GCSE Maths Takeaway – Simultaneous Equation Word Problems

If I had to choose one thing you should have as a takeaway from GCSE maths i.e. one ‘poster’ topic to represent GCSE maths, I think it should be simultaneous equations – in particular word problems leading to simultaneous equations ¹.

This is because I think if you can solve a sufficiently varied set of simultaneous equation word problems, you must have a good understanding, for a sufficiently involved type of problem, of the complete ‘mathematical process’:

1. Translating a ‘real world’ problem into a mathematical model
2. Solving a mathematical model
3. Interpreting the solution(s) in the context of the ‘real world’.

Most maths questions just cover stage (2) processes of formally computing things by applying rules, often according to a set procedure, and while I think such skills are important to have and understand, it is stages (1) and (3) I’m most interested in as these are the parts that can’t easily be done by a computer algorithm, and they are also what makes maths useful.

Simultaneous equation word problems are great because they can be genuine problems that exist outside the world of formal mathematics, and yet a sophisticated formal mathematical process is required to solve them, so they give one of the first glimpses into the power of formal mathematics.

To understand something well it is important to have a good stock of examples, that is ‘complete’ in some sense. So I have tried to curate a list of my favourite simultaneous equation word problems. I have tried to choose questions such that:

• Ideally, it is possible to imagine being faced with the problem or a similar problem
• Ideally, the question is memorable and/or short
• Ideally, the solution is not obvious without formally solving the equations
• Any equations to be solved are linear
• The calculations involved should be fairly straightforward (at least possible to do by hand)
• Any relationship the problem requires use of should be knowable to the intuition or well known
• Ideally, the question itself shouldn’t mention too many numbers
• The set of questions as a whole is in some sense ‘complete’ i.e. covers ‘all kinds’ of linear simultaneous equation word problems ²
• No two questions are of the same ‘formula’
• There are no more than 15 questions
• There are questions of varying difficulty

Here is the list of questions I’ve come up with. Some of them I have taken from books and the internet and others I have written myself ³. The questions are roughly in order of increasing difficulty of forming the equations (difficulty of solving the equations is random), with some starred questions at the end. While the starred questions do not require any specific knowledge outside of GCSE maths, they may be more challenging.

1. A bat and a ball costs £1.10. The bat costs one pound more than the ball. How much does the ball cost?
2. At a community swimming pool, the sign saying the price of children’s admission and adult’s admission has disappeared and no-one remembers what the prices were. It is recorded that on Monday the pool had 12 children and 33 adults, which brought in £177, and that same week on Tuesday 42 children and 33 adults came to the pool, which brought in £207. What are the admission prices for children and adults?
3. It takes 11 hrs to knit 2 hats and 3 jumpers. It takes 4 hours to knit 1 hat and 1 jumper. How long would it take to knit 5 hats and 6 jumpers?
4. Warmsnug calculate the prices of their windows according to the area of glass used and the length of frame needed. These are two example windows and their prices:
   3 x 5 costs £470,
   2 x 4 costs £320.
   What would be the quote to do a square window of width 3?
5. A sports pitch has length 30m and width 15m. It is required that the dimensions be changed so that the length is at least three times the width and the perimeter is increased by at least 70m. What are the minimum new dimensions of the pitch?
6. A taxi company has a cost structure which is an initial fee plus a fee proportional to the distance travelled. I did one journey that costs £9.50; then I remembered I had forgotten something important and had to get another taxi home and back again (same company). The second journey cost £17.50. How much is the initial fee?
7. In a certain family, each girl has just as many sisters as brothers but each boy has twice as many sisters as brothers. How many children are there?
8. When I was 14 my mother was 41, and she is now twice as old as I am; how old am I?
9. 1. In a two digit number, the units digit is thrice the tens digit and if 36 is added to the number the digits are interchanged. Find the number.
   2. Is there a two digit number such that the units and tens digit add up to 9, and when 36 is added to the number the digits are interchanged?
   3. Is there a number such that when you add it to any two digit number where the units digit is 2 higher than the tens digit, the digits are interchanged?
10. It takes a motor boat 2 minutes to travel 2 km when going with the current. When the boat is going against the current it takes 4 minutes. The current is always constant. How long would it take the boat to do the same journey in slack water, when there is no current at all?
11. There are two neighbouring countries Foo and Bar. Fooish is the primary language of Foo and Barish is the primary language of Bar, but both languages are spoken across both countries (no other language is spoken in these countries and no other country speaks these languages). Half of people who speak Barish can also speak Fooish. Only 5 percent of people who speak Fooish can also speak Barish. The total population of Foo and Bar is 21 million. How many of these people only speak Fooish?
12. * In a food-eating competition it took me 5 minutes to eat 7 burgers and 2 hotdogs. I know it takes me at least 20 seconds to eat a hotdog. Is it possible for me to eat 12 burgers and 8 hotdogs within 10 minutes (without any more practice)?
13. * Bill and Ben’s combined age is 91. Bill is now twice as old as Ben was when Bill was as old as Ben is now. How old are they?
14. * Two horsemen started out at daybreak. They travelled the same distance and arrived at their destination at the same time. A rode twice as long as B rested and B rode for three times long as A rested for. Who rode the fastest?