National 5 Maths
Percentages

In this 3-mark question, we are told that \(80\%\) of the total equals \(480\,000\small.\)

Our task is to find what \(100\%\) equals.

There are two good methods for reverse percentage questions. Both are shown below. We prefer Method 2 (direct proportion), but you are free to choose whichever you have been taught, or that you prefer.

Method 1: Algebra equation

Let's use \(x\) for the total available tickets.

Note that \(80\%\,=80\!\div\!100 = \,0.8\small.\)

$$ \begin{eqnarray} \small\textsf{number sold}\normalsize &=& 80\% \small\textsf{ of the total}\normalsize \\[6pt] 480\,000 &=& 0.8x \\[6pt] x &=& \frac{480\,000}{0.8} \\[6pt] x &=& \frac{4\,800\,000}{8}\:\small\textsf{ (x10 top and bottom)}\normalsize \\[6pt] x &=& 600\,000 \end{eqnarray} $$

So the total number of available tickets was \(600\,000\small.\)

Method 2: Direct proportion

\(80\) isn't a factor of \(100,\) so we can't do this in just one step.

This is a non-calculator question, so we need to think of an easy number to use as our middle step. The trick here is to choose a number that is a factor of both \(80\) and \(100\small.\) The highest common factor always works well.

In this case, there are actually two good options that keep the arithmetic simple: \(10\) or \(20\small.\)

The working below will use \(20\%\) for the middle step. You might like to try it for yourself with \(10\%\) instead.

$$ \begin{eqnarray} 80\%\ \small\textsf{of the available tickets}\normalsize &=& 480\,000 \\[6pt] \textcolor{red}{\small\div\textsf{4}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: &\ & \:\:\:\textcolor{red}{\small\div\textsf{4}}\\[6pt] 20\%\ \small\textsf{of the available tickets}\normalsize &=& 120\,000 \\[6pt] \textcolor{red}{\small \times \textsf{5}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: &\ & \:\:\:\textcolor{red}{\small\times\textsf{5}}\\[6pt] 100\%\ \small\textsf{of the available tickets}\normalsize &=& 600\,000 \\[6pt] \end{eqnarray} $$

So the total number of available tickets was \(600\,000\small.\)

Video solution by Clelland Maths 
Detailed breakdown by ChatGPT 

This 3-mark non-calculator question tells us that \(75\%\) of an unknown number equals \(2400\small.\)

There are two good methods for reverse percentage. Both are shown below. Choose whichever you prefer.

Method 1: Algebra equation

Let \(x\) be the number of Glasgow tickets sold, and note that \(75\%\,=\large\frac34\small.\)

$$ \begin{gather} \small\textsf{Edinburgh tickets}\normalsize=75\% \small\textsf{ of Glasgow tickets}\normalsize \\[6pt] 2400=\small\frac34\normalsize x\\[6pt] 2400\times 4=3x\\[6pt] 9600=3x\\[6pt] x=\small\frac{9600}{3}\\[6pt] x=3200 \end{gather} $$

So the band sold \(3200\) tickets for their Glasgow gig.

Method 2: Direct proportion

The best common factor of \(75\) and \(100\) to use for the middle step is \(25\small.\) To keep the arithmetic as simple as possible, we should usually try to think of the highest common factor.

$$ \begin{eqnarray} 75\%\ \small\textsf{of the Glasgow tickets}\normalsize &=& 2400 \\[6pt] \textcolor{red}{\small\div\textsf{3}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: &\ & \textcolor{red}{\small \div \textsf{3}}\\[6pt] 25\%\ \small\textsf{of the Glasgow tickets}\normalsize &=& 800 \\[6pt] \textcolor{red}{\small \times \textsf{4}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: &\ & \textcolor{red}{\small \times \textsf{4}}\\[6pt] 100\%\ \small\textsf{of the Glasgow tickets}\normalsize &=& 3200 \\[6pt] \end{eqnarray} $$

So the band sold \(3200\) tickets for their Glasgow gig.

Note: Method 1 was quite fiddly for this question. We prefer Method 2, and most students also find it easier.

Video solution by Clelland Maths 
Detailed breakdown by ChatGPT 

We think of the original price as \(100\%\small.\)

So the sale price is \(100\%-20\%=80\%\) of the original price.

Method 1: Algebra equation

Let's use \(x\) for the original price.

$$ \begin{eqnarray} \small\textsf{sale price}\normalsize &=& 80\% \small\textsf{ of original price}\normalsize \\[6pt] 48 &=& 0.8x \\[6pt] x &=& \frac{48}{0.8} \\[6pt] x &=& \frac{480}{8}\:\small\textsf{ (x10 top and bottom)}\normalsize \\[6pt] x &=& 60 \end{eqnarray} $$

So the original price was \(\text{£}60\small.\)

Method 2: Direct proportion

This is a non-calculator question, so we choose the highest common factor of \(80\%\) and \(100\%\small,\) which is \(20\%\small.\) Note that \(10\%\) would also work well.

$$ \begin{eqnarray} 80\%\ \small\textsf{of the original price}\normalsize &=& £48 \\[6pt] \textcolor{red}{\small\div\textsf{4}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: &\ & \textcolor{red}{\small \div \textsf{4}}\\[6pt] 20\%\ \small\textsf{of the original price}\normalsize &=& £12 \\[6pt] \textcolor{red}{\small \times \textsf{5}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: &\ & \textcolor{red}{\small \times \textsf{5}}\\[6pt] 100\%\ \small\textsf{of the original price}\normalsize &=& £60 \\[6pt] \end{eqnarray} $$

So the original price was \(\text{£}60\small.\)

Video solution by Clelland Maths 
Detailed breakdown by ChatGPT 

Again, we think of the original number as \(100\%\).

This has increased by \(15\%\), so it's now \(100\%+15\%=115\%\) of the original number.

Method 1: Algebra equation

Let's use \(x\) for the original number.

$$ \begin{eqnarray} \small\textsf{new number}\normalsize &=& 115\% \small\textsf{ of original number}\normalsize \\[6pt] 161 &=& 1.15x \\[6pt] x &=& \frac{161}{1.15} \\[6pt] x &=& 140 \end{eqnarray} $$

So the firm had \(140\) clients before their marketing campaign.

Method 2: Direct proportion

Because this is a calculator question, we should just use \(1\%\) as the middle step.

$$ \begin{eqnarray} 115\%\ \small\textsf{of the original number}\normalsize &=& 161 \\[6pt] \textcolor{red}{\small\div\textsf{115}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: &\ & \textcolor{red}{\small \div \textsf{115}}\\[6pt] 1\%\ \small\textsf{of the original number}\normalsize &=& 1.4 \\[6pt] \textcolor{red}{\small \times \textsf{100}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: &\ & \textcolor{red}{\small \times \textsf{100}}\\[6pt] 100\%\ \small\textsf{of the original number}\normalsize &=& 140 \\[6pt] \end{eqnarray} $$

So the firm had \(140\) clients before their marketing campaign.

Video solution by Clelland Maths 
Detailed breakdown by ChatGPT 

As always, we think of the original value as \(100\%\).

This has increased by \(8\%\), so it's now \(100\%+8\%=108\%\) of the original value.

Method 1: Algebra equation

Let's use \(x\) for the original number.

Note that \(108\%\,=108\!\div\!100 = \,1.08\small.\)

$$ \begin{eqnarray} \small\textsf{new value}\normalsize &=& 108\% \small\textsf{ of original value}\normalsize \\[6pt] 94\,500 &=& 1.08x \\[6pt] x &=& \frac{94\,500}{1.08} \\[6pt] x &=& 87\,500 \end{eqnarray} $$

So Nadim paid \(\text{£}87\,500\) for his flat.

Method 2: Direct proportion

Because this is a calculator question, we should just use \(1\%\) as the middle step.

$$ \begin{eqnarray} 108\%\ \small\textsf{of the original value}\normalsize &=& £94\,500 \\[6pt] \textcolor{red}{\small\div\textsf{108}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: &\ & \textcolor{red}{\small\div\textsf{108}}\\[6pt] 1\%\ \small\textsf{of the original value}\normalsize &=& £875 \\[6pt] \textcolor{red}{\small \times \textsf{100}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: &\ & \textcolor{red}{\small\times\textsf{100}}\\[6pt] 100\%\ \small\textsf{of the original value}\normalsize &=& £87\,500\\[6pt] \end{eqnarray} $$

So Nadim paid \(\text{£}87\,500\) for his flat.

Video solution by Clelland Maths 
Detailed breakdown by ChatGPT 

For a \(2.5\%\) annual increase, we multiply by \(100\%+2.5\%=102.5\%\)

As a decimal, \(102.5\%=102.5 \div 100=1.025\)

So the final balance will be \( 20\small\ \normalsize 000\times 1.025^3 = £21\small\ \normalsize 537.81\)

The question asks for the interest, so this is \(£21\small\ \normalsize 537.81-£20\small\ \normalsize 000=£1537.81\)

Video solution by Clelland Maths 
Detailed breakdown by ChatGPT 

The word "depreciates" means that the value is decreasing.

For a \(15\%\) annual decrease, we multiply by \(100\%-15\%=85\%=0.85\)

So the value after four years will be \( 25\small\ \normalsize 000\times 0.85^4 = £13\small\ \normalsize 050.15\)

Rounded to 3 significant figures, this is \(£13\small\ \normalsize 100\).

Video solution by Clelland Maths 
Detailed breakdown by ChatGPT 

\(15\%\) decrease: \(100\%-15\%=85\%=0.85\)

So the number after three years will be \( 964\times 0.85^3 = 592.0165\)

Rounded to the nearest \(10\small,\) this is \(590\) pupils.

Video solution by Clelland Maths 
Detailed breakdown by ChatGPT 

\(15\%\) increase: \(100\%+15\%=115\%=1.15\)

So the number after three years will be \( 80\small\ \normalsize 000\times 1.15^3 = 121\small\ \normalsize 670\)

Video solution by Clelland Maths 
Detailed breakdown by ChatGPT 

\(4\%\) increase: \(100\%+4\%=104\%=1.04\)

So the value after two years will be \( 250\small\ \normalsize 000\times 1.04^2 = £270\small\ \normalsize 400\)

Video solution by Clelland Maths 
Detailed breakdown by ChatGPT 

We think of the original price as \(100\%\).

So the discounted price is \(100\%-30\%=70\%\) of the price before discount.

Method 1: Algebra equation

Let's use \(x\) for the original price.

Note that \(70\%\,=\,0.7\small.\)

$$ \begin{eqnarray} \small\textsf{discounted price}\normalsize &=& 70\% \small\textsf{ of original price}\normalsize \\[6pt] 16.10 &=& 0.7x \\[6pt] x &=& \frac{16.1}{0.7} \\[6pt] x &=& \frac{161}{7}\:\small\textsf{ (x10 top and bottom)}\normalsize \\[6pt] x &=& 23 \end{eqnarray} $$

So the original price was \(\text{£}23\small.\)

Method 2: Direct proportion

This method is more commonly used than the first.

The highest common factor of \(70\) and \(100\) is \(10\small,\) so let's use \(10\%\) for the middle step.

$$ \begin{eqnarray} 70\%\ \small\textsf{of the original price}\normalsize &=& £16.10 \\[6pt] \textcolor{red}{\small\div\textsf{7}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: &\ & \textcolor{red}{\small\div\textsf{7}}\\[6pt] 10\%\ \small\textsf{of the original price}\normalsize &=& £2.30 \\[6pt] \textcolor{red}{\small \times \textsf{10}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: &\ & \textcolor{red}{\small\times\textsf{10}}\\[6pt] 100\%\ \small\textsf{of the original price}\normalsize &=& £23 \\[6pt] \end{eqnarray} $$

So the original price was \(\text{£}23\small.\)

Video solution by Clelland Maths 
Detailed breakdown by ChatGPT 

\(11\%\) depreciation: \(100\%-11\%=89\%=0.89\)

\(6\%\) depreciation: \(100\%-6\%=94\%=0.94\)

So the value after three years will be \( 20\small\ \normalsize 000\times 0.89\times 0.94^2 = £15\small\ \normalsize 728.08\)

Note: This is a silly level of accuracy, but the question didn't tell us to round it, so we should leave it as it is.

Video solution by Clelland Maths 
Detailed breakdown by ChatGPT 

\(26\%\) decrease: \(100\%-26\%=74\%=0.74\)

So the value after three years will be \( 460\times 0.74^3 = £186.40\small,\) rounded to the nearest penny.

Video solution by Clelland Maths 
Detailed breakdown by ChatGPT 

We think of the cost last year as \(100\%\).

This has increased by \(16\%\), so it's now \(100\%+16\%=116\%\) of last year's cost.

Method 1: Algebra equation

Let's use \(x\) for the original cost.

$$ \begin{eqnarray} \small\textsf{new cost}\normalsize &=& 116\% \small\textsf{ of original cost}\normalsize \\[6pt] 278.40 &=& 1.16x \\[6pt] x &=& \frac{278.40}{1.16}\normalsize \\[6pt] x &=& 240 \end{eqnarray} $$

So Charley's car insurance cost £240.00 last year.

Method 2: Direct propotion

This is a calculator question, so we can just use \(1%\) for the middle step.

$$ \begin{eqnarray} 116\%\ \small\textsf{of last year's cost}\normalsize &=& £278.40 \\[6pt] \textcolor{red}{\small\div\textsf{116}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: &\ & \textcolor{red}{\small\div\textsf{116}}\\[6pt] 1\%\ \small\textsf{of last year's cost}\normalsize &=& £2.40 \\[6pt] \textcolor{red}{\small \times \textsf{100}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: &\ & \textcolor{red}{\small\times\textsf{100}}\\[6pt] 100\%\ \small\textsf{of last year's cost}\normalsize &=& £240 \\[6pt] \end{eqnarray} $$

Video solution by Clelland Maths 
Detailed breakdown by ChatGPT 

We should think of the price before the sale as \(100\%\).

The price has decreased by \(20\%\), so the sale price is \(100\%-20\%=80\%\) of the original price.

Method 1: Algebra equation

Let's use \(x\) for the original price.

Note that \(80\%\,=\,0.8\small.\)

$$ \begin{eqnarray} \small\textsf{sale price}\normalsize &=& 80\% \small\textsf{ of original price}\normalsize \\[6pt] 720 &=& 0.8x \\[6pt] x &=& \frac{720}{0.8} \\[6pt] x &=& \frac{7200}{8}\:\small\textsf{ (x10 top and bottom)}\normalsize \\[6pt] x &=& 900 \end{eqnarray} $$

So the price before the sale was \(\text{£}900\small.\)

Method 2: Direct proportion

Because this is a non-calculator question, we need to keep the arithmetic simple.

Choosing the highest common factor of \(80\%\) and \(100\%\) will work well for the middle step.

The highest common factor is \(20\%\small.\) Note that \(10\%\) would also have worked well.

$$ \begin{eqnarray} 80\%\ \small\textsf{of the original price}\normalsize &=& £720 \\[6pt] \textcolor{red}{\small\div\textsf{4}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: &\ & \textcolor{red}{\small\div\textsf{4}}\\[6pt] 20\%\ \small\textsf{of the original price}\normalsize &=& £180 \\[6pt] \textcolor{red}{\small \times \textsf{5}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: &\ & \textcolor{red}{\small\times\textsf{5}}\\[6pt] 100\%\ \small\textsf{of the original price}\normalsize &=& £900 \\[6pt] \end{eqnarray} $$

So the price before the sale was \(\text{£}900\small.\)

Video solution by Clelland Maths 
Detailed breakdown by ChatGPT 

\(4\%\) increase: \(100\%+4\%=104\%=1.04\)

So the expected number is \( 118\,750\times 1.04^2 = 128\,440\small.\)

Video solution by Clelland Maths 
Detailed breakdown by ChatGPT 

The number of miles that an electric car can travel on a fully-charged battery is known as its range, but Qualifications Scotland presumably decided not to use that word, in case some students didn't know it.

We should think of the full range as \(100\%\).

So we know that \(60\%\) of the range equals \(180\) miles.

Method 1: Algebra equation

Let's use \(x\) for the range.

Note that \(60\%\,=\,0.6\small.\)

$$ \begin{eqnarray} 180 &=& 60\% \small\textsf{ of the range}\normalsize \\[6pt] 180 &=& 0.6x \\[6pt] x &=& \frac{180}{0.6} \\[6pt] x &=& \frac{1800}{6}\:\small\textsf{ (x10 top and bottom)}\normalsize \\[6pt] x &=& 300 \end{eqnarray} $$

So the full range is \(300\) miles.

Method 2: Direct proportion

Because this is a non-calculator question, we need to keep the arithmetic simple.

Choosing the highest common factor of \(60\%\) and \(100\%\) will work well for the middle step.

The highest common factor is \(20\%\small.\) Note that \(10\%\) would also have worked well.

$$ \begin{eqnarray} 60\%\ \small\textsf{of the range}\normalsize &=& 180 \\[6pt] \textcolor{red}{\small\div\textsf{3}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: &\ & \textcolor{red}{\small\div\textsf{3}}\\[6pt] 20\%\ \small\textsf{of range}\normalsize &=& 60 \\[6pt] \textcolor{red}{\small \times \textsf{5}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: &\ & \textcolor{red}{\small\times\textsf{5}}\\[6pt] 100\%\ \small\textsf{of the range}\normalsize &=& 300 \\[6pt] \end{eqnarray} $$

So the full range is \(300\) miles.

Video solution by Clelland Maths 
Detailed breakdown by ChatGPT 

\(28\%\) decrease: \(100\%-28\%=72\%=0.72\)

So the value after three years is \( 22\,600\times 0.72^3 = \text{£}8\,435.40\small.\)

Rounded to \(3\) significant figures, this is \(\text{£}8\,440\small.\)

Video solution by Clelland Maths 
Detailed breakdown by ChatGPT