NCERT Class 8 Mathematics
Chapter 7: Comparing Quantities
Welcome to Chapter 7 of your Class 8 Mathematics NCERT textbook! This chapter introduces the important concepts of comparing quantities using ratios, percentages, profit and loss, and simple interest. These concepts are essential for understanding everyday financial transactions and making informed decisions.
What are Ratios?
A ratio is a comparison of two quantities of the same kind by division. It shows how many times one quantity is contained in another.
Ratio of a to b is written as a:b or \(\frac{a}{b}\)
For example, if there are 20 boys and 30 girls in a class:
Ratio of boys to girls = 20:30 = \(\frac{20}{30}\) = \(\frac{20^{20\div10}}{30^{30 \div 3}}\) = \(\frac{2}{3}\) = 2:3
Ratio of girls to boys = 30:20 = \(\frac{30}{20}\) = \(\frac{30^{30 \div 10}}{20^{20 \div 10}}\) = \(\frac{3}{2}\) = 3:2
Ratios are used to compare quantities in various real-life situations like recipes, maps, and financial statements.
Understanding Percentages
A percentage is a fraction with denominator 100. The word "percent" means "per hundred".
Percentage = \(\frac{\text{Part}}{\text{Whole}} \times 100\)
For example, if 15 out of 20 students passed an exam:
Percentage of students who passed = \(\frac{15}{20} \times 100 = 75\%\)
Percentages are widely used in everyday life for calculating discounts, interest rates, and many other comparisons.
Profit and Loss
When we buy and sell goods, we encounter the concepts of profit and loss:
- Cost Price (CP): The price at which an article is purchased
- Selling Price (SP): The price at which an article is sold
- Profit: When SP > CP, Profit = SP - CP
- Loss: When CP > SP, Loss = CP - SP
Profit % = \(\frac{\text{Profit}}{\text{CP}} \times 100\)
Loss % = \(\frac{\text{Loss}}{\text{CP}} \times 100\)
Simple Interest
Simple Interest is the extra money paid by borrowers to lenders for using their money.
Simple Interest (SI) = \(\frac{P \times R \times T}{100}\)
Where:
P = Principal (initial amount)
R = Rate of interest per annum
T = Time in years
For example, if you borrow ₹1000 at 5% interest for 2 years:
Here: P (Principal)= 1000, R (Rate) = 5, T (Time) = 2
SI = \( \frac {1000 \times 5 \times 2}{100} \) = \( \frac {10000}{100} \) = ₹100
Amount to be returned = Principal + Interest = ₹1000 + ₹100 = ₹1100
Applications in Real Life
The concepts of comparing quantities have practical applications in various fields:
- Shopping: Calculating discounts and sale prices
- Banking: Understanding interest rates on loans and deposits
- Business: Analyzing profit and loss statements
- Cooking: Adjusting recipe quantities using ratios
- Statistics: Interpreting data presented in percentages
What You Will Learn
In this chapter, you will learn how to:
- Convert ratios to percentages and vice versa
- Solve problems involving profit and loss
- Calculate simple interest
- Find discounts and sales tax
- Solve problems involving compound interest (introductory)
- Apply these concepts to real-life situations
Key Points to Remember
- Ratios compare two quantities of the same kind
- Percentages are fractions with denominator 100
- Profit occurs when selling price > cost price
- Loss occurs when cost price > selling price
- Simple interest is calculated only on the principal amount
- Discount is a reduction in the marked price of an article
- Sales tax is an additional amount added to the bill
As you progress through this chapter, you'll discover how these mathematical concepts are applied in everyday life. Understanding comparing quantities will help you make better financial decisions and become a more informed consumer!