Algebra · Ordinary Level
Algebraic Fractions
Ordinary Level · Class 4 of 8 · Tap NEXT to begin
Section 1 of 2
Adding Algebraic Fractions
Just like $\tfrac12 + \tfrac13$ needs a common bottom, algebraic fractions do too. The numerators are expressions in $x$, but the recipe doesn’t change.
The recipe
$\dfrac{a}{b} + \dfrac{c}{d} = \dfrac{ad + bc}{bd}$. Multiply the two bottoms for the common denominator, cross-multiply each top by the other bottom, add, expand and tidy.
Worked example — adding two fractions
Simplify $\dfrac{x+1}{3} + \dfrac{x+5}{2}$.
$= \dfrac{2(x+1) + 3(x+5)}{6}$
$= \dfrac{2x + 2 + 3x + 15}{6}$
$\dfrac{5x + 17}{6}$
Worked example — with a coefficient on x
Simplify $\dfrac{2x-1}{7} + \dfrac{x+3}{6}$.
$= \dfrac{6(2x-1) + 7(x+3)}{42}$
$= \dfrac{12x - 6 + 7x + 21}{42}$
$\dfrac{19x + 15}{42}$
You try
Simplify $\dfrac{x+5}{3} + \dfrac{x+7}{5}$.
Common bottom $15$. Cross-multiply: $5(x+5) + 3(x+7)$, then tidy.
$= \dfrac{5(x+5) + 3(x+7)}{15}$
$= \dfrac{5x + 25 + 3x + 21}{15}$
$\dfrac{8x + 46}{15}$
Section 2 of 2
Subtracting Algebraic Fractions
Subtraction — the trap
The minus in front of the second fraction multiplies every term in that numerator. Use brackets: $(3x-5) - (2x-1) = 3x - 5 - 2x + 1$ — the minus flips both signs.
Worked example — subtraction
Simplify $\dfrac{3x-5}{7} - \dfrac{2x-1}{10}$.
$= \dfrac{10(3x-5) - 7(2x-1)}{70}$
$= \dfrac{30x - 50 - 14x + 7}{70}$
$\dfrac{16x - 43}{70}$
You try
Simplify $\dfrac{7x-3}{2} - \dfrac{x-7}{7}$.
Brackets on both tops. The minus flips both signs in $-(x-7) = -x + 7$.
$= \dfrac{7(7x-3) - 2(x-7)}{14}$
$= \dfrac{49x - 21 - 2x + 14}{14}$
$\dfrac{47x - 7}{14}$
You try
Simplify $\dfrac{3x-1}{6} - \dfrac{x-3}{5}$.
Common bottom $30$. Watch the double sign flip: $-6(x-3) = -6x + 18$.
$= \dfrac{5(3x-1) - 6(x-3)}{30}$
$= \dfrac{15x - 5 - 6x + 18}{30}$
$\dfrac{9x + 13}{30}$
That’s Class 4.
Multiply the bottoms, cross-multiply the tops (with brackets), expand and tidy — and mind the double sign flip when subtracting. Class 5: long division and rearranging formulae.