A particularly tricky topic for the new 9-1 GCSE maths qualification is simplifying and manipulating algebraic fractions.
Success with these types of questions requires students to fully understand what they are doing when they simplify and manipulate numerical fractions.
For example, let's add together two very easy fractions:
The reason this example is so easy is because the denominators of both fractions are the same, and so only the numerators need to be added.
A more difficult example is where the denominators are not equal. In this example, it is necessary to make both denominators the same first by finding the lowest common denominator, and ensuring we multiply the numerator by whatever we multiplied the denominator by:
The same principle, although conceptually harder, applies to adding algebraic fractions. For example:
Here, n+1 and n are not the same terms and so we can't simply add the numerators as in the first example. It will therefore be necessary to find a common denominator.
But what is the common denominator of n+1 and n? As we don't know what n is, we need to find an expression that both n+1 and n will go into (in a similar way to how we found that both 4 and 5 go into 20 in the second example).
The simplest way is to multiply both n+1 and n together, as both terms go into n(n+1), making sure the numerator is multiplied by whatever we multiplied the denominator by:
So in this example, we multiplied the denominator of the first fraction by n, and made sure to multiply the numerator by n too. Similarly, for the second fraction, we multiplied the denominator by n+1, so made sure to multiply the numerator by n+1 too.
Now that the denominators are the same, we can subtract the fractions to create a single fraction:
We can see if this algebraic fraction simplifies further by expanding the brackets and collecting any resulting like terms:
Note that the nm and mn terms cancel as they are the same.
Here is another example. Solve the following equation:
The first step is to combine both of these fractions into a single fraction. As before we need to find a common denominator. What do both 3x and 2x go into? If we multiply them together, both the 3x and 2x will go into this. However, we must remember to multiply the numerators by whatever we multiplied the denominators by. We therefore have:
We can now add these two fractions together to give:
We can now expand the brackets in the numerators to give:
Now, collect the like terms in the numerator to give:
Multiply both sides by the denominator to 'get rid' of the fraction:
Next, subtract both sides by the first term in order to collect like terms:
Add the 2x term to both sides to give:
Now, take out x as a factor:
In order for the right hand side to equal 0, either the x term is equal to 0, or the 13x+2 term is equal to 0:
Solving the second equation for x:
The final answer therefore has two solutions: