NCERT Solutions Class 8 Maths Chapter 07 Proportional Reasoning 1

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Question. Which images look similar and which ones look different?
Answer: Images A, C, and D look similar because they all have the same height - width ratio, so their appearance stays normal and consistent with each other. Images B and E look different because image B appears stretched or wider, while image E appears taller or thinner.
In simple words: Three images (A, C, D) have the same shape because their width and height match the same pattern. Two images (B, E) have different shapes because one is too wide and the other is too tall.

Exam Tip: When comparing figures for similarity, always check if the ratio of width to height is the same - if the ratios match, the shapes are similar.

Question. Do images B and E look like the other three images? Why?
Answer: No, images B and E do not look like the other three images. Images A, C, and D look similar in shape because their height - width ratio is the same. However, in image B, the tiger looks a bit wide and stretched. In image E, the tiger looks a little tall and thin. So their shapes do not match the proportion of the other three, which is why B and E look different.
In simple words: A, C, and D have matching shapes because their width to height ratio is the same. B and E have different shapes because their ratios changed.

Exam Tip: Always simplify the ratio and compare - if the simplest forms match, the shapes are similar regardless of their actual size.

Question. What makes images A, C, and D appear similar, and B and E different?
Answer: The height to width ratio of images A, C, and D come out to be the same after simplifying. Because their ratios match, the tiger in these images keeps the same shape - not stretched, not compressed - so they look similar. Image B has a height - width ratio that becomes smaller, which means it looks wider sideways. Image E has a ratio that becomes larger, so it looks taller vertically. Since their ratios do not match the ratio of A, C, and D, the shapes get distorted, which makes B and E look different.
In simple words: When two ratios are the same, shapes look the same. When ratios are different, shapes look stretched or squashed.

Exam Tip: Use the simplified form of ratios to compare - two ratios are equal only if their simplest forms match exactly.

Question. Can you check by what factors the width and height of image D change as compared to image A? Are the factors the same?
Answer: Width of A = 60 mm
Width of D = 90 mm
Width factor = 90 ÷ 60 = 3/2 = 1.5

Height of A = 40 mm
Height of D = 60 mm
Height factor = 60 ÷ 40 = 3/2 = 1.5

Both factors are the same (3/2). So, the width and height of image D change by the same factor when compared to image A.
In simple words: When you make an image bigger or smaller, both width and height must grow or shrink by the same amount to keep the shape the same.

Exam Tip: For similar figures, the scaling factor must be identical for all dimensions - if width scales by a factor but height scales by a different factor, the figure will be distorted.

Question. By what factor should we multiply the ratio 60 : 40 (image A) to get 90 : 60 (image D)?
Answer: Ratio of image A = 60 : 40
Ratio of image D = 90 : 60

To find the factor, compare any one pair of corresponding terms:
Width factor = 90 ÷ 60 = 3/2
Height factor = 60 ÷ 40 = 3/2

Since both give the same value: Factor = 3/2 (which is 1.5)

So, we should multiply both terms of 60 : 40 by 3/2 to get 90 : 60.
In simple words: To get a new ratio, multiply both numbers in the original ratio by the same factor.

Exam Tip: When finding the scaling factor, divide any term in the new ratio by its corresponding term in the original ratio - the answer will tell you what to multiply by.

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Question. Example 1: Are the ratios 3 : 4 and 72 : 96 proportional?
Answer: Yes, the ratios 3 : 4 and 72 : 96 are proportional.

First ratio = 3 : 4 (already in simplest form)

Second ratio = 72 : 96
Find the HCF of 72 and 96:
HCF = 24

Now divide both terms by 24:
72 ÷ 24 = 3
96 ÷ 24 = 4

So, the simplest form of 72 : 96 is also 3 : 4.
Both ratios become 3 : 4 in simplest form.
In simple words: To check if two ratios are equal, reduce both to their simplest form using HCF - if they match, the ratios are proportional.

Exam Tip: Always simplify both ratios completely before comparing them - use HCF to divide both terms.

Question. What is the HCF of 72 and 96?
Answer: Factors of 72 → 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
Factors of 96 → 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96
Common factors → 1, 2, 3, 4, 6, 8, 12, 24
The largest of these is 24.
So, HCF of 72 and 96 = 24.
In simple words: List all factors of both numbers, find which ones match, and pick the biggest one.

Exam Tip: To find HCF quickly, identify common prime factors and multiply them together - this method is faster than listing all factors.

Question. Example 2: Kesang wanted to make lemonade for a celebration. She made 6 glasses of lemonade in a vessel and added 10 spoons of sugar to the drink. Her father expected more people to join the celebration. So he asked her to make 18 more glasses of lemonade. To make the lemonade with the same sweetness, how many spoons of sugar should she add?
Answer: Kesang's original mixture: 6 glasses of lemonade → 10 spoons of sugar

Now she needs 18 more glasses of lemonade.

To keep the same sweetness, the ratios must be proportional:
6 : 10 :: 18 : ?

First term changes from 6 to 18.
Factor = 18 ÷ 6 = 3

Multiply sugar by the same factor:
10 × 3 = 30

So, she should add 30 spoons of sugar to make 18 more glasses of lemonade with the same sweetness.
In simple words: When you make more lemonade, multiply the amount of sugar by the same factor you used for the glasses.

Exam Tip: Always find the factor of change first by dividing the new quantity by the old quantity, then apply the same factor to all other ingredients.

Question. How can we find the factor of change in the ratio?
Answer: To find the factor of change in a ratio, we compare how one term changes to the corresponding term in the new ratio. Take the first terms of the two ratios and divide:

Factor of change = (new first term) ÷ (old first term)

Example from the lemonade problem:
Old ratio → 6 : 10
New ratio → 18 : ?
To find the factor: 18 ÷ 6 = 3
This number (3) is the factor of change.

Then we multiply the second term by the same factor.
So, we find the factor of change by dividing the new value by the original value.
In simple words: Divide the new amount by the old amount to get the factor of change - then multiply every other quantity by this factor.

Exam Tip: The factor of change works for any ratio problem - always apply it to all corresponding terms to maintain proportionality.

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Question. Example 3: Nitin and Hari were constructing a compound wall around their house. Nitin was building the longer side, 60 ft in length, and Hari was building the shorter side, 40 ft in length. Nitin used 3 bags of cement but Hari used only 2 bags of cement. Nitin was worried that the wall Hari built would not be as strong as the wall he built because she used less cement. Is Nitin correct in his thinking?
Answer: Given:
Nitin's wall → 60 ft, 3 bags of cement
Hari's wall → 40 ft, 2 bags of cement

Nitin thinks Hari used less cement, so her wall might be weaker. But we must check whether both used cement in the same proportion.

Nitin's ratio (length : cement): 60 : 3
Hari's ratio (length : cement): 40 : 2

For Nitin: 60 ÷ 3 = 20, so 60 : 3 = 20 ft per bag
For Hari: 40 ÷ 2 = 20, so 40 : 2 = 20 ft per bag

Both ratios give 20 ft per 1 bag of cement. This means both used cement at the same rate.

No, Nitin is not correct. Hari used the same proportion of cement as Nitin. So her wall will be just as strong as his wall.
In simple words: Hari used cement in the same ratio as Nitin, so even though she used fewer total bags, the strength of her wall will be the same.

Exam Tip: Always check the ratio or rate, not just the total amount - two quantities may look different in size but be proportionally equal.

Question. Example 4: In my school, there are 5 teachers and 170 students. The ratio of teachers to students in my school is 5 : 170. Count the number of teachers and students in your school. What is the ratio of teachers to students in your school? Write it below.
Answer: In my school:
Number of teachers = 20
Number of students = 600
So the ratio of teachers to students is 20 : 600

Divide both by 20:
20 ÷ 20 = 1
600 ÷ 20 = 30

So the simplest form of the ratio is 1 : 30
Therefore, Teachers : Students = 1 : 30.
In simple words: Find the teacher to student ratio in your school, then simplify it by dividing both numbers by their greatest common factor.

Exam Tip: Always reduce ratios to their simplest form for easier comparison and clarity.

Question. Is the teacher-to-student ratio in your school proportional to the one in my school?
Answer: Given school's ratio:
Teachers : Students = 5 : 170
Divide both by 5 → 1 : 34
So given school's ratio = 1 : 34

My school's ratio:
Teachers : Students = 1 : 30

These are not the same.
In simple words: Compare the simplified ratios - if they are different, the ratios are not proportional.

Exam Tip: Two ratios are proportional only if their simplified forms are identical - use HCF to reduce both before comparing.

Question. Example 5: Measure the width and height (to the nearest cm) of the blackboard in your classroom. What is the ratio of width to height of the blackboard?
Answer: Width of blackboard: 240 cm
Height of blackboard: 120 cm

Now the ratio of width to height is 240 : 120
Simplify by dividing both terms by 120:
240 ÷ 120 = 2
120 ÷ 120 = 1

So the simplest form of the ratio is Width : Height = 2 : 1.
In simple words: Measure both dimensions, then divide both by the largest number that divides into both evenly.

Exam Tip: When expressing a ratio, always reduce it to simplest form by dividing both numbers by their HCF.

Question. Can you draw a rectangle in your notebook whose width and height are proportional to the ratio of the blackboard?
Answer: Since my blackboard ratio is 2 : 1, I can draw any rectangle whose width is twice the height.

For example, in my notebook I draw:
Width = 10 cm
Height = 5 cm

Because 10 : 5 = 2 : 1, the same ratio as the blackboard.
In simple words: Pick any two numbers that have the same ratio as the original - make the first number twice the second.

Exam Tip: You can draw rectangles of any size as long as the width to height ratio stays the same - all such rectangles will look identical in shape.

Question. Compare the rectangle you have drawn to those drawn by your classmates. Do they all look the same?
Answer: When I compare my rectangle with my classmates' rectangles:

Some may draw 8 cm × 4 cm
Some may draw 6 cm × 3 cm
Some may draw 12 cm × 6 cm

Even though the sizes are different, the shape is the same because all of them follow the 2 : 1 proportional ratio. So, yes, all rectangles drawn with the same ratio (2 : 1) will look the same in shape, even if the sizes are different. Only the scale changes, not the shape.
In simple words: When the ratio is the same, all shapes look alike even if they are bigger or smaller.

Exam Tip: Similar figures have the same shape but different sizes - their corresponding sides always have the same ratio.

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Question. Example 6: When Neelima was 3 years old, her mother's age was 10 times her age. What is the ratio of Neelima's age to her mother's age? What would be the ratio of their ages when Neelima is 12 years old? Would it remain the same?
Answer: When Neelima was 3 years old:
Mother's age = 10 × Neelima's age
Mother's age = 10 × 3 = 30 years
So the ratio of their ages is:
Neelima : Mother = 3 : 30

Simplify by dividing both by 3:
3 : 30 = 1 : 10

When Neelima is 12 years old:
Since Neelima was 3 earlier, she has grown by: 12 - 3 = 9 years
Her mother will also increase by 9 years:
Mother's earlier age = 30
Mother's new age = 30 + 9 = 39
So new ratio: Neelima : Mother = 12 : 39

Simplify by dividing both by 3:
12 ÷ 3 = 4
39 ÷ 3 = 13
So the simplified ratio is 4 : 13.

Earlier ratio = 1 : 10
Later ratio = 4 : 13
These are not the same.
In simple words: As people grow older, their age ratio changes because both people age by the same number of years, not the same factor.

Exam Tip: When working with age ratios, remember that adding the same number of years to both ages changes the ratio - this is different from multiplying by a factor.

Question. Example 7: Fill in the missing numbers for the following ratios that are proportional to 14 : 21. _____ : 42 6 : _____ 2 : _____
Answer: Given ratio 14 : 21.
Divide both by 7:
14 ÷ 7 = 2
21 ÷ 7 = 3
So 14 : 21 = 2 : 3

Now we have to use the simplified ratio 2 : 3 to fill the blanks.

▶ ___ : 42
We want the second term to be 42.
The original second term is 21.
Factor = 42 ÷ 21 = 2
Multiply the first term (14) by 2:
14 × 2 = 28
So the ratio is 28 : 42

▶ 6 : ____
We want the first term to be 6.
The original first term is 14.
Factor = 6 ÷ 14 = 3/7
Multiply the second term (21) by 3/7:
21 × 3/7 = 3 × 3 = 9
So the ratio is 6 : 9

▶ 2 : ____
We use the simplified ratio 2 : 3.
If the first term is 2, the second term must be 3.
So the ratio is 2 : 3.
In simple words: To fill blanks in proportional ratios, find the factor by dividing any term in the new ratio by its matching term in the original, then apply that factor to find the missing number.

Exam Tip: Always simplify the given ratio first, then use it as a reference to fill in blanks - this ensures all ratios remain proportional.

Question. What factor should we multiply 14 by to get 6? Can it be an integer? Or should it be a fraction?
Answer: We want to find the factor such that: 14 × (factor) = 6
So, factor = 6 ÷ 14 = 3 ÷ 7 = 3/7

3/7 is not an integer. There is no integer you can multiply 14 by to get 6. It must be a fraction and that fraction is 3/7.
In simple words: Not all factors are whole numbers - sometimes you need a fraction to scale a number to a different value.

Exam Tip: Factors of change can be fractions, decimals, or whole numbers - always divide the new value by the old value to find the correct factor.

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Question. Why is this coffee stronger?
Answer: Normal coffee:
Coffee decoction : Milk = 15 : 35
(Simplest form → divide by 5 → 3 : 7)

Stronger coffee:
Coffee decoction : Milk = 20 : 30
(Simplest form → divide by 10 → 2 : 3)

The second coffee is stronger because in the stronger coffee, the amount of decoction is more compared to milk.

Let's compare the simplified ratios:
Normal coffee → 3 parts decoction for 7 parts milk
Stronger coffee → 2 parts decoction for 3 parts milk

Normal coffee: Total parts = 3 + 7 = 10
Decoction part = 3/10 = 30%

Stronger coffee: Total parts = 2 + 3 = 5
Decoction part = 2/5 = 40%

So the stronger coffee has 40% decoction, while the normal one has 30%.
In simple words: When there is more coffee decoction compared to milk, the coffee tastes stronger because the coffee flavor is more concentrated.

Exam Tip: To compare mixtures, convert ratios to percentages - the ingredient with a higher percentage makes the mixture stronger or more intense.

Question. Why is this coffee lighter?
Answer: Normal coffee ratio:
Decoction : Milk = 15 : 35
Simplest form → 3 : 7
Decoction = 30%

Lighter coffee (more milk):
If he increases milk or reduces decoction, the ratio becomes something like:
Decoction : Milk = 10 : 40 (just an example)
Simplest form → 1 : 4
Decoction = 20%

This coffee becomes lighter because the amount of milk is much more compared to decoction and the percentage of decoction decreases. This makes the coffee taste milder, thinner, and less strong.
In simple words: When you add more milk or use less coffee decoction, the coffee flavor becomes weaker and tastes lighter.

Exam Tip: In any mixture, the ingredient with a lower percentage becomes less noticeable - reducing the proportion of the main ingredient makes the mixture milder or lighter in flavor.

Question. 1. Circle the following statements of proportion that are true. (i) 4 : 7 :: 12 : 21 (ii) 8 : 3 :: 24 : 6 (iii) 7 : 12 :: 12 : 7 (iv) 21 : 6 :: 35 : 10 (v) 12 : 18 :: 28 : 12 (vi) 24 : 8 :: 9 : 3
Answer: (i) 4 : 7 :: 12 : 21
4 : 7 (already simplest) → 4 : 7
12 : 21 → divide by 3 → 4 : 7
True

(ii) 8 : 3 :: 24 : 6
8 : 3 (already simplest) → 8 : 3
24 : 6 → divide by 6 → 4 : 1
8 : 3 ≠ 4 : 1
False

(iii) 7 : 12 :: 12 : 7
7 : 12 (simplest) → 7 : 12
12 : 7 (simplest) → 12 : 7
7 : 12 ≠ 12 : 7
False

(iv) 21 : 6 :: 35 : 10
21 : 6 → divide by 3 → 7 : 2
35 : 10 → divide by 5 → 7 : 2
True

(v) 12 : 18 :: 28 : 12
12 : 18 → divide by 6 → 2 : 3
28 : 12 → divide by 4 → 7 : 3
2 : 3 ≠ 7 : 3
False

(vi) 24 : 8 :: 9 : 3
24 : 8 → divide by 8 → 3 : 1
9 : 3 → divide by 3 → 3 : 1
True
In simple words: To check if two ratios are proportional, simplify both and see if they match.

Exam Tip: Always reduce each ratio to its simplest form using HCF before comparing - this is the only reliable way to check proportionality.

Question. 2. Give 3 ratios that are proportional to 4 : 9. ______ : ______ ______ : ______ ______ : ______
Answer: ▶ 1st ratio (multiply by 2)
4 × 2 = 8
9 × 2 = 18
8 : 18

▶ 2nd ratio (multiply by 3)
4 × 3 = 12
9 × 3 = 27
12 : 27

▶ 3rd ratio (multiply by 5)
4 × 5 = 20
9 × 5 = 45
20 : 45
In simple words: To create ratios that are proportional to a given ratio, multiply both numbers by the same factor.

Exam Tip: You can generate unlimited proportional ratios by multiplying both terms by any number - all will be equally correct.

Question. 3. Fill in the missing numbers for these ratios that are proportional to 18 : 24. 3 : ______ 12 : ______ 20 : ______ 27 : ______
Answer: We want all ratios to be proportional to 18 : 24.
18 ÷ 6 = 3
24 ÷ 6 = 4

So 18 : 24 = 3 : 4

Now we use the ratio 3 : 4 to fill in each blank.

▶ 3 : ______
This already matches the first term of 3 : 4.
So the second term must be 4.
3 : 4

▶ 12 : ______
Compare with 3 : 4.
Find factor: 12 ÷ 3 = 4
Multiply second term: 4 × 4 = 16
12 : 16

▶ 20 : ______
Find factor: 20 ÷ 3 = 20/3 (not an integer but allowed)
Multiply second term: 4 × (20/3) = 80/3
20 : 80/3

▶ 27 : ______
Find factor: 27 ÷ 3 = 9
Multiply second term: 4 × 9 = 36
27 : 36
In simple words: Find the simplified ratio, then use it to figure out the missing number in each blank.

Exam Tip: When finding the missing term, always check if the factor you calculate is correct by verifying that the new ratio simplifies back to the original simplified form.

Question. 4. Look at the following rectangles. Which rectangles are similar to each other? You can verify this by measuring the width and height using a scale and comparing their ratios.
Answer: Rectangles that look similar in shape: C and D. Both are long, stretched rectangles with the same kind of proportion (long width compared to height). Even though they are rotated, rotation does not change similarity.

Rectangles that also look similar to each other: A and E. Both are tall, narrow rectangles with almost the same width - to - height ratio. E is just tilted, but tilting also does not affect similarity.

Rectangle B looks like a more regular, broad rectangle. Its ratio does not visually match A, C, D or E.
In simple words: Two shapes are similar if their width to height ratio is the same, no matter how big they are or which way they are turned.

Exam Tip: Similar figures have identical ratios of corresponding sides - orientation (rotation or tilting) and size do not matter for similarity.

Question. 5. Look at the following rectangle. Can you draw a smaller rectangle and a bigger rectangle with the same width to height ratio in your notebooks? Compare your rectangles with your classmates' drawings. Are all of them the same? If they are different from yours, can you think why? Are they wrong?
Answer: We have to draw a smaller rectangle and a bigger rectangle. Both must have the same width to height ratio as the original rectangle. That means all rectangles should be similar (same shape, different size).

Even though all students use the same ratio, their rectangles will have these differences:
▶ shorter, taller, wider or longer
▶ drawn at different scales
▶ drawn in different orientations (vertical or horizontal)

So the rectangles may look different in size, but their shape will be the same if the ratio is correct. A rectangle with the same width to height ratio can be drawn in many different sizes. For example, if the ratio of the original rectangle is 4 : 3, then these are all correct: 8 : 6 or 12 : 9 or 20 : 15 or 40 : 30 and more. Different numbers, same ratio → all are correct. Only if someone does not keep the ratio the same, then the rectangle shape will change → that would be wrong.
In simple words: You can draw many rectangles with the same ratio - they will all look the same shape even if they are different sizes.

Exam Tip: For similar figures, maintain the exact ratio - if the ratio changes, even slightly, the shape will be distorted.

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.
Answer: (a) Red bricks - There are 3 red clusters, each cluster having:
3 red bricks on top
2 red bricks below
1 red brick at bottom
So each cluster = 3 + 2 + 1 = 6 red bricks

Grey bricks - Now count the grey bricks in the same portion:
2 grey bricks on top
3 grey bricks below
4 grey bricks at bottom
So each cluster = 2 + 3 + 4 = 9 grey bricks

Grey : Red = 9 : 6
Simplify by dividing both by 3:
9 ÷ 3 = 3
6 ÷ 3 = 2
So, the ratio for (a) = 3 : 2

(b) Grey bricks in each cluster = 16
Brown bricks in each cluster = 12
So, Grey : Brown = 16 : 12
Simplify by dividing both by 4:
16 ÷ 4 = 4
12 ÷ 4 = 3
Therefore, the ratio for (b): 4 : 3
In simple words: Count all the bricks of each color in one repeating unit, write them as a ratio, then simplify.

Exam Tip: When dealing with patterns, always identify and count one complete repeating unit - this ensures your ratio is accurate for the entire pattern.

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.
Answer: ▶ head : torso
Head = 20 cm
Torso = 40 cm
Ratio = 20 : 40 = 1 : 2

▶ torso : arms
Torso = 40 cm
Arms = 50 cm
Ratio = 40 : 50 = 4 : 5

▶ torso : legs
Torso = 40 cm
Legs = 80 cm
Ratio = 40 : 80 = 1 : 2
In simple words: Take two measurements, put them in a ratio, and simplify by dividing both by their largest common factor.

Exam Tip: When measuring body proportions, be consistent with your units and always simplify the final ratio to show the true relationship between the measurements.

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Question. Does the drawing look more realistic if the ratios are proportional? Why? Why not?
Answer: Yes, the drawing definitely looks more realistic when the body - part ratios are proportional.

A real human body has natural proportions. For example: Head is usually smaller than torso, Arms and legs have fixed proportional lengths, and Torso is roughly twice the head, etc. When you draw your figure using ratios that match real - life proportions, the parts fit together correctly.

So the drawing looks balanced, natural, and human - like. It looks unrealistic when ratios are not proportional - if the ratios are wrong, for example: Head is too big or too small, Arms are too long, and Legs are too short. Then the drawing looks distorted or cartoonish, because the parts no longer match real human proportions.
In simple words: Drawing with correct proportions makes figures look natural and real. Wrong proportions make them look weird or cartoon - like.

Exam Tip: Understanding proportions is key to realistic drawings - real human figures always follow specific ratio patterns that can be measured and applied.

Question. Example 8: For the mid - day meal in a school with 120 students, the cook usually makes 15 kg of rice. On a rainy day, only 80 students came to school. How many kilograms of rice should the cook make so that the food is not wasted?
Answer: Given:
120 students → 15 kg rice
80 students → ? kg rice

Let the rice for 80 students = k kg
So, number of students : Rice = 120 : 15 :: 80 : k

\[ \Rightarrow \frac{120}{15} = \frac{80}{k} \]

\[ \Rightarrow 120k = 80 \times 15 \]

\[ \Rightarrow 120k = 1200 \]

\[ \Rightarrow k = \frac{1200}{120} = 10 \]

So the cook should make 10 kg of rice for 80 students.
In simple words: Set up the proportion with students on one side and rice on the other, cross multiply, then solve for the unknown value.

Exam Tip: In proportion problems, always set up the ratio pairs correctly - numerator should correspond to numerator and denominator to denominator.

Question. What is the factor of change in the first term?
Answer: The factor of change in the first term is 80 ÷ 120 = 2/3

So the factor of change is 2/3.
In simple words: Divide the new value by the old value to get the factor of change.

Exam Tip: The factor of change can always be found by dividing the new quantity by the original quantity - apply this factor to all related quantities.

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Question. Example 9: A car travels 90 km in 150 minutes. If it continues at the same speed, what distance will it cover in 4 hours?
Answer: 4 hours = 4 × 60 = 240 minutes
90 km is covered in 150 minutes.
We have to find distance in 240 minutes.

So, 150 : 90 :: 240 : x

\[ \Rightarrow x = 90 \times \frac{240}{150} \]

\[ \Rightarrow x = 90 \times \frac{8}{5} \]

\[ \Rightarrow x = 18 \times 8 \]

\[ \Rightarrow x = 144 \]

So the car will cover 144 km in 4 hours.
In simple words: First convert hours to minutes so both quantities have the same unit, then set up the proportion and solve.

Exam Tip: Always ensure units are consistent before setting up a proportion - convert hours to minutes or minutes to hours as needed.

Question. Is this the right way to formulate the question?
Answer: Yes, this is the right way to formulate the question.

Distance increases when time increases, if speed stays the same. So we compare the two situations like this:
150 minutes → 90 km
240 minutes → x km

This is the correct proportional method because distance is directly proportional to time when the speed does not change.
In simple words: In a direct proportion, when one quantity grows, the other grows by the same factor - distance and time are directly proportional at constant speed.

Exam Tip: Identify whether quantities are directly proportional (both increase/decrease together) or inversely proportional (one increases while the other decreases) before setting up the equation.

Question 1. How can you find the distance covered in 240 minutes?
Answer: The car travels 90 km in 150 minutes. First, calculate the factor of change: 240 ÷ 150 = 8/5. Then, multiply the distance by this same factor: 90 × 8/5 = 144. So the distance covered in 240 minutes is 144 km.
In simple words: Find how many times bigger 240 is than 150, then multiply the distance by that number.

Exam Tip: Always identify which quantity is changing (time, distance, speed) and find the scaling factor to solve proportion problems efficiently.

Question 2. A small farmer in Himachal Pradesh sells each 200 g packet of tea for Rs.200. A large estate in Meghalaya sells each 1 kg packet of tea for Rs.800. Are the weight-to-price ratios in both places proportional? Which tea is more expensive?
Answer: For the Himachal farmer: Price per gram = 200 ÷ 200 = Rs.1 per gram. For the Meghalaya estate: First convert 1 kg to 1000 g, then Price per gram = 800 ÷ 1000 = Rs.0.8 per gram. The Himachal ratio is 200 g : Rs.200 = 1 g : Rs.1, while the Meghalaya ratio is 1000 g : Rs.800 = 1 g : Rs.0.8. Since these ratios are not the same, the weight-to-price ratios are not proportional. The Himachal tea is more expensive because it costs Rs.1 per gram, compared to Rs.0.8 per gram in Meghalaya.
In simple words: The price per gram is different for each tea, so they are not proportional. Himachal tea costs more money for the same weight.

Exam Tip: Always calculate the unit cost (price per gram, per litre, etc.) to compare whether two ratios are proportional - equal unit costs mean proportional ratios.

Question 3. The Earth travels approximately 940 million kilometres around the Sun in a year. How many kilometres will it travel in a week?
Answer: The Earth travels 940 million km in 1 year. Since 1 year equals 52 weeks, we can find the distance per week by dividing: 940 million ÷ 52 ≈ 18.07 million kilometres. So the distance travelled in one week is approximately 18.07 million kilometres.
In simple words: Divide the yearly distance by the number of weeks in a year to get the weekly distance.

Exam Tip: When converting between time periods, identify the conversion factor first (52 weeks = 1 year) and then divide or multiply accordingly.

Question 4. 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.
Answer: From the diagram, the house is a rectangle 15 ft long and 12 ft wide with one inner wall running from top to bottom dividing it into two rooms. The outer walls form the perimeter: 2 × (15 + 12) = 2 × 27 = 54 ft. The inner wall runs vertically = 12 ft. So the total wall length = 54 + 12 = 66 ft. Since 10 ft of wall needs 1450 bricks, the rate is 1450 ÷ 10 = 145 bricks per foot. For 66 ft, the number of bricks = 66 × 145 = 9570 bricks.
In simple words: Find the total length of all walls, then multiply by the number of bricks needed per foot.

Exam Tip: Always account for all walls - both outer (perimeter) and inner - when calculating total wall length in construction problems.

Question 5. Puneeth's father went from Lucknow to Kanpur in 2 hours by riding his motorcycle at a speed of 50 km/h. If he drives at 75 km/h, how long will it take him to reach Kanpur? Can we form this problem as a proportion - 50 : 2 :: 75 : __?
Answer: The distance from Lucknow to Kanpur is the same in both cases. Speed and time are inversely related: higher speed means less time, and lower speed means more time. So they are inversely proportional, not directly proportional. The proportion 50 : 2 :: 75 : __ is not correct because it assumes speed and time move together, but here speed increases while time decreases. This is inverse proportion, so: Speed₁ × Time₁ = Speed₂ × Time₂. Therefore: 50 × 2 = 75 × t
\implies 100 = 75t
\implies t = 100 ÷ 75 = 4/3 hours = 1 hour 20 minutes.
In simple words: When speed goes up, time goes down - they move in opposite directions, so this is inverse proportion, not direct proportion.

Exam Tip: Recognise inverse proportion situations: if one quantity increases and the other decreases (like speed and time for a fixed distance), use multiplication equality, not ratio equality.

Question 6. Go to the market and collect the prices of different sizes of shampoo containers of the same shampoo and create a table. See if the volume of shampoo is proportional to the price.
Answer: To check if volume and price are proportional, calculate the price per mL for each container size. For the given example: Sachet (6 mL): 2 ÷ 6 = 0.33 Rs./mL; Small Bottle (180 mL): 154 ÷ 180 ≈ 0.86 Rs./mL; Medium Bottle (340 mL): 276 ÷ 340 ≈ 0.81 Rs./mL; Large Bottle (1000 mL): 540 ÷ 1000 = 0.54 Rs./mL. If volume and price were proportional, the price per mL would be identical for all sizes. Since the values are 0.33, 0.86, 0.81, and 0.54 Rs./mL, they are not equal. Therefore, volume is not proportional to price.
In simple words: Check if the cost per unit is the same for all sizes - if not, volume and price are not proportional.

Exam Tip: When testing proportionality, always calculate the unit value (price per mL, distance per hour, etc.) - if these are equal, the quantities are proportional.

Question 7. Why do you think that the ratio of the prices is not proportional to the ratio of the volumes?
Answer: The price per mL differs for each container because smaller pouches and bottles typically have higher costs per unit of product, while larger bottles have lower costs per unit. This happens due to extra expenses for packaging materials, design, and convenience that companies add. A sachet needs more packaging material per mL than a 1-litre bottle, making small amounts costlier. These additional packaging and marketing costs mean the price does not increase at the same rate as the volume, so the ratios remain disproportionate.
In simple words: Small containers cost more per mL because packaging takes up more of the product's value than it does for large bottles.

Exam Tip: Remember that real-world pricing involves overhead costs that affect smaller units more heavily - this is why bulk purchases are usually cheaper per unit.

Question 8. Form a pair. Collect 12 countable objects or counters. Now, share them between the two of you in different ways. If you divide them equally, what is the ratio of the number of counters with each of you?
Answer: When 12 counters are divided equally between two people, each person receives 12 ÷ 2 = 6 counters. The ratio of counters with each person is 6 : 6. When simplified, this becomes 1 : 1, meaning each person has an equal share.
In simple words: If 12 things are split evenly between 2 people, each gets 6, so the ratio is 1 : 1.

Exam Tip: When simplifying ratios, divide both parts by their greatest common divisor to get the simplest form.

Question 9. If your partner gets 5 counters, how many objects will you get? What is the ratio of the counters?
Answer: With a total of 12 counters, if your partner receives 5 counters, then you get 12 - 5 = 7 counters. The ratio of counters (you : partner) is 7 : 5.
In simple words: Subtract your partner's share from the total to find your share, then write the ratio.

Exam Tip: Always verify that the two parts of your ratio add up to the total quantity given.

Question 10. Now, if you want to share the counters between the two of you in the ratio of 3 : 1, how many counters would each of you get?
Answer: To divide 12 counters in the ratio 3 : 1, first add the ratio parts: 3 + 1 = 4 parts. Find the value of 1 part: 12 ÷ 4 = 3 counters per part. Then multiply: Person 1 gets 3 parts = 3 × 3 = 9 counters; Person 2 gets 1 part = 1 × 3 = 3 counters. Therefore, the counters are shared as 9 and 3.
In simple words: Add the ratio numbers to get total parts, divide the amount by the number of parts, then multiply by each person's share.

Exam Tip: Always verify: add the final shares to check they equal the total amount (9 + 3 = 12 ✓).

Question 11. Now, if you want to share 42 counters between the two of you in the ratio of 4 : 3, how will you do it?
Answer: To divide 42 counters in the ratio 4 : 3, add the ratio parts: 4 + 3 = 7 parts. Divide 42 counters by 7 parts: 42 ÷ 7 = 6 counters per part. Then calculate each person's share: First person gets 4 parts = 4 × 6 = 24 counters; Second person gets 3 parts = 3 × 6 = 18 counters.
In simple words: Split the total into the number of ratio parts, then multiply each part by the ratio number for that person.

Exam Tip: Check your work by adding the shares: 24 + 18 = 42 ✓

Question 12. What is the size of each group?
Answer: When dividing 42 counters in the ratio 4 : 3, the total number of parts is 4 + 3 = 7. The size of each group (or each part) is 42 ÷ 7 = 6 counters.
In simple words: Each part or group has 6 counters.

Exam Tip: The size of each group is found by dividing the total by the sum of the ratio parts - this value applies to every part in the ratio.

Question 13. Prashanti and Bhuvan started a food cart business near their school. Prashanti invested Rs.75,000 and Bhuvan invested Rs.25,000. At the end of the first month, they gained a profit of Rs.4,000. They decided that they would share the profit in the same ratio as that of their investment. What is each person's share of the profit?
Answer: The investment ratio is Prashanti : Bhuvan = 75,000 : 25,000. Dividing both by 25,000 gives: 75,000 ÷ 25,000 = 3 and 25,000 ÷ 25,000 = 1, so the ratio simplifies to 3 : 1. The total profit is Rs.4,000 and the total ratio parts = 3 + 1 = 4 parts. Therefore, 1 part = 4,000 ÷ 4 = Rs.1,000. Prashanti receives 3 parts = 3 × 1,000 = Rs.3,000; Bhuvan receives 1 part = 1 × 1,000 = Rs.1,000.
In simple words: Simplify the investment ratio, then divide the profit by the number of parts and multiply by each person's share.

Exam Tip: In profit-sharing problems, always match the profit-sharing ratio exactly to the investment ratio unless told otherwise.

Question 14. A mixture of 40 kg contains sand and cement in the ratio of 3 : 1. How much cement should be added to the mixture to make the ratio of sand to cement 5 : 2?
Answer: With a ratio of 3 : 1, the total parts = 3 + 1 = 4. Each part = 40 ÷ 4 = 10 kg. So Sand = 3 × 10 = 30 kg and Cement = 1 × 10 = 10 kg. Let x = the amount of cement to be added. The new cement amount = 10 + x, while sand stays at 30 kg. For the new ratio to be 5 : 2, we set up: 30 : (10 + x) = 5 : 2. Cross-multiplying: 30 × 2 = 5 × (10 + x)
\implies 60 = 50 + 5x
\implies 10 = 5x
\implies x = 2. Therefore, 2 kg of cement must be added to achieve the 5 : 2 ratio.
In simple words: Find the original amounts, then add an unknown amount to cement and solve using ratio equality.

Exam Tip: When one component stays fixed and another changes, use the ratio equality method and cross-multiply to find the unknown.

Question 15. Divide Rs.4,500 into two parts in the ratio 2 : 3.
Answer: The total amount is Rs.4,500 and the ratio is 2 : 3. The total ratio parts = 2 + 3 = 5. The value of one part = 4,500 ÷ 5 = 900. The first share = 2 × 900 = Rs.1,800; The second share = 3 × 900 = Rs.2,700. Therefore, the two parts are Rs.1,800 and Rs.2,700.
In simple words: Divide the total into the number of parts in the ratio, then multiply each part by the ratio numbers.

Exam Tip: Verify by adding: 1,800 + 2,700 = 4,500 ✓

Question 16. 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?
Answer: The ratio of acid to water is 1 : 5, so the total parts = 1 + 5 = 6 parts. The total solution volume is 240 mL. Therefore, 1 part = 240 ÷ 6 = 40 mL. Acid = 1 part = 40 mL; Water = 5 parts = 5 × 40 = 200 mL. So the solution contains 40 mL of acid and 200 mL of water.
In simple words: Use the ratio to find how many equal parts make up the total, then calculate each component.

Exam Tip: Always verify: 40 + 200 = 240 mL ✓

Question 17. 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?
Answer: The original ratio is 3 : 5, so the total parts = 3 + 5 = 8. For 40 mL, 1 part = 40 ÷ 8 = 5 mL. Blue = 3 parts = 3 × 5 = 15 mL; Yellow = 5 parts = 5 × 5 = 25 mL. So for 40 mL of green paint, we need 15 mL of blue and 25 mL of yellow. After adding 20 mL of yellow: Original yellow = 25 mL, Added yellow = 20 mL, New yellow = 25 + 20 = 45 mL. Blue stays the same = 15 mL. The new ratio = 15 : 45, which simplifies to 1 : 3 by dividing both by 15.
In simple words: First find the amounts needed using the original ratio, then recalculate the ratio after adding more of one colour.

Exam Tip: When a component changes after initial mixing, recalculate and simplify the new ratio by finding the greatest common divisor.

Question 18. 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?
Answer: The ratio of rice to urad dal is 2 : 1, so the total parts = 2 + 1 = 3. We need 6 cups of the mixture, so 1 part = 6 ÷ 3 = 2 cups. Rice = 2 parts = 2 × 2 = 4 cups; Urad dal = 1 part = 1 × 2 = 2 cups. Therefore, we need 4 cups of rice and 2 cups of urad dal.
In simple words: Apply the ratio to the total amount needed to find each ingredient's amount.

Exam Tip: Verify: 4 + 2 = 6 cups ✓

Question 19. 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?
Answer: The original ratio of red : yellow is 3 : 5, which means Red = 3 parts and Yellow = 5 parts. When another full bucket of yellow (equal to 5 parts) is added, the new amounts become: Red = 3 parts (unchanged), Yellow = 5 + 5 = 10 parts. Therefore, the new ratio of red to yellow is 3 : 10.
In simple words: When you add more of one colour, keep the other unchanged and recalculate the ratio.

Exam Tip: In mixing problems where one component is added to an existing mixture, clearly separate the "before" and "after" amounts to avoid confusion.

Question 20. 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.
Answer: The ratio is 600 : 900. To simplify, divide both numbers by their greatest common divisor, which is 300: 600 ÷ 300 = 2 and 900 ÷ 300 = 3. Therefore, the ratio in simplest form is 2 : 3.
In simple words: Find the largest number that divides both amounts evenly, then divide both by it.

Exam Tip: Always express ratios in simplest form by dividing both parts by their greatest common divisor.

Question 21. 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?
Answer: Last year, 3 buses carried 162 people, so the capacity per bus = 162 ÷ 3 = 54 people per bus. This year with 204 students, we need to determine how many buses are required: 204 ÷ 54. Since 54 × 3 = 162 and 54 × 4 = 216, three buses are insufficient. So we need 4 buses. The total capacity of 4 buses = 4 × 54 = 216 people. With 204 people, the number of empty seats = 216 - 204 = 12. Therefore, we need 4 buses, but they will not all be completely full.
In simple words: Find the capacity per bus from last year's data, then divide this year's number by that capacity to see how many buses are needed.

Exam Tip: When the total is not evenly divisible by the unit capacity, you must round up to the next whole number of units (buses, trucks, etc.).

Question 22. 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?
Answer: For Delhi: Area = 1,484 sq km, Population = 30 million. Population density = 30,000,000 ÷ 1,484 ≈ 20,216 people per sq km. For Mumbai: Area = 550 sq km, Population = 20 million. Population density = 20,000,000 ÷ 550 ≈ 36,363 people per sq km. Comparing the densities, Mumbai has approximately 36,363 people per sq km while Delhi has approximately 20,216 people per sq km. Since Mumbai has more people living in each square kilometre, Mumbai is more crowded. This is because population density (people per unit area) is the proper measure of how crowded a place is, not just the total population.
In simple words: Mumbai is more crowded because more people live in each square kilometre, even though Delhi has more total people.

Exam Tip: Population density is the key comparison metric for crowdedness - always calculate people per unit area rather than relying on total population alone.

Question 23. 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?
Answer: The ratio of neck : rest of body is 4 : 6. The total parts = 4 + 6 = 10. If your height is 150 cm, then 1 part = 150 ÷ 10 = 15 cm. Your neck = 4 parts = 4 × 15 = 60 cm. Therefore, if you maintained this ratio, your neck would be 60 cm tall.
In simple words: Apply the ratio to your own height to calculate how tall your neck would be at that proportion.

Exam Tip: When applying a ratio from one context to another, use the same process: find the value per part, then multiply by the required ratio numbers.

Question 24. Let us try an ancient problem from Lilavati. At that time weights were measured in a unit named palas and niskas was a unit of money. "If 2½ palas of saffron costs 3/7 niskas, O expert businessman! tell me quickly what quantity of saffron can be bought for 9 niskas?"
Answer: We know that 2½ palas of saffron costs 3/7 niskas, which we write as 5/2 palas costs 3/7 niskas. To find how much saffron costs 1 niska, divide both sides by 3/7: (5/2) ÷ (3/7). Since dividing by a fraction means multiplying by its reciprocal: (5/2) × (7/3) = 35/6 palas per niska. Therefore, for each niska, we can buy 35/6 palas. For 9 niskas, we multiply: (35/6) × 9 = (35 × 9) ÷ 6 = 315 ÷ 6 = 52.5 palas. So for 9 niskas, one can purchase 52.5 palas of saffron.
In simple words: Find the amount per unit cost (palas per niska), then multiply by the total cost (9 niskas).

Exam Tip: When working with fraction ratios, convert mixed numbers to improper fractions and use reciprocal multiplication for division.

Question 25. 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?
Answer: Let Harmain's future age = x years. Currently, Harmain is 1 year old and her brother is 5 years old. The age difference = 5 - 1 = 4 years, which remains constant. We need the ratio of Harmain's age to her brother's age to equal 1 : 2. When Harmain is x years old, her brother will be x + 4 years old. Setting up the ratio: x : (x + 4) = 1 : 2. This gives x ÷ (x + 4) = 1 ÷ 2. Cross-multiplying: 2x = x + 4
\implies 2x - x = 4
\implies x = 4. Therefore, Harmain will be 4 years old when the ratio becomes 1 : 2 (her brother will be 8 years old).
In simple words: The age gap never changes, so set up an equation using the ratio and solve for the unknown age.

Exam Tip: In age-ratio problems, always remember that the difference between two people's ages stays constant over time.

Question 26. 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?
Answer: The ratio of masses (gold : water) is 37 : 2. This means for equal volumes, gold has 37 parts of mass and water has 2 parts. Since 1 litre of water weighs 1 kg, we have 2 parts = 1 kg. Therefore, 1 part = 1 ÷ 2 = 0.5 kg. Gold has 37 parts, so the mass of 1 litre of gold = 37 × 0.5 = 18.5 kg.
In simple words: If water's mass is 2 parts and equals 1 kg, find the value of 1 part, then multiply by 37 for gold's mass.

Exam Tip: Always convert the known quantity to find the value per part, then use that to calculate the unknown quantity.

Question 27. 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?
Answer: The plot dimensions are 200 ft by 500 ft, so the area = 200 × 500 = 100,000 sq ft. Since 1 acre = 43,560 sq ft, we convert: Area in acres = 100,000 ÷ 43,560 ≈ 2.295 acres (approximately 2.3 acres). The recommended practice is 10 tonnes per acre, so the manure required = 10 × 2.295 ≈ 22.95 tonnes. Therefore, the farmer should purchase approximately 23 tonnes of manure.
In simple words: Convert the plot area to acres, then multiply by the manure needed per acre.

Exam Tip: Remember the conversion: 1 acre = 43,560 sq ft. Always convert to the same units before multiplying.

Question 28. 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?
Answer: The tap fills 500 mL in 15 seconds, so the flow rate = 500 ÷ 15 = 100/3 mL per second (approximately 33.33 mL per second). The bucket capacity is 10 litres = 10,000 mL. Using Time = Volume ÷ Flow rate: Time = 10,000 ÷ (100/3) = 10,000 × (3/100) = 300 seconds. Converting to minutes: 300 ÷ 60 = 5 minutes. Therefore, the tap takes 5 minutes to fill the 10-litre bucket.
In simple words: Calculate the flow rate (volume per second), then divide the bucket volume by the flow rate to get the time.

Exam Tip: Always convert all volumes to the same unit before performing calculations with rates.

Question 29. One acre of land costs Rs.15,00,000. What is the cost of 2,400 square feet of the same land?
Answer: Since 1 acre = 43,560 sq ft and 1 acre costs Rs.1,500,000, the cost per sq ft = 1,500,000 ÷ 43,560 ≈ Rs.34.43 per sq ft. For 2,400 sq ft, the cost = 2,400 × 34.43 ≈ Rs.82,632. Therefore, the cost of 2,400 square feet is approximately Rs.82,632.
In simple words: Find the cost per square foot, then multiply by the area in square feet.

Exam Tip: Use unit pricing (cost per sq ft, per kg, etc.) to scale costs up or down for different quantities.

Question 30. 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 him if he decides to use a tractor instead?
Answer: A pair of oxen takes 6 hours per acre. For a 20-acre field, the total time using oxen = 20 × 6 = 120 hours. A tractor is 4 times faster than oxen. If oxen take 6 hours per acre, a tractor takes 6 ÷ 4 = 1.5 hours per acre. For 20 acres with a tractor, the time = 20 × 1.5 = 30 hours. Therefore, using oxen takes 120 hours and using a tractor takes 30 hours.
In simple words: Calculate time for oxen by multiplying hours per acre by total acres. For the tractor, divide the oxen time by 4 (since it is 4 times faster).

Exam Tip: When one tool is a multiple faster than another, divide the time by that multiple - faster speed means less time.

Question 31. The Rs.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 Rs.906 per kg and the cost of nickel is Rs.1,341 per kg, what is the cost of these metals in a Rs.10 coin?
Answer: The ratio of copper to nickel is 3 : 1, so total parts = 3 + 1 = 4. The coin mass is 7.74 g, so 1 part = 7.74 ÷ 4 = 1.935 g. Copper mass = 3 parts = 3 × 1.935 = 5.805 g; Nickel mass = 1 part = 1.935 g. Converting to kg: Copper = 5.805 ÷ 1000 = 0.005805 kg; Nickel = 1.935 ÷ 1000 = 0.001935 kg. Cost of copper = 0.005805 × 906 ≈ Rs.5.26; Cost of nickel = 0.001935 × 1,341 ≈ Rs.2.59. Total cost of metals = Rs.5.26 + Rs.2.59 = Rs.7.85.
In simple words: Find each metal's mass using the ratio, convert to kilograms, then multiply by the per-kg cost.

Exam Tip: Always convert mass units (grams to kilograms) before multiplying by unit prices to ensure correct answers.