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

This week, we’re exploring:

### Question 1: Box Plots

A box plot, sometimes called a box and whisker diagram, is a way of displaying data. Five **vertical** lines are marked; at the minimum value, the lower quartile, the median, the upper quartile and the maximum value.

For a reminder on how to find the upper and lower quartile, take a look at week 2. We will use the same set of values in the example below.

**Draw a box plot for the following data.**

**2.7 3.2 4.5 4.7 5.1 5.6**

First, we need to identify the median and the quartiles.

median = (4.5 + 4.7) ÷ 2

= 4.6

lower quartile = 3.2

upper quartile = 5.1

Now, we mark these three values and the maximum, 5.6, and minimum, 2.7, on a scale:

Then, we use horizontal lines to connect the central 3 lines to make a box and the maximum and minimum to the centre of the box:

This type of diagram is sometimes called a **box and whisker **plot – the centre area is the box and the lines either side are the whiskers.

### Question 2: Simultaneous Equations

Solving simultaneous equations is a tricky bit of algebra. However, there is a series of steps you can follow.

Make sure you set this work out neatly – it’ll be easier for both you and whoever is marking your work.

**Let’s solve the simultaneous equations below.**

To be able to solve these equations, we need to eliminate one of the variables. We do this by multiplying the equations to get a pair of coefficients that are the same, ignoring the sign. In this case, we will get the -coefficients to be the same.

The lowest common multiple of 3 and 5 is 15. So, we will multiply both equations to get a common -coefficient of 15, again ignoring the sign.

Now, we can add the two equations together. This will eliminate the -variable. Make sure you are careful with negatives.

Then, we solve the equation to find the value of .

Finally, we substitute this value of into one of the equations to find the value of :

So, the solution to our simultaneous equations is and .

Sometimes, you will get equations where the sign in the middle is the same. In this case, the process is almost identical – you just **subtract** the equations instead of adding them.

### Question 3: The Difference of Two Squares

In week 2, we looked at factorising expressions of the form . The difference of two squares is an extension of this – the expressions are of the form .

When factorising , we looked for a pair of numbers that multiplied together to give and added together to give .

When the expression is in the form , we are looking for a pair of numbers that multiply together to give and add together to give 0.

**Let’s factorise .**

We want to find a pair of numbers that **multiply** together to give -196 and add together to give 0. The only such pair is +14 and -14.

Expressions of this form will **always** factorise into this form. It’s just a case of spotting that it is a difference of two squares! Knowing the first 15 square numbers can really help with this.

### Question 4: Circle Theorems

These are the two circle theorems that are relevant this week.

**Given that AC is a diameter of the circle, find the size of angles ABC and BCD.**

ABCD is a cyclic quadrilateral so the opposite angles will add up to 180°.

BCD = 180 – 102

**= 78°**

If AC is a diameter, then ABC is the angle in a semicircle – which is a right angle.

ABC = **90° **

### Question 5: Two-Way Tables

Two-way tables are an excellent way to organise information. We carefully split up the information into its component parts – it’s all about setting it out in a clear, usable way.

**100 people were asked to choose an activity (swimming or running) and a dessert (cake or pie). One person is chosen at random.** **Given that they chose swimming, find the probability that they also chose cake.**

We are given that the person chose swimming. That means we are choosing from 70 people – those that chose swimming. Of these 70 people, 64 of them chose cake.

So, the probability that someone chose cake, given they chose swimming, is .

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