A rectangular path has perimeter of 240m. If the rectangle is split lengthways, two paths of 160m are formed. Work out the lengths of the sides of the original path.

Firstly, make sure to draw a picture! Worded questions are used to catch people out. Drawing a diagram makes solving certain problems much easier.

Secondly, notice that we are asked for two pieces of information: the lengths of the sides of a rectangle. This suggests that the problem we may be seeing is a simultaneous equations problem.

The first bit of information of the question lets us draw out a rectangle with two sides of different lengths. Call these lengths 'a' and 'b' as we don't know their values yet.

The second bit of information draws out the same rectange cut in half along one of the sides. One of the sides of the path is the same i.e. 'a', but now the other side is half of 'b', i.e. 'b/2'.

Recall that the perimeter of a rectangle is the sum of the lengths of all sides of the rectangle. In the first case:

-240m = a + b + a + b ===> 2a + 2b = 240m ===> a + b = 120m _____ 1)

In the second case:

-160m = a + b/2 + a + b/2 ===> 2a + b = 160m _____ 2)

We realise that we have two equations involving two 'variables', 'a' and 'b' - this is a simultaneous equations problem and a very common kind of question to come up in GCSE maths papers of both tiers.

In order to solve these equations, lets label them 1) and 2) respectively.

We firstly need to get one variable on its own so that we can determine its value before we can determine the second. This involves what we call 'eliminating' a variable, which is essentially making either 'a' or 'b' disappear.

Notice that both 1) and 2) have 'b', while 1) has 'a' and 2) has '2a'. If we subtract 1) from 2), b - b = 0, 2a - a = a for the left hand side, 160m - 120m = 40m for the right hand side. This leaves us with a = 40m... an answer has popped out!

Knowing our value of a = 40m, we can substitute this into either 1) or 2). This allows us to determine the value of b). Substituting into 1):

40m + b = 120m

Subtracting 40m from both sides:

b = 80m

We have worked out the answers: a = 40m, b = 80m!