Eight Standard >> Rational numbers | addition and subtraction

Addition and Subtraction of rational numbers

Let's dive into more detail about addition and subtraction of rational numbers, both when the denominators are the same and when they are different.

Addition of Rational Numbers:

a) Addition with the same denominator: When adding rational numbers with the same denominator, you can simply add their numerators while keeping the common denominator unchanged. The reason is that since the denominators are the same, it means the fractions represent the same-sized parts.

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

Step 1: The denominators are the same (both are 5).

Step 2: Add the numerators: 2 + 3 = 5

Step 3: Keep the denominator unchanged: 5

Result:\(\frac{2}{5}\) +\(\frac{3}{5}\) =\(\frac{5}{5}\) = 1

b) Addition with different denominators: When adding rational numbers with different denominators, you need to find a common denominator before performing the addition. The common denominator is the least common multiple (LCM) of the individual denominators.

Example: Add \(\frac{1}{3}\) and \(\frac{1}{4}\).

Step 1: Find the LCM of 3 and 4, which is 12.

Step 2: Convert both fractions to have the common denominator of 12. (\(\frac{1}{3}\)) * (\(\frac{4}{4}\)) = \(\frac{4}{12}\) (\(\frac{1}{4}\)) * (\(\frac{3}{3}\)) = \(\frac{3}{12}\)

Step 3: Add the fractions with the same denominator: \(\frac{4}{12}\) + \(\frac{3}{12}\) = \(\frac{7}{12}\)

So, \(\frac{1}{3}\) + \(\frac{1}{4}\) = \(\frac{7}{12}\)

Subtraction of Rational Numbers:

a) Subtraction with the same denominator: When subtracting rational numbers with the same denominator, you can directly subtract their numerators while keeping the common denominator unchanged.

Example: Subtract \(\frac{1}{6}\) from \(\frac{2}{6}\).

Step 1: The denominators are the same (both are 6).

Step 2: Subtract the numerators: 2 - 1 = 1

Step 3: Keep the denominator unchanged: 6

Result: \(\frac{2}{6}\) - \(\frac{1}{6}\) = \(\frac{1}{6}\)

b) Subtraction with different denominators: When subtracting rational numbers with different denominators, you also need to find a common denominator before performing the subtraction.

Example: Subtract \(\frac{2}{5}\) from \(\frac{3}{7}\).

Step 1: Find the LCM of 5 and 7, which is 35.

Step 2: Convert both fractions to have the common denominator of 35. ( \(\frac{3}{7}\)) * (\(\frac{5}{5}\)) = \(\frac{15}{37}\) (\(\frac{2}{5}\)) * (\(\frac{7}{7}\)) = \(\frac{14}{35}\)

Step 3: Subtract the fractions with the same denominator: \(\frac{15}{35}\) - \(\frac{14}{35}\) = \(\frac{1}{35}\)

So, \(\frac{3}{7}\) - \(\frac{2}{5}\)= \(\frac{1}{35}\)