Proportional Reasoning-2 Class 8 Mathematics NCERT Solutions

Q1Figure it Out (Section 3.4)

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?

Solution

Given: Total duration of a practice session = 150 minutes. The ratio of time for warm-up/cool-down : batting : bowling : fielding is

3:4:3:5

.

To Find: Time spent on each activity.

Solution: First, we find the sum of the terms in the ratio: Sum of ratio parts =

3 + 4 + 3 + 5 = 15

.

Now, we can calculate the time for each activity by dividing the total time into 15 parts and then distributing them according to the ratio. One part of time =

\frac{\text{Total time}}{\text{Sum of ratio parts}} = \frac{150}{15} = 10

minutes.

Time for each activity:

Time for warm-up/cool-down = $3 \times 10 = 30$ minutes.
Time for batting = $4 \times 10 = 40$ minutes.
Time for bowling = $3 \times 10 = 30$ minutes.
Time for fielding = $5 \times 10 = 50$ minutes.

Check: Total time =

30 + 40 + 30 + 50 = 150

minutes. This matches the given total time.

Final Answer: The time spent on each activity is:

Warm-up/cool-down: 30 minutes
Batting: 40 minutes
Bowling: 30 minutes
Fielding: 50 minutes

Q2Figure it Out (Section 3.4)

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?

Solution

Given: The ratio of Odiya books : Hindi books : English books is

3:2:1

. The number of Odiya books is 288.

To Find: The number of Hindi and English books in the library.

Solution: Let the common factor for the ratio be

x

. Then:

Number of Odiya books = $3x$
Number of Hindi books = $2x$
Number of English books = $1x = x$

We are given that the number of Odiya books is 288.

3x = 288

x = \frac{288}{3} = 96

Now we can find the number of Hindi and English books:

Number of Hindi books = $2x = 2 \times 96 = 192$ books.
Number of English books = $x = 96$ books.

Final Answer: The library has 192 Hindi books and 96 English books.

Q3Figure it Out (Section 3.4)

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?

Solution

Given: Total number of coins = 100. The ratio of the number of coins is: ₹10 coins : ₹5 coins : ₹2 coins : ₹1 coins ::

4:3:2:1

.

To Find: The total amount of money in coins.

Solution: First, we find the number of each type of coin. Sum of ratio parts =

4 + 3 + 2 + 1 = 10

.

Number of each type of coin:

Number of ₹10 coins = $\frac{4}{10} \times 100 = 40$ coins.
Number of ₹5 coins = $\frac{3}{10} \times 100 = 30$ coins.
Number of ₹2 coins = $\frac{2}{10} \times 100 = 20$ coins.
Number of ₹1 coins = $\frac{1}{10} \times 100 = 10$ coins.

Next, we calculate the total value of the money.

Value of ₹10 coins = $40 \times ₹10 = ₹400$ .
Value of ₹5 coins = $30 \times ₹5 = ₹150$ .
Value of ₹2 coins = $20 \times ₹2 = ₹40$ .
Value of ₹1 coins = $10 \times ₹1 = ₹10$ .

Total money = Value of all coins =

₹400 + ₹150 + ₹40 + ₹10 = ₹600

.

Final Answer: I have ₹600 in coins.

Q4Figure it Out (Section 3.4)

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?

Solution

To Construct: A triangle with side lengths in the ratio

3:4:5

.

Steps of Construction: We can choose a scaling factor, for example, let the sides be

3 \text{ cm}, 4 \text{ cm}

, and

5 \text{ cm}

.

Draw a line segment BC of length $4 \text{ cm}$ .
With B as the center and a radius of $3 \text{ cm}$ , draw an arc.
With C as the center and a radius of $5 \text{ cm}$ , draw another arc that intersects the first arc at point A.
Join AB and AC. $\ riangle ABC$ is the required triangle with sides $AB = 3 \text{ cm}$ , $BC = 4 \text{ cm}$ , and $AC = 5 \text{ cm}$ .

Will all triangles be congruent? No, not all triangles drawn with this ratio of side lengths will be congruent to each other.

Reason: Congruent triangles must have corresponding sides of equal length. Triangles with side lengths in the ratio

3:4:5

are all similar to each other by the SSS (Side-Side-Side) similarity criterion. However, they are not necessarily congruent. For example, a triangle with sides

3 \text{ cm}, 4 \text{ cm}, 5 \text{ cm}

is not congruent to a triangle with sides

6 \text{ cm}, 8 \text{ cm}, 10 \text{ cm}

, even though both have side lengths in the ratio

3:4:5

. The second triangle is an enlargement of the first one.

Q5Figure it Out (Section 3.4)

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

Solution

Question: Can a triangle be constructed with side lengths in the ratio

1:3:5

?

Solution: No, a triangle cannot be constructed with side lengths in this ratio.

Reason: This is based on the Triangle Inequality Theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Let the side lengths be

k, 3k

, and

5k

for some positive value

k

. Let's check the condition:

Sum of the two smaller sides = $k + 3k = 4k$ .
The length of the third (longest) side is $5k$ .

According to the theorem, for a triangle to be possible,

k + 3k > 5k

. However,

4k

is not greater than

5k

(since

k > 0

,

4k < 5k

).

Because the sum of the two shorter sides is less than the longest side, the sides cannot form a closed triangle.

Final Answer: No, because the sum of the lengths of the two smaller sides (

1+3=4

parts) is not greater than the length of the longest side (5 parts), which violates the Triangle Inequality Theorem.

Q1Figure it Out (Section 3.5)

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.

Solution

Given: Total number of people = 360. Number of people who like summer = 90. Number of people who like rainy season = 120.

To Do: Draw a pie chart to represent this information.

Solution: Step 1: Find the number of people who like winter. Number of people who like winter = Total people - (people who like summer + people who like rainy) Number of people who like winter =

360 - (90 + 120) = 360 - 210 = 150

.

Step 2: Calculate the central angle for each season. The total angle in a pie chart is

360^\circ

. Angle for a category =

(\frac{\text{Number in category}}{\text{Total number}}) \times 360^\circ

.

Angle for Summer = $(\frac{90}{360}) \times 360^\circ = 90^\circ$ .
Angle for Rainy = $(\frac{120}{360}) \times 360^\circ = 120^\circ$ .
Angle for Winter = $(\frac{150}{360}) \times 360^\circ = 150^\circ$ .

Check: Total angle =

90^\circ + 120^\circ + 150^\circ = 360^\circ

.

Step 3: Draw the pie chart.

Draw a circle of any convenient radius.
Draw a horizontal radius.
Using a protractor, measure an angle of $90^\circ$ from the radius to represent 'Summer'.
From this new line, measure an angle of $120^\circ$ to represent 'Rainy'.
The remaining sector will have an angle of $150^\circ$ and will represent 'Winter'.
Label each sector with the season it represents.

Q2Figure it Out (Section 3.5)

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

Solution

Given: Percentage of viewers for favourite TV channels:

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

To Do: Draw a pie chart to represent this information.

Solution: Step 1: Calculate the central angle for each channel type. The total angle in a pie chart is

360^\circ

. Angle for a category =

(\frac{\text{Percentage}}{100}) \times 360^\circ

.

Angle for Entertainment = $(\frac{50}{100}) \times 360^\circ = 0.5 \times 360^\circ = 180^\circ$ .
Angle for Sports = $(\frac{25}{100}) \times 360^\circ = 0.25 \times 360^\circ = 90^\circ$ .
Angle for News = $(\frac{15}{100}) \times 360^\circ = 0.15 \times 360^\circ = 54^\circ$ .
Angle for Information = $(\frac{10}{100}) \times 360^\circ = 0.1 \times 360^\circ = 36^\circ$ .

Check: Total angle =

180^\circ + 90^\circ + 54^\circ + 36^\circ = 360^\circ

.

Step 2: Draw the pie chart.

Draw a circle of any convenient radius.
Draw a diameter. The semicircle on one side represents the $180^\circ$ angle for 'Entertainment'.
From the center along the diameter, use a protractor to measure an angle of $90^\circ$ to create the sector for 'Sports'.
From the new line, measure an angle of $54^\circ$ to create the sector for 'News'.
The remaining sector will have an angle of $36^\circ$ and will represent 'Information'.
Label each sector with the channel type it represents.

Q3Figure it Out (Section 3.5)

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:
Subject Language Arts Education Vocational Education Social Science Physical Education Maths Science
Number of Students

Solution

Objective: To create a pie chart representing the favourite subjects of students in a class.

Procedure: This is an activity-based question. The steps to complete it are as follows:

Step 1: Collect Data

Survey your classmates. Ask each student to choose only one favourite subject from the given list: Language, Arts Education, Vocational Education, Social Science, Physical Education, Maths, Science.
Record the number of students who chose each subject in a table.

Step 2: Tabulate the Data (Example)

Let's assume you have a class of 40 students and you collected the following data:

Subject	Number of Students
Language	6
Arts Education	5
Vocational Education	3
Social Science	7
Physical Education	8
Maths	7
Science	4
Total	40

Step 3: Calculate the Central Angle for Each Subject

The total angle in a pie chart is $360^\circ$ .
The formula is: Angle = $(\frac{\text{Number of students for a subject}}{\text{Total number of students}}) \times 360^\circ$ .
Using the example data:
- Language: $(\frac{6}{40}) \times 360^\circ = 54^\circ$
- Arts Education: $(\frac{5}{40}) \times 360^\circ = 45^\circ$
- Vocational Education: $(\frac{3}{40}) \times 360^\circ = 27^\circ$
- Social Science: $(\frac{7}{40}) \times 360^\circ = 63^\circ$
- Physical Education: $(\frac{8}{40}) \times 360^\circ = 72^\circ$
- Maths: $(\frac{7}{40}) \times 360^\circ = 63^\circ$
- Science: $(\frac{4}{40}) \times 360^\circ = 36^\circ$
- Total Angle: $54+45+27+63+72+63+36 = 360^\circ$

Step 4: Construct the Pie Chart

Draw a circle with a suitable radius.
Draw a radius to start.
Use a protractor to draw sectors for each subject according to the angles calculated in Step 3.
Label each sector with the subject name and, if desired, the number of students or percentage.
Give the pie chart a title, such as 'Favourite Subjects of Students'.

Q1Figure it Out (Section 3.6, part 1)

Which of these are in inverse proportion?

(i)

| $x$ | 40 | 80 | 25 | 16 | |---|---|---|---|---| | $y$ | 20 | 10 | 32 | 50 |

(ii)

| $x$ | 40 | 80 | 25 | 16 | |---|---|---|---|---| | $y$ | 20 | 10 | 12.5 | 8 |

(iii)

| $x$ | 30 | 90 | 150 | 10 | |---|---|---|---|---| | $y$ | 15 | 5 | 3 | 45 |

Solution

To Find: Which of the given tables show quantities in inverse proportion.

Principle: Two quantities

x

and

y

are in inverse proportion if their product,

xy

, is a constant (

k

). That is,

xy = k

.

Solution: We will check the product

xy

for each pair of values in each table.

(i)

$40 \times 20 = 800$
$80 \times 10 = 800$
$25 \times 32 = 800$
$16 \times 50 = 800$ Since the product $xy$ is constant (800) for all pairs, the quantities in table (i) are in inverse proportion.

(ii)

$40 \times 20 = 800$
$80 \times 10 = 800$
$25 \times 12.5 = 312.5$
$16 \times 8 = 128$ Since the product $xy$ is not constant, the quantities in table (ii) are not in inverse proportion.

(iii)

$30 \times 15 = 450$
$90 \times 5 = 450$
$150 \times 3 = 450$
$10 \times 45 = 450$ Since the product $xy$ is constant (450) for all pairs, the quantities in table (iii) are in inverse proportion.

Final Answer: The pairs of quantities in tables (i) and (iii) are in inverse proportion.

Q2Figure it Out (Section 3.6, part 1)

Fill in the empty cells if $x$ and $y$ are in inverse proportion.
$x$ 16 12 36
$y$ 9 48

$x$	16	12		36
$y$	9		48

Solution

Given: Quantities

x

and

y

are in inverse proportion. A table with some missing values.

To Find: The missing values in the table.

Principle: If

x

and

y

are in inverse proportion, their product

xy

is a constant,

k

.

Solution: Step 1: Find the constant of proportionality, $k$ . Using the first column where both

x

and

y

are given:

x_1 = 16, y_1 = 9

k = x_1 \times y_1 = 16 \times 9 = 144

.

Step 2: Find the missing values. For every column, the product

xy

must be 144.

Second column: $x_2 = 12$ . Let the missing value be $y_2$ . $12 \times y_2 = 144$ $y_2 = \frac{144}{12} = 12$ .
Third column: $y_3 = 48$ . Let the missing value be $x_3$ . $x_3 \times 48 = 144$ $x_3 = \frac{144}{48} = 3$ .
Fourth column: $x_4 = 36$ . Let the missing value be $y_4$ . $36 \times y_4 = 144$ $y_4 = \frac{144}{36} = 4$ .

Final Answer: The completed table is:

$x$	16	12	3	36
$y$	9	12	48	4

Q1Figure it Out (Section 3.6, part 2)

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

To Find: Which pairs of quantities are in inverse proportion.

Principle: Two quantities are in inverse proportion if an increase in one quantity causes a proportional decrease in the other, and vice-versa.

Analysis:

(i)

Taps and Time: If you increase the number of taps, the time taken to fill the tank decreases. This is inverse proportion.

(ii)

Painters and Days: If you hire more painters, the number of days needed to paint the wall decreases. This is inverse proportion.

(iii)

Distance and Petrol: If you have more petrol, the car can travel a greater distance. Both quantities increase together. This is direct proportion, not inverse.

(iv)

Speed and Time (fixed route): If you increase the speed, the time taken to cover the fixed distance decreases. This is inverse proportion.

(v)

Cloth Length and Price: If you buy more cloth, you pay a higher price. Both quantities increase together. This is direct proportion, not inverse.

(vi)

Pages and Reading Time: If a book has more pages, it will take more time to read it (at a fixed speed). Both quantities increase together. This is direct proportion, not inverse.

Final Answer: The pairs of quantities in inverse proportion are (i), (ii), and (iv).

Q2Figure it Out (Section 3.6, part 2)

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

Solution

Given: Cost of 24 pencils = ₹120.

To Find: The cost of 20 pencils.

Solution: This is a case of direct proportion, as more pencils cost more money.

Method 1: Unitary Method First, find the cost of one pencil. Cost of 1 pencil =

\frac{\text{Total cost}}{\text{Number of pencils}} = \frac{₹120}{24} = ₹5

.

Now, find the cost of 20 pencils. Cost of 20 pencils =

20 \times (\text{cost of 1 pencil}) = 20 \times ₹5 = ₹100

.

Method 2: Ratio and Proportion Let the cost of 20 pencils be

x

. The ratio of the number of pencils is equal to the ratio of their costs.

\frac{24}{20} = \frac{120}{x}

24 \times x = 20 \times 120

x = \frac{20 \times 120}{24}

x = \frac{2400}{24} = 100

Final Answer: 20 such pencils will cost ₹100.

Q3Figure it Out (Section 3.6, part 2)

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?

Solution

Given: Initial number of families (

x_1

) = 20. Number of days the water lasts (

y_1

) = 6 days. 10 more families move in.

To Find: How long the water will last for the new total number of families.

Solution: This is a case of inverse proportion because if the number of families increases, the water supply will last for fewer days.

New number of families (

x_2

) =

20 + 10 = 30

. Let the new number of days be

y_2

.

For inverse proportion, the product of the quantities remains constant:

x_1 y_1 = x_2 y_2

.

20 \times 6 = 30 \times y_2

120 = 30 \times y_2

y_2 = \frac{120}{30} = 4

The water will last for 4 days.

Assumptions: To solve this problem, we need to make the following assumption:

The rate of water consumption per family remains constant. Every family consumes the same amount of water each day, and this does not change after the new families move in.

Final Answer: The water will last for 4 days. The key assumption is that all families consume water at the same constant average rate.

Q4Figure it Out (Section 3.6, part 2)

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

Given: A list of sleep durations: 15, 2.5, 20, 8, 3.5, 13, 10.5, 18 hours. A list of living beings: Cow, Python, Giraffe, Cat, Human, Horse, Elephant, Sloth.

To Do: Match each living being with its average daily sleep duration from the given list.

Solution: We will match the hours from the list to the animals based on general biological knowledge, from longest to shortest sleep times.

Sloth: Known for sleeping very long hours. The longest duration in the list is 20 hours.
Python: Also known for long periods of sleep. The next longest is 18 hours.
Cat: Domestic cats sleep a lot. 15 hours is a reasonable value from the list.
Human: A typical value is 8 hours.
Cow: Ruminants sleep for short periods. 3.5 hours is a plausible value.
Horse: Horses also sleep very little, often standing up. 2.5 hours is the shortest time in the list.

This leaves the numbers 13 and 10.5, and the animals Giraffe and Elephant. Both giraffes and elephants are known to be very short sleepers (2-5 hours). The remaining numbers in the list (13, 10.5) do not accurately reflect this. However, to complete the question as asked, we must assign these remaining values.

Elephant: We will assign one of the remaining values, e.g., 10.5 hours.
Giraffe: We will assign the last remaining value, 13 hours.

Note: The assigned values for Elephant and Giraffe are not biologically accurate but are required to exhaust the given list.

Final Answer: The matched list is as follows:

Sloth: 20 hours
Python: 18 hours
Cat: 15 hours
Giraffe: 13 hours
Elephant: 10.5 hours
Human: 8 hours
Cow: 3.5 hours
Horse: 2.5 hours

Q5Figure it Out (Section 3.6, part 2)

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

Given: A pie chart representing modes of transport for school children. The central angles for the sectors are as follows:

Walk: $90^\circ$
Bus: $120^\circ$
Cycle: $60^\circ$
Car and Taxi have equal-sized sectors. The remaining angle is $360^\circ - (90^\circ + 120^\circ + 60^\circ) = 360^\circ - 270^\circ = 90^\circ$ . Since Car and Taxi are equal, each has an angle of $90^\circ / 2 = 45^\circ$ .

Solution:

(i) What is the most common mode of transport? The most common mode corresponds to the largest sector, which has the largest central angle. The largest angle is

120^\circ

, which corresponds to the Bus.

(ii) What fraction of children travel by car? The angle for the car sector is

45^\circ

. The total angle is

360^\circ

. The fraction is

\frac{\text{Angle for Car}}{\text{Total Angle}} = \frac{45^\circ}{360^\circ} = \frac{1}{8}

. So, 1/8 of the 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? Let the total number of children be

N

. We know that the fraction of children travelling by car is

\frac{1}{8}

. So,

\frac{1}{8} \times N = 18

.

N = 18 \times 8 = 144

. 144 children took part in the survey.

The taxi sector also has an angle of

45^\circ

, so the same fraction of children use taxis. Number of children using taxis =

\frac{1}{8} \times 144 = 18

. 18 children use taxis.

(iv) By which two modes of transport are equal numbers of children travelling? Equal numbers of children correspond to sectors with equal angles. The sectors for Car and Taxi both have an angle of

45^\circ

.

Final Answer:

(i)

Bus

(ii)

\frac{1}{8}

(iii)

144 children took part in the survey; 18 children use taxis.

(iv)

Car and Taxi.

Q6Figure it Out (Section 3.6, part 2)

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?

Solution

Given: Initial number of workers (

x_1

) = 3. Time taken (

y_1

) = 4 days. One more worker joins the team.

To Find: How many days it will take the new team to finish the work.

Solution: This is a case of inverse proportion. If the number of workers increases, the time taken to complete the work decreases.

New number of workers (

x_2

) =

3 + 1 = 4

. Let the new time taken be

y_2

.

For inverse proportion,

x_1 y_1 = x_2 y_2

.

3 \times 4 = 4 \times y_2

12 = 4 \times y_2

y_2 = \frac{12}{4} = 3

It will take the team 3 days to finish the work.

Assumptions:

All workers work at the same constant rate.
The efficiency of the work does not change with the addition of a new worker (i.e., they don't get in each other's way).

Final Answer: It will take them 3 days to finish the work. The assumption is that all workers work at the same constant pace.

Q7Figure it Out (Section 3.6, part 2)

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?

Solution

Given: Time to fill 2 tanks = 6 hours.

To Find: The time it will take to fill 5 such tanks.

Solution: This is a case of direct proportion. If the number of tanks to be filled increases, the time taken will also increase.

Method 1: Unitary Method First, find the time taken to fill one tank. Time for 1 tank =

\frac{\text{Time for 2 tanks}}{2} = \frac{6 \text{ hours}}{2} = 3

hours.

Now, find the time taken to fill 5 tanks. Time for 5 tanks =

5 \times (\text{time for 1 tank}) = 5 \times 3 = 15

hours.

Method 2: Ratio and Proportion Let the time to fill 5 tanks be

x

hours. The ratio of the number of tanks is equal to the ratio of the time taken.

\frac{2}{5} = \frac{6}{x}

2 \times x = 5 \times 6

2x = 30

x = \frac{30}{2} = 15

Final Answer: It will take 15 hours to fill 5 such tanks.

Q8Figure it Out (Section 3.6, part 2)

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?

Solution

Given: Initial arrangement: 25 rows with 12 chairs per row. New arrangement: 20 chairs per row.

To Find: The number of rows in the new arrangement.

Solution: The total number of chairs is constant. This is a case of inverse proportion: if the number of chairs per row increases, the number of rows must decrease.

Step 1: Find the total number of chairs. Total chairs = (Number of rows)

\times

(Chairs per row) Total chairs =

25 \times 12 = 300

.

Step 2: Find the new number of rows. Let the new number of rows be

R

.

R \times (\text{New chairs per row}) = \text{Total chairs}

R \times 20 = 300

R = \frac{300}{20} = 15

.

Final Answer: The new arrangement has 15 rows.

Q9Figure it Out (Section 3.6, part 2)

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?

Solution

Given: Initial schedule: 8 periods, each of 45 minutes duration. New schedule: 9 periods a day. The total school hours per day remain the same.

To Find: The duration of each period in the new schedule.

Solution: This is a case of inverse proportion. Since the total school time is constant, if the number of periods increases, the duration of each period must decrease.

Step 1: Find the total duration of the school day in minutes. Total time = (Number of periods)

\times

(Duration of each period) Total time =

8 \times 45 = 360

minutes.

Step 2: Find the new duration of each period. Let the new duration be

D

minutes. (New number of periods)

\times D = \text{Total time}

9 \times D = 360

D = \frac{360}{9} = 40

minutes.

Final Answer: If the school has 9 periods a day, each period will be 40 minutes long.

Q10Figure it Out (Section 3.6, part 2)

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?

Solution

Given: Time taken by small pump to fill the tank = 3 hours. Time taken by large pump to fill the tank = 2 hours.

To Find: The time taken to fill the tank if both pumps are used together.

Solution: We can solve this by considering the rate of work of each pump. Let the total work of filling one tank be 1 unit.

Step 1: Find the rate of each pump.

Rate of small pump = $\frac{\text{Work}}{\text{Time}} = \frac{1 \text{ tank}}{3 \text{ hours}} = \frac{1}{3}$ of the tank per hour.
Rate of large pump = $\frac{1 \text{ tank}}{2 \text{ hours}} = \frac{1}{2}$ of the tank per hour.

Step 2: Find the combined rate. When both pumps work together, their rates add up. Combined rate = (Rate of small pump) + (Rate of large pump) Combined rate =

\frac{1}{3} + \frac{1}{2} = \frac{2+3}{6} = \frac{5}{6}

of the tank per hour.

Step 3: Find the time taken together. Time =

\frac{\text{Total Work}}{\text{Combined Rate}}

Time =

\frac{1}{\frac{5}{6}} = \frac{6}{5}

hours.

To convert this to hours and minutes:

\frac{6}{5}

hours =

1.2

hours = 1 hour and

0.2 \times 60

minutes = 1 hour and 12 minutes.

Final Answer: If both pumps are used together, the tank will take

\frac{6}{5}

hours (or 1.2 hours, or 1 hour and 12 minutes) to fill.

Q11Figure it Out (Section 3.6, part 2)

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?

Solution

Given: To produce a certain number of toys: Number of machines (

x_1

) = 42. Time taken (

y_1

) = 63 days. New time taken (

y_2

) = 54 days.

To Find: The number of machines required (

x_2

) to produce the same number of toys in 54 days.

Solution: This is a case of inverse proportion. To produce the same number of toys in fewer days, more machines will be required.

For inverse proportion,

x_1 y_1 = x_2 y_2

.

42 \times 63 = x_2 \times 54

x_2 = \frac{42 \times 63}{54}

We can simplify the fraction:

x_2 = \frac{42 \times 63}{54} = \frac{(6 \times 7) \times (9 \times 7)}{6 \times 9}

x_2 = 7 \times 7 = 49

.

Final Answer: 49 machines are required to produce the same number of toys in 54 days.

Q12Figure it Out (Section 3.6, part 2)

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 $80 \text{ km/h}$ ?

Solution

Given: Initial speed (

x_1

) =

60 \text{ km/h}

. Initial time taken (

y_1

) = 2 hours. New speed (

x_2

) =

80 \text{ km/h}

.

To Find: The time taken (

y_2

) if the car travels at the new speed.

Solution: This is a case of inverse proportion. Since the distance to the destination is fixed, if the speed increases, the time taken will decrease.

For inverse proportion,

x_1 y_1 = x_2 y_2

.

60 \times 2 = 80 \times y_2

120 = 80 \times y_2

y_2 = \frac{120}{80} = \frac{12}{8} = \frac{3}{2} = 1.5

1.5 hours is equal to 1 hour and 30 minutes.

Final Answer: The car will take 1.5 hours (or 1 hour and 30 minutes) to reach the destination.

Proportional Reasoning-2Class 8 Mathematics NCERT Solutions

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 :: 3:4:3:53:4:3:53:4:3:5. If each session is 150 minutes long, how much time is spent on each activity?

Solution

A school library has books in different languages in the following ratio no. of Odiya books : no. of Hindi books : no. of English books :: 3:2:13:2:13:2:1. If the library has 288 Odiya books, how many Hindi and English books does it have?

Solution

I have 100 coins in the ratio - no. of ₹10 coins : no. of ₹5 coins : no. of ₹2 coins : no. of ₹1 coins : 4:3:2:14:3:2:14:3:2:1. How much money do I have in coins?

Solution

Construct a triangle with sidelengths in the ratio 3:4:53:4:53:4:5. Will all the triangles drawn with this ratio of sidelengths be congruent to each other? Why or why not?

Solution

Can you construct a triangle with sidelengths in the ratio 1:3:51:3:51:3:5? Why or why not?

Solution

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.

Solution

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

Solution

Solution

Which of these are in inverse proportion?

| xxx | 40 | 80 | 25 | 16 | |---|---|---|---|---| | yyy | 20 | 10 | 32 | 50 |

| xxx | 40 | 80 | 25 | 16 | |---|---|---|---|---| | yyy | 20 | 10 | 12.5 | 8 |

| xxx | 30 | 90 | 150 | 10 | |---|---|---|---|---| | yyy | 15 | 5 | 3 | 45 |

Solution

Fill in the empty cells if xxx and yyy are in inverse proportion. xxx161236yyy948

Solution

Which of the following pairs of quantities are in inverse proportion?

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

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

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

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

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

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

Solution

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

Solution

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?

Solution

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

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.

What is the most common mode of transport?

What fraction of children travel by car?

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

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

Solution

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?

Solution

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?

Solution

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?

Solution

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?

Solution

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?

Solution

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?

Solution

A car takes 2 hours to reach a destination, travelling at a speed of 60 km/h60 \text{ km/h}60 km/h. How long will the car take if it travels at a speed of 80 km/h80 \text{ km/h}80 km/h?

Solution

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 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?

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?

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?

| $x$ | 40 | 80 | 25 | 16 | |---|---|---|---|---| | $y$ | 20 | 10 | 32 | 50 |

| $x$ | 40 | 80 | 25 | 16 | |---|---|---|---|---| | $y$ | 20 | 10 | 12.5 | 8 |

| $x$ | 30 | 90 | 150 | 10 | |---|---|---|---|---| | $y$ | 15 | 5 | 3 | 45 |

Fill in the empty cells if $x$ and $y$ are in inverse proportion.
$x$ 16 12 36
$y$ 9 48

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 $80 \text{ km/h}$ ?