# Ratio Foundation Revision Card Answers

## 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

QUESTION: Wes is making pancakes using the recipe outlined below. Work out the number of eggs, and the amounts of flour and milk needed to make 21 pancakes.

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 $182$cm 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 $175$cm.

## 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.

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

QUESTION: A pressure of $40 N/M^2$ is exerted evenly on the area of the triangle shown here. Calculate the force being applied to exert this amount of pressure.

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

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

QUESTION: Two different brands are selling bottles of squash as outlined in the table. A study has shown that Brand 1 will provide 3 drinks per $100$ml, whilst Brand 2 will provide 7 drinks per $200$ml. Determine which brand has the best value.

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

Brand 1

Each bottle is $600$ml and provides 3 drinks per $100$ml. $600\div 100=6$, 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 $1,300$ml and provides 7 drinks per $200$ml. $1,300\div 200=6.5$, 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.