Subtraction of Two Rational Numbers

Subtracting two rational numbers involves finding the difference between two fractions or integers.

It requires having a common denominator for the fractions involved and then subtracting the numerators.

There are three components of a rational number:

Numerator: This is the top part of a fraction, representing the number of parts considered.

Denominator: This is the bottom part of a fraction, representing the total number of equal parts.

Operator: For subtraction, the operator is the minus (-) symbol.

When subtracting rational numbers, students should follow a set of rules:

Find a common denominator: Ensure both fractions have the same denominator before subtracting the numerators.

Subtract the numerators: Only after obtaining a common denominator, subtract the numerators of the fractions.

Simplify the result: After subtracting the numerators, simplify the resulting fraction to its simplest form.

The following are methods of subtracting rational numbers:

Method 1: Common Denominator Method

To use the common denominator method for subtraction of rational numbers, follow these steps.

Step 1: Find a common denominator for both fractions.

Step 2: Adjust the fractions accordingly to have the same denominator.

Step 3: Subtract the numerators and simplify the result.

Let’s apply these steps to an example:

Question: Subtract 1/4 from 3/5

Step 1: The common denominator of 4 and 5 is 20.

Step 2: Convert the fractions: 1/4 = 5/20 and 3/5 = 12/20.

Step 3: Subtract the numerators: 12/20 - 5/20 = 7/20.

Answer: 7/20

Method 2: Mixed Numbers Method

When subtracting rational numbers that are mixed numbers, convert them to improper fractions first, then proceed as with fractions.

For example,

Subtract 2 1/3 from 3 1/2

Solution: Convert to improper fractions: 2 1/3 = 7/3 and 3 1/2 = 7/2.

Find a common denominator and subtract: 7/2 = 21/6, 7/3 = 14/6. 21/6 - 14/6 = 7/6.

Therefore, the result is 7/6 or 1 1/6 when converted back to a mixed number.

Subtraction of rational numbers has some characteristic properties. These properties are listed below:

Subtraction is not commutative In subtraction, changing the order of the numbers changes the result, i.e., A - B ≠ B - A.

Subtraction is not associative Unlike addition, we cannot regroup in subtraction.

When three or more numbers are involved, changing the grouping changes the result.

(A − B) − C ≠ A − (B − C)

Subtraction is the addition of the opposite sign Subtracting a number is the same as adding its opposite, so to make calculations easier, you can convert subtraction into addition by changing the signs of the second term.

A − B = A + (−B) Subtracting zero from a number leaves the number as is Subtracting zero from any number results in the same rational number: A - 0 = A.

Tips and tricks are useful for students to efficiently deal with the subtraction of rational numbers. Some helpful tips are listed below:

Tip 1: Always find a common denominator before subtracting fractions.

Tip 2: If dealing with mixed numbers, convert them into improper fractions for easier calculation.

Tip 3: Simplify the result by dividing the numerator and denominator by their greatest common divisor (GCD).

Subtraction of rational numbers can be challenging, often leading to common mistakes. However, being aware of these errors can help students avoid them.