Proportional Reasoning - 2 | FIO

Question 11

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.

Solution

Understand the Question

Two quantities $x$ and $y$ are in direct proportion if an increase in one leads to a proportional increase in the other (their ratio $\dfrac{x}{y} = k$ remains constant).
Two quantities $x$ and $y$ are in inverse proportion if an increase in one leads to a proportional decrease in the other (their product $x \times y = k$ remains constant).

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

Step 1 · Analyze the Relationship

Let $N$ be the number of taps and $T$ be the time taken to fill the tank.

Increasing the number of taps reduces the time required to fill the tank proportionally ( $N \times T = \text{constant}$ ).

Therefore, these quantities are in inverse proportion.

Answer

(i) Inverse proportion

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

Step 1 · Analyze the Relationship

Let $P$ be the number of painters and $D$ be the days needed.

Increasing the number of painters reduces the number of days needed to complete the fixed work ( $P \times D = \text{constant}$ ).

Therefore, these quantities are in inverse proportion.

Answer

(ii) Inverse proportion

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

Step 1 · Analyze the Relationship

Let $D$ be the distance and $A$ be the amount of petrol.

More petrol allows the car to travel a greater distance ( $\dfrac{D}{A} = \text{constant}$ ).

Therefore, these quantities are in direct proportion.

Answer

(iii) Direct proportion

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

Step 1 · Analyze the Relationship

Let $S$ be the speed and $T$ be the time taken.

For a fixed distance, a higher speed results in less time taken ( $S \times T = \text{Distance} = \text{constant}$ ).

Therefore, these quantities are in inverse proportion.

Answer

(iv) Inverse proportion

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

Step 1 · Analyze the Relationship

Let $L$ be the length of cloth and $P$ be the price paid.

Buying more cloth increases the total cost proportionally at a fixed rate per metre ( $\dfrac{P}{L} = \text{Rate} = \text{constant}$ ).

Therefore, these quantities are in direct proportion.

Answer

(v) Direct proportion

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

Step 1 · Analyze the Relationship

Let $N$ be the number of pages and $T$ be the reading time.

Reading more pages at a constant speed requires more time ( $\dfrac{T}{N} = \text{constant}$ ).

Therefore, these quantities are in direct proportion.

Answer

(vi) Direct proportion

Common Mistakes

Confusing Direct and Inverse Proportion: In direct proportion, both quantities increase or decrease together ( $x/y = k$ ). In inverse proportion, as one quantity increases, the other decreases ( $x \cdot y = k$ ).
Ignoring Fixed Constraints: Proportionality depends on constant parameters (such as a fixed distance, fixed rate, or fixed work). Always check what stays constant in the problem.

More questions in FIO

A cricket coach schedules practice sessions that include different activities in a specific ratio — time for warm-up/cool-down : time for batting : time for bowling : time for fielding :: $3 : 4 : 3 : 5$ . If each session is 150 minutes long, how much time is spent on each activity?

A school library has books in different languages in the following ratio — $\text{no. of Odiya books} : \text{no. of Hindi books} : \text{no. of English books} :: 3 : 2 : 1$ . If the library has 288 Odiya books, how many Hindi and English books does it have?

I have $100$ coins in the ratio — no. of $\text{₹}10$ coins : no. of $\text{₹}5$ coins : no. of $\text{₹}2$ coins : no. of $\text{₹}1$ coins $:: 4 : 3 : 2 : 1$ . How much money do I have in coins?

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?

A group of 360 people were asked to vote for their favourite season from the three seasons — rainy, winter and summer. 90 liked the summer season, 120 liked the rainy season, and the rest liked the winter. Draw a pie chart to show this information.

Draw a pie chart based on the following information about viewers’ favourite type of TV channel: Entertainment — $50\%$ , Sports — $25\%$ , News — $15\%$ , Information — $10\%$ .

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:

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 $80 \text{ km/h}$ ?