## What you need to know

### Conversions

This topic is linked to conversion graphs

A unit is a standard measurement of a particular quantity. For example, metres and kilometres are both units for measuring distance, and seconds, minutes, and hours are all units for measuring time. You will need to be familiar with metric and imperial unit conversions as well.

You will also need to know how to write different units. For example, one unit of speed is “metres per second” (written like \text{m/s} or \text{ms}^{-1}), and one unit for density is “grams per cubic metre” (written like \text{g/m}^3 or \text{gm}^{-3}).

You do need to know a number of different unit conversions. Fortunately, if you can remember a few prefixes then it makes life a lot easier. Let’s consider units of length:

1\text{ metre } = 1,000\textbf{ milli}\text{metres}

1\text{ metre} = 100\textbf{ centi}\text{metres}

1,000\text{ metres} = 1\textbf{ kilo}\text{metres}

This means that to convert from metres into millimetres for example, all you need to do is multiply by 1,000 (this is effectively the scale factor).

### Take Note:

You don’t need to memorise metric to imperial unit conversions as in the GCSE maths exams you will be given the the conversion, but you will need to apply it to the question so do spend time practising this.

### Conversion Graphs

Another way you might be expected to convert between units, typically metric and imperial, is by using a conversion graph. It’ll be a straight-line graph with one unit on the x-axis and another on the y-axis, and the line will be our guide for converting between the two units. Below shows an example of a miles and kilometres conversion graph, for more on conversion graphs visit the dedicated page.

### Example 1: Unit Conversion Distance and Time

A car is driving at 50 km per hour. What is the speed of the car in metres per second?

We know that 1km is 1,000m. This implies that 50km is 50 \times 1000 = 50,000 m. So, the speed of the car is 50,000 metres per hour.

Next, we know that there are 60 seconds in a minute and 60 minutes in an hour, so there is a total of 60 \times 60 = 3,600 seconds in 1 hour. In other words, it will take this particular car 3,600 seconds to travel 50,000 metres, so the number of metres they travel every second will be 50,000\div 3,600 = 13.9 (1 dp). Thus, the speed of the car is 13.9 metres per second.

### Example 2: Metric to Imperial Conversion

A baby is born weighing 4.2kg. What is the mass of the baby in pounds to 3sf? Use the conversion rate 1\text{ pound } = 0.4536\text{ kg}.

We have to convert a value from kilograms into pounds. That conversion rate tells us that to get from pounds to kilograms, we would need to multiply by 0.4536. Therefore, in order for us to convert kilograms to pounds we must do the opposite and __divide__ by 0.4536.

\text{Mass of baby } = 4.2 \div 0.4536 = 9.26\text{ pounds (3 sf)}

### Example Questions

1) Selina wishes to convert £500 into dollars for her holiday. There is a 3% fee for converting the currency. Work out how many dollars she will have to spend on her holiday. Use the conversion rate: \pounds 1 = \$ 1.56.

We can either deduct the 3% commission from the amount in pounds at the start or from the amount in dollars at the end (it won’t make a difference); I’d recommend getting it out of the way now. A 3% decrease means she will have 97% of her money left over, so

\text{Money left } = \dfrac{97}{100} \times 500 = \pounds 485

If £1 is worth $1.56 dollars, and we are converting £485, this means that we are converting 485 times the number of pounds given to us in the exchange rate. This means that she will receive 485 times the dollar figure from the exchange rate. Therefore, the number of dollars she receives is:

485 \times 1.56 = \$756.60

2) Diana measures her morning walk to work to be 2.3 kilometres. What is this distance of Diana’s walk to the nearest foot? Use the conversion rate: 1\text{ foot} = 0.3048\text{ metres}.

The conversion rate is between metres and feet, so first we need to convert 2.3km into metres. We know that 1km is 1,000m, so:

2.3\text{km} = 2.3 \times 1,000 \text{m} = 2,300\text{m}.

If 1 foot is equal to 0.3048 metres, that means we would have to multiply a distance in feet by 0.3048 in order to get the equivalent distance in metres. So, to go from metres to feet, we will have to divide by 0.3048. This means Diana’s walk in feet is:

2,300 \text{ metres} \div 0.3048 \text{ metres}= 7,546\text{ feet (to nearest foot)}.

3) Using the conversion graph below find approximate values for:

a) the number of litres in 4 pints

b) the number of pints in 5 litres

a) For this first conversion, we are converting litres to pints. Since pints is on the horizontal x axis, locate 4 on this axis and go up until you touch the blue line. Then go across to the corresponding value on the vertical y axis. We can see that this line touches between 2.2 litres and 2.4 litres, so we can say that 4 pints is approximately equal to 2.3 litres.

b) For the second conversion, we are converting pints to litres. Since litres is on the vertical y axis, locate 5 on this axis and go across to the right until you touch the blue line. Then go down to the corresponding value on the horizontal x axis. We can see that this line touches between 8.6 pints and 8.8 pints so we can say that 5 litres is approximately equal to 8.7 pints.

4) If a half marathon is 13.1 miles, how long will it take a runner to complete the distance running at an average pace of 5 minutes and 30 seconds per kilometre? Give your answer in hours and minutes, rounding your answer to the nearest minute.

Use the conversion rate: 1 \text{ mile} = 1.61 \text{ kilometres}

Since we have been given a pace in minutes per kilometre, we will need to convert the race distance of 13.1 miles into kilometres. We can convert 13.1 miles into kilometres as follows:

13.1 \text{ miles} \times 1.61 \text{ kilometres} = 21.091 \text{ kilometres}

If it takes the runner 5 minutes and 30 seconds to run one kilometre, then we simply need to multiply 21.091 kilometres by the runner’s pace. However, the pace of the runner is in mixed units, so we will need to convert it. It makes sense to keep the pace in minutes rather than convert to seconds per kilometre:

5 \text{ minutes } 30 \text{ seconds} \equiv 5.5 \text{ minutes}

(Remember that 30 seconds is \frac{1}{2} a minute = 0.5 minutes)

The total time to complete the half marathon at a pace of 5.5 minutes per kilometre is therefore:

21.091 \text{ kilometres} \times 5.5 \text{ minutes per kilometre} = 116.0005 \text{ minutes}

The question says the we can round the answer to the nearest minute, so the half marathon will take 116 minutes.

We now need to convert 116 minutes into hours and minutes:

116 \text{ minutes} \equiv 1 \text{ hour } 56 \text{ minutes}

5) A businessman buys £5,000 of euros in January which he then trades back to pounds in February. To the nearest pound, what profit has the businessman made?

January exchange rate: \pounds1 = €1.16

February exchange rate: \pounds1 = €1.15

If the businessman exchanges £5,000 into euros in January, we need to calculate how many euros he receives. If £1 equates to €1.16, then £5,000 can be converted as follows:

\pounds5,000 \times €1.16 = €5,800

If he trades the €5,800 back into pounds in February, we need to calculate how many pounds he will receive. The exchange rate is €1.15 to the pound, so we need to see how many times €1.15 goes into €5,800:

€5,800 \div €1.15 = \pounds5,043 (to the nearest pound).

If the businessman started with £5,000 and ended up with £5043, he has made a profit of £43.

### Worksheets and Exam Questions

#### (NEW) Conversions Exam Style Solutions - MME

Level 4-5#### (NEW) Unit Conversions Exam Style Solutions - MME

Level 1-3#### Conversion Graphs

Level 1-3### Videos

#### Conversions Q1

GCSE MATHS#### Conversions Q2

GCSE MATHS#### Conversions Q3

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