Daily Maths Revision – Week 8 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. We’ve not included the formula for the volume or surface area of a sphere because these are given in the exam (and on the booklet).

Compound Interest
original amount × (multiplier)number of years

\text{area} = \frac{\theta}{360} \times \pi r^2

\text{gradient} = \frac{\text{change in } y}{\text{change in } x}

This week, we’re exploring:

  1. Spheres
  2. Compound Interest
  3. Sectors
  4. Gradient
  5. Direct Proportion

Question 1: Spheres

You are given the formula for both the surface area and the volume of a sphere. So finding these values is mostly about substitution. Just make sure you are using the radius – the distance from the centre to the surface of the sphere.

Volume of a sphere = \frac{4}{3} \pi r^3

Surface area of a sphere = 4\pi r^2

Let’s find the volume and surface area of the sphere below. We will give both answers to 1 decimal place.

Volume and surface area of a sphere

14cm is the diameter and we want the radius. So, we simply divide it by 2.

\begin{aligned} r &= 14 \div 2 \\ &= 7cm \end{aligned}

\begin{aligned} V &= \frac{4}{3} \pi r^3 \\ &= \frac{4}{3} \pi \times 7^3 \\ &= \frac{1372}{3}\pi \\ &= \boldsymbol{1436.8 \text{cm}^3} \end{aligned}

\begin{aligned} SA &= 4 \pi r^2 \\ &= 4\pi \times 7^2 \\ &= 196\pi \\ &= \boldsymbol{615.8\text{cm}^2} \end{aligned}

Your calculator will, at first, give your answer in terms of \pi . It’s always a good idea to write this version down as well as a rounded version.

Question 2: Compound Interest

The key to calculating compound interest is percentage multipliers. We covered this right back in Week 3 – have a look at this blog if you want a quick reminder.

Let’s say a bank account containing £1000 earns compound interest of 1.5% a year. How much will be in the bank account after 4 years?

Each year, the amount will increase by 1.5%. If we are increasing by 1.5%, we are finding 101.5%. To find the multiplier, we divide this by 100:

(100 + 1.5) ÷ 100 = 1.015

If your calculator has a percentage button, you can use 101.5% for this calculation. However, not all calculators have this function, so we are going to use the method that always works.

After year 1, there will be 1000 × 1.015 (= £1015) in the account. In year 2, the additional £15 is also included when calculating the interest. So, after year 2, there will be 1015 × 1.015 (= £1030.23) in the account.

We can keep multiplying by 1.015 until we get to year 4.

Year 3: 1030.225 × 1.015 = £1045.68
Year 4: 1045.678… × 1.015 = £1061.36

Remember, money is always given to 2 decimal places. However, you shouldn’t round your answer until the end of your work as this could lead to a rounding error.

There is a quicker way. If you are good at remembering formulae, you can remember:

total amount = original amount × (multiplier)number of years

Let’s have a look at the same example with this formula.

Original amount= £1000
Percentage multiplier = 1.015
Number of years = 4

Total amount = 1000 × 1.0154
                        = £1061.36

Question 3: Area of a Sector

A sector is simply a fraction of a circle – you can think of it as a pizza slice. To find the area of a sector, we need to find what fraction of the whole circle it is. This means you need to be confident with finding the area of a circle – we covered this in week 2 so it might be worth heading to that blog for a quick refresher.

Let’s find the area of the sector below. We will give our answer to 2 decimal places.

Area of a sector

There 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 area of the whole circle, then find the area of a 1° slice and finally the area of the sector itself.

Sector AngleArea
360°π × 9.22 = 265.9…
÷ 360÷ 360
265.9… ÷ 360 = 0.7386…
× 117 × 117
117°0.738… × 117 = 86.41…

So, the area of the sector is 86.42cm2 to 2 decimal places.

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}}
r = 9.2 \text{cm}

\begin{aligned} A &= \frac{\theta}{360} \times \pi r^2 \\ &= \frac{117}{360} \times \pi \times 9.2^2 \\ &= 86.42 \text{cm}^2 &&(2\text{d.p.}) \end{aligned}

Question 4: Gradient

The gradient of a line gives information about how steep the line is. It is the change in y divided by the change in x. If we have an equation in the form y=mx+c, the gradient is the coefficient of x (m). 

Let’s find the gradient of the line that passes through the points with coordinates (3, 4) and (7, 2).

Make sure you are consistent with which coordinates you use first – you are finding the change not the difference so your answer could be negative.

Change in y :    4 – 2 = 2
Change in x :    3 – 7 = -4

\begin{aligned} \text{gradient} &= \frac{2}{\text{-}4} \\ &= -0.5 \end{aligned}

Question 5: Direct Proportion

We first looked at proportion in week 5. Two (or more) values are directly proportional if they change at the same rate. If one doubles, the other also doubles.

For example, the time taken to fill some bottles and the number of bottles to fill. Assuming everything else stays the same, if there are twice as many bottles, it will take twice as long to fill them!

Let’s say \boldsymbol{A} is directly proportional to \boldsymbol{B} . \boldsymbol{A} and \boldsymbol{B} are linked by the equation \boldsymbol{A=\frac{6}{7}B} . Find the value of \boldsymbol{A} when \boldsymbol{B=91} .

To solve this, we simply substitute B=91 into our equation.

\begin{aligned} A &= \frac{6}{7} \times 91 \\ &= \boldsymbol{78} \end{aligned}

Don’t forget to read even more of our blogs here and you can find our main Daily Maths Revision Page here! You can also subscribe to Beyond for access to thousands of secondary teaching resources. You can sign up for a free account here and take a look around at our free resources before you subscribe too.

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