### Very Hard GCSE Maths Problems Made Easy

Everyone knows that sinking feeling when you turn over the page and spot a mind-bendingly difficult maths problem. Encountering this in an exam is a GCSE student’s **worst nightmare**!

The only way to make sure that this doesn’t happen to you is to **practice** a **variety of questions **from across the syllabus and to complete as many past papers as possible. Although there’s really **no shortcut**, we have got a little helping hand for you in this blog post!

Our** Maths guru** Simon has cherry-picked a handful of the **toughest problems** from the most recent Edexcel GCSE Maths papers, and he’s written a **step-by-step guide** to solving each and every one.

Remember that in GCSE exams, question types are frequently** repeated**, so if you follow his methods carefully, you’ll be ready to ace these formats when they pop up in future papers!

So if you do just **one thing today** to benefit your Maths revision, make sure that it’s reading this post.

**Proportionality**

June 2018 Paper 1H **(question 14) **

*y is inversely proportional to .*

*When d =10, y =4*

*d is directly proportional to *

*When x = 2, d = 24 *

*Find a formula for y in terms of x . Give your answer in its simplest form. *

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As soon as you clock that it’s a **proportion question**, the first thing you should aim to do is to find the both of the **constants of proportionality (k)**.

Let’s start with the first case, *y is inversely proportional to* , which we can write as:

y= (with k being the** constant of proportionality**).

Because we are told that y=4 when d=10, this means that we can substitute these numbers into out equation, giving us

This means that **k is 400**.

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Looking at the second case, * * (note, here the proportionality is **direct** rather than inverse), we can write d=k, then substitute the given values to get 24=4k

So, **k = 6 in this case.**

Phew, **first stage complete! **

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Now we have two equations, and

But we need an equation for **x in terms of y. **

That means that we need to replace the **d in the first equation** with its value in the second equation.

Which we can **simplify** to:

and again:

which is **our answer. **

*Need some more guidance? Check out the video solution. *

**Quadratic inequalities **

June 2018 Paper 1H ** (question 20)**

*N is an integer such that 3n+2 14 and > 1 *

*Find all the possible values of n. *

**First things first:** remember that an integer is a whole number.

Now we can start by rearranging the first inequality:

3n + 2 14

3n 12

n 4

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**Next** let’s turn to the second inequality, and try to solve that to find the n values.

First multiply both sides by + 5

6n > +5

Now **rearrange to give a quadratic**

0 > – 6n + 5

0 > (n-1)(n-5)

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We can then **sketch the graph** of this quadratic – a U-shaped parabola which cuts the x axis at 1 and 5.

1 < n < 5

We combine this with the first inequality’s rule, that n 4 , to conclude that **n can be 2, 3, or 4.**

*Want to go through this again? Watch the video solution. *

**Bounds**

June 2018 Paper 2H **(question 21)**

*Jackson is trying to find the density, in , of a block of wood. The block of wood is in the same shape of a cuboid. *

*He measures the length as 13.2cm (correct to nearest mm), the width as 16.0cm (correct to nearest mm), and the height as 21.7cm (correct to the nearest mm). *

*He measures the mass as 1970g, correct to the nearest 5g. *

*By considering bounds, work out the density of the wood. Give your answer to a suitable degree of accuracy. **You must show all your working and give a reason for your final answer. *

Whoa, what a lot of information to process! Let’s **break this down** and attack it in parts.

We know that the **overall aim** is to calculate the density of the wood in *.*

Let’s start by calculating the upper bound (UB) and lower bound (LB) of the **volume** of the block.

**Volume = length x width x height **

Length UB = 13.15

Length LB = 13.25

Width UB= 15.95

Width LB= 16.05

Height UB= 21.65

Height LB= 21.75

**Volume UB = 4625.409375**

**Volume LB= 4540.925125**

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We know the mass of the block is 1970g to the nearest 5g, meaning that the** UB of the mass is 1972.5g** and the** LB is 1967.5g.**

so,

Density UB= 0.43438285…

Density LB= 0.4253677…

When we write out the full answers, we see that they converge when rounded to **2 significant figures**. This is why we must give our final answer as **0.43** to 2 s.f. (this is a ‘suitable degree of accuracy’)

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*Check out the video solution to this problem here*

*Find our revision video on ‘Calculations using Bounds’ **here*

**Trigonometry**

June 2018 Paper 3H (question 19)

*Here are two right-angled triangles. *

*Given that tan e = tan f *

*Find the value of x. You must show all your working. *

To solve this problem correctly, we must apply techniques from trigonometry **and **algebra.

Since we know than we know that and

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We can **equate **the two, because we are told that tan e = tan f

Then it’s simply a matter of **rearranging to solve for x**:

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Finally, we must use the **quadratic formula** to arrive at our final answer:

This gives an x value of (remember, lengths **can’t be negative**, so we ignore the negative x value)

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*If you’d like to see the video solution, you’ll find it **here**. *

We hope you’ve found these worked examples helpful – keep your eyes peeled for similar questions on your exam paper, or other opportunities to put these **skills** and **techniques **into practice.

Would you like to see **more blog posts like this**? Let us know!

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*If you’re looking to continue your past paper practice session, you’ll find many more challenging questions from REAL past papers (accompanied by more step-by-step answer guides) here. *

*If you’re looking for our Maths revision advice, you’ll find it* *here** and* *here. *