Introduction to Comparison

Comparison is the act of determining which of two or more numbers is larger, smaller, or if they are equal. This concept is fundamental not only in mathematics but also in everyday life - for example, comparing prices, ages, distances, or quantities. In the Indian context, consider comparing amounts in rupees (Rs.), or measuring lengths in metres and centimetres; such comparisons help us make informed decisions efficiently.

We encounter different types of numbers when comparing values: integers (whole numbers which can be positive, negative, or zero), fractions (parts of a whole), and decimals (numbers expressed with a decimal point). Understanding how to compare these various types accurately is essential, especially for competitive exams where speed and correctness both matter.

Understanding Comparison of Numbers

To compare numbers, it is important to understand some commonly used symbols:

• Greater than ( > ): The number on the left is bigger than the one on the right. Example: 7 > 3.
• Less than ( < ): The number on the left is smaller than the one on the right. Example: 2 < 4.
• Equal to ( = ): Both numbers are the same. Example: 5 = 5.

A helpful way to visualize these comparisons is the number line. It is a straight line with numbers placed at intervals where smaller numbers are on the left and larger numbers are to the right.

From the number line above, you can see that numbers increase as we move from left to right. For example, -3 is less than 0, and 2 is greater than 0.5.

When comparing decimals, the key is to look at place values - starting from the digits before the decimal and moving rightwards to digits after the decimal point, comparing digit by digit.

Techniques to Compare Fractions

Comparing fractions like \(\frac{3}{7}\) and \(\frac{4}{9}\) directly can be complex because their denominators differ. There are two common ways to compare fractions:

1. Cross Multiplication Method: Multiply numerator of first fraction by denominator of the second and vice versa, then compare these products.
2. Convert to decimals: Divide numerator by denominator to convert fractions into decimals and compare the decimal values.

The cross multiplication method is often quicker and avoids finding a common denominator.

Worked Example: Comparing Two Integers Using Number Line

Example: Compare -5 and 3

Which number is greater: -5 or 3?

Step 1: Locate -5 and 3 on the number line.

Step 2: Since 3 lies to the right of -5, it is greater.

Answer: \(3 > -5\)

Worked Example: Comparing Fractions Using Cross Multiplication

Example: Compare \(\frac{3}{7}\) and \(\frac{4}{9}\)

Step 1: Cross multiply:

Left fraction numerator x Right fraction denominator = \(3 \times 9 = 27\)

Right fraction numerator x Left fraction denominator = \(4 \times 7 = 28\)

Step 2: Compare the two products:

Since 27 < 28, \(\frac{3}{7}\) is less than \(\frac{4}{9}\).

Answer: \(\frac{3}{7} < \frac{4}{9}\)

Worked Example: Comparing Decimals

Example: Compare 0.504 and 0.54

Step 1: Align decimal places and compare digit by digit:

• Units place: Both are 0 (equal)
• First decimal place: 5 in both (equal)
• Second decimal place: 0 (0.504) vs 4 (0.54)
  Since 0 < 4, 0.504 < 0.54

Answer: \(0.504 < 0.54\)

Worked Example: Application in Currency Comparison

Example: Compare Rs.150.75 and Rs.149.85

Step 1: Look at rupees part first:

• 150 > 149, so Rs.150.75 > Rs.149.85.

Step 2: If rupees were equal, compare paise (decimal part).

Practical importance: Helps in knowing which price is costlier or offers more value.

Answer: Rs.150.75 is greater than Rs.149.85.

Worked Example: Comparing Mixed Fractions

Example: Compare \(2 \frac{3}{5}\) and \(2 \frac{2}{3}\)

Step 1: Convert mixed fractions to improper fractions:

• \(2 \frac{3}{5} = \frac{2 \times 5 + 3}{5} = \frac{13}{5}\)
• \(2 \frac{2}{3} = \frac{2 \times 3 + 2}{3} = \frac{8}{3}\)

Step 2: Use cross multiplication to compare \(\frac{13}{5}\) and \(\frac{8}{3}\):

\(13 \times 3 = 39\); \(8 \times 5 = 40\)

Since 39 < 40, \(\frac{13}{5} < \frac{8}{3}\) thus \(2 \frac{3}{5} < 2 \frac{2}{3}\).

Answer: \(2 \frac{3}{5} < 2 \frac{2}{3}\)