# Daily Maths Revision – Week 3 Walkthrough

If you’re struggling with this week’s questions, this walkthrough should help you get the skills you need.

If it’s a formula you need, here are the two that are relevant this week. In the 2023 exams, you will be given these formulae.

Pythagoras’ Theorem
$a^2 + b^2 = c^2$

Volume of a Prism
volume = area of the cross-section × length

This week, we’re exploring:

### Question 1: Pythagoras’ Theorem

For a right-angled triangle, if we square the two shorter sides and add the results together, this gives the square of the longer side. This is known as Pythagoras’ theorem and is commonly written as $a^2 + b^2 = c^2$ where $c$ is the longest side (the hypotenuse).

We can use Pythagoras’ theorem to find the missing side of a right-angled triangle if we know two of the sides.

Let’s say we want to find the hypotenuse of this triangle:

The hypotenuse is the longest side and always opposite the right-angle – in this case it’s marked $x$. So, we know that we should get an answer larger than 0.2cm and 1.2cm. We can use this knowledge to check our answer.

We start by simply substituting these values into Pythagoras’ theorem. It doesn’t matter which way round $a$ and $b$ are as long as $c$ is the hypotenuse. We’ll use $a = 0.2$ and $b = 1.2$.

$c^2 = a^2 + b^2$
$c^2 = 0.2^2 + 1.2^2$
$c^2 = 1.48$
$c = \sqrt{1.48}$
$c = \boldsymbol{1.22}$ (2d.p.)

As we expected, our value is larger than 1.2cm.

Let’s say we want to find the length side marked $\boldsymbol{a}$ instead:

This time, we are looking for one of the shorter sides. We know that it will be less than 3.8cm because the hypotenuse is always the longest side. We are going to use a slightly different form of Pythagoras’ Theorem. When we are looking for a shorter side, we subtract:

$a^2 = c^2 - b^2$

Since $c$ is the longest side, $c=3.8$ and $b = 2.1$.

$a^2 = c^2 - b^2$
$a^2 = 3.8^2 - 2.1^2$
$a^2 = 10.03$
$a = \sqrt{10.03}$
$a = \boldsymbol{3.17}$ (2d.p.)

As we expected, our value is smaller than 3.8cm.

### Question 2: Percentage Increase and Decrease

The key to increasing or decreasing by a percentage is finding the multiplier. This way, we can find the right percentage in only a couple steps. However, you can also find the percentage and either add it on or take it off the original amount.

Let’s increase 180 by 12%.

If we are increasing 180 by 12%, we are essentially adding 12% onto the original amount, 100%. So, we are finding 112% of 180.

If your calculator has a percentage button, you can simply input 112% × 180 into your calculator.

However, if you don’t have a percentage button, you need to write 112% as a decimal. We do this by dividing by 100:

112% = 112 ÷ 100
= 1.12

We call this number (1.12) the multiplier. Now, we can multiply this by the amount:

1.12 × 180 = 201.6

If we are decreasing by a percentage, we subtract this from 100% instead of adding it. For example, if we were decreasing by 12% we would find 100% – 12% which is 88%; our multiplier would be 0.88.

### Question 3: Volume of a Prism

A prism is any 3D shape with straight edges that has a consistent cross-section. To find the volume of a prism we simply find the area of the cross-section and multiply it by the length of the prism.

Let’s find the volume of the triangular prism below.

First, we need to find the area of the cross-section. In this case, that’s the triangle. In some cases, you might be given this area – if this is the case you can skip this step.

The height of this triangle is 2cm and the base is 6cm. So, using the formula for the area of a triangle:

area = $\frac{1}{2}$ × base × height
= $\frac{1}{2}$ × 2 × 10
= 6cm2

Now, we simply multiply this area by the length of the prism; 10cm.

Volume = 6 × 10
= 60cm3

Best buy questions, sometimes called “value for money”, are about getting the most of something for the least possible amount of money. This is definitely a helpful life skill!

Let’s say a shop sells lemonade in two sized bottles. We want 12l of lemonade; which size bottle should we buy?

To work out the best value, we should work out how much 12l will cost in each case.

2l for £2:      In this case, we will need 12 ÷ 2 = 6 bottles.

The third bottle is free, so the cost won’t be the same for every bottle – let’s write out each of the costs.

Bottle 1 = £2
Bottle 2 = £2
Bottle 3 = £0
Bottle 4 = £2
Bottle 5 = £2
Bottle 6 = £0

Total cost = 4 × 2
= £8.00

1l for £1.50:      In this case, we will need 12 bottles.

Every other bottle is free, so the cost won’t be the same for every bottle – let’s write out each of the costs.

Bottle 1 = £1.50
Bottle 2 = £0
Bottle 3 = £1.50
Bottle 4 = £0
Bottle 5 = £1.50
Bottle 6 = £0
Bottle 7 = £1.50
Bottle 8 = £0
Bottle 9 = £1.50
Bottle 10 = £0
Bottle 11 = £1.50
Bottle 12 = £0

Total cost = 6 × 1.50
= £9.00

So, it is cheaper to get six 2l bottles.

On questions like this, make sure you come to a conclusion. You need to make it clear which one is the best choice to get full marks.

### Question 5: Factorising into Single Brackets

The key to factorising into single brackets is to make sure you take out the largest possible factor.

Let’s fully factorise:

$\boldsymbol{8a^2+4ab}$.

Start by identifying the highest common factor of the terms. In this case, it’s $4a$. We write this on the outside of a set of brackets:

$8a^2+4ab = 4a(\phantom{2a+b})$

Now, we consider what we multiply by $4a$ to get each of the terms.

To get $8a^2$, we multiply $4a$ by $2a$.

To get $4ab$ we multiple $4a$ by $b$. So, the expression in the brackets is $2a+b$:

$8a^2 + 4ab = \boldsymbol{4a(2a + b)}$

It’s worth noting that we could have taken out just $4$ or $a$. However, this wouldn’t be fully factorising because it’s not the largest factor. In an exam, taking out $4$ or $a$ wouldn’t get you full marks.

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