Addition of Fraction

Steps for addition of fractions:
Step 1: Add the numerators.
Step 2: Retain the (common) denominator.
Step 3: Write the fraction as: ${\displaystyle \frac{\text{Result of step 1}}{\text{Result of step 2}}}$
1. Proper Fraction + Proper Fraction:
Add ${\displaystyle \frac{\text{3}}{\text{7}}}$ + ${\displaystyle \frac{\text{2}}{\text{7}}}$
We have ${\displaystyle \frac{\text{3}}{\text{7}}}$ + ${\displaystyle \frac{\text{2}}{\text{7}}}$ = ${\displaystyle \frac{\text{3+2}}{\text{7}}}$ = ${\displaystyle \frac{\text{5}}{\text{7}}}$.

2. Improper Fraction + Improper Fraction:
Add ${\displaystyle \frac{\text{2}}{\text{5}}}$+ ${\displaystyle \frac{\text{1}}{\text{3}}}$
=${\displaystyle \frac{\text{(2 × 3)}}{\text{(5 × 3)}}}$ + ${\displaystyle \frac{\text{(1 × 5)}}{\text{(3 × 5)}}}$
= ${\displaystyle \frac{\text{6}}{\text{15}}}$ + ${\displaystyle \frac{\text{5}}{\text{15}}}$
= ${\displaystyle \frac{\text{11}}{\text{15}}}$

3. Mixed Fraction + Mixed Fraction:

• Mixed fractions can be written either as a whole part plus a proper fraction or entirely as an improper fraction.
• One way to add mixed fractions is to do the operation separately for the whole parts and the other way is to write the mixed fractions as improper fractions and then directly add them.

Add ${\displaystyle 2\frac{4}{5}\text{and}3\frac{5}{6}}$.
Solution: ${\displaystyle 2\frac{4}{5}+3\frac{5}{6}=2+\frac{4}{5}+3+\frac{5}{6}=5+\frac{4}{5}+\frac{5}{6}}$

Now, ${\displaystyle \frac{4}{5}+\frac{5}{6}=\frac{4\times 6}{5\times 6}+\frac{5\times 5}{6\times 5}}$.......(Since LCM of 5 and 6 = 30)

= ${\displaystyle \frac{24}{30}+\frac{25}{30}=\frac{49}{30}=\frac{30+19}{30}=1+\frac{19}{30}}$

Thus, ${\displaystyle 5+\frac{4}{5}+\frac{5}{6}=5+1+\frac{19}{30}=6+\frac{19}{30}=6\frac{19}{30}}$

And, therefore, ${\displaystyle 2\frac{4}{5}+3\frac{5}{6}=6\frac{19}{30}}$.