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.

You aren’t given this one, so you will need to remember it.

**Speed**

In the 2023 exams, you will be given this formula but it is still worth trying to memorise it.

**Area of a Circle**

This week, we’re exploring:

### Question 1: Solving Equations

Solving equations means finding the specific value the satisfies that equation. Usually, this means finding the value of a particular letter.

**Let’s solve:**

We need to start by expanding the brackets on the right-hand side. The easiest way to do this is by using the grid method – this was covered in Week 1. We place the term on the outside (2) on the left and the terms inside the brackets across the top. Then, we multiply the term in each row by each column.

So, our equation is now:

Now, we want to solve the equation by collecting all the variables (the terms with letters) on one side:

### Question 2: Sharing an Amount in a Ratio

Questions about sharing an amount in a ratio can take many different forms. Take a ratio that splits an amount between two people. You will sometimes be given the total amount and asked to calculate how much each person receives and you will sometimes be given how much one person gets and need to calculate how much the other person gets. You could also be given the difference between how much each person gets and use this to calculate the amounts.

Whatever form the question takes, the easiest way to approach it is using the **bar model** – sometimes called the bucket method. The two methods are based on the same principle and are just set out slightly differently.

**Let’s say Robert and Prav share £136 in the ratio 3:5. Calculate how much each person receives. **

The principle of the bar model is that we set up two bars – one for each person. Robert’s will be 3 squares long and Prav’s will be 5 squares long.

Since we are sharing the total amount of money, we count the total number of boxes – in this case, 8. We share the total amount (£136) evenly across each of the boxes. In other words, we calculate 136 ÷ 8 = 17.

Now, we place £17 in each box and calculate the total amount for each person.

Robert gets 17 + 17 + 17 = **£51**.

Prav gets 17 + 17 + 17 + 17 + 17 = **£85**.

You should always check your answer by adding the values together to see if you get the original amount: 51 + 85 = 136.

If the question gives you one person’s amount, then you simply share this amount between that person’s boxes.

For example, if Robert gets £57, we would find 57 ÷ 3 (= 19) and then write this amount in each box.

If the question gives you the difference between the amounts, you find the difference in the number of boxes and divide by this amount.

For example, if Prav gets £23 more than Robert, as Prav has 2 more boxes than Robert, we would calculate 23 ÷ 2 (= 11.5) and write this amount in each box.

### Question 3: Substitution

Substitution means replacing the letters in an expression or formula with their corresponding values. It’s really important that you remember your **order of operations** (BIDMAS) and are particularly careful with **negatives**.

**Let’s say , and . Work out the value of the expression .**

Start by replacing each of the variables (letters) with its value. If the number is negative, place brackets around it. Remember, .

= (-1) × 7 + (-2)^{2} | |

= (-1) × 7 + 4 | Calculate the indices first. |

= -7 + 4 | Then the multiplication. |

= -3 | Finally, the addition. |

If this is on a calculator paper, you can just use your calculator – just make sure you include the brackets.

### Question 4: Speed

To find the speed of something, we divide the distance travelled by the time taken:

You can also use a formula triangle to help organise the formula – this is particularly helpful when you need to find the distance or time rather than the speed.

**Let’s find the time taken for Jonathon to travel 500m at 6m/s.**

In this case, we want to find the time taken so we cover this up in the formula triangle and use the other two variables. Since distance is over speed, we divide the distance by the speed:

time = distance ÷ speed

= 500 ÷ 6

= 83.33….

**= 83.3 seconds (1d.p.)**

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

If the question asked for the distance travelled, we would use distance = speed × time. If the question asked for the speed, we would use speed = distance ÷ time.

### Question 5: The Area of a Circle

The formula for the area of a circle is area = where is the radius of the circle. Sometimes, you will be given the diameter of the circle so remember that the diameter is twice the radius.

**Let’s find the area of the circle below.**

The radius of this circle is 5cm, so we substitute = 5 into the formula:

Area = π × 5^{2}

= 25π

**= 78.5cm ^{2} **(1d.p.)

Your calculator will probably give the answer in the form 25π at first. The question may ask you to leave your answer in terms of π or give it as a decimal. Either way, you should write down the initial calculator answer (25π). If you need the answer as a decimal, use the S⇔D button to change the answer to a decimal.