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

This week, we’re exploring:

1. Dividing Fractions
2. Vector Addition
3. Conversions and Units
4. Standard Form
5. Frequency Trees

Question 1: Dividing Fractions

The key to dividing fractions is inverses. We use the inverse operation (we multiply instead of dividing) 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:

We will give the answer as a mixed number in its simplest form.

We start by rewriting the calculation. The first fraction stays the same, the divide changes to a multiply and the second fraction is “flipped”.

Now, we can carry on with the calculation as a multiplication. We covered this in detail on Week 4. To multiply fractions, we simply multiply the numerators and multiply the denominators.

Next, we simplify. Both numbers are even, so we can divide them by 2.

To write this as a mixed number, we divide 35 by 12.

35 ÷ 12 = 2 remainder 11

Question 2: Vector Addition

Vectors can be added using diagrams or just numerically. Remember, the top part of the vector gives information about how far to the right a vector goes. The lower part of the vector is how far up the vector goes.

Let’s find:

To do this question numerically, we simply multiply the first vector by two, then add the second vector. We treat the top number and bottom numbers completely separately.

This method is quite abstract – you might find it easier to draw the vectors out. We’ll start by drawing the vectors on a grid. The first vector is 3 units to the right and 1 unit down:

We need two of the first vector – one after the other. Then, we need the second vector. This will be 7 units to the left and 1 unit down. 

The resultant vector (the answer to our addition) is the vector from our starting point to the finish point. It is drawn below in colour.

This is the vector:

Question 3: Conversions and Units

The metric units are all based around multiples of 10. To convert between the different measurements, we multiply or divide by 10, 100 or 1000.

This is how you convert between different units of length:

This is how you convert between different units of weight:

This is how you convert between different units of capacity:

This system is very similar for all three, but you do need to learn them.

Convert 3.4km into centimetres.

We will start by converting from kilometres to metres. We do this by multiplying by 1000:

3.4km = 3.4 × 1000
            = 3400m

Now, we’ll convert this into centimetres by multiplying buy 100.

3400m = 3400 × 100
              = 340 000cm

You could also be asked to convert between imperial units (old measures such as feet and inches) and metric units (the measures we use now). You don’t need to know the conversion rate for these; you’ll always be given it.

Tammam walks 3 miles. Given that 1 mile ≈ 1.6 km, determine how far he walked in kilometres.

1 mile is roughly 1.6 kilometres.

So, 3 miles will simply be 3 × 1.6 = 4.8km

Question 4: Standard Form

Standard form is a way to write very large or very small numbers more efficiently. We use powers of 10 to replace long lists of zeros.

Write 275 000 in standard form.

We look at the first significant figure, 2 – this represents 200 000. This is the same as 2 × 10⁵.

So, 275 000 is equivalent to 2.75 × 10⁵.

Write 0.00782 in standard form.

We look at the first significant figure again – this represents 0.007. This is the same as 7 × 10^-3.

So, 0.00782 is equivalent to 7.82 × 10^-3.

Question 5: Frequency Trees

Frequency trees are a 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 desert (cake or pie). 30 people chose running. Of the 74 people who chose cake, 10 also chose running.
Use this information to complete the frequency diagram below.

Start by placing the information we are given in the diagram. Highlighting the information can help with this.

The only information we haven’t used here is that 74 people chose cake. We know that 10 of these also chose running so 74 – 10 = 64 people chose cake and swimming.

Now, each of the branches will add up the box they come from. So, swimming and running will add together to give 100 as everyone choses one of the activities.

Swimming:                    100 – 30 = 70
Swimming and pie:        70 – 64 = 6
Running and pie:           30 – 10 = 20

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