Algebraic Fractions with Numerical Denominators

Algebraic fractions work exactly like number fractions. To add x over three and x over four, you need a common denominator. The lowest common denominator of three and four is twelve. x over three is four x over twelve, and x over four is three x over twelve. Now the bottoms match, so add the tops: seven x over twelve.

When a numerator has two terms, treat it as one whole thing in a bracket. For x plus two over three, plus x minus one over four, the common denominator is twelve. The first fraction is scaled by four: four times the bracket x plus two is four x plus eight. The second is scaled by three: three times the bracket x minus one is three x minus three. Add the numerators over twelve and collect: seven x plus five, all over twelve.

Subtraction needs care. Take x plus two over three, minus x minus one over four. The first numerator becomes four x plus eight, as before. The second fraction is being subtracted, so you subtract three times the whole bracket x minus one. Negative three times x is negative three x, and negative three times negative one is positive three. Combine: four x plus eight minus three x plus three is x plus eleven, all over twelve.

Use the lowest common denominator, not just the product of the denominators. For five x over six minus three x over eight, list the multiples: six, twelve, eighteen, twenty-four, and eight, sixteen, twenty-four. The lowest common denominator is twenty-four. Five x over six becomes twenty x over twenty-four, and three x over eight becomes nine x over twenty-four. Subtract the tops: eleven x over twenty-four.

To simplify, factorise the numerator first and then cancel a common factor. Six x plus nine over three: the top is three times the bracket two x plus three, so the threes cancel and you get two x plus three. Four x plus ten over two is two times the bracket two x plus five over two, which is two x plus five. But x plus six over six cannot be simplified, because six is not a factor of the whole top. You can only cancel factors, never single terms. Quiz and exercises now.