# Daily Maths Revision – Week 9 Walkthrough

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

If you need a formula, these are the relevant ones this week.

Arc Length
$\text{arc length} = \frac{\theta}{360} \times \pi d$

Density
$\text{density} = \frac{\text{mass}}{\text{volume}}$

This week, we’re exploring:

### Question 1: Highest Common Factor with Prime Factors

For large numbers, finding the prime factors is the easiest way to find the highest common factor. We use a Venn diagram to organise the factors and see which are common.

Find the prime factors of 180 and 460 and, hence, find the higher common factor of 180 and 460.

Let’s start by finding the prime factors. This was covered in week 1 so you should be relatively confident with it. Otherwise, go back and have a look at that week’s blog.

Now, we place each of the factors in a Venn diagram. If a factor is in both numbers, then we place it in the middle (the intersection). Otherwise, we just put it on the relevant side:

The highest common factor is the product of the numbers in the intersection (the bit in the middle).

highest common factor = 2 × 2 × 5
= 22 × 5
= 20

Any of these three answers would get full marks on a question like this. The first two are as the product of prime factors.

### Question 2: Arcs

A sector is simply a fraction of a circle – you can think of it as a pizza slice. The arc length is the curved section of this sector – you can think about it like the crust of the pizza slice! To find the arc length, we need to find what fraction of the whole circle it is.

Let’s find the perimeter of this sector. We will give our answer to 1 decimal places.

To find the arc length, we need the diameter. Remember, this is double the radius.

Diameter = 2 × 9.2
= 18.4cm

Here are two methods – the first follows a simple procedure and the second jumps straight in with a formula. In an exam, either of the methods will get you the marks.

For the first method, we will find the diameter of the whole circle, then find the arc length of a 1° slice and finally, find the arc length of the sector itself.

So, the arc length is 18.8cm to 1 decimal place. However, the question asks for the perimeter, not the arc length. So, we need to add the two straight edges to this.

Perimeter = 18.7… + 9.2 + 9.2
= 37.2cm (1d.p.)

If you prefer to use the formula, you need to start by identifying the different variables. Remember, $\theta$ represents the angle of the sector.

$\theta = 117 ^\text{o}$
$\theta = 18.4\text{cm}$

\begin{aligned} \text{Arc length} &= \frac{\theta}{360}\times\pi d \\ &= \frac{117}{360} \times \pi \times 18.4 \\ &= 18.7...\text{cm} \end{aligned}

Again, we need to add this to the straight edges to find the perimeter.

\begin{aligned} \text{Perimeter} &= 18.7... + 9.2 + 9.2 \\ &= 37.2\text{cm (1d.p.)} \end{aligned}

### Question 3: Density

The density of an object is the mass per unit volume; usually g/cm3 or kg/m3. You aren’t given the formula for density, so you need to learn it.

$\text{density} = \frac{\text{mass}}{\text{volume}}$

You can also write this as a formula triangle:

Let’s find the mass of an object of volume 30cm3 that has a density of 0.8g/cm3.

In this case, we want to find the mass so we cover this up in the formula and use the other two variables. Since density and volume are on the same level, we multiply them together.

mass = density × volume
= 0.8 × 30
= 24g

Make sure you include units in your answer – there is often a mark for this.

If the question asked for the density, we would use density = mass ÷ volume. If the question asked for the volume, we would use volume = mass ÷ density.

### Question 4: Factorising into Double Brackets

Factorising is the opposite of expanding. It’s the easiest way to solve a quadratic equation, although you do also get questions that are just on factorising.

This week, we are looking at questions in the form $x^2 + bx + c$. These will factorise into two brackets.

When this is the case, we look for two numbers that multiply together to give $c$ and add together to give $b$. These are the two numbers that will go into our brackets.

Let’s factorise $\boldsymbol{x^2+7x+10}$.

In this case, $b=7$ and $c=10$. So, we are looking for two numbers that multiply to give 10 and add to give 7. You might be able to spot it straight away but, if you can’t, write out the factors of 10:

1          10
2          5

Now, we can see that the numbers we want are 2 and 5. We write these into our brackets:

$x^2+7x+10=\boldsymbol{(x+2)(x+5)}$

It doesn’t matter which way round the two brackets are; $(x+5)(x+2)$ would also be correct for this question.

Now, let’s factorise $\boldsymbol{x^2-5x-14}$.

As you can see, we have negative numbers this time. This doesn’t change the process though; we are still looking for two numbers that multiply together to give -14 and add together to give -5. Make sure you include the signs when you find the values of $b$ and $c$.

Let’s start by writing out the factors of -14. Since $c$ is negative, the signs of our two numbers must be different.

1          -14
2          -7
7          -2
14        -1

The only pair that add together to give -5 is 2 and -7. So, these are what we write in our brackets:

$x^2-5x-14=\boldsymbol{(x+2)(x-7)}$

### Question 5: Angles in Parallel Lines

There are three rules you need to know for angles in parallel lines:

You might also need to use these rules – they are often used in questions that include angles in parallel lines.

Let’s find the labelled missing angles in the shape below.

The best way to start is to write down any angles you can see. For example, you might spot that 83° and $c$ are on a straight line. Label $c$ on the diagram and then write out your working and the reason.

$c$ = 180 – 83
= 97°             Angles on a straight line add up to 180°.

Write as much detail as you can in the reason – most exam boards have keywords they are looking for, so it is worth looking at mark schemes to see what these are.

Next, we might spot that $b$ and 37° are corresponding angles. Corresponding angles make an F shape on the diagram.

$b$ = 37°             Corresponding angles are equal.

Finally, you could spot that $a$ and 83° are alternate angles. Alternate angles make a Z shape on the diagram.

$a$ = 83°             Alternate angles are equal.

You could also have also spotted that $a$ and $c$ are co-interior angles which add up to 180°. Co-interior angles make a C shape on the diagram.

There is always more than one way to answer a question like this. The reasons are worth a large proportion of the marks so if you can see more than one solution, you should use the one that you can explain better.

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