Comparing Quantities | Grade 7th - Mathematics Chapter Summary & NCERT CBSE Study Resources

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7th

7th - Mathematics

Comparing Quantities

Chapter Overview: Comparing Quantities

This chapter introduces students to the concept of comparing quantities using ratios, percentages, and simple interest. It helps in understanding real-life applications of these mathematical concepts.

Key Concepts

Ratio: A ratio compares two quantities of the same unit by division. It is expressed in the simplest form.

Percentage: A percentage represents a part per hundred. It is denoted by the symbol '%'.

Simple Interest: Simple interest is calculated on the principal amount for a given time period at a fixed rate.

Ratio and Proportion

Ratios are used to compare quantities. Two ratios are said to be in proportion if they are equal.

Example: If the ratio of boys to girls in a class is 3:2, and there are 15 boys, the number of girls can be found using proportion.

Percentage

Percentage is a way to express a number as a fraction of 100. It is useful for comparing quantities.

Example: If a student scores 45 out of 50, the percentage is calculated as (45/50) × 100 = 90%.

Profit and Loss

Profit occurs when the selling price is higher than the cost price, while loss occurs when the selling price is lower.

Example: If an item is bought for ₹200 and sold for ₹250, the profit is ₹50.

Simple Interest

Simple interest is calculated using the formula: SI = (P × R × T)/100, where P is principal, R is rate, and T is time.

Example: For a principal of ₹1000 at 5% per annum for 2 years, the simple interest is ₹100.

Applications in Real Life

These concepts are widely used in everyday situations like shopping, banking, and data interpretation.

All Question Types with Solutions – CBSE Exam Pattern

Explore a complete set of CBSE-style questions with detailed solutions, categorized by marks and question types. Ideal for exam preparation, revision and practice.

Very Short Answer (1 Mark) – with Solutions (CBSE Pattern)

These are 1-mark questions requiring direct, concise answers. Ideal for quick recall and concept clarity.

Question 1:

Convert the ratio 3:4 into a percentage.

Answer:

75%

Question 2:

If the cost price of a book is ₹200 and the selling price is ₹250, find the profit.

Answer:

₹50

Question 3:

Express 25% as a fraction in simplest form.

Answer:

1/4

Question 4:

Find the simple interest on ₹1000 at 5% per annum for 2 years.

Answer:

₹100

Question 5:

A shirt is sold for ₹400 at a loss of 20%. Find its cost price.

Answer:

₹500

Question 6:

Convert 0.75 into a percentage.

Answer:

75%

Question 7:

If 40% of a number is 80, find the number.

Answer:

200

Question 8:

A cycle is bought for ₹1200 and sold for ₹1500. Calculate the profit percentage.

Answer:

25%

Question 9:

Write 5:8 as a decimal.

Answer:

0.625

Question 10:

Find the amount if ₹5000 is invested at 8% per annum for 3 years (Simple Interest).

Answer:

₹6200

Question 11:

A shopkeeper gives a discount of 10% on a ₹300 toy. Find the selling price.

Answer:

₹270

Question 12:

If 15% of students in a class are absent and 34 are present, find the total students.

Answer:

Question 13:

Convert the ratio 3:5 into percentage.

Answer:

To convert the ratio 3:5 into percentage:
Step 1: Add the parts of the ratio (3 + 5 = 8).
Step 2: Divide each part by the total (3/8 and 5/8).
Step 3: Multiply by 100 to get percentage (3/8 × 100 = 37.5% and 5/8 × 100 = 62.5%).
So, the ratio 3:5 is equivalent to 37.5% : 62.5%.

Question 14:

If the cost price of a book is ₹250 and the selling price is ₹300, find the profit percentage.

Answer:

To find the profit percentage:
Step 1: Calculate profit (Selling Price - Cost Price = ₹300 - ₹250 = ₹50).
Step 2: Divide profit by cost price (₹50 / ₹250 = 0.2).
Step 3: Multiply by 100 to get percentage (0.2 × 100 = 20%).
So, the profit percentage is 20%.

Question 15:

Express 75% as a fraction in its simplest form.

Answer:

To express 75% as a fraction:
Step 1: Write 75% as 75/100.
Step 2: Simplify by dividing numerator and denominator by 25 (75 ÷ 25 = 3, 100 ÷ 25 = 4).
So, 75% as a fraction in simplest form is 3/4.

Question 16:

A shirt is sold at a discount of 15%. If the marked price is ₹800, find the selling price.

Answer:

To find the selling price after a 15% discount:
Step 1: Calculate discount amount (15% of ₹800 = 0.15 × 800 = ₹120).
Step 2: Subtract discount from marked price (₹800 - ₹120 = ₹680).
So, the selling price is ₹680.

Question 17:

If 20% of a number is 50, find the number.

Answer:

To find the original number:
Step 1: Let the number be x.
Step 2: 20% of x = 50 ⇒ 0.20 × x = 50.
Step 3: Solve for x (x = 50 / 0.20 = 250).
So, the number is 250.

Question 18:

Convert the fraction 4/5 into percentage.

Answer:

To convert the fraction 4/5 into percentage:
Step 1: Divide numerator by denominator (4 ÷ 5 = 0.8).
Step 2: Multiply by 100 (0.8 × 100 = 80%).
So, 4/5 as a percentage is 80%.

Question 19:

Find the simple interest on ₹2000 at 5% per annum for 3 years.

Answer:

To calculate simple interest:
Step 1: Use the formula SI = (P × R × T) / 100.
Step 2: Substitute values (P = ₹2000, R = 5%, T = 3 years).
Step 3: Calculate (2000 × 5 × 3) / 100 = ₹300.
So, the simple interest is ₹300.

Question 20:

If the selling price of an article is ₹450 and the loss is 10%, find the cost price.

Answer:

To find the cost price when loss is 10%:
Step 1: Let cost price be CP.
Step 2: Selling Price = CP - Loss ⇒ ₹450 = CP - (10% of CP).
Step 3: ₹450 = 0.90 × CP ⇒ CP = ₹450 / 0.90 = ₹500.
So, the cost price is ₹500.

Question 21:

A shopkeeper offers a discount of 25% on a pair of shoes marked at ₹1200. Find the discount amount.

Answer:

To find the discount amount:
Step 1: Calculate 25% of marked price (0.25 × ₹1200 = ₹300).
So, the discount amount is ₹300.

Question 22:

If the population of a town increases from 50,000 to 55,000, find the percentage increase.

Answer:

To find the percentage increase:
Step 1: Calculate increase (55,000 - 50,000 = 5,000).
Step 2: Divide increase by original population (5,000 / 50,000 = 0.10).
Step 3: Multiply by 100 (0.10 × 100 = 10%).
So, the percentage increase is 10%.

Very Short Answer (2 Marks) – with Solutions (CBSE Pattern)

These 2-mark questions test key concepts in a brief format. Answers are expected to be accurate and slightly descriptive.

Question 1:

If the cost price of a book is ₹250 and it is sold at a profit of 20%, find its selling price.

Answer:

To find the selling price (SP) of the book:
Step 1: Calculate the profit amount (20% of ₹250).
Profit = (20/100) × 250 = ₹50.
Step 2: Add the profit to the cost price (CP).
SP = CP + Profit = ₹250 + ₹50 = ₹300.
Thus, the selling price is ₹300.

Question 2:

Express 0.75 as a percentage.

Answer:

To convert 0.75 into a percentage:
Multiply the decimal by 100.
0.75 × 100 = 75%.
Thus, 0.75 is equivalent to 75%.

Question 3:

A shirt is marked at ₹800 and sold at a discount of 15%. Find the discount amount.

Answer:

To calculate the discount amount:
Step 1: Find 15% of the marked price (₹800).
Discount = (15/100) × 800 = ₹120.
Thus, the discount amount is ₹120.

Question 4:

If 25% of a number is 50, find the number.

Answer:

Let the number be x.
Given: 25% of x = 50.
Step 1: Convert percentage to decimal (25% = 0.25).
Step 2: Divide both sides by 0.25.
x = 50 ÷ 0.25 = 200.
Thus, the number is 200.

Question 5:

Calculate the simple interest on ₹2000 at 5% per annum for 3 years.

Answer:

Simple Interest (SI) formula: SI = (P × R × T)/100.
Given: P = ₹2000, R = 5%, T = 3 years.
Step 1: Substitute the values into the formula.
SI = (2000 × 5 × 3)/100 = 30000/100 = ₹300.
Thus, the simple interest is ₹300.

Question 6:

A bag contains ₹1, ₹2, and ₹5 coins in the ratio 3:2:1. If the total amount is ₹60, find the number of ₹5 coins.

Answer:

Let the number of ₹1, ₹2, and ₹5 coins be 3x, 2x, and x respectively.
Total amount = (3x × 1) + (2x × 2) + (x × 5) = 3x + 4x + 5x = 12x.
Given: 12x = ₹60 ⇒ x = 5.
Thus, the number of ₹5 coins is 5.

Question 7:

If the selling price of an article is ₹450 and the profit is 12.5%, find the cost price.

Answer:

Let the cost price (CP) be ₹x.
Given: Profit = 12.5% of CP, SP = ₹450.
SP = CP + Profit ⇒ 450 = x + (12.5/100)x ⇒ 450 = 1.125x.
x = 450 ÷ 1.125 = ₹400.
Thus, the cost price is ₹400.

Question 8:

Convert the fraction 4/5 into a percentage.

Answer:

To convert 4/5 into a percentage:
Step 1: Divide numerator by denominator (4 ÷ 5 = 0.8).
Step 2: Multiply by 100 (0.8 × 100 = 80%).
Thus, 4/5 is equivalent to 80%.

Question 9:

A shopkeeper offers a discount of 10% on a pair of shoes marked at ₹1200. Find the selling price after the discount.

Answer:

To find the selling price (SP) after a 10% discount:
Step 1: Calculate the discount amount (10% of ₹1200).
Discount = (10/100) × 1200 = ₹120.
Step 2: Subtract the discount from the marked price.
SP = ₹1200 - ₹120 = ₹1080.
Thus, the selling price is ₹1080.

Short Answer (3 Marks) – with Solutions (CBSE Pattern)

These 3-mark questions require brief explanations and help assess understanding and application of concepts.

Question 1:

A shirt is marked at ₹500 with a discount of 20%. Calculate the selling price after the discount.

Answer:

To find the selling price after a 20% discount:

Original price = ₹500
Discount = 20% of ₹500 = (20/100) × 500 = ₹100
Selling price = Original price - Discount = ₹500 - ₹100 = ₹400

Question 2:

If the cost price of a book is ₹250 and it is sold at a profit of 15%, what is the selling price?

Answer:

To calculate the selling price with a 15% profit:

Cost price = ₹250
Profit = 15% of ₹250 = (15/100) × 250 = ₹37.50
Selling price = Cost price + Profit = ₹250 + ₹37.50 = ₹287.50

Question 3:

Convert the ratio 3:5 into a percentage.

Answer:

To convert the ratio 3:5 into a percentage:

Total parts = 3 + 5 = 8
Percentage of first part = (3/8) × 100 = 37.5%
Percentage of second part = (5/8) × 100 = 62.5%

Question 4:

A shopkeeper bought a cycle for ₹1200 and sold it for ₹1500. Calculate the profit percentage.

Answer:

To find the profit percentage:

Cost price = ₹1200
Selling price = ₹1500
Profit = Selling price - Cost price = ₹1500 - ₹1200 = ₹300
Profit percentage = (Profit/Cost price) × 100 = (300/1200) × 100 = 25%

Question 5:

If the simple interest on a sum of ₹2000 for 2 years is ₹400, what is the rate of interest per annum?

Answer:

To calculate the rate of interest per annum:

Principal (P) = ₹2000
Time (T) = 2 years
Simple Interest (SI) = ₹400
Using the formula: SI = (P × R × T)/100
400 = (2000 × R × 2)/100
400 = 40 × R
R = 400/40 = 10% per annum

Question 6:

If the cost price of a book is ₹250 and it is sold for ₹300, calculate the profit percentage.

Answer:

To calculate the profit percentage:

Cost Price (CP) = ₹250
Selling Price (SP) = ₹300
Profit = SP - CP = ₹300 - ₹250 = ₹50
Profit Percentage = (Profit/CP) × 100 = (50/250) × 100 = 20%

Question 7:

A shopkeeper offers a 15% discount on a pair of shoes priced at ₹1200. What is the amount the customer pays?

Answer:

To find the amount paid after a 15% discount:

Original price = ₹1200
Discount amount = 15% of ₹1200 = (15/100) × 1200 = ₹180
Amount paid = Original price - Discount = ₹1200 - ₹180 = ₹1020

Long Answer (5 Marks) – with Solutions (CBSE Pattern)

These 5-mark questions are descriptive and require detailed, structured answers with proper explanation and examples.

Question 1:

A shopkeeper offers a discount of 15% on a pair of shoes marked at ₹1200. Calculate the selling price and explain the steps involved.

Answer:

Introduction

We studied that discounts reduce the marked price of an item. Here, we calculate the selling price after a 15% discount.

Argument 1

Marked Price (MP) = ₹1200
Discount = 15% of MP = (15/100) × 1200 = ₹180

Argument 2

Selling Price (SP) = MP - Discount = 1200 - 180 = ₹1020

Conclusion

Our textbook shows similar problems. The selling price after a 15% discount is ₹1020.

Question 2:

Convert the ratio 3:5 into a percentage and explain the process with an example from daily life.

Answer:

Introduction

Ratios compare quantities, and percentages represent parts per hundred. We convert 3:5 into a percentage.

Argument 1

Total parts = 3 + 5 = 8
Fraction of first part = 3/8

Argument 2

Percentage = (3/8) × 100 = 37.5%
Example: If 3 out of 8 students like cricket, 37.5% prefer cricket.

Conclusion

Our textbook explains such conversions. The ratio 3:5 equals 37.5%.

Question 3:

If the simple interest on ₹5000 for 2 years is ₹600, find the rate of interest per annum. Show the steps.

Answer:

Introduction

Simple interest (SI) is calculated using the formula SI = (P×R×T)/100. We find the rate (R) here.

Argument 1

Given: P = ₹5000, T = 2 years, SI = ₹600
Formula: R = (SI × 100)/(P × T)

Argument 2

R = (600 × 100)/(5000 × 2) = 6%

Conclusion

Our textbook shows similar problems. The rate of interest is 6% per annum.

Question 4:

A shopkeeper marks a dress at ₹800 and offers a discount of 15%. Calculate the selling price and explain the steps.

Answer:

Introduction

We studied how discounts reduce the marked price. Here, we calculate the selling price after a 15% discount.

Argument 1

Marked Price (MP) = ₹800
Discount = 15% of MP = (15/100) × 800 = ₹120

Argument 2

Selling Price (SP) = MP - Discount = ₹800 - ₹120 = ₹680
Our textbook shows similar problems like Example 8.2.

Conclusion

The selling price is ₹680. Discounts help customers save money.

Question 5:

Convert the ratio 3:5 into a percentage and explain the steps with a real-life example.

Answer:

Introduction

Ratios compare quantities. We convert 3:5 into a percentage to understand proportions better.

Argument 1

Total parts = 3 + 5 = 8
Fraction for first term = 3/8

Argument 2

Percentage = (3/8) × 100 = 37.5%
In real life, if 3 out of 8 students like math, 37.5% prefer it.

Conclusion

The ratio 3:5 as a percentage is 37.5%. Percentages make comparisons easier.

Question 6:

If the principal is ₹5000, the rate is 8% per annum, and the time is 2 years, calculate the simple interest.

Answer:

Introduction

Simple interest is calculated using the formula SI = (P×R×T)/100. Here, we find SI for given values.

Argument 1

Principal (P) = ₹5000
Rate (R) = 8% per annum

Argument 2

Time (T) = 2 years
SI = (5000×8×2)/100 = ₹800
Our textbook shows this in Example 7.3.

Conclusion

The simple interest is ₹800. Banks use this to calculate loans.

Question 7:

A shopkeeper offers a discount of 20% on a pair of shoes marked at ₹1200. Calculate the selling price and explain the steps using the concept of percentage decrease.

Answer:

Introduction

We studied that discounts reduce the marked price of an item. Here, we calculate the selling price after a 20% discount.

Argument 1

Marked Price (MP) = ₹1200
Discount = 20% of MP = (20/100) × 1200 = ₹240

Argument 2

Selling Price (SP) = MP - Discount = ₹1200 - ₹240 = ₹960
Our textbook shows similar problems where percentage decrease is applied.

Conclusion

The selling price is ₹960. This method helps compare prices during sales.

Question 8:

Convert the ratio 3:5 into a percentage and explain the steps. How is this useful in real-life scenarios like mixing solutions?

Answer:

Introduction

Ratios compare quantities, and percentages make them easier to understand. We convert 3:5 into a percentage.

Argument 1

Total parts = 3 + 5 = 8
First part (3) as percentage = (3/8) × 100 = 37.5%

Argument 2

Second part (5) as percentage = (5/8) × 100 = 62.5%
In real life, like mixing solutions, ratios help maintain proportions.

Conclusion

The ratio 3:5 converts to 37.5% and 62.5%. This is useful in cooking or science experiments.

Question 9:

If the simple interest on ₹5000 for 2 years is ₹600, find the rate of interest per annum. Show the formula and steps.

Answer:

Introduction

Simple interest (SI) is calculated using the formula SI = (P×R×T)/100. We find the rate (R) for given values.

Argument 1

Principal (P) = ₹5000, Time (T) = 2 years, SI = ₹600
Formula: R = (SI × 100)/(P × T)

Argument 2

R = (600 × 100)/(5000 × 2) = 6%
Our textbook shows similar problems for practice.

Conclusion

The rate of interest is 6% per annum. This helps in planning loans or savings.

Question 10:

Convert the ratio 3:5 into a percentage and explain the steps with an example from daily life.

Answer:

Introduction

We learned that ratios can be converted to percentages. Here, we convert 3:5 into a percentage.

Argument 1

Total parts = 3 + 5 = 8
Percentage of first part = (3/8) × 100 = 37.5%

Argument 2

Percentage of second part = (5/8) × 100 = 62.5%
Example: In a class of 8 students, 3 are boys (37.5%) and 5 are girls (62.5%).

Conclusion

Our textbook shows how ratios help in real-life comparisons. The ratio 3:5 converts to 37.5% and 62.5%.

Question 11:

If the principal is ₹2000, the rate is 5% per annum, and the time is 2 years, calculate the simple interest and total amount.

Answer:

Introduction

We studied simple interest (SI) as interest calculated only on the principal. Here, we find SI and total amount.

Argument 1

Principal (P) = ₹2000, Rate (R) = 5%, Time (T) = 2 years
SI = (P × R × T)/100 = (2000 × 5 × 2)/100 = ₹200

Argument 2

Total Amount = P + SI = ₹2000 + ₹200 = ₹2200
Example: Our textbook shows similar banking problems.

Conclusion

The simple interest is ₹200, and the total amount after 2 years is ₹2200.

Question 12:

A shopkeeper offers a discount of 20% on a pair of shoes marked at ₹2000. The customer also has a coupon for an additional 10% off the discounted price. Calculate the final amount paid by the customer and explain the steps involved.

Answer:

To find the final amount paid by the customer, follow these steps:

Step 1: Calculate the first discount of 20% on the marked price.
Discount = 20% of ₹2000 = (20/100) × 2000 = ₹400.
Price after first discount = ₹2000 - ₹400 = ₹1600.

Step 2: Apply the additional 10% coupon discount on the discounted price.
Additional Discount = 10% of ₹1600 = (10/100) × 1600 = ₹160.
Final Price = ₹1600 - ₹160 = ₹1440.

The customer pays ₹1440 after both discounts. Always remember to apply discounts sequentially, not additively.

Question 13:

The population of a town increased from 50,000 to 55,000 in a year. Calculate the percentage increase in population and explain the formula used.

Answer:

The percentage increase in population is calculated using the formula:

Percentage Increase = [(New Value - Original Value) / Original Value] × 100.

Step 1: Find the increase in population.
Increase = 55,000 - 50,000 = 5,000.

Step 2: Apply the formula.
Percentage Increase = (5,000 / 50,000) × 100 = 10%.

The population increased by 10%. This formula helps compare growth across different quantities.

Question 14:

A man bought a bicycle for ₹3,000 and sold it at a profit of 15%. Calculate the selling price and explain how profit percentage is derived.

Answer:

To find the selling price with a 15% profit:

Step 1: Calculate the profit amount.
Profit = 15% of ₹3,000 = (15/100) × 3000 = ₹450.

Step 2: Add the profit to the cost price.
Selling Price = ₹3,000 + ₹450 = ₹3,450.

The selling price is ₹3,450. Profit percentage is always calculated on the cost price, not the selling price.

Question 15:

A shopkeeper offers a discount of 15% on a pair of shoes marked at ₹1200. The customer also has a coupon for an additional 5% discount on the already discounted price. Calculate the final amount the customer pays for the shoes. Show all steps clearly.

Answer:

To find the final amount paid by the customer, we follow these steps:

Step 1: Calculate the first discount (15%)
Original price = ₹1200
Discount = 15% of ₹1200
= (15/100) × 1200
= ₹180
Price after first discount = ₹1200 - ₹180 = ₹1020

Step 2: Apply the additional coupon discount (5%)
Additional discount = 5% of ₹1020
= (5/100) × 1020
= ₹51
Final price = ₹1020 - ₹51 = ₹969

The customer pays a final amount of ₹969 for the shoes.

Question 16:

The population of a town increased from 50,000 to 55,000 in one year. Calculate the percentage increase in population. Also, explain why understanding percentage change is important in real-life situations.

Answer:

Step 1: Calculate the increase in population
Initial population = 50,000
New population = 55,000
Increase = 55,000 - 50,000 = 5,000

Step 2: Calculate the percentage increase
Percentage increase = (Increase / Initial population) × 100
= (5,000 / 50,000) × 100
= 10%

The population increased by 10% in one year.

Importance of percentage change in real life:

Helps in comparing growth or decline in quantities like population, prices, or profits.
Used in financial decisions, such as calculating interest rates or discounts.
Essential for analyzing data trends in business, economics, and science.

Question 17:

A shopkeeper offers a discount of 20% on a pair of shoes marked at ₹2000. During a festive season, he gives an additional discount of 10% on the already discounted price.

Calculate the final selling price of the shoes and explain the concept of successive discounts with another example.

Answer:

First, calculate the initial discount of 20% on the marked price of ₹2000.

20% of ₹2000 = (20/100) × 2000 = ₹400
Selling price after first discount = ₹2000 - ₹400 = ₹1600

Now, apply the additional discount of 10% on ₹1600.

10% of ₹1600 = (10/100) × 1600 = ₹160
Final selling price = ₹1600 - ₹160 = ₹1440

The final selling price of the shoes is ₹1440.

Successive discounts mean applying multiple discounts one after another on the reduced price, not the original price. For example, if a shirt costs ₹500 with discounts of 15% and then 5%, the calculation would be:

First discount: 15% of ₹500 = ₹75 → New price = ₹425
Second discount: 5% of ₹425 = ₹21.25 → Final price = ₹403.75

Question 18:

The population of a town increased from 50,000 to 55,000 in one year.

(a) Calculate the percentage increase in population.

(b) If the population continues to grow at the same rate, what will be the population after one more year? Explain the concept of percentage growth in real-life scenarios.

Answer:

(a) To find the percentage increase in population:

Increase in population = 55,000 - 50,000 = 5,000
Percentage increase = (Increase / Original population) × 100
= (5,000 / 50,000) × 100 = 10%

(b) If the population grows at the same rate (10%), the population after another year will be:

10% of 55,000 = (10/100) × 55,000 = 5,500
New population = 55,000 + 5,500 = 60,500

Percentage growth helps us understand changes over time, like population, prices, or investments. For example:

A 5% annual price rise means an item costing ₹100 will cost ₹105 next year.
If a bank offers 7% interest, ₹1,000 becomes ₹1,070 after a year.

Question 19:

The population of a town increased from 25,000 to 27,500 in a year. Calculate the percentage increase in population. Also, explain why understanding percentage change is useful in real-life scenarios.

Answer:

Step 1: Calculate the increase in population
Initial population = 25,000
Final population = 27,500
Increase = 27,500 - 25,000 = 2,500

Step 2: Calculate the percentage increase
Percentage increase = (Increase / Initial population) × 100
= (2,500 / 25,000) × 100
= 10%

The population increased by 10% in a year.

Real-life application: Understanding percentage change helps in analyzing trends, such as price hikes, population growth, or profit/loss in business. For example, a shopkeeper can compare monthly sales, or a student can track improvement in marks using percentage change.

Question 20:

A shopkeeper offers a discount of 15% on a pair of shoes marked at ₹1200. However, during a festive sale, an additional discount of 5% is given on the already discounted price.

Calculate the final selling price of the shoes after both discounts. Also, determine the total discount percentage offered compared to the original marked price.

Answer:

To find the final selling price after both discounts, follow these steps:

Step 1: Calculate the first discount (15%)
Discount = 15% of ₹1200
= (15/100) × 1200
= ₹180
Selling price after first discount = ₹1200 - ₹180 = ₹1020

Step 2: Calculate the second discount (5%) on the new price
Additional discount = 5% of ₹1020
= (5/100) × 1020
= ₹51
Final selling price = ₹1020 - ₹51 = ₹969

Step 3: Calculate the total discount percentage
Total discount amount = ₹1200 - ₹969 = ₹231
Total discount percentage = (231/1200) × 100
= 19.25%

Thus, the final selling price is ₹969, and the total discount offered is 19.25% compared to the original marked price.

Question 21:

The population of a town increased from 50,000 to 55,000 in one year.

Calculate the percentage increase in population. If this growth rate continues for the next year, what will be the population after two years? Explain the concept of compound growth in this context.

Answer:

Step 1: Calculate the percentage increase in population for the first year
Increase in population = 55,000 - 50,000 = 5,000
Percentage increase = (5,000 / 50,000) × 100
= 10%

Step 2: Project the population for the second year using the same growth rate
Population after second year = 55,000 + (10% of 55,000)
= 55,000 + (10/100 × 55,000)
= 55,000 + 5,500
= 60,500

Compound growth refers to the process where the increase in each period is calculated on the updated value (including previous increases) rather than the original value. Here, the 10% growth in the second year is applied to the new population (55,000), not the original (50,000). This leads to a higher absolute increase in the second year (₹5,500) compared to the first year (₹5,000).

Case-based Questions (4 Marks) – with Solutions (CBSE Pattern)

These 4-mark case-based questions assess analytical skills through real-life scenarios. Answers must be based on the case study provided.

Question 1:

A shopkeeper offers a discount of 15% on a pair of shoes marked at ₹1200. A customer also has a coupon for an additional 5% off.
Problem Interpretation: What is the final price the customer pays?

Answer:

Problem Interpretation: We need to find the final price after two successive discounts.
Mathematical Modeling: First discount = 15% of ₹1200, second discount = 5% of the reduced price.
Solution:

First discount = (15/100) × 1200 = ₹180. New price = ₹1200 - ₹180 = ₹1020.
Second discount = (5/100) × 1020 = ₹51. Final price = ₹1020 - ₹51 = ₹969.

The customer pays ₹969.

Question 2:

In a school, 40% of students are girls. If there are 240 boys,
Problem Interpretation: What is the total number of students?

Answer:

Problem Interpretation: We need to find the total students given the percentage of girls and number of boys.
Mathematical Modeling: If 40% are girls, 60% must be boys. Let total students be x.
Solution:

60% of x = 240 → (60/100) × x = 240.
x = 240 × (100/60) = 400.

The total number of students is 400.

Question 3:

A shopkeeper offers a discount of 20% on a pair of shoes marked at ₹1,200. Calculate the selling price after the discount. Also, find the profit percentage if the cost price was ₹800.

Answer:

Problem Interpretation

We need to find the selling price after a 20% discount and then calculate the profit percentage.

Mathematical Modeling

Discount = 20% of ₹1,200 = ₹240
Selling Price = ₹1,200 - ₹240 = ₹960

Solution

Profit = ₹960 - ₹800 = ₹160
Profit Percentage = (160/800) × 100 = 20%

Question 4:

In a class of 40 students, 60% are girls. How many boys are there? If 25% of the girls wear glasses, find the number of girls wearing glasses.

Answer:

Problem Interpretation

We need to find the number of boys and the number of girls wearing glasses.

Mathematical Modeling

Number of girls = 60% of 40 = 24
Number of boys = 40 - 24 = 16

Solution

Girls wearing glasses = 25% of 24 = 6

Question 5:

A shopkeeper offers a discount of 20% on a pair of shoes marked at ₹1200. A customer also has a coupon for an additional 5% off. Find the final price the customer pays.

Answer:

Problem Interpretation

We need to calculate the final price after applying two successive discounts.

Mathematical Modeling

First discount: 20% of ₹1200 = ₹240
Price after first discount: ₹1200 - ₹240 = ₹960
Second discount: 5% of ₹960 = ₹48

Solution

Final price = ₹960 - ₹48 = ₹912. The customer pays ₹912.

Question 6:

In a class of 40 students, 60% are girls. If 25% of the girls and 40% of the boys wear spectacles, find the total number of students wearing spectacles.

Answer:

Problem Interpretation

We need to find the total number of students wearing spectacles from given percentages.

Mathematical Modeling

Number of girls: 60% of 40 = 24
Number of boys: 40 - 24 = 16
Girls wearing spectacles: 25% of 24 = 6
Boys wearing spectacles: 40% of 16 = 6.4 ≈ 6 (since students can't be in fractions)

Solution

Total students wearing spectacles = 6 (girls) + 6 (boys) = 12.

Question 7:

A shopkeeper offers a discount of 20% on a pair of shoes marked at ₹1200. Calculate the selling price after discount and compare it with the original price.

Answer:

Problem Interpretation

We need to find the selling price after a 20% discount on ₹1200 and compare it with the original price.

Mathematical Modeling

Discount = 20% of ₹1200
Selling Price = Original Price - Discount

Solution

Discount = (20/100) × 1200 = ₹240. Selling Price = 1200 - 240 = ₹960. The selling price is ₹240 less than the original price.

Question 8:

A fruit seller bought 50 kg of apples at ₹40 per kg. Due to spoilage, 5 kg were wasted. He sold the remaining at ₹50 per kg. Find the profit or loss percentage.

Answer:

Problem Interpretation

We need to calculate the profit or loss percentage after selling the remaining apples at ₹50/kg.

Mathematical Modeling

Total Cost Price = 50 kg × ₹40/kg
Total Selling Price = (50 kg - 5 kg) × ₹50/kg

Solution

Cost Price = ₹2000. Selling Price = 45 × 50 = ₹2250. Profit = ₹250. Profit % = (250/2000) × 100 = 12.5%.

Question 9:

A shopkeeper offers a discount of 15% on a school bag priced at ₹800. Calculate the selling price after the discount. Also, determine the amount saved by the customer.

Answer:

Problem Interpretation

We need to find the selling price after a 15% discount and the amount saved on a bag costing ₹800.

Mathematical Modeling

Discount = 15% of ₹800
Selling Price = Original Price - Discount

Solution

Discount = (15/100) × 800 = ₹120
Selling Price = 800 - 120 = ₹680
Amount Saved = ₹120

Question 10:

In a class of 50 students, 30% are girls. Find the number of girls and boys separately. If 10 more girls join the class, what will be the new percentage of girls?

Answer:

Problem Interpretation

We need to find the number of girls and boys in a class of 50, where 30% are girls. Then, calculate the new percentage if 10 more girls join.

Mathematical Modeling

Girls = 30% of 50
Boys = Total - Girls
New Percentage = (New Girls / New Total) × 100

Solution

Girls = (30/100) × 50 = 15
Boys = 50 - 15 = 35
New Girls = 15 + 10 = 25
New Percentage = (25/60) × 100 ≈ 41.67%

Question 11:

A shopkeeper offers a discount of 20% on a pair of shoes marked at ₹1200. Calculate the selling price after the discount. Also, find the profit percentage if the cost price was ₹800.

Answer:

Problem Interpretation

We need to find the selling price after a 20% discount and then calculate the profit percentage.

Mathematical Modeling

Discount = 20% of ₹1200 = ₹240
Selling Price = ₹1200 - ₹240 = ₹960

Solution

Profit = ₹960 - ₹800 = ₹160
Profit Percentage = (160/800) × 100 = 20%

Question 12:

Rahul bought a bicycle for ₹2500 and sold it for ₹3000. His friend Riya bought the same bicycle but sold it at a loss of 10%. Compare their profit/loss percentages.

Answer:

Problem Interpretation

We compare Rahul's profit and Riya's loss percentages for the same bicycle.

Mathematical Modeling

Rahul's Profit = ₹3000 - ₹2500 = ₹500
Rahul's Profit Percentage = (500/2500) × 100 = 20%

Solution

Riya's Selling Price = ₹2500 - (10% of ₹2500) = ₹2250
Riya's Loss Percentage = 10% (given)

Question 13:

Rahul bought a bicycle for ₹2,500 and sold it at a profit of 12%. Later, he bought another bicycle for ₹3,000 and sold it at a loss of 8%.

Calculate his overall profit or loss amount and percentage.

Answer:

To find Rahul's overall profit or loss, let's break it down step by step:

First Bicycle:
Cost Price (CP) = ₹2,500
Profit = 12% of CP = (12/100) × 2,500 = ₹300
Selling Price (SP) = CP + Profit = 2,500 + 300 = ₹2,800

Second Bicycle:
Cost Price (CP) = ₹3,000
Loss = 8% of CP = (8/100) × 3,000 = ₹240
Selling Price (SP) = CP - Loss = 3,000 - 240 = ₹2,760

Total Calculations:
Total CP = 2,500 + 3,000 = ₹5,500
Total SP = 2,800 + 2,760 = ₹5,560
Overall Profit = Total SP - Total CP = 5,560 - 5,500 = ₹60

Profit Percentage:
Profit % = (Profit / Total CP) × 100 = (60 / 5,500) × 100 ≈ 1.09%

Rahul made an overall profit of ₹60, which is approximately 1.09%.

Question 14:

A shopkeeper marks the price of a toy 30% above its cost price but allows a discount of 10% on the marked price.

If the cost price of the toy is ₹800, find the selling price and the actual profit percentage earned by the shopkeeper.

Answer:

Let's solve this step by step:

Step 1: Calculate Marked Price (MP)
CP = ₹800
Markup = 30% of CP = (30/100) × 800 = ₹240
MP = CP + Markup = 800 + 240 = ₹1,040

Step 2: Calculate Discount
Discount = 10% of MP = (10/100) × 1,040 = ₹104
Selling Price (SP) = MP - Discount = 1,040 - 104 = ₹936

Step 3: Calculate Profit and Profit Percentage
Profit = SP - CP = 936 - 800 = ₹136
Profit % = (Profit / CP) × 100 = (136 / 800) × 100 = 17%

The selling price is ₹936, and the shopkeeper earns a profit of 17%.

Question 15:

A shopkeeper offers a discount of 15% on a pair of shoes marked at ₹1200. During a festive season, he gives an additional discount of 5% on the already discounted price.

Calculate the final selling price of the shoes after both discounts.

Answer:

First, calculate the price after the initial 15% discount:

Original price = ₹1200
Discount = 15% of ₹1200 = (15/100) × 1200 = ₹180
Price after first discount = ₹1200 - ₹180 = ₹1020

Now, apply the additional 5% discount on ₹1020:

Additional discount = 5% of ₹1020 = (5/100) × 1020 = ₹51
Final selling price = ₹1020 - ₹51 = ₹969

Thus, the customer pays ₹969 after both discounts.

Question 16:

Rahul invests ₹8000 in a savings scheme that offers 8% simple interest per annum.

Calculate the total amount he will receive after 3 years. Also, explain the difference between simple interest and compound interest with an example.

Answer:

First, calculate the simple interest for 3 years:

Principal (P) = ₹8000
Rate (R) = 8% per annum
Time (T) = 3 years
Simple Interest (SI) = (P × R × T)/100 = (8000 × 8 × 3)/100 = ₹1920
Total amount = P + SI = ₹8000 + ₹1920 = ₹9920

Difference between simple interest and compound interest:

Simple interest is calculated only on the principal amount throughout the time period.
Compound interest is calculated on the principal as well as the accumulated interest of previous periods.

Example: If ₹1000 is invested at 10% for 2 years:
Simple Interest = ₹1000 × 10 × 2 / 100 = ₹200
Compound Interest = ₹1000 × (1 + 10/100)² - ₹1000 = ₹210

Question 17:

Rahul bought a bicycle for ₹ 3,500 and sold it at a profit of 15%. Later, he realized he could have sold it for ₹ 4,200 if he waited.

Based on this case:

Calculate the actual selling price.
Find the profit percentage he missed by selling early.

Answer:

Step 1: Calculate the actual selling price.

Profit = 15% of Cost Price (CP)
CP = ₹ 3,500
Profit = (15/100) × 3500 = ₹ 525
Selling Price (SP) = CP + Profit = 3500 + 525 = ₹ 4,025

Step 2: Find the missed profit percentage.

Potential SP = ₹ 4,200
Potential Profit = 4200 - 3500 = ₹ 700
Missed Profit Percentage = (700/3500) × 100 = 20%

Key Insight: Rahul earned a 15% profit but missed an additional 5% (20% - 15%) by selling early.

Question 18:

A shopkeeper marks a pair of shoes at ₹ 1,800 and offers a 10% discount. During a sale, he gives an extra 5% discount on the already discounted price.

Based on this case:

Calculate the final selling price after both discounts.
Determine the equivalent single discount percentage.

Answer:

Step 1: Calculate the first discount.

Marked Price (MP) = ₹ 1,800
First Discount = 10% of MP = (10/100) × 1800 = ₹ 180
Price after first discount = 1800 - 180 = ₹ 1,620

Step 2: Calculate the second discount.

Second Discount = 5% of ₹ 1,620 = (5/100) × 1620 = ₹ 81
Final Selling Price = 1620 - 81 = ₹ 1,539

Step 3: Find the equivalent single discount.

Total Discount Amount = 1800 - 1539 = ₹ 261
Equivalent Discount Percentage = (261/1800) × 100 = 14.5%

Key Insight: Sequential discounts of 10% and 5% are equivalent to a single discount of 14.5%, not 15%.

Question 19:

Rahul bought a bicycle for ₹3,500 and sold it at a profit of 12%. Later, he realized he could have sold it for ₹4,200. Did he gain or lose the opportunity to earn more? Calculate the difference.

Answer:

First, calculate Rahul's profit on the bicycle:

Cost Price (CP) = ₹3,500
Profit Percentage = 12%
Profit Amount = (12/100) × 3,500 = ₹420
Selling Price (SP) = CP + Profit = 3,500 + 420 = ₹3,920

Now, compare it with the missed opportunity:

Potential SP = ₹4,200
Difference = 4,200 - 3,920 = ₹280

Rahul lost the opportunity to earn ₹280 more.

Question 20:

A shopkeeper marks a pair of shoes at ₹1,800 and offers a discount of 15%. A customer bargains for an additional 5% discount. Calculate the final price paid by the customer and the total discount percentage.

Answer:

Step 1: Calculate the first discount:

Marked Price (MP) = ₹1,800
First Discount = 15% of 1,800 = (15/100) × 1,800 = ₹270
Price after first discount = 1,800 - 270 = ₹1,530

Step 2: Calculate the additional discount:

Additional Discount = 5% of 1,530 = (5/100) × 1,530 = ₹76.50
Final Price = 1,530 - 76.50 = ₹1,453.50

Step 3: Calculate total discount percentage:

Total Discount Amount = 270 + 76.50 = ₹346.50
Total Discount Percentage = (346.50 / 1,800) × 100 = 19.25%

Question 21:

Rahul bought a bicycle for ₹2,500 and sold it for ₹3,000 after a year.

Based on this information:

Calculate the profit Rahul made.
Find the profit percentage.

Answer:

To solve this problem, we need to calculate the profit and profit percentage.

Step 1: Calculate Profit
Profit = Selling Price - Cost Price
Profit = ₹3,000 - ₹2,500 = ₹500

Step 2: Calculate Profit Percentage
Profit Percentage = (Profit / Cost Price) × 100
Profit Percentage = (₹500 / ₹2,500) × 100 = 20%

Rahul made a profit of ₹500 with a profit percentage of 20%.

Question 22:

A shopkeeper marks the price of a toy at ₹800. During a sale, he offers a discount of 15%.

Based on this information:

Find the discount amount.
Calculate the selling price after the discount.

Answer:

To solve this problem, we need to calculate the discount amount and the selling price after the discount.

Step 1: Calculate Discount Amount
Discount Amount = Marked Price × (Discount Percentage / 100)
Discount Amount = ₹800 × (15 / 100) = ₹120

Step 2: Calculate Selling Price
Selling Price = Marked Price - Discount Amount
Selling Price = ₹800 - ₹120 = ₹680

The discount amount is ₹120, and the selling price after the discount is ₹680.

Comparing Quantities – CBSE NCERT Study Resources

Chapter Overview: Comparing Quantities

Key Concepts

Ratio and Proportion

Percentage

Profit and Loss

Simple Interest

Applications in Real Life

All Question Types with Solutions – CBSE Exam Pattern

Very Short Answer (1 Mark) – with Solutions (CBSE Pattern)

Very Short Answer (2 Marks) – with Solutions (CBSE Pattern)

Short Answer (3 Marks) – with Solutions (CBSE Pattern)

Long Answer (5 Marks) – with Solutions (CBSE Pattern)

Case-based Questions (4 Marks) – with Solutions (CBSE Pattern)