Subtraction of Rational Numbers

Subtraction of rational numbers involves finding the difference between two rational numbers. A rational number is a number that can be expressed as a fraction a/b, where a and b are integers, and b≠0.

Steps to Subtract Rational Numbers:

1. Find a Common Denominator:

• If the denominators of the rational numbers are different, find the least common denominator (LCD).
• Rewrite each rational number as an equivalent fraction with the LCD as the denominator.

2. Rewrite the Fractions:

• Adjust the numerators of each fraction according to the new common denominator.

3. Subtract the Numerators:

• Subtract the numerators of the adjusted fractions.

4. Simplify the Result:

• Simplify the resulting fraction if possible (i.e., reduce it to its lowest terms).

Example 1: Subtracting with Common Denominators

Subtract 5/8 and 3/8 :

5\8−3/8

• Since the denominators are the same, simply subtract the numerators:

(5−3)/8= 2/8

• Simplify the fraction:

2/8=1/4

So, 5/8-3/8=1/4.

Example 2: Subtracting with Different Denominators

Subtract 3/4 and 2/5:

Find the Common Denominator:

The denominators are 4 and 5.

The least common denominator (LCD) is 20.

Rewrite the Fractions:

Convert 3/4 to an equivalent fraction with denominator 20:

3/4 = (3×5)/(4×5) = 15/20

Convert 2/5 to an equivalent fraction with denominator 20:

2/5 = (2×4)/(5×4)​ = 8/20

Subtract the Numerators:

Subtract the numerators:

(15/20)-(8/20) = (15−8)/20=7/20

Simplify the Result:

The fraction 7/20 is already in its simplest form.

So, 3/4−2/5=7/20
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Example 3: Subtracting a Negative Rational Number
Subtract 1/3 and −2/9:

Rewrite the Problem:

Subtracting a negative is the same as adding its positive:

1/3 − (−2/9)= 1/3+2/9

Find the Common Denominator:

The denominators are 3 and 9. The least common denominator (LCD) is 9.

Rewrite the Fractions:

Convert 1/3 to an equivalent fraction with denominator 9:

(1×3)/(3 x 3) = 3/9 =1/3

Add the Numerators:

Add the numerators: 3/9+2/9=5/9

Simplify the Result:

The fraction 5/9 is already in its simplest form.

So, 1/3−(−2/9) = 5/9

Summary:

• Ensure the denominators are the same.
• Rewrite each fraction with the common denominator.
• Subtract the numerators.
• Simplify the result.
• Understanding and following these steps will help in accurately subtracting any pair of rational numbers.