Problem solving is a major part of the new GCSE format. The most important point is that, if you see a question that is unlike anything you have seen before, don’t panic or assume that you don’t know how to answer it! You may well know all of the maths content needed to solve the problem but the challenge is working out what you have to use. Join Beyond’s Maths Team as we explore GCSE Maths problem solving questions using Algebra…

## First thing’s first: don’t panic!

- Think about what the question has presented to you and if this links in with any of the topics you’ve covered. You may wish to underline the important information in the question, so that you can focus on that and not the irrelevant things.
- Sometimes, a question will give you lots of information in a very roundabout way. Can you think of a better way to present the information, to make it easier to process? Perhaps you could use a two-way table or a Venn diagram.

## Sample GCSE Maths problem solving questions

At a party, guests are offered chicken, lasagne or quiche to eat. 7 of the adults choose quiche. There are 100 guests altogether. 16 of the 48 people who choose chicken are children. 28 people eat lasagne and a quarter of these are children. How many children are at the party?

A two-way table really helps with this:

Chicken | Lasagne | Quiche | Total | |

Children | 16 | 21 | 17 | 54 |

Adults | 32 | 7 | 7 | 46 |

Total | 48 | 28 | 24 | 100 |

- Using algebra is often a good way to solve problems. If the information in the question suggests a sum, where you have the answer, but not the numbers that go into the sum, using algebra would be a sensible approach.

For example, ‘Aeron is 3 years older than Lily. Heather is 3 times as old as Aeron. The sum of their ages is 67. How old is Lily?’

The calculation that this suggests is: Aeron’s age + Lily’s age + Heather’s age = 67 We don’t know any of the ages but we do know the result of adding them together. Let’s say that Aeron is a years old. In that case, Lily is *a* – 3 years old and Heather is 3*a* years old.

Therefore, the sum of their ages is *a* + (*a* – 3) + 3*a* = 67.

Now we can find *a* by solving the equation and find the ages of the people.

- Coming up with an algebraic expression can be tricky. Say you buy
*c*cartons of juice at*x*pence each and a hat for £*y*; how much change would you get from £*m*? If you can’t see the answer, make the letters be numbers for a moment and then come up with the sum. Let’s make*c*be 5,*x*be 30,*y*be 2 and*m*be 20. The cost of the juices would be 5 x 30 pence; with the hat that would be a total cost of (200 + 5 x 30) pence. The change, in pence, from £20 would be 2000 – (200 + 5 x 30). Putting letters back in gives 100*m*– (100*y*+*cx*). - When a question asks you to ‘show’ something, don’t focus on the final answer; focus on the information that has been given to you, up to that point. Try to build an equation from that information, then try to simplify to give the result.

The perimeter of the shape is 24cm, *y* = 3*x* and *z* = 2*x*, show that *x* = 2

Perimeter = horizontal sides + vertical sides

(*x* + *y* + *x* + *y*)+ (*z* + *z*) = 24

Substituting in *y* and *z* in terms of *x* gives:

*x* + 3*x* + *x* + 3*x* + * *2*x* + 2*x* = 24

12*x* = 24

*x* = 2

- Be comfortable about using methods in reverse. For example, you can use a Venn diagram and prime factors to find the HCF and LCM of two numbers. If you know the HCF and LCM, starting with the Venn Diagram, you can work backwards to find the numbers.

## Beyond resources for Maths problem solving questions

GCSE Probability and Statistics Problem Solving Worksheets

GCSE Algebra Problem Solving Worksheets

Long Multiplication Word Problems Differentiated Worksheets

Edexcel Foundation Specimen Papers

You can find our other Maths blogs here – you might find our Maths Mastery Blogs particularly useful! While we’ve got you, don’t forget to subscribe to Beyond for access to thousands of secondary teaching resources. You can sign up for a free account here and take a look around at our free resources before you subscribe too. Happy revising from the Beyond Maths Team.