## FR1 – Ratio

**QUESTION: **The ratio of the ages of Kemah compared to Bob and Deborah is 1∶2∶4. Deborah is 21 years older than Kemah. Work out the ages of Kemah, Bob, and Deborah.

**ANSWER:** We know that difference between Deborah’s and Kemah’s ages is 21. Looking at the ratio, Kemah has 1 part and Deborah has 4, meaning that the difference between them (21 years) constitutes 3 parts in the ratio. Therefore, we get that

\text{1 part }=21\div 3=7

Kemah, Bob, and Deborah have 1, 2, and 4 parts in the ratio respectively. So, we get that

\text{Kemah’s age }=1\times 7=7

\text{Bob’s age }=2\times 7=14

\text{Deborah’s age }=4\times 7=28

## FR2 – Proportionality

**ANSWER:** This recipe makes 6 pancakes, but Wes wants to make 21.

21\div 6=3.5

Therefore, he needs to 3.5 times as much of every ingredient. So, we get

\text{flour: }100\times 3.5=350\text{ g}

\text{eggs: }2\times 3.5=7\text{ eggs}

\text{milk: }300\times 3.5=1,050\text{ ml}

## FR3 – Percentage Change

**QUESTION:**Matt buys a TV for \pounds 550, and a year later sells it to his friend Dave for 32\% less. Calculate how much Dave purchased the TV for.

**ANSWER:** This is a 32\% decrease, so the multiplier for a 32\% decrease is

1-\dfrac{32}{100}=0.68

Therefore, multiplying this by the original value Matt bought the TV for, we get the price that Dave purchased it for to be

550\times 0.68=\pounds 374

## FR4 – Reverse Percentage

**QUESTION:** Tom measures himself to be 182cm tall and calculates that this new height is a 4\% increase on his height 2 years ago. Work out how tall Tom was 2 years ago.

**ANSWER:** We need to consider how we would calculate a 4\% increase. We know that 4\%=0.04, so we get the multiplier for a 4\% increase to be

1+0.04=1.04

Let H be Tom’s height from two years ago. We know that the result of multiplying H by 1.04 must be 182. We can write this as an equation:

H\times 1.04=182

Then, if we divide both sides by 1.04 we get

H=182\div 1.04=175

So, Tom’s height two years ago was 175cm.

## FR5 – Growth & Decay

**QUESTION:** Bacteria are being grown in a lab. Initially, there are 480 bacteria in a dish, and the number increases by 40\% every day. The scientist estimates that after one week, there will be over 5,000 bacteria in the dish. Show that the scientist’s estimate is correct.

**ANSWER:** This is a case of compound growth. Firstly, the multiplier for a 40\% increase is

1+0.40=1.4

We are looking at the number of bacteria after one week, which means SEVEN 40\% increases. Therefore, our calculation is

480\times (1.4)^7=5,060\text{ (to nearest whole number)}

5,060 is clearly bigger than 5,000, so the scientist’s estimate is correct.

## FR6 – Speed, Distance, Time

**QUESTION:**An aeroplane travels at an average speed of 230 kilometres per hour for a total journey time of 414 minutes. Work out the total distance covered by this aeroplane.

**ANSWER:**

If we cover up d in the triangle, we see that we will have to multiply s by t to get our answer. However, the units don’t match up – we need to convert the minutes to hours, which we will do by dividing it by 60.

t=414\div 60=6.9\text{ hours}

Now, we can do the multiplication:

\text{distance covered }=d=230\times 6.9=1,587\text{ km}

## FR7 – Pressure & Density

ANSWER: Covering up F on the triangle,

We see that we need to multiply p (pressure) by A (area). To do this, we need to find the area of the triangle above. Before that, however, notice that the sides of the triangle are measured in **centimetres** which doesn’t match up with the “Newtons per **metres **squared” in the question. So, we will convert the dimensions of the triangle into metres by dividing by 100:

\text{height }=80\div 100=0.8\text{ m}

\text{base }=150\div 100=1.5\text{ m}

Now we can calculate the area of the triangle:

\text{area }=\dfrac{1}{2}bh=\dfrac{1}{2}\times0.8\times1.5=0.6\text{ m}^2

Therefore, we get that the force being applied is

F=p\times A=40\times 0.6=24\text{ N}

## FR8 – Best Buys

**ANSWER:** We will work out the cost per drink for each brand.

__Brand 1__

Each bottle is 600ml and provides 3 drinks per 100ml. 600\div 100=6, so we get

\text{drinks per bottle }=3\times 6=18

The cost of a bottle of Brand 1 squash is \pounds 1.89, therefore we get

\text{cost per drink }=1.89\div 18=\pounds 0.105

__Brand 2__

Each bottle is 1,300ml and provides 7 drinks per 200ml. 1,300\div 200=6.5, so we get

\text{drinks per bottle }=7\times 6.5=45.5

The cost of a bottle of Brand 2 squash is \pounds 5.10, therefore we get

\text{cost per drink }=5.10\div 45.5=\pounds 0.112...

The cost per drink is lower for Brand 2, therefore Brand 1 is better value.