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

This week, we’re exploring:

- Dividing fractions
- Plans and elevations
- Sharing in a ratio
- The 𝒏
^{th}term of a linear sequence - Negative and fractional indices

### Question 1: Dividing Fractions

The key to dividing fractions is using the **inverses**. We use the inverse operation (we multiply instead of divide) with the inverse (reciprocal) fraction. This is often remembered as ‘Keep Change Flip’: keep the first fraction, change the operation to multiply and flip the last fraction.

**Let’s calculate \(1\frac{7}{8} \div 2\frac{3}{5}\).**

We start by writing the mixed numbers as improper fractions. To do this, multiply the whole number by the denominator and then add the numerator. We then write the answer to this over the original denominator.

\(\begin{aligned}1\frac{7}{8} &= \frac{1\times8+7}{8}\\&=\frac{15}{8} \\ 2\frac{3}{5} &= \frac{2\times5+3}{5}\\&=\frac{13}{5}\end{aligned}\)

Now, we need to rewrite the calculation. The first fraction stays the same, the divide changes to a multiply and the second fraction is “flipped”.

\(\begin{aligned}1\frac{7}{8} &= \frac{15}{8} \div \frac{13}{5} \\ &= \frac{15}{8} \times \frac{5}{13}\end{aligned}\)

Now, we can carry on with the calculation as a multiplication. To multiply fractions, we simply multiply the numerators and multiply the denominators.

\(\begin{aligned} \frac{15}{8} \times \frac{5}{13} &= \frac{75}{104}\end{aligned}\)

### Question 2: Plans and Elevations

Plans and elevations are 2-dimensional representations of 3-dimensional shapes. The **plan view** is how the shape looks from directly above. The **front **and **side elevations** are how the shape looks from directly in front and from the side. You can think of the plans and elevations as shadows of the shape.

**Draw the plan view, front elevation and side elevation of the shape below. The arrow shows the front view.**

From above, you won’t be able to tell how tall the shape is – you will only be able to see 2 squares. From the front, you will see a pile of three squares with an extra one at the right on the bottom. From the side, you will see a pile of three squares. Using a ruler, we can draw these views accurately.

### Question 3: Sharing 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 4: Linear Sequences

A linear sequence is one that goes up by the same amount each time. We use the \(n\)^{th} term of a sequence to describe the different terms in relation to their position in the sequence, \(n\). For the first term, \(n=1\); for the second term, \(n=2\) and so on.

**Let’s say we want to find the first 3 terms of the sequence with the \(n\) ^{th} term \(3n+1\).**

We simply substitute in \(n=1\), \(n=2\) and \(n=3\).

\(\begin{aligned} n&=1: &3(1)+1&=4 \\ n&=2: &3(2)+1&=7 \\ n&=3: &3(3)+1&=10 \\ \end{aligned}\)

So, the first 3 terms of the sequence are 4, 7 and 10.

What if we are given the start of a sequence and we want to find the \(n\)^{th} term? This is a common question and the process is relatively simple to learn.

**Let’s find the \(n\)^{th} term of the sequence that starts 2, 9, 16, 23, 30….**

We start by finding the common difference of the terms. This simply means finding the amount we add each time:

We are adding 7 each time – this means the sequence is related to the **7 times table**. The \(n\)^{th} term will start with \(7n\).

Next, we find the 0th term; this is the term that would come before 2 in our sequence. This term gives us the amount to add or subtract from \(7n \) to get our sequence. It works because, for this term, \(n=0\) and \(7n\) is also 0. This means the amount we add or subtract will be the term itself.

2 – 7 = -5, so -5 would come before 2 in our sequence.

Therefore, our complete \(n\)^{th} term is \(7n-5\).

### Question 5: Negative and Fractional Indices

Let’s start with the general rules and then we will look at how we **apply** those rules:

\( \begin{aligned} a^{\text{-}m} &= \frac{1}{a^m} \\ a^{\frac{1}{m}} &= \sqrt[m]{a}\end{aligned}\)

We looked at negative indices in week 1 and fractional indices in week 4. So, we will specifically look at when these are **combined**.

**Let’s find the value of \(\frac{4}{25}^{-\frac{3}{2}}\).**

Let’s break down each section. The denominator of the fraction, 2, tells us to **square root**; the numerator, 3, tells us to **cube **and the negative sign tells us to find the reciprocal. We do each of these one by one:

\( \begin{aligned} \frac{4}{25}^{-\frac{3}{2}} &= \frac{2}{5}^{-3} \\&=\frac{8}{125}^{-1} \\& = \frac{125}{8}\end{aligned}\)

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