Fractions operations — addition, subtraction, multiplication, division algorithm

Question

How do we add, subtract, multiply, and divide fractions? Solve: $\frac{2}{3} + \frac{3}{4}$, $\frac{5}{6} - \frac{1}{4}$, $\frac{2}{5} \times \frac{3}{7}$, and $\frac{4}{9} \div \frac{2}{3}$.

(CBSE 6-7 Board pattern)

Solution — Step by Step

flowchart TD
    A["Fraction Problem"] --> B{"Which operation?"}
    B -- "Addition or Subtraction" --> C["Find LCM of denominators"]
    C --> D["Convert to equivalent fractions"]
    D --> E["Add or subtract numerators"]
    E --> F["Simplify the result"]
    B -- "Multiplication" --> G["Multiply numerators together"]
    G --> H["Multiply denominators together"]
    H --> F
    B -- "Division" --> I["Flip the second fraction"]
    I --> G

Why This Works

We can only add or subtract fractions with the same denominator because the denominator tells us the “size of each piece.” $\frac{2}{3}$ means 2 pieces where each piece is $\frac{1}{3}$ of the whole. $\frac{3}{4}$ means 3 pieces of size $\frac{1}{4}$. We cannot add these directly because the pieces are different sizes. Making the denominator the same (using LCM) makes all pieces the same size.

Multiplication is simpler: $\frac{2}{5}$ of $\frac{3}{7}$ means we take $\frac{3}{7}$ and split it into 5 parts and take 2. This naturally gives $\frac{2 \times 3}{5 \times 7}$.

Division by a fraction asks “how many times does one fraction fit into another?” Flipping and multiplying is a shortcut for this counting process.

Alternative Method

For addition/subtraction, instead of finding the LCM, you can use the cross-multiplication shortcut:

For $\frac{2}{3} + \frac{3}{4}$: $\frac{2 \times 4 + 3 \times 3}{3 \times 4} = \frac{8 + 9}{12} = \frac{17}{12}$

This always works but may give a larger denominator that needs simplification.

Common Mistake