NCERT Solutions for Class 8 Maths Part 2 Chapter 3 - Proportional Reasoning - 2

Class 8 Maths Part 2 Chapter 3 Solutions PDF Free Download - Ganita Prakash Book 2

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NCERT Solutions for Class 8 Maths Part 2 Chapter 3 - Exercise Questions and Answers

Here are the NCERT Solutions for Class 8 Maths Ganita Prakash Part 2 Chapter 3 Proportional Reasoning - 2 question answers with clear and detailed solutions.

Question 1. 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?

Answer:

Given: Warm-up/Cool-down : Batting : Bowling : Fielding $=3: 4: 3: 5$
Total parts $=3+4+3+5=15$
Total session time = 150 minutes
So, time for 1 part $=\frac{150}{15}=10$ minutes
Therefore,
Time spent on Warm-up/Cool-down $=3 \times 10=30$ minutes
Time spent on Batting $=4 \times 10=40$ minutes
Time spent on Bowling $=3 \times 10=30$ minutes
Time spent on Fielding $=5 \times 10=50$ minutes

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

Answer: 192 and 96

Explanation:

Given:
Number of Odiya books : Number of Hindi books : Number of English books $= 3: 2: 1$
Total Odiya books = 288, which is equal to 3 parts.
So, 1 part = $\frac{288}3=96$ books
Therefore,
Total Hindi books $=2\times96=192$
Total English books $=1\times96=96$

Question 3. I have 100 coins in the ratio - no. of ₹10 coins : no. of ₹5 coins : no. of ₹2 coins : no. of ₹1 coins $:: 4: 3: 2: 1$. How much money do I have in coins?

Answer: ₹600

Explanation:

₹10 coins : ₹5 coins: ₹2 coins: ₹1 coins $=4: 3: 2: 1$
Total parts $=4+3+2+1=10$
Total coins $=100$
So, value of 1 part $=\frac{100}{10}=10$
Therefore,
Number of ₹10 coins = $4\times10=40$
Number of ₹5 coins = $3\times10=30$
Number of ₹2 coins = $2\times10=20$
Number of ₹1 coins = $1\times10=10$
$\therefore$ Total value of these coins
$=40 \times 10+30 \times 5+20 \times 2+10 \times 1$
$=400+150+40+10$
$=600$

Hence, you have ₹600 in coins.

Question 4. 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?

Answer:

Let the common factor be $x$.
Then the sides are $3x,4x,$ and $5x$.
When $x=1$, three sides of the triangle are: 3, 4, and 5.
When $x=2$, three sides of the triangle are: 6, 8, and 10.
When $x=3$, three sides of the triangle are: 9, 12, and 15.
and so on.

No, triangles with side lengths in the ratio $3: 4: 5$ are similar, i.e., the same shape, but they need not have the same size. Since congruent triangles must have exactly equal corresponding sides, triangles such as $3,4,5$ and 6,8,10 are not congruent.
Hence, they are similar but not necessarily congruent because their sizes may differ.

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

Answer: No

Explanation:

Let the sides of the triangles be $x,3x$ and $5x$.
We know that for a triangle, the sum of any two sides must be greater than the third side.
But $x+3x<5x$
Since the sum of the two smaller sides is less than the largest side, the triangle inequality is violated.

Hence, no such triangle can be constructed with side lengths in the ratio $1: 3: 5$.

Question 1. 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.

Answer:

Total number of people = 360

People liking the summer season = 90

People liking rainy season = 120

People liking winter season $=360-(90+120)=150$

Therefore,

Summer season's angle $=\frac{90}{360} \times 360=90^{\circ}$

Rainy season's angle $=\frac{120}{360} \times 360=120^{\circ}$

Winter season's angle $=\frac{150}{360} \times 360=150^{\circ}$

Question 2. Draw a pie chart based on the following information about viewers' favourite type of TV channel: Entertainment - 50%, Sports - 25%, News - 15%, Information - 10%.

Answer:

Given:

Entertainment: 50%, Sports: 25%, News: 15%, Information: 10%

Total angle = 360°

Part of Entertainment in the pie chart $=\frac{50}{100}\times360=180^\circ$

Part of Sports in the pie chart $=\frac{25}{100}\times360=90^\circ$

Part of News in the pie chart $=\frac{15}{100}\times360=54^\circ$

Part of Information in the pie chart $=\frac{10}{100}\times360=36^\circ$

Question 3. 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:

Answer:

Let the total number of students be 100.

Number of students in the Language subject = 15

Number of students in the Arts Education subject = 20

Number of students in the Vocational Education subject = 8

Number of students in the Social Science subject = 12

Number of students in the Physical Education subject = 10

Number of students in the Maths subject = 20

Number of students in the Science subject = 15

Total angle = 360°

Part of the students in the Language subject in the pie chart $=\frac{15}{100}\times360=54^\circ$

Part of the students in the Arts Education subject in the pie chart $=\frac{20}{100}\times360=72^\circ$

Part of the students in the Vocational Education subject in the pie chart $=\frac{8}{100}\times360\approx 29^\circ$

Part of the students in the Social Science subject in the pie chart $=\frac{12}{100}\times360\approx 43^\circ$

Part of the students in the Physical Education subject in the pie chart $=\frac{10}{100}\times360=36^\circ$

Part of the students in the Maths subject in the pie chart $=\frac{20}{100}\times360=72^\circ$

Part of the students in the Science subject in the pie chart $=\frac{15}{100}\times360=54^\circ$

Question 1. Which of these are in inverse proportion?

Answer:

We know that Quantities are inversely proportional if, when one quantity changes by a factor $n$, the other quantity changes by the inverse $\frac{1}{n}$.

That means if $x$ and $y$ are in inverse proportion, then the product of $x$ and $y$ will be the same in all cases.

(i)
When $x_1=40$ and $y_1=20$, $x_1y_1=40\times20=800$
When $x_2=80$ and $y_2=10$, $x_2y_2=80\times10=800$
When $x_3=25$ and $y_3=32$, $x_3y_3=25\times32=800$
When $x_4=16$ and $y_4=50$, $x_4y_4=16\times50=800$

Since the product of $x$ and $y$ is constant, $x$ and $y$ are in inverse proportion.

(ii)

When $x_1=40$ and $y_1=20$, $x_1y_1=40\times20=800$

When $x_2=80$ and $y_2=10$, $x_2y_2=80\times10=800$

When $x_3=25$ and $y_3=12.5$, $x_3y_3=25\times12.5=312.5$

When $x_4=16$ and $y_4=8$, $x_4y_4=16\times8=128$

Since the product of $x$ and $y$ is not constant, $x$ and $y$ are not in inverse proportion.

(iii)

When $x_1=30$ and $y_1=15$, $x_1y_1=30\times15=450$

When $x_2=90$ and $y_2=5$, $x_2y_2=90\times5=450$

When $x_3=150$ and $y_3=8$, $x_3y_3=150\times8=1200$

When $x_4=10$ and $y_4=45$, $x_4y_4=10\times45=450$

Since the product of $x$ and $y$ is not constant, $x$ and $y$ are not in inverse proportion.

Question 2. Fill in the empty cells if x and y are in inverse proportion.

Answer:

If $x$ and $y$ are in inverse proportion, then the product of $x$ and $y$ will be the same in all cases.

When $x_1=16$ and $y_1=9$, $x_1y_1=16\times9=144$
When $x_2=12$, then $y_2=\frac{144}{12}=12$
When $y_3=48$, then $y_3=\frac{144}{48}=3$
When $x_4=36$, then $y_4=\frac{144}{36}=4$

Question 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.

Answer:

Two quantities are in inverse proportion if increasing one causes the other to decrease so that their product remains constant.

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

Answer: ₹100

Explanation:

24 pencils cost ₹120.
So, 1 pencil costs $\frac{120}{24}=5$
Therefore, the cost of 20 such pencils $20\times5=100$

Question 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?

Answer: 4 days

Explanation:

A tank can supply 20 families for 6 days.
Total water available = 120 × 6 = 120 family days
Total family after addition = 20 + 10 = 30
$\therefore$ Number of days the water will last $=\frac{120}{30}=4$ days

Assumptions:

1. Every family uses the same amount of water per day.

2. Water consumption remains constant throughout.

3. No additional water is added to the tank.

4. There is no wastage or leakage of water.

Question 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.

Answer:

Question 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?

Answer:

Total = 360°
Walk = 90°
Bus = 120°
Cycle = 60°
Two-wheeler = 60°
Car = 360° − (90° + 120° + 60° + 60°) = 30°

5 (i): Bus

Explanation:
Bus = 120°, which is the maximum.

Hence, bus is the most common mode of transport.

5 (ii): $\frac{1}{12}$

Explanation:

Children travel by car = 30°
$\therefore$ Required fraction = $\frac{30°}{360°}=\frac1{12}$

(iii) 600 and 50

Explanation:

Given: 18 children travel by car, which equals 30°.
So, 1° is equivalent to $\frac{30}{18}=\frac53$
So, the total number of children that took part in the survey $=\frac53\times360=600$

Number of students who took taxis $=\frac53\times30=50$

(iv) Cycles and two-wheelers

Explanation:

Cycles and two-wheelers have equal numbers of children travelling, as both have a part in the pie chart, i.e., 60°.

Question 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?

Answer: 3 days

Explanation:

3 workers can paint a fence in 4 days.
We know that Total Work = Workers × Days $=3\times4=12$ days
Therefore, (3 + 1) = 4 workers can paint a fence in
$\text{Days} = \frac{\text{Total Work}}{\text{Total Workers}} = \frac{12}{4} = 3 \text{ days}$

Hence, it will take 3 days to finish the work.

Assumptions:

• All workers work at the same rate.

• Each worker works for the same number of hours each day.

• No time is lost due to interruptions or coordination issues.

• The amount of work, i.e., painting the fence, remains unchanged.

Question 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?

Answer: 15 hours

Explanation:

It takes 6 hours to fill 2 tanks of the same size with a pump.
So, the time to fill 1 tank $=\frac62=3$ hours
Then, the time to fill 5 such tanks $=5\times3=15$ hours

Hence, it will take 15 hours to fill 5 tanks.

Question 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?

Answer: 15 rows

Explanation:

Total chairs = Number of rows × Number of chairs in each row = 25 × 12 = 300
Now, there are 20 chairs in each row.
$\therefore$ Number of rows = $\frac{300}{20}=15$

Hence, there are 15 rows in this new arrangement.

Question 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?

Answer: 40 minutes

Explanation:

Total class hours = Number of periods × Duration of a period = 8 × 45 = 360 minutes
Now, there are 9 periods.
$\therefore$ Duration of each period = $\frac{360}9=40$ minutes

Hence, 40 minutes is the duration of a new period.

Question 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?

Answer: 72 minutes or 1 hour 12 minutes

Explanation:

A small pump can fill a tank in 3 hours.
In 1 hour, it will fill $\frac13$ part of the tank.
A large pump can fill the same tank in 2 hours.
In 1 hour, it will fill $\frac12$ part of the tank.
Together in 1 hour, they will fill $\frac13+\frac12=\frac56$
Therefore, together they will take $\frac1{\frac56}=\frac65$ hours to fill the tank.
$\frac65$ hours = $\frac65\times60=72$ minutes or 1 hour 12 minutes

Hence, both pumps together will take 1 hour 12 minutes or 72 minutes to fill the tank.

Question 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?

Answer: 49

Explanation:

We know that $M_1D_1=M_2D_2$, where $M_1, M_2$ are the number of machines produced, and $D_1, D_2$ are the number of required days.
So, $42\times63=M_2\times54$
$\therefore M_2=49$

Hence, 49 machines are required to produce the same number of toys in 54 days.

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

Answer: 1.5 hours

Explanation:

We know that Distance = Speed × Time
So, total distance = 60 × 2 = 120 km
Now, the speed is 80 km/hr.
Time $=\frac{\text { Distance }}{\text { Speed }}=\frac{120}{80}=\frac32=1.5$ hours or 1 hour 30 minutes

The car will take 1.5 hours or 1 hour and 30 minutes to reach the destination.

Key Concepts Explained in Proportional Reasoning - 2: Ganita Prakash Book 2

Topics you will learn in NCERT Solutions for Class 8 Maths Part 2 Chapter 3 Proportional Reasoning - 2 include:

• 3.1 Proportionality - A Quick Recap

• 3.2 Ratios in Maps

• 3.3 Ratios with More than 2 Terms

• 3.4 Dividing a Whole in a Given Ratio

• 3.5 A Slice of the Pie

• 3.6 Inverse Proportions

Extra Questions for Class 8 Maths Part 2 Chapter 3 Proportional Reasoning - 2

Question 1:

A camp of soldiers has food for 200 days. After 20 days, 100 more soldiers join the camp, and the food can now last for 80 days. What was the initial number of soldiers in the camp?

Answer:

Let the initial number of soldiers be $x$ and the initial number of days the food could last for 200
Food Supply = the number of soldiers × the number of days = $x×200$
After 20 days, food for (200 – 20) = 180 days remained for $x$ soldiers.
So, $x×180 = (x+100)×80$
⇒ $100x = 800$
⇒ $x = 80$
Hence, the correct answer is 80.

Question 2:

Pipes P and Q can fill a tank in 12 hours and 36 hours, respectively. If both pipes are opened, then how much time (in hours) will it take to fill the tank?

Answer:

We have,
Pipes P and Q can fill a tank in 12 hours and 36 hours.
Part filled by P in 1 hour = $\frac{1}{12}$
Part filled by Q in 1 hour = $\frac{1}{36}$
Part filled by both P and Q together in 1 hour = $\frac{1}{12}+\frac{1}{36}$ = $\frac{3+1}{36}$ = $\frac{4}{36}$ = $\frac{1}{9}$
The total time needed to fill the tank by P and Q is 9 hours.
Hence, the correct answer is 9.

Question 3:

Three numbers are in the ratio of 1 : 2 : 3. The product of the three numbers is 1296. What is the sum of the three numbers?

Answer:

The ratio of three numbers = 1 : 2 : 3
Product of three numbers = 1296
Let the numbers be $x, 2x$ and $3x$.
Product of three numbers $= x × 2x × 3x = 6x^3$
So, 6x$^3$ = 1296
⇒ $x^3 = \frac{1296}{6} = 216$
⇒ $x = 6$
So, First number $= x = 6$
Second Number $= 2x = 2 × 6 = 12$
Third Number $= 3x = 3 × 6 = 18$
So, the sum of all numbers = 6 + 12 + 18 = 36
Hence, the correct answer is 36.

Question 4:

The following pie chart shows the percentage number of students who play different sports.

If the total number of students is 500, then how many students play cricket?

Answer:

Given: The total number of students is 500.
The number of students who play cricket is 21% of 500 = $\frac{21}{100}$ × 500 = 105
Hence, the correct answer is 105.

Question 5:

The following pie chart shows results from a survey of 160 kids about their favourite desserts.

What is the number of kids who like ice cream?

Answer:

Total kids in the survey = 160
Kids who likes icecream = 25%
So, the number of kids who like ice cream $= \frac{25}{100}\times160 = 40$
Hence, the correct answer is 40.

NCERT Solutions for Class 8 Maths Chapter Wise

We at Careers360 compiled all the latest NCERT Class 8 Maths solutions based on latest textbook in one place for easy student reference. The following links will allow you to access them.

NCERT Books and NCERT Syllabus

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• NCERT Books Class 8 Maths

• NCERT Syllabus Class 8 Maths

• NCERT Books Class 8

• NCERT Syllabus Class 8

Chapter Summary: NCERT Solutions for Class 8 Maths Ganita Prakash Part 2 Chapter 3

Number of Exercises: 4

Total number of questions: 22

Expert Review of NCERT Solutions for Class 8 Maths Part 2 Chapter 3

Here are the concepts and topics that are related to the latest NCERT Solutions for Class 8 Maths Ganita Prakash Part 2 Chapter 3 in higher classes.

Class 9-10

• Direct and inverse proportion

• Ratio and proportion

• Variation and unitary method

• Speed, time and distance

• Work and time

• Scaling and map problems

• Applications of proportional reasoning in geometry and mensuration

• Real-life word problems involving proportional relationships

Class 11-12

• Variation and functional relationships

• Linear equations and graphs

• Coordinate geometry applications

• Rates of change and proportional models

• Trigonometric ratios and proportionality

• Applications in physics and statistics

JEE (Main & Advanced)

• Ratio, proportion, and variation concepts

• Time and work problems

• Speed, time and distance applications

• Functional relationships and graphs

• Trigonometric proportionality

• Coordinate geometry applications

• Mathematical modelling and problem-solving

• Quantitative aptitude and data interpretation based on proportional reasoning