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Proportional Reasoning-2 Exercise 3.4 Class 8 Maths - Chapter 3 NCERT Solutions

Detailed solutions for Class 8 Maths Chapter 3, Proportional Reasoning-2 Exercise 3.4, prepared by the ALLEN expert team, are provided below. They help students understand the questions, follow the correct steps to solve them, and become familiar with the question format before their examinations.The exercise includes identifying pairs of quantities as direct or inverse proportion and solving questions based on cost, water supply, workers, seating arrangements, school periods, pumps, machines, and speed. It also includes questions that require students to apply proportional relationships to find unknown values.

1.0Download Class 8 Maths Chapter 3 Ex 3.4 NCERT Solutions PDF

Download the free PDF of Class 8 Maths Exercise 3.4 detailed solutions for learning, practice, and revision.

Chapter 3 : Proportional Reasoning - 2 Exercise 3.4

2.0Detailed NCERT Solutions on Proportional Reasoning-2 Exercise 3.4 Class 8 Maths (Ganita Prakash Term 2)

1. 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.

Sol. (i) If the number of taps increases, then the time taken to fill the tank decreases. If the number of taps decreases, then the time taken to fill the tank increases. The quantities change in opposite directions by the same factor. If we double the no. of taps, the time taken becomes half.

Hence, they are in inverse proportion. (ii) If the number of painters increases, then the number of days needed decreases.

If the number of painters decreases, then the number of days needed increases.

If we double the no. of painters, the work gets done in half the time.

Hence, they are in inverse proportion. (iii) If the distance increases, then the amount of petrol also increases. If the distance decreases, then the amount of petrol also increases. So, they are not in inverse proportion. (iv) If the speed increases, then the time taken decreases. If the speed decreases, then the time taken increases. For a fixed distance, if speed doubles, time becomes half. Hence, they are in inverse proportion. (v) If the amount of cloth increases, then the price to be paid increases. If the amount of cloth decreases, then the price to be paid decreases. Both quantities decrease together and increase together, so they are in direct proportion. (vi) If the number of pages increases, then the time taken to read increases. If the number of pages decreases, then the time taken to read decreases. Both quantities decrease together and increase together, so they are in direct proportion.

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

Sol. The number of pencils and the cost of pencils are in direct proportion. If x is the required cost, then $\frac{24}{20} = \frac{120}{x}$ $\Rightarrow x \times 24 = 120 \times 20$ $\Rightarrow x = 100$ So, the cost of 20 such pencils, is ₹ 100

3. 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?

Sol. The number of families and the number of days is in inverse proportion.

Assumptions needed (i) All families consume equal amounts of water. (ii) The daily water consumption per family remains fixed. (iii) No additional water is supplied to the tank.

Let the water last for x days. So, $20 \times 6 = 30 \times x$ $\Rightarrow x = 4$ So, the water will last for 4 days.

4. 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 .

Sol. Common Sleep Patterns: Average no. of hours a giraffe sleeps = 2.5 hours

Average number of hours an elephant sleeps $= 3.5$ hours

Average number of hours a boy sleeps = 8 hours

Average number of hours a dog sleeps = 10.5 hours

Average no. of hours a cat sleeps = 13 hours Average no. of hours a squirrel sleeps = 15 hours

Average no. of hours a snake sleeps $= 18$ hours

Average no. of hours a bat sleeps $= 20$ hours

5. 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? Sol. (i) The largest angle is $12 0^{\circ}$ , which corresponds to the Bus. (ii) Angle corresponding to car $= 36 0^{\circ} - (9 0^{\circ} + 12 0^{\circ} + 6 0^{\circ} + 6 0^{\circ}) = 3 0^{\circ}$ The fraction of children who travel by $car \frac{36}{360} = \frac{1}{12}$ (iii) Let x be the total number of children who took part in the survey. Then $18 = \frac{1}{12} \times x$ $\Rightarrow x = 18 \times 12 = 216$ The number of children using taxis is 0 , as taxis are not a category in this chart. (iv) Cycle and two-wheeler ( $6 0^{\circ}$ each)

6. 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?

Sol. When the number of workers increases, the number of days needed to paint the fence decreases.

Car30

^{\circ}

Assumptions Needed

(i) All workers work at the same speed/rate. (ii) The work is uniformly distributed among all workers. (iii) All workers work for the same number of hours each day. So, the number of workers and the number of days, is in inverse proportion. Let x be the no. of days taken. $3 \times 4 = 4 \times x \to x = 3$

So, they will take 3 days to finish the work.

7. 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?

Sol. No. of hours and no. of tanks are in direct proportion. Let 5 such tanks take x hours. $\frac{6}{2} = \frac{x}{5}$ $\Rightarrow x = 15$ So, 5 tanks will take 15 hours.

8. 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?

Sol. No. of rows and no. of chairs in each row are in inverse proportion.

Let the new arrangement has $x$ rows. $25 \times 12 = x \times 20$ $\Rightarrow x = 15$ So, the new arrangement has 15 rows.

9. 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?

Sol. No. of periods and the duration of each period are in inverse proportion. Let each period be of $x$ minutes. $8 \times 45 = 9 \times x$ $\Rightarrow x = 40$ So, each period is 40 minutes.

10. 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?

Sol. Consider the work done to fill the tank as 1 unit of work. Let us find the work done by each pump in 1 hour.

Small pump fills the tank in 3 hours, so in 1 hour it does $\frac{1}{3}$ unit of work.
Large pump fills the tank in 2 hours, so in 1 hour it does $\frac{1}{2}$ unit of work.

So, the work done by both pumps in 1 hour is $\frac{1}{3} + \frac{1}{2} = \frac{5}{6}$ units of work.

Therefore, to complete $\frac{5}{6}$ units of work, it takes 1 hour when both work together. How much time will it take to complete 1 unit of work?

The quantity of work and time taken are directly proportional.

So, $\frac{5}{6} : 1 :: 1 : x$ Since it is a direct proportion,

$\frac{5}{6} \times x = 1 x = \frac{6}{5}$

Therefore, both pumps together will fill the tank in $\frac{6}{5}$ hours (or 1.2 hours).

11. 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?

Sol. Let the number of machines be x

Number of Machines	42	X
Number of Days	63	54

Number of machines and no. of days are in inverse proportion. So, $42 \times 63 = x \times 54$ $\Rightarrow x = \frac{42 \times 63}{54} = 7 \times 7$ or $x = 49$ So, 49 machines are required.

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

Sol. Let the car take $t$ hours.

Speed $(km / hr)$	60	$80 ↑$
Time $(hrs)$	2	$t ↓$

The speed of the car and the time taken are in inverse proportion. So, $2 \times 60 = t \times 80$ $t = \frac{2 \times 60}{30} = \frac{3}{2}$ $\Rightarrow t = 1.5$ hours So, the car will take 1.5 hours.

3.0Key Concepts of Chapter 3 Proportional Reasoning-2 Exercise 3.4

Direct and inverse proportion in situations involving taps, painters, cyclists, cloth, and reading.
Cost of 20 pencils when the price of 24 pencils is given.
Change in the duration of a water supply when the number of families increases.
Effect of changing the number of workers on the time needed to paint a fence.
Relationship between tanks, rows of chairs, and school periods when one quantity changes.
Combined rate of two pumps filling the same tank.
Effect of changing the number of machines or the speed of a car on the time required.

4.0Benefits of NCERT Solutions Class 8 Maths Chapter 3 Exercise 3.4

Explains direct and inverse proportion through the practical situations given in the exercise.
Clarifies how to identify the type of proportion from the relationship between two quantities.
Provides step-by-step solutions for questions involving workers, families, pumps, machines, and speed.
Shows the application of proportional reasoning to situations involving water supply, pencil costs, and school periods.
Makes multi-step questions involving combined pump rates easier to understand.
Provides a clear approach to questions that require finding unknown values from changing quantities.
Supports revision of different applications of proportional reasoning covered in Exercise 3.4.

NCERT Class 8 Maths Ch. 3 Proportional Reasoning - 2 Other Exercises:-

Exercise 3.1

Exercise 3.2

Exercise 3.3

Exercise 3.4

NCERT Solutions Class 8 Maths All Chapters:-

Chapter 1 : Fractions in Disguise