Adding two fractions: the common denominator without hunting

The formula to remember

$\dfrac{a}{b} + \dfrac{c}{d} = \dfrac{ad + bc}{bd}$

Numerator: cross products added. Denominator: product of the two.

Example

$\dfrac{2}{3} + \dfrac{4}{5}$:

• Numerator: $2 \times 5 + 3 \times 4 = 10 + 12 = 22$.
• Denominator: $3 \times 5 = 15$.
• Result: $\dfrac{22}{15}$.

Simplify at the end

The common denominator you get isn’t necessarily the smallest. After the computation, look for the GCD of numerator and denominator.

$\dfrac{1}{4} + \dfrac{1}{6} = \dfrac{6 + 4}{24} = \dfrac{10}{24} = \dfrac{5}{12}$ (dividing by 2 top and bottom).

Subtraction: same method

$\dfrac{a}{b} - \dfrac{c}{d} = \dfrac{ad - bc}{bd}$

Watch the sign of the cross product.

?Your turn

What is 3/4 + 2/5?

A.5/9B.6/20C.23/20D.5/20

Check