Start practicing the Class 8 Ganita Prakash Solutions and Part 2 Chapter 3 Proportional Reasoning 2 Class 8 Question Answer to consolidate your knowledge effectively.
Class 8 Maths Ganita Prakash Part 2 Chapter 3 Solutions
Class 8 Maths Proportional Reasoning 2 Solutions
Class 8 Ganita Prakash Part 2 Chapter 3 Solutions Proportional Reasoning 2
Ratios in Maps (NCERT Page 56-57)
Question 1.
Convert 60,00,000 cm to kilometres. It is 60 km. Verify this.
Solution:
Given, 6000000 cm
We know that 1 km = 1000 m
1 m = 100 cm
So, 1 km = 1000 × 100 = 100000 cm = 105 cm
Now, 6000000 cm = 6000000 + 100000 = 60 km
Hence, 6000000 cm = 60 km
Question 2.
Using the map above, can you find the geographical distance between Bengaluru and Chennai? Also, find the geographical distance between Mangaluru and Chennai.
[Hint Use a ruler to find the distance between the cities on the map. Then, use the ratio given on the map to find the actual geographical distance.]
Solution:
Do yourself
Question 3.
Try to find the distances between the same two pairs of cities with different maps that have different scales (ratios). Do they all give the same geographical distance, approximately?
Solution:
Do yourself
Figure it Out (NCERT Page 60)
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?
Solution:
Given ratio,
Warm-up : Batting : Bowling : Fielding:: 3: 4 : 3 : 5
and total session time =150 minutes
∵ Sum of ratio parts = 3 + 4 + 3 + 5 = 15
Each part = 150 + 15 = 10 minutes
So, time spent on
Warm-up = 3 × 10 = 30 minutes
Batting = 4 × 10 = 40 minutes
Bowling = 3 × 10 = 30 minutes
Fielding = 5 × 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?
Solution:
Given ratio, Odiya: Hindi: English:: 3 : 2 : 1
There are 288 Odiya books, which corresponds to 3 ratio parts.
So, the value of one ratio part = \(\frac{288}{3}\) = 96 books/ part
Now, Hindi 2 parts = 2 × 96 = 192 books
English = 1 part = 1 × 96 = 96 books
So, the library has 192 Hindi books and 96 English books.
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?
Solution:
Given ratio for coins,
₹ 10 coins: ₹ 5 coins: ₹ 2 coins : ₹ 1 coins:: 4 : 3 : 2 : 1 and total coins = 100
Then, total ratio parts = 4 + 3 + 2 + 1 = 10 parts
So, 1 part = 100 + 10 = 10 coins
Now number of coins
₹ 10 coins = 4 × 10 = 40 coins
₹ 5 coins = 3 × 10 = 30 coins
₹ 2 coins = 2 × 10 = 20 coins
₹ 1 coins = 1 × 10 = 10 coins
Now total money = (40 × 10) + (30 × 5) + (20 × 2) + (10 × 1) = 400 + 150 + 40 + 10 = ₹ 600
So, I have total ₹ 600.
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?
Solution:
All triangles drawn with side lengths in the ratio 3:4:5 will not be congruent, they will be congruent only if their sizes (scale) are the same.
Question 5.
Can you construct a triangle with sidelengths in the ratio 1:3:5? Why or why not?
Solution:
By triangle inequality rule,
Sum of smaller two sides > largest side.
Now, 1 + 3 = 4
But 4 is not greater than 5, so triangle cannot be formed.
So, triangle with ratio 1:3: 5 cannot be constructed.
Figure it Out (NCERT Page 62-63)
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.
Solution:
Given, total number of people = 360
Number of people who like summer season = 90
Number of people who like rainy season = 120
Number of people who like winter = 360 – (90 + 120) = 150
Since, the total angle of a circle is 360°.
So, angle for summer season
= \(\frac{\text { Number of people who like summer }}{\text { Total people }}\) × 360°
= \(\frac{90}{360^{\circ}}\) × 360° = 90°
angle for rainy season
= \(\frac{\text { Number of people who like rainy season }}{\text { Total people }}\) × 360°
= \(\frac{120}{360^{\circ}}\) × 360° = 120°
angle for winter
= \(\frac{\text { Number of people who like winter }}{\text { Total people }}\) × 360° = 120°
= \(\frac{150}{360^{\circ}}\) × 360° = 120°
Hence, here is the pie chart representation.
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%.
Solution:
Given, entertainment—50% Sports—25%
News—15%
Information—10%
Since, total angle of a circle is 360°.
So, angle for entertainment = 50% of 360°
= \(\frac{50}{100}\) × 360° = 180°
angle for sports = 25% of 360° = \(\frac{25}{100}\) × 360° = 90°
angle for news = 15% of 360° = \(\frac{15}{100}\) × 360° = 54°
angle for information = 10% of 360° = \(\frac{10}{100}\) × 360° = 36°
Hence, here is the pie chart representation.
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.
Solution:
Do yourself
Figure it Out (NCERT Page 65)
Question 1.
Which of these are in inverse proportion?
Solution:
We know that two quantities are in inverse proportion, if x × y = constant (k)
(i) On checking the product of x and y, we get
- 40 × 20 = 800
- 80 × 10 = 800
- 25 × 32 = 800
- 16 × 50 = 800
Since, product is constant (k = 800).
Hence, this is in inverse proportion.
(ii) On checking the products of x and y, we get
- 40 × 20 = 800
- 80 × 10 = 800
- 25 × 125 = 3125
- 16 × 8 = 128
Here, the products are not constant.
Hence, this is not in inverse proportion.
(iii) On checking the products of x and y, we get
- 30 × 15 = 450
- 90 × 5 = 450
- 150 × 3 = 450
- 10 × 45 = 450
Here, product is constant (k = 450).
Hence, this is inverse proportion.
Question 2.
Fill in the empty cells, if x and y are in inverse proportion.
Solution:
From the first pair
x × y = 16 × 9 = 144
So, constant k = 144
(a) When x = 12
12 × y = 144
⇒ y = \(\frac{144}{12}\) = 12
(b) When y = 48
x × 48 = 144
⇒ x = \(\frac{144}{48}\)
(c) When x = 36
36 × y = 144 [from Eq. (i)]
⇒ y = \(\frac{144}{36}\) = 4
Hence, here is the complete table.
Figure it Out (NCERT Page 67-68)
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.
Solution:
We know that two quantities are in inverse proportion when if one increases, the other decreases and the product of both quantities remains constant or when one
changes by a factor n, the other changes by
(i) Statement Number of taps filling a water tank and the time taken to fill it.
Reasoning
More taps → tank fills faster —Hess time.
If taps double, time becomes half.
Hence, this pair of quantities is in inverse proportion.
(ii) Statement Number of painters hired and days needed to paint a wall of fixed size.
Reasoning
More painters → fewer days.
If painters increase by factor n, days decrease by factor.
Hence, this pair of quantities is in inverse proportion.
(iii) Statement Distance a car can travel and the amount of petrol in the tank.
Reasoning
More petrol → more distance.
Both increase in the same direction (directly proportional), not opposite.
Hence, this pair of quantities is not in inverse proportion.
(iv) Statement Speed of a cyclist and the time taken to cover a fixed route.
Reasoning
Higher speed → less time.
Speed × Time = Distance (constant)
Hence, this pair of quantities is in inverse proportion.
(v) Statement Length of cloth bought and price paid at fixed rate per metre.
Reasoning
More length → more cost.
Both increase together (direct proportion).
Hence, this pair of quantities is not in inverse proportion.
(vi) Statement Number of pages in a book and the time required to read it at fixed speed
Reasoning
More pages → more time needed to read.
Both increase in same direction (direct proportion).
Hence, this pair of quantities is not in inverse proportion.
Question 2.
If 24 pencils cost ₹ 120, how much will 20 such pencils cost?
Solution:
Given, the cost of 24 pencils is ₹ 120.
Let the cost of 20 pencils be ₹ x
This is a case of direct proportion because
More pencils → more cost
Fewer pencils → less cost
So, \(\frac{\text { Cost }}{\text { Quantity }}\) = constant
Using direct proportion, we get
\(\frac{120}{24}=\frac{x}{20}\)
⇒ 120 × 20 = 24x
⇒ 2400 = 24x
So, x = \(\frac{2400}{24}\) = 100
Hence, the cost of 20 pencils will be ₹ 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?
Solution:
Given, the number of family increases from 20 to 30.
Let now the water will last for x days.
This is a case of inverse proportion because,
More families → Fewer days water lasts Using inverse proportion, we get
20 × 6 = 30 × x
⇒ 120 = 30x
⇒ x = = 4
Hence, the water will last for 4 days.
The assumptions needed are that each family consumes water at the same constant rate, the rate does not change and the tank is not refilled.
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.
Solution:
We know that there are 24 hours in a day.
By matching the visual proportional of the shaded areas to the list of provided hours (15, 2.5, 20, 8, 3.5, 13, 10.5, 18), we get the following results.
| Living Being | Sleep (hours/day) |
| Giraffe | 2.5 hours |
| Elephant | 3.5 hours |
| Human | 8 hours |
| Dog | 10.5 hours |
| Cat | 13 hours |
| Sloth | 15 hours |
| Snake | 18 hours |
| Bat | 20 hours |
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?
Solution:
(i) The mode with the largest angle represents the most common transport.
From the pie chart the largest angle is 120°, which represents to bus.
So, most common mode of transport is bus.
(ii) The angle of car = 360° – (90° + 60° + 60° + 120°)
= 30°
So, fraction = \(\frac{\text { Angle for car }}{360^{\circ}}\)
= \(\frac{30^{\circ}}{360^{\circ}}=\frac{1}{12}\)
(iii) We have,
Fraction for Car = \(\frac{1}{12}\)
Let total children = x
So, \(\frac{1}{12}\) x = 18
⇒ x = 18 × 12 = 216
So, total children in the survey is 216.
There is no category for taxi in the provided pie chart. So, not a single child use taxi to travel to school.
(iv) From the pie chart,
Angle of cycle = Angle of two wheeler = 60°
Hence, cycle and two-wheeler are the two modes of transport by which an equal number of children are travelling.
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?
Solution:
Given,
Workers = 3 and days = 4
New workers = 4
Let required days = x
Using inverse proportion, we get 3 × 4 = 4 × x
⇒ 12 = 4x
⇒ x = \(\frac{12}{4}\) = 3
Hence, the work will be completed in 3 days
Assumptions
- All workers work at same rate.
- Work done is uniform every day.
- No worker is absent.
- No improvement or reduction in efficiency.
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?
Solution:
Given, tanks = 2 and time = 6 hours
Now, tanks = 5
Let the required time be x hours.
More tanks → More hours (direct proportion)
Using direct proportion, we get
\(\frac{2}{6}=\frac{5}{x}\)
⇒ x = 6 × \(\frac{5}{2}\)
⇒ x = 6 × 2.5 = 15
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?
Solution:
Given, number of rows = 25,
chairs in each row = 12 and
rearranged chairs in each row = 20
Let the number of rows in the new arrangement be x.
If the chairs per row increases, the number of rows decreases.
So, these quantities are inversely proportional.
∴ 25 × 12 =20 × x
⇒ 300 = 20x
⇒ x = \(\frac{300}{20}\) = 15
Hence, the new arrangement has 15 rows.
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?
Solution:
Since, total school time is fixed.
∵ Number of periods T duration of each period ↓
So, these two quantities are inversely proportional.
Let new period duration be x.
⇒ 8 × 45 = 9 × x
⇒ 360 = 9x
⇒ x = \(\frac{360}{9}\) = 40
Hence, the duration of each period is 40 minutes.
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?
Solution:
Given, time taken by small pump to fill tank = 3 hours
and time taken by large pump to fill tank = 2 hours
Let the time taken to fill the tank using both pumps together be x hours.
Consider the task of filling the tank as 1 unit of work.
Work done by small pump in 1 hour = \(\frac{1}{3}\)
Work done by large pump in 1 hour = \(\frac{1}{2}\)
Work done together in 1 hour = \(\frac{1}{3}+\frac{1}{2}=\frac{2}{6}+\frac{3}{6}=\frac{5}{6}\)
This means \(\frac{5}{6}\) of the tank is filled in 1 hour.
Time taken to fill the whole tank x = \(\frac{1}{(5 / 6)}=\frac{6}{5}\) hours
Hence, time taken to fill the tank if both pumps are used together = 1.2 hours
= 1 hour 12 minutes
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?
Solution:
Given, machines = 42
Days = 63
New days = 54
Let required number of machines = x
Machines ↑ Days ↓ (inverse proportion)
So, 42 × 63 = x × 54
⇒ 2646 = 54x
⇒ x = \(\frac{2646}{54}\)
⇒ x = 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 km/h. How long will the car take if it travels at a speed of 80 km/h?
Solution:
Given,
Speed of car = 60 km/h
Time taken = 2 hours
New speed = 80 km/h
Let required time = x hours
∵ Speed ↑ Time ↓ (inverse proportion)
∴ 60 × 2 = 80 × x
⇒ 120 = 80x
⇒ x = \(\frac{120}{80}\)
⇒ x = 1.5 hours
= 1.5 hours + 0.5 hour
= 1 hour + \(\frac{1}{2}\) hour
= 1 hour 30 minutes
Hence, the car will take 1 hour and 30 minutes to reach the destination at a speed of 80 km/h.