Algebra · Junior Cert
Algebraic Fractions
Junior Cert Higher · Common denominators, tops & signs · Tap NEXT to begin
Section 1 of 4
Number Fractions First
To add fractions you need a common denominator (a common bottom). Rewrite each fraction over it, then add the tops.
Worked example
Simplify $\dfrac{1}{2}+\dfrac{1}{3}$.
Common denominator $6$: $\dfrac{3+2}{6}$
$\dfrac{5}{6}$
Worked example
Simplify $\dfrac{2}{5}+\dfrac{3}{4}$.
$\dfrac{2}{5}=\dfrac{8}{20},\quad \dfrac{3}{4}=\dfrac{15}{20}$
$\dfrac{8+15}{20}$
$\dfrac{23}{20}$
You try
Simplify $\dfrac{1}{5}+\dfrac{1}{7}$.
Pen and paper out — try it before you reveal.
$\dfrac{1}{5}=\dfrac{7}{35},\ \dfrac{1}{7}=\dfrac{5}{35}$
$\dfrac{7+5}{35}$
$\dfrac{12}{35}$
$\dfrac{12}{35}$
Section 2 of 4
Letters on Top
Same idea, but now the tops are little brackets. Multiply out each top carefully — and watch a minus sign in front of a bracket.
Worked example
Simplify $\dfrac{x+5}{2}+\dfrac{x+1}{3}$.
$\dfrac{3(x+5)+2(x+1)}{6}$
$\dfrac{3x+15+2x+2}{6}$
$\dfrac{5x+17}{6}$
Worked example — a subtraction
Simplify $\dfrac{2x+1}{3}-\dfrac{x-1}{5}$.
$\dfrac{5(2x+1)-3(x-1)}{15}$
$\dfrac{10x+5-3x+3}{15}$
$\dfrac{7x+8}{15}$
Worked example
Simplify $\dfrac{x+1}{3}+\dfrac{2x+5}{4}$.
$\dfrac{4(x+1)+3(2x+5)}{12}$
$\dfrac{4x+4+6x+15}{12}$
$\dfrac{10x+19}{12}$
Worked example
Simplify $\dfrac{3x+5}{7}+\dfrac{x-1}{3}$.
$\dfrac{3(3x+5)+7(x-1)}{21}$
$\dfrac{9x+15+7x-7}{21}$
$\dfrac{16x+8}{21}$
Worked example — a minus in the middle
Simplify $\dfrac{5x-3}{5}-\dfrac{x-7}{6}$.
$\dfrac{6(5x-3)-5(x-7)}{30}$
$\dfrac{30x-18-5x+35}{30}$
$\dfrac{25x+17}{30}$
You try
Simplify $\dfrac{5x-1}{4}-\dfrac{x-7}{6}$.
Pen and paper out — try it before you reveal.
$\dfrac{3(5x-1)-2(x-7)}{12}$
$\dfrac{15x-3-2x+14}{12}$
$\dfrac{13x+11}{12}$
$\dfrac{13x+11}{12}$
Worked example — three fractions
Simplify $\dfrac{2x-1}{3}+\dfrac{x-5}{4}-\dfrac{3x-4}{6}$.
$\dfrac{4(2x-1)+3(x-5)-2(3x-4)}{12}$
$\dfrac{8x-4+3x-15-6x+8}{12}$
$\dfrac{5x-11}{12}$
You try
Simplify $\dfrac{2x-7}{4}-\dfrac{x-1}{5}-\dfrac{5x-1}{10}$.
Pen and paper out — try it before you reveal.
$\dfrac{5(2x-7)-4(x-1)-2(5x-1)}{20}$
$\dfrac{10x-35-4x+4-10x+2}{20}$
$\dfrac{-4x-29}{20}$
$\dfrac{-4x-29}{20}$
Worked example
Simplify $\dfrac{5x+1}{3}+\dfrac{x+7}{4}$.
$\dfrac{4(5x+1)+3(x+7)}{12}$
$\dfrac{20x+4+3x+21}{12}$
$\dfrac{23x+25}{12}$
Section 3 of 4
Letters on the Bottom
When the denominators contain $x$, the common denominator is the two brackets multiplied together. Leave the bottom as a product.
Worked example
Simplify $\dfrac{5}{x+1}+\dfrac{7}{2x+3}$.
$\dfrac{5(2x+3)+7(x+1)}{(x+1)(2x+3)}$
$\dfrac{10x+15+7x+7}{(x+1)(2x+3)}$
$\dfrac{17x+22}{(x+1)(2x+3)}$
Worked example
Simplify $\dfrac{5}{2x+7}+\dfrac{6}{x+9}$.
$\dfrac{5(x+9)+6(2x+7)}{(2x+7)(x+9)}$
$\dfrac{5x+45+12x+42}{(2x+7)(x+9)}$
$\dfrac{17x+87}{(2x+7)(x+9)}$
You try
Simplify $\dfrac{5}{3x+1}+\dfrac{4}{x+8}$.
Pen and paper out — try it before you reveal.
$\dfrac{5(x+8)+4(3x+1)}{(3x+1)(x+8)}$
$\dfrac{5x+40+12x+4}{(3x+1)(x+8)}$
$\dfrac{17x+44}{(3x+1)(x+8)}$
$\dfrac{17x+44}{(3x+1)(x+8)}$
Worked example — a subtraction
Simplify $\dfrac{5}{2x-1}-\dfrac{3}{x-4}$.
$\dfrac{5(x-4)-3(2x-1)}{(2x-1)(x-4)}$
$\dfrac{5x-20-6x+3}{(2x-1)(x-4)}$
$\dfrac{-x-17}{(2x-1)(x-4)}$
Worked example
Simplify $\dfrac{2}{x-5}-\dfrac{3}{4x-7}$.
$\dfrac{2(4x-7)-3(x-5)}{(x-5)(4x-7)}$
$\dfrac{8x-14-3x+15}{(x-5)(4x-7)}$
$\dfrac{5x+1}{(x-5)(4x-7)}$
You try
Simplify $\dfrac{3}{2x-1}-\dfrac{5}{x-7}$.
Pen and paper out — try it before you reveal.
$\dfrac{3(x-7)-5(2x-1)}{(2x-1)(x-7)}$
$\dfrac{3x-21-10x+5}{(2x-1)(x-7)}$
$\dfrac{-7x-16}{(2x-1)(x-7)}$
$\dfrac{-7x-16}{(2x-1)(x-7)}$
You try
Simplify $\dfrac{8}{3x-2}-\dfrac{5}{x-1}$.
Pen and paper out — try it before you reveal.
$\dfrac{8(x-1)-5(3x-2)}{(3x-2)(x-1)}$
$\dfrac{8x-8-15x+10}{(3x-2)(x-1)}$
$\dfrac{-7x+2}{(3x-2)(x-1)}$
$\dfrac{-7x+2}{(3x-2)(x-1)}$
Worked example — one bottom is just a number
Simplify $\dfrac{5}{2x-1}-\dfrac{3}{7}$.
Common denominator $7(2x-1)$: $\dfrac{5(7)-3(2x-1)}{7(2x-1)}$
$\dfrac{35-6x+3}{7(2x-1)}$
$\dfrac{38-6x}{7(2x-1)}$
Worked example
Simplify $\dfrac{3}{2x-1}-\dfrac{5}{5x-7}$.
$\dfrac{3(5x-7)-5(2x-1)}{(2x-1)(5x-7)}$
$\dfrac{15x-21-10x+5}{(2x-1)(5x-7)}$
$\dfrac{5x-16}{(2x-1)(5x-7)}$
Worked example — a bracket on top too
Simplify $\dfrac{5}{3x-1}-\dfrac{2x-1}{4}$.
$\dfrac{5(4)-(2x-1)(3x-1)}{4(3x-1)}$
$(2x-1)(3x-1) = 6x^2-5x+1$
$\dfrac{20-(6x^2-5x+1)}{4(3x-1)} = \dfrac{20-6x^2+5x-1}{4(3x-1)}$
$\dfrac{-6x^2+5x+19}{4(3x-1)}$
You try
Simplify $\dfrac{3x-1}{x-1}-\dfrac{x-5}{2x-3}$.
Pen and paper out — try it before you reveal.
$\dfrac{(3x-1)(2x-3)-(x-5)(x-1)}{(x-1)(2x-3)}$
$(3x-1)(2x-3)=6x^2-11x+3$, $(x-5)(x-1)=x^2-6x+5$
$\dfrac{6x^2-11x+3-x^2+6x-5}{(x-1)(2x-3)}$
$\dfrac{5x^2-5x-2}{(x-1)(2x-3)}$
$\dfrac{5x^2-5x-2}{(x-1)(2x-3)}$
Section 4 of 4
Simplify by Factorising
If the top and bottom can be factorised, factor both and cancel the matching bracket. (This uses the difference of two squares — see the Factorising class.)
Worked example
Simplify $\dfrac{x^2+14x+49}{x^2-49}$.
Top: $x^2+14x+49 = (x+7)(x+7)$
Bottom: $x^2-49 = (x-7)(x+7)$
$\dfrac{(x+7)(x+7)}{(x-7)(x+7)}$ — cancel the $(x+7)$
$\dfrac{x+7}{x-7}$
That’s fractions.
Common denominator, multiply out the tops, mind the minus — leave $x$-denominators as a product, and cancel when top and bottom factorise.