Want help or practice with GCSE Maths Problem Solving ?

As a current Further Maths A Level student, I spend a decent amount of my spare time helping others with GCSE and IGCSE Maths. A lot of students i've worked with struggle with the problem solving nature of a lot of questions. Which is fair enough. When you get tons of information thrown at you (or relatively little, as in the case of some of the most difficult problems), under exam time pressure, it is so easy to get thrown and give up on the question before even starting it or trying to process the available information. Even if the question may not be conceptually challenging as such. 30% of GCSE Maths Higher Tier is problem solving and proof focused. The grade 9 questions are almost exclusively focused on it (and occasionally the odd top end concept such as quadratic turning points).


To give practice, I have, in approximate difficulty order, problem solving questions that range from about grade 4 - 5 standard all the way up to stuff that is right at the top end of grade 9 and is unlikely to ever be asked in a GCSE Maths exam, but makes for extremely good practice nonetheless. All the other questions are non-calculator. I designed them to be accessible without one and with only gcse knowledge (well, calculus might be helpful for the last one but it certainly isn't the easiest method for a solution). Key thing about formulae below. Note that I have arranged them as per MY perceived difficulty. That doesn't necessarily mean for example that you'll find the algebraic probability proof easier than the MIT Geometry Question. The difficulty is only indicative, not definitive.


Do not use without proof list:
The quadratic formula and any related points
Any statement about the division or multiplication of surds.
The formula for the sum of n terms of an arithmetic sequence

Grade 4-5ish
A square with sides of length 20m is inscribed in a circle.
Calculate the area of the region outside the square but within the circle.
Give your answer in the form a(π-b) 〖 m〗^2. (5)

Grade 4
55% of people attending a concert are women.
3/5 of them are aged between 13 and 25 years old. Nobody at the concert is younger than 13 years old. There are 1100 women older than 25 at the concert.
There are 3 men older than 25 for every 7 men younger than 25.
Calculate the total number of men aged less than 25 at the concert. (5)

Grade 6 - 7 (not sure where this fits, it's kinda tricky but not particularly so. I think the wording might be iffy, quote me in if you need a clarification.)
There are k marbles in a bag. 12 of these are blue. Marbles are drawn and then replaced . There are only Purple and Blue marbles in the bag. . The probability of picking out two blue marbles in a row is 1/9. The ratio of purple to brown marbles is 3:5. How many purple marbles are there in the bag? (5)

Grade 7-8 (not sure where this goes in the scale either, but I'd say it is unlikely to be one of the final questions. Edexcel asked a much more structured, wordy and confusing version of this in 2017 as the second from last question. This should be much easier to do.)

A trapezium has a line of symmetry that bisects its parallel sides.
Prove that the diagonals of the quadrilateral are of equal length. (5)

Grade 7-8 (kinda tricky blend of algebraic proof and probability into one question.)
Two randomly selected integers are multiplied together. Prove that there is a 12.5% probability that the product of the two integers is an odd positive integer. (5)

Grade 8 (MIT set a very similar question to this on one of their entrance exams about 150 years ago. It's fiddly to get right but conceptually not that difficult. Make sure your statements are fully justified.)
A right-angled triangle has hypotenuse AB of length 71 metres. A line BD which is perpendicular to the hypotenuse splits the hypotenuse AB such the ratio of the length of AB to the length of AD is 25:9.
Calculate the length of BD in metres. Give your answer to 3 significant figures. (5)

Grade 8 - 9 (This question is pretty tough, but only because the first step is hard. Each step is easier than the one before it though )
A triangle which is the base of a fully filled container containing oil of density 800kg/m^3 has perimeter 10+5√3 cm and angles 5x-15,x+21 and 13x+3 degrees. The container is 1.6 metres tall.
Calculate the mass of all the oil in the container in metric tonnes. (10)

Grade 9+(the individual concepts aren't difficult. But linking all of them together is quite messy. The equation is shown on the Youtube Channel MindYourDecisions. Don't look at that till AT LEAST after you've solved the equation)
The solutions of (〖x^2-7x+11)〗^(x^2-13x+42)=1 form part of a decreasing sequence where S_n=(n(51-n))/2. The smallest solution of the equation is the 24th term.
Determine the two values of n where S_n is at its maximum value. (2)
Prove fully that S_n=(n(51-n))/2. Fully justify your solution. (8)

KEY TIP: You should justify all of your statements that your proof relies on. For example, you cannot just state that the second difference of a quadratic sequence is twice the coefficient of n^2. You should prove it in a generalised form. How you would do this particular thing is part of the job for your mathematician's mind to figure out

If you want me to create and add more questions here, do tell me. And especially if you want a specific difficulty of questions, do tell me also. I am also aware there is a gap of middle difficulty questions here as there is a noticeable jump from the 2nd to the 3rd shorter question for example. I will also check solutions for you and tell you if you're on the right track (quote me in). But PLEASE post your solutions in spoiler tags so others can answer the questions first.

nice idea, but i can see an issue in most of the questions.

eg.

grade 7-8 qn : the answer is not exactly 12.5% since there is the chance that at least one of the integers chosen is zero.

grade 8 question : doesn't make sense. BD is perpendicular to AB, yet somehow splits AB ?

Grade 8-9 question : is the container a right prism, or a pyramid, or some other type of solid?

I have made edits to the question wording to fix the very points you just mentioned

1. The hypotenuse question still contains the error.
How exactly can a line BD, perpendicular to AB, actually split AB?

2. The oil container question: you really need to say that the container is a RIGHT prism. (you do know what a right prism is?)

have a look at this thread:

3 grade 9 questions