \dfrac{18}{24} = \dfrac{3}{4} = 0.75$
For $20\text{ hours}$ : $F = \dfrac{20}{24} = \dfrac{5}{6} \approx 0.833$

Step 2 · Match sleep hours to each living being

Comparing the blue sector of each circular chart with the calculated fractions in ascending order:

Giraffe: Smallest blue sector $\rightarrow \mathbf{2.5\text{ hours}}$ ( $0.104$ )
Elephant: Slightly larger than giraffe's sector $\rightarrow \mathbf{3.5\text{ hours}}$ ( $0.146$ )
Boy: Approximately one-third of the circle $\rightarrow \mathbf{8\text{ hours}}$ ( $0.333$ )
Dog: Less than half but more than one-third $\rightarrow \mathbf{10.5\text{ hours}}$ ( $0.4375$ )
Cat: Slightly more than half of the circle $\rightarrow \mathbf{13\text{ hours}}$ ( $0.542$ )
Squirrel: About five-eighths of the circle $\rightarrow \mathbf{15\text{ hours}}$ ( $0.625$ )
Snake: Exactly three-quarters of the circle $\rightarrow \mathbf{18\text{ hours}}$ ( $0.75$ )
Bat: Largest sector, about five-sixths of the circle $\rightarrow \mathbf{20\text{ hours}}$ ( $0.833$ )

Answer

Giraffe: $2.5\text{ hours}$

Elephant: $3.5\text{ hours}$
Boy: $8\text{ hours}$
Dog: $10.5\text{ hours}$
Cat: $13\text{ hours}$
Squirrel: $15\text{ hours}$
Snake: $18\text{ hours}$
Bat: $20\text{ hours}$

Common Mistakes

Assuming a 12-Hour Clock: Mistakenly treating the full circle as $12\text{ hours}$ instead of a full $24\text{ hours}$ day, leading to halving all calculated fractions.
Ordering Confusion: Mixing up adjacent values such as $2.5\text{ hours}$ and $3.5\text{ hours}$ , or $13\text{ hours}$ and $15\text{ hours}$ , without systematically sorting the visual sector sizes from smallest to largest.

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?

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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 $80 \text{ km/h}$ ?