👁 Preview — Study, Practice and Revise are open; mock tests and the rest of the syllabus unlock on subscription. Unlock all · ₹4,999
← Back to Arithmetic
Study mode

Comparison

Study Practice Revise 🔒 Test

Introduction to Comparison in Arithmetic

Comparison in arithmetic means determining the relationship between two or more numbers or quantities - figuring out which is greater, smaller, or if they are equal. This skill is fundamental, especially in competitive exams, where numbers appear in different forms such as whole numbers, fractions, decimals, and percentages. Understanding how to compare these different types of numbers quickly and accurately enables you to solve problems efficiently.

Why is comparison important? Imagine deciding which product is cheaper, which interest rate is better, or which measurement is longer - all require comparison. In exams, many questions test your ability to compare numbers presented in various forms. Since these forms represent the same or similar values but appear differently, a key strategy is to convert numbers into a common format before comparing, often using inequalities such as > (greater than), < (less than), and = (equal to).

In this chapter, we will learn how to compare:

  • Whole numbers - based on their place value
  • Fractions - by finding common denominators or converting to decimals
  • Decimals - by comparing digit by digit
  • Percentages - by converting to decimals/fractions or applying to the same base

Carefully following these methods will not only help you answer comparison questions correctly but also save valuable time in competitive exams.

Comparison of Whole Numbers

Whole numbers are numbers without fractions or decimals, such as 458, 12, or 1000. To compare whole numbers, the best approach is to examine their place values starting from the leftmost digit because the left digit represents the highest place value.

For example, compare 458 and 485:

  • Look at the hundreds place: both have '4', so they are equal.
  • Move to the tens place: 5 in 458 and 8 in 485. Since 8 > 5, 485 > 458.

We use inequality symbols to express this: 485 > 458. The symbols you must know are:

  • > means "greater than"
  • < means "less than"
  • = means "equal to"
458 485 458 < 485 Place Value 458 485 Hundreds 4 4 Tens 5 8 Ones 8 5

Comparison of Fractions

Fractions represent parts of a whole, for example, \( \frac{3}{8} \) or \( \frac{5}{12} \). Comparing fractions can be tricky because their denominators (bottom numbers) might be different. Two main methods help us:

  • Convert to common denominators: Find a shared denominator to make fractions easy to compare.
  • Convert to decimals: Divide numerator by denominator to get a decimal equivalent.

Consider these fractions:

FractionCommon DenominatorDecimal Equivalent
\( \frac{3}{8} \)\( \frac{9}{24} \)0.375
\( \frac{5}{12} \)\( \frac{10}{24} \)0.4167

Since \( \frac{9}{24} < \frac{10}{24} \), \( \frac{3}{8} < \frac{5}{12} \). Similarly, 0.375 < 0.4167 confirms the same.

Comparison of Decimals

Decimals are numbers with a decimal point, such as 0.507 or 0.57. The key to comparing decimals is to align digits by the decimal point and compare digit by digit from left to right:

  • Compare tenths place (first digit after decimal)
  • If equal, compare hundredths place
  • Continue comparing further places as needed

Trailing zeros do not change the value but can cause confusion. For example, 0.5 and 0.50 are equal because trailing zeros represent no extra value.

0 . 5 0 7 0 . 5 7 0 Tenths Hundredths Compare digits left to right

Comparison of Percentages

Percentages express parts per hundred, like 25%, 42%, or 100%. To compare percentages effectively, convert them to decimals by dividing by 100, or to fractions:

  • Convert 25% = \( \frac{25}{100} = 0.25 \)
  • Convert 42% = 0.42

This helps in easy comparison or applying percentages to a common base. For example, comparing discounts on an INR 1000 product:

  • 25% discount = 0.25 x 1000 = INR 250
  • 30% discount = 0.30 x 1000 = INR 300

So, 30% discount is greater than 25% because the amount saved in INR is higher.

Summary of Comparison Methods

Key strategies for effective comparison:

  • For whole numbers, analyze digits from left (highest place value) to right.
  • For fractions, bring to a common denominator or convert to decimals.
  • For decimals, compare digit by digit, aligning decimal points.
  • For percentages, convert to decimals or fractions for meaningful comparison.

Worked Example 1: Comparing Whole Numbers on Number Line (Easy)

Example 1: Comparing 458 and 485 Easy
Which number is larger: 458 or 485? Use place value and the number line.

Step 1: Compare the hundreds digit: both have 4, so they're equal here.

Step 2: Compare the tens digit: 5 in 458 vs 8 in 485. Since 8 > 5, 485 is larger.

Step 3: Therefore, 458 < 485.

Answer: 485 is greater than 458.

Worked Example 2: Comparing Fractions by Common Denominator (Medium)

Example 2: Compare \( \frac{3}{8} \) and \( \frac{5}{12} \) Medium
Determine which is greater: \( \frac{3}{8} \) or \( \frac{5}{12} \).

Step 1: Find the least common denominator (LCD) of 8 and 12.

Prime factors: 8 = 2³, 12 = 2² x 3

LCD = \( 2^3 x 3 = 24 \).

Step 2: Convert both fractions to denominators of 24:

\( \frac{3}{8} = \frac{3 x 3}{8 x 3} = \frac{9}{24} \),

\( \frac{5}{12} = \frac{5 x 2}{12 x 2} = \frac{10}{24} \).

Step 3: Compare numerators 9 and 10; since 10 > 9, \( \frac{5}{12} \) is greater.

Answer: \( \frac{3}{8} < \frac{5}{12} \).

Worked Example 3: Comparing Decimals with Different Digits (Medium)

Example 3: Compare 0.507 and 0.57 Medium
Which decimal is larger: 0.507 or 0.57?

Step 1: Write decimals aligned by decimal points:

0.507

0.570 (adding a trailing zero)

Step 2: Compare digit by digit:

  • At tenths place: 5 vs 5 - equal.
  • At hundredths place: 0 vs 7 - 7 > 0, so 0.57 > 0.507.

Answer: 0.57 > 0.507.

Worked Example 4: Comparing Percentages in Financial Context (Medium)

Example 4: Which is more? 25% discount or 0.3 decimal discount on 1000 INR Medium
A product priced at 1000 INR offers a 25% discount. Another product offers a 0.3 decimal discount. Which discount is greater?

Step 1: Convert 25% to decimal: \( \frac{25}{100} = 0.25 \).

Step 2: Calculate actual discounts:

25% discount = 0.25 x 1000 = 250 INR.

0.3 decimal discount = 0.30 x 1000 = 300 INR.

Step 3: Compare amounts: 300 INR > 250 INR.

Answer: 0.3 decimal discount is greater than 25% discount on 1000 INR.

Worked Example 5: Mixed Comparison - Fraction, Decimal, and Percentage (Hard)

Example 5: Compare 0.45, \( \frac{7}{16} \), and 42% Hard
Determine the order of 0.45 (decimal), \( \frac{7}{16} \) (fraction), and 42% (percentage) from smallest to largest.

Step 1: Convert all to decimals for easy comparison.

\( 0.45 \) is already decimal.

\( \frac{7}{16} = \frac{7}{16} = 0.4375 \) (divide 7 by 16).

42% = \( \frac{42}{100} = 0.42 \).

Step 2: List decimals:

0.42 (42%), 0.4375 (\( \frac{7}{16} \)), 0.45 (decimal given).

Step 3: Order from smallest to largest:

0.42 < 0.4375 < 0.45

Answer: 42% < \( \frac{7}{16} \) < 0.45.

Formula Bank

Fraction to Decimal Conversion
\[ \text{Decimal} = \frac{\text{Numerator}}{\text{Denominator}} \]
where: Numerator = top part of fraction, Denominator = bottom part of fraction
Percentage to Decimal Conversion
\[ \text{Decimal} = \frac{\text{Percentage}}{100} \]
where: Percentage = given value in %
Common Denominator for Fractions
\[ \text{Common denominator} = \operatorname{LCM}(d_1, d_2, \ldots) \]
where: \( d_1, d_2, \ldots \) = denominators of fractions

Tips & Tricks

Tip: Convert all numbers to decimals for quick and uniform comparison.

When to use: Comparing fractions, decimals, and percentages together.

Tip: For two fractions with small denominators, cross-multiply instead of finding LCM.

When to use: Quickly compare two fractions without lengthy calculations.

Tip: Use rounding and estimation to simplify decimal comparison during exams.

When to use: When exact comparison is not crucial or under time pressure.

Tip: Align decimal points vertically when comparing decimals to avoid digit confusion.

When to use: Comparing decimals of different lengths.

Tip: Remember 100% equals 1 in decimals.

When to use: Converting percentages to decimals or fractions.

Common Mistakes to Avoid

❌ Comparing only the numerators of fractions without considering denominators.
✓ Always convert fractions to a common denominator or decimal form before comparing.
Why: The numerator alone does not reflect fraction size without its denominator context.
❌ Assuming 0.5 < 0.45 because 5 is less than 45.
✓ Compare decimal places left to right, digit by digit (tenths, hundredths, etc.).
Why: Decimal place-values determine size not just the digits themselves.
❌ Believing the bigger percentage always means a larger amount without context.
✓ Compare percentages after converting to decimals or apply them to the same base value.
Why: Percentage value depends on the quantity it is applied to.
❌ Confusing inequality signs > and < when writing answers.
✓ Remember > means "greater than", < means "less than". Use number lines to confirm.
Why: Confusion happens due to similar symbols, especially under time pressure.
❌ Ignoring trailing zeros in decimals and treating 0.5 and 0.50 as different.
✓ Treat trailing zeros as equal; align decimal places to confirm equality.
Why: Trailing zeros do not affect decimal value but lead to digit count confusion.
Curated videos per subtopic
Top YouTube explainers, AI-ranked for your exam and language. Unlocks with subscription.
Unlock

Try Practice next.

Progress tracking is paywalled — subscribe to mark subtopics as understood and save your streak.

Go to practice →
Ask a doubt
Comparison · 10 free messages
Ask me anything about this subtopic. You have 10 free messages this session — chat history isn't saved in preview.