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In everyday life and mathematics, we often need to find out which number or quantity is bigger, smaller, or if two values are the same. This process is called comparison. By comparing numbers or quantities, we make decisions such as which product is cheaper, which distance is longer, or which bank offers a better interest rate.
For example, when shopping in INR (Indian Rupees), you decide between two products costing Rs.250 and Rs.275 by comparing the amounts. Similarly, when measuring lengths in meters or weights in kilograms, comparing helps us understand the relationships between different quantities.
In this chapter, we will learn how to compare numbers, fractions, decimals, and percentages using clear symbols and methods. We will also explore how to apply these comparisons to real-world problems like discounts and loans.
To compare numbers, we use special symbols that show their relationship:
For example, 5 > 3 means 5 is greater than 3, and 2 < 4 means 2 is less than 4. If two numbers are 7 and 7, we write 7 = 7.
A helpful way to visualize these comparisons is on a number line, where numbers increase as we move right.
When comparing fractions, it can be tricky because their denominators (the numbers below the line) might be different. Two main methods help us compare fractions easily:
For decimals, comparing place values from left to right helps determine which is bigger or smaller.
| Fraction | Equivalent Decimal |
|---|---|
| 1/2 | 0.5 |
| 3/4 | 0.75 |
| 5/8 | 0.625 |
| 7/10 | 0.7 |
| 9/20 | 0.45 |
A percentage is a special way of expressing a number as a part of 100, written with the symbol %. Since percentages compare parts of a whole, they are often easier to understand when dealing with discounts, interest rates, or data comparisons.
However, when comparing percentages, always consider the base value (the original total) because the same percentage on different totals represents different amounts.
For example, a 10% discount on Rs.2000 is Rs.200 off, but 10% on Rs.1500 is only Rs.150 off. Therefore, one must calculate the actual amounts to find which is better.
Step 1: Convert each fraction to a decimal:
\( \frac{3}{4} = 0.75 \) and \( \frac{5}{8} = 0.625 \)
Since 0.75 > 0.625, we conclude \( \frac{3}{4} > \frac{5}{8} \).
Alternatively, Step 2: Find a common denominator:
LCM of 4 and 8 is 8.
\( \frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8} \).
Compare \( \frac{6}{8} \) and \( \frac{5}{8} \). Since 6 > 5, \( \frac{3}{4} > \frac{5}{8} \).
Answer: \( \frac{3}{4} > \frac{5}{8} \).
Step 1: Calculate discount for each product.
Product 1 discount: \( 1500 \times \frac{10}{100} = 150 \) INR.
Product 2 discount: \( 1400 \times \frac{15}{100} = 210 \) INR.
Step 2: Calculate final prices after discount.
Price of Product 1 after discount: \( 1500 - 150 = 1350 \) INR.
Price of Product 2 after discount: \( 1400 - 210 = 1190 \) INR.
Answer: Product 2 is cheaper at Rs.1190 after discount.
Step 1: Calculate simple interest for both loans using \( SI = \frac{P \times R \times T}{100} \).
Loan 1: \( SI = \frac{10000 \times 5 \times 2}{100} = Rs.1000 \).
Loan 2: \( SI = \frac{9000 \times 6 \times 2}{100} = Rs.1080 \).
Step 2: Compare interest amounts.
Loan 2 provides Rs.80 more interest than Loan 1.
Answer: Loan 2 offers better returns.
Step 1: Write the ratio as \( \frac{3}{4} = \frac{600}{x} \), where \(x\) is the flour needed.
Step 2: Cross-multiply:
\( 3 \times x = 4 \times 600 \)
\( 3x = 2400 \)
\( x = \frac{2400}{3} = 800 \) grams.
Answer: You need 800g of flour.
Step 1: Look at the decimal parts carefully.
2.35 m means 2 meters and 35 centimeters, 2.5 m means 2 meters and 50 centimeters.
Step 2: Since 0.5 (which is 50 cm) > 0.35 (which is 35 cm), 2.5 m > 2.35 m.
Answer: 2.5 meters is longer.
When to use: When denominators are different or large, making manual LCM calculation slow.
When to use: When comparing measurements like meters and centimeters or INR with paise included.
When to use: During exams when exact calculation isn't needed but to eliminate options quickly.
When to use: To reduce errors under timed exam conditions.
When to use: When comparing discount amounts or interest across different principal values.
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