MATHEMATICS CLASS- 8
CHAPTER-7 (PROPORTIONAL REASONING -1 )
Figure it out
1. Circle the following statements of proportion that are true.
(i) 4 : 7 :: 12 : 21
4
7
=
12
21
=
4
7
True ✓
(ii) 8 : 3 :: 24 : 6
8
3
≠
24
6
False ✗
(iii) 7 : 12 :: 12 : 7
7
12
≠
12
7
False ✗
(iv) 21 : 6 :: 35 : 10
21
6
=
35
10
=
7
2
True ✓
(v) 12 : 18 :: 28 : 12
12
18
=
2
3
28
12
=
7
3
Since
2
3
≠
7
3
False ✗
(vi) 24 : 8 :: 9 : 3
24
8
=
9
3
= 3
True ✓
Answer: The true statements of proportion are (i), (iv) and (vi).
2. Give 3 ratios that are proportional to 4 : 9.
Multiply both terms of 4 : 9 by the same number.
4 : 9 × 2 = 8 : 18
4 : 9 × 3 = 12 : 27
4 : 9 × 4 = 16 : 36
Answer: 8 : 18, 12 : 27, 16 : 36
3. Fill in the missing numbers for these ratios that are proportional to 18 : 24.
18 : 24 = 18 24 = 3 4
3 : 4
12 : 16
20 :
80
3
27 : 36
Answer:
3 : 4
12 : 16
20 :
80
3
27 : 36
Question 4. Look at the following rectangles. Which rectangles are similar to each other?
To find whether two rectangles are similar, compare their width : height ratios.
Measure the width and height of each rectangle using a ruler. Then write the ratio of width to height for each rectangle.
If two rectangles have the same width : height ratio, they are similar rectangles. If the ratios are different, the rectangles are not similar.
After measuring and comparing the ratios, we observe that rectangles A and E have the same ratio. Similarly, rectangles C and D also have the same ratio.
Rectangle B has a different ratio from the others, so it does not belong to either group.
Therefore, one group of similar rectangles is A and E, and another group is C and D.
Question 5. Look at the following rectangle. Can you draw a smaller rectangle and a bigger rectangle with the same width to height ratio?
Yes. A smaller rectangle and a bigger rectangle can be drawn by keeping the width : height ratio the same.
For example, if a rectangle has width : height = 4 : 3, then:
Smaller rectangle: 8 cm × 6 cm
Given rectangle: 12 cm × 9 cm
Bigger rectangle: 16 cm × 12 cm
In all three rectangles, the ratio remains 4 : 3. Therefore, they are similar rectangles.
When you compare your drawings with those of your classmates, all drawings may not look exactly the same. This is because different students may choose different dimensions.
However, the drawings are not wrong if the width : height ratio remains the same. Rectangles of different sizes can still be similar as long as their corresponding sides are proportional.
Thus, many correct answers are possible for this question because there can be many rectangles having the same width : height ratio.
Question 6. The following figure shows a small portion of a long brick wall with patterns made using coloured bricks. Each wall continues this pattern throughout the wall. What is the ratio of grey bricks to coloured bricks? Try to give the ratios in their simplest form.
(a)
Observe one complete repeating pattern in the wall.
In one pattern, the coloured design is made using 6 red bricks. Counting the grey bricks in the same repeating section gives 18 grey bricks.
Therefore,
Grey bricks : Coloured bricks = 18 : 6
Divide both terms by 6.
Grey bricks : Coloured bricks = 3 : 1
Hence, the ratio of grey bricks to coloured bricks in the simplest form is 3 : 1.
(b)
Again, look at one complete repeating pattern of the design.
The coloured pattern contains 12 coloured bricks. Counting the grey bricks in the corresponding repeating section gives 24 grey bricks.
Therefore,
Grey bricks : Coloured bricks = 24 : 12
Divide both terms by 12.
Grey bricks : Coloured bricks = 2 : 1
Hence, the ratio of grey bricks to coloured bricks in the simplest form is 2 : 1.
Answer:
(a) Grey bricks : Coloured bricks = 3 : 1
(b) Grey bricks : Coloured bricks = 2 : 1
Question 7. Let us draw some human figures. Measure your friend's body—the lengths of their head, torso, arms, and legs. Write the ratios as mentioned below. Now draw a figure with head, torso, arms, and legs with equivalent ratios as above.
This is an activity-based question. The measurements will vary from one person to another. Therefore, different students may get different answers.
Let us take an example. Suppose the measurements are:
Head = 12 cm
Torso = 24 cm
Arms = 18 cm
Legs = 30 cm
Then,
Head : Torso = 12 : 24 = 1 : 2
Torso : Arms = 24 : 18 = 4 : 3
Torso : Legs = 24 : 30 = 4 : 5
Using these ratios, we can draw a larger or smaller human figure. While drawing, we must keep the same proportions between the head, torso, arms, and legs.
For example, if every measurement is doubled, the new figure may have:
Head = 24 cm
Torso = 48 cm
Arms = 36 cm
Legs = 60 cm
The ratios remain:
Head : Torso = 1 : 2
Torso : Arms = 4 : 3
Torso : Legs = 4 : 5
Since the ratios are unchanged, the new figure is similar to the original figure. Different students may obtain different measurements and ratios depending on the person they choose to measure.
Figure it out
Question 1. The Earth travels approximately 940 million kilometres around the Sun in a year. How many kilometres will it travel in a week?
The Earth travels approximately 940 million kilometres in 1 year.
We know that 1 year = 52 weeks.
Therefore, the distance travelled in 1 week is:
Distance in 1 week = 940,000,000 ÷ 52
= 18,076,923.08 km (approximately)
Thus, the Earth travels approximately 18.08 million kilometres in a week.
Answer: The Earth travels approximately 18.08 million kilometres in a week.
Question 2. A mason is building a house in the shape shown in the diagram. He needs to construct both the outer walls and the inner wall that separates two rooms. To build a wall of 10 feet, he requires approximately 1450 bricks. How many bricks would he need to build the house? Assume all walls are of the same height and thickness.
To find the number of bricks required, we first calculate the total length of all the walls.
The house consists of:
- Outer walls
- One inner wall that separates the two rooms
From the figure:
Width of the main rectangular part = 9 ft + 15 ft = 24 ft
Height of the main rectangular part = 12 ft
Length of the lower rectangular extension = 9 ft
Width of the lower rectangular extension = 6 ft
First, find the total length of the outer boundary.
Top side = 24 ft
Right side of main rectangle = 12 ft
Bottom of main rectangle excluding extension = 24 − 6 = 18 ft
Two vertical sides of extension = 9 ft + 9 ft = 18 ft
Bottom side of extension = 6 ft
Left side of main rectangle = 12 ft
Therefore,
Outer wall length = 24 + 12 + 18 + 18 + 6 + 12
= 90 ft
Now add the inner wall.
Length of inner wall = 12 ft
Total wall length = 90 + 12
= 102 ft
We are given that a wall of 10 ft requires 1450 bricks.
Therefore, a wall of 1 ft requires
1450 ÷ 10 = 145 bricks
Hence, a wall of 102 ft requires
102 × 145
= 14790 bricks
Figure it out
Question 1. Divide ₹4,500 into two parts in the ratio 2 : 3.
The ratio of the two parts is 2 : 3.
Total parts = 2 + 3 = 5
Value of one part = ₹4500 ÷ 5 = ₹900
First part = 2 × ₹900 = ₹1800
Second part = 3 × ₹900 = ₹2700
Answer: The two parts are ₹1800 and ₹2700.
Question 2. In a science lab, acid and water are mixed in the ratio of 1 : 5 to make a solution. In a bottle that has 240 mL of the solution, how much acid and water does the solution contain?
Acid : Water = 1 : 5
Total parts = 1 + 5 = 6
Value of one part = 240 ÷ 6 = 40 mL
Amount of acid = 1 × 40 = 40 mL
Amount of water = 5 × 40 = 200 mL
Answer: The solution contains 40 mL of acid and 200 mL of water.
Question 3. Blue and yellow paints are mixed in the ratio of 3 : 5 to produce green paint. To produce 40 mL of green paint, how much of these two colours are needed? To make the paint a lighter shade of green, I added 20 mL of yellow to the mixture. What is the new ratio of blue and yellow in the paint?
Blue : Yellow = 3 : 5
Total parts = 3 + 5 = 8
Value of one part = 40 ÷ 8 = 5 mL
Amount of blue paint = 3 × 5 = 15 mL
Amount of yellow paint = 5 × 5 = 25 mL
Therefore, 40 mL of green paint requires 15 mL blue paint and 25 mL yellow paint.
Now, 20 mL of yellow paint is added.
New amount of yellow paint = 25 + 20 = 45 mL
Blue paint remains 15 mL.
New ratio of blue : yellow = 15 : 45
Dividing both terms by 15,
15 : 45 = 1 : 3
Answer:
Blue paint = 15 mL
Yellow paint = 25 mL
New ratio of blue : yellow = 1 : 3
Question 4. To make soft idlis, you need to mix rice and urad dal in the ratio of 2 : 1. If you need 6 cups of this mixture to make idlis tomorrow morning, how many cups of rice and urad dal will you need?
Rice : Urad dal = 2 : 1
Total parts = 2 + 1 = 3
Value of one part = 6 ÷ 3 = 2 cups
Rice required = 2 × 2 = 4 cups
Urad dal required = 1 × 2 = 2 cups
Answer: To make 6 cups of mixture, we need 4 cups of rice and 2 cups of urad dal.
Question 5. I have one bucket of orange paint that I made by mixing red and yellow paints in the ratio of 3 : 5. I added another bucket of yellow paint to this mixture. What is the ratio of red paint to yellow paint in the new mixture?
Initially, red : yellow = 3 : 5
Assume one bucket contains 8 equal parts.
Then red paint = 3 parts and yellow paint = 5 parts.
One more bucket of yellow paint is added. This bucket has the same quantity as the original bucket, that is, 8 parts.
New amount of yellow paint = 5 + 8 = 13 parts
Red paint remains 3 parts.
Therefore, the new ratio of red paint : yellow paint is
3 : 13
Answer: The ratio of red paint to yellow paint in the new mixture is 3 : 13.
Answer: The mason would need approximately 14,790 bricks to build all the outer walls and the inner wall of the house.
Figure it out
Question 1. Anagh mixes 600 mL of orange juice with 900 mL of apple juice to make a fruit drink. Write the ratio of orange juice to apple juice in its simplest form.
Orange juice : Apple juice = 600 : 900
To simplify the ratio, divide both numbers by their greatest common factor, 300.
600 ÷ 300 = 2
900 ÷ 300 = 3
Therefore,
Orange juice : Apple juice = 2 : 3
Answer: The simplest ratio of orange juice to apple juice is 2 : 3.
Question 2. Last year, we hired 3 buses for the school trip. We had a total of 162 students and teachers who went on that trip and all the buses were full. This year we have 204 students. How many buses will we need? Will all the buses be full?
Last year, 162 people travelled in 3 buses.
Number of people per bus = 162 3 = 54
Therefore, each bus can carry 54 people.
This year there are 204 students.
Number of buses required = 204 54 ≈ 3.78
Since a fraction of a bus cannot be hired, we need 4 buses.
Capacity of 4 buses = 4 × 54 = 216 people.
Empty seats = 216 − 204 = 12
Answer: We need 4 buses. All the buses will not be full because 12 seats will remain vacant.
Question 3. The area of Delhi is 1,484 sq. km and the area of Mumbai is 550 sq. km. The population of Delhi is approximately 30 million and that of Mumbai is 20 million people. Which city is more crowded? Why do you say so?
To compare how crowded the cities are, we calculate the population per square kilometre.
For Delhi:
Population density = 30,000,000 1484 ≈ 20,216 persons per sq. km
For Mumbai:
Population density = 20,000,000 550 ≈ 36,364 persons per sq. km
Mumbai has more people living in each square kilometre than Delhi.
Therefore, Mumbai is more crowded.
Answer: Mumbai is more crowded because its population density is higher than that of Delhi.
Question 4. A crane of height 155 cm has its neck and the rest of its body in the ratio 4 : 6. For your height, if your neck and the rest of the body also had this ratio, how tall would your neck be?
The ratio of neck : rest of body is 4 : 6.
Total parts = 4 + 6 = 10
This means the neck is 4 10 of the total height.
The neck therefore forms 2 5 of the total height.
Suppose your height is H cm.
Then the length of your neck would be
H × 2 5 cm
For example, if your height is 150 cm,
Neck length = 150 × 2 5 = 60 cm
Answer: Your neck would be 2 5 of your total height .
Question 5. Let us try an ancient problem from Lilavati. If 2½ palas of saffron costs 3⁄7 niskas, what quantity of saffron can be bought for 9 niskas?
Given:
2½ palas of saffron cost 3 7 niskas.
2½ palas = 5 2 palas
Quantity of saffron for 1 niska:
= 5 2 ÷ 3 7
= 5 2 × 7 3
= 35 6 palas
For 9 niskas:
9 × 35 6 = 315 6 = 52½ palas
Answer: For 9 niskas, one can buy 52½ palas of saffron.
Question 6. Harmain is a 1-year-old girl. Her elder brother is 5 years old. What will be Harmain's age when the ratio of her age to her brother's age is 1 : 2?
At present:
Harmain's age = 1 year
Brother's age = 5 years
Let after x years, the ratio becomes 1 : 2.
Then,
Harmain's age = (1 + x) years
Brother's age = (5 + x) years
According to the question,
(1 + x) : (5 + x) = 1 : 2
2(1 + x) = 5 + x
2 + 2x = 5 + x
x = 3
Therefore, after 3 years,
Harmain's age = 1 + 3 = 4 years
Brother's age = 5 + 3 = 8 years
The ratio becomes 4 : 8 = 1 : 2.
Answer: Harmain will be 4 years old when the ratio of her age to her brother's age becomes 1 : 2.
Question 7. The mass of equal volumes of gold and water are in the ratio 37 : 2. If 1 litre of water is 1 kg in mass, what is the mass of 1 litre of gold?
Equal volumes of gold and water have masses in the ratio 37 : 2.
Gold : Water = 37 : 2
We are given that 1 litre of water has a mass of 1 kg.
Therefore, 2 parts correspond to 1 kg.
Mass corresponding to 1 part = 1 2 kg
Mass of gold = 37 parts
= 37 × 1 2 kg
= 37 2 kg
= 18.5 kg
Answer: The mass of 1 litre of gold is 18.5 kg.
Question 8. It is good farming practice to apply 10 tonnes of cow manure for 1 acre of land. A farmer is planning to grow tomatoes in a plot of size 200 ft by 500 ft. How much manure should he buy?
Area of the plot
= 200 ft × 500 ft
= 100000 square feet
We know that
1 acre = 43560 square feet
Therefore, area of the plot in acres
= 100000 43560
≈ 2.295 acres
For 1 acre, manure required = 10 tonnes
For 2.295 acres,
Manure required = 2.295 × 10
= 22.95 tonnes
Answer: The farmer should buy approximately 23 tonnes of cow manure.
Question 9. A tap takes 15 seconds to fill a mug of water. The volume of the mug is 500 mL. How much time does the same tap take to fill a bucket of water if the bucket has a 10-litre capacity?
Volume of the mug = 500 mL
Time taken to fill the mug = 15 seconds
Bucket capacity = 10 litres
= 10000 mL
The bucket can hold
= 10000 500
= 20 mugs of water
If one mug takes 15 seconds, then 20 mugs will take
20 × 15 = 300 seconds
Convert seconds into minutes:
300 seconds = 5 minutes
Answer: The tap will take 5 minutes to fill the bucket.
Question 10. One acre of land costs ₹15,00,000. What is the cost of 2,400 square feet of the same land?
We know that
1 acre = 43,560 square feet
Cost of 43,560 square feet = ₹15,00,000
Therefore, cost of 1 square foot
= 1500000 43560
≈ ₹34.44
Cost of 2,400 square feet
= 2400 × 34.44
≈ ₹82,644.63
Answer: The cost of 2,400 square feet of land is approximately ₹82,645.
Question 11. A tractor can plough the same area of a field 4 times faster than a pair of oxen. A farmer wants to plough his 20-acre field. A pair of oxen takes 6 hours to plough an acre of land. How much time would it take if the farmer used a pair of oxen to plough the field? How much time would it take if he decides to use a tractor instead?
A pair of oxen takes 6 hours to plough 1 acre.
For 20 acres,
Time taken by oxen = 20 × 6
= 120 hours
Therefore, the pair of oxen will take 120 hours to plough the entire field.
The tractor is 4 times faster than the oxen.
Hence, the tractor will take
= 120 4
= 30 hours
Answer:
Using a pair of oxen: 120 hours
Using a tractor: 30 hours
Question 12. The ₹10 coin is an alloy of copper and nickel called 'cupro-nickel'. Copper and nickel are mixed in a 3 : 1 ratio to get this alloy. The mass of the coin is 7.74 grams. If the cost of copper is ₹906 per kg and the cost of nickel is ₹1,341 per kg, what is the cost of these metals in a ₹10 coin?
Copper : Nickel = 3 : 1
Total parts = 3 + 1 = 4
Mass of the coin = 7.74 g
Mass of copper
= 3 4 × 7.74
= 5.805 g
Mass of nickel
= 1 4 × 7.74
= 1.935 g
Convert grams into kilograms:
Copper = 0.005805 kg
Nickel = 0.001935 kg
Cost of copper
= 0.005805 × 906
≈ ₹5.26
Cost of nickel
= 0.001935 × 1341
≈ ₹2.60
Total cost of metals
= ₹5.26 + ₹2.60
≈ ₹7.86
Answer: The approximate cost of copper and nickel present in the ₹10 coin is ₹7.86.