## What you need to know

A conversion graph is a tool that we use to convert between different units. Any conversion graph you see will be a straight-line graph. You have to know how to construct a conversion graph given a conversion rate, and also how to use a conversion graph.

Example: Shown is a conversion graph for converting between litres and fluid ounces. Using the graph, convert

i) 2.5 litres to fluid ounces;

ii) 24 fluid ounces to litres.

As we can see, the graph shown has litres on the y-axis and fluid ounces on the x-axis. When using these kinds of graphs, always be wary of the scales on both axes.

So, in order to use the graph to convert the values given, we need to draw on the graph. To do this (in this case to find a conversion for 3 litres) we

1. Starting from 3 on the y-axis, draw a horizontal line until you meet the graph (the blue line).

2. From the point where your line meets the graph, draw a vertical line down until it meets the x-axis.

3. Read the value where your line met the x-axis – trying to be as accurate as possible – and you have the result of the conversion.

Then, when we have to convert in the other direction, we simply reverse the process: draw up to the graph first, then across, and this time read the result off the y-axis. For both conversions, i) and ii), the graph would look like this:

i) We can see that the 2.5-litre line ends up at 88 on the x-axis, so we get

$\mathbf{2.5 \text{ litres }=88\text{ fluid ounces}}$

ii) We can also see that the 24-fluid-ounce line ends up between 0.6 and 0.8 on the y-axis. It looks to be about halfway between the two small squares, so it’s safe to go for 0.7. So, we get

$\mathbf{24 \text{ fluid ounces }\equiv 0.7 \text{ litres }}$

Next, we will see how to make our own conversion graph.

Example: a) Draw a conversion graph for ‘miles per hour’ and ‘metres per second’ using the conversion rate below.

$1\text{ mile per hour }\equiv 0.447\text{ metres per second }$

b) Use your conversion graph to convert 1.5 m/s into m/h.

To draw a straight-line graph that represents this conversion, we need some points to plot. We will assume here that the axes are given with m/h on the x-axis whilst m/s on the y-axis.

We know that 1 m/h is the same as 0.447 m/s. This means that the line on our conversion graph must pass through the coordinate (1, 0.447) – this is our first point. To find others, think of the conversion as a ratio. The ratio of “m/h : m/s” is “1 : 0.447”. Now, a ratio can be scaled up, i.e. we can times both values by the same number without changing it. Multiplying both parts by 2 we get

$1 : 0.447 = 2 : 0.894$

This means we have a second coordinate that the line must pass through: (2, 0.894). Lastly, if we multiply both values in the ratio by 2 again, we get

$2 : 0.894 = 4 : 1.788$

So, our third coordinate is (4, 1.788). You may be thinking that these points look unpleasant to plot, and that is true. We just have to be as accurate as we can. The good news is the decimals are quite close to simpler ones, e.g. when plotting (1, 0.447), we can get away with plotting (1, 0.45); the difference is small enough to not be a problem.

Now, we must plot these three points and draw the most accurate straight line through them that we can. The result should look like the graph shown here.

Now we’ve got the graph we can use it!

b) Drawing a line across for 1.5 on the ‘metres per second’ axis and following the process for using a conversion graph should look like the graph shown below.

Therefore, we get the answer to be

$1.5 \text{ m/s } \equiv 3.3 \text{ m/h}$

## Example Questions

For a) we must draw a line up from 3 on the x-axis to meet the graph, and then across from the graph to the y-axis.

For b) we must draw a line across from 3.8 on the y-axis to meet the graph, and then down from the graph to the x-axis.

The result should look like the graph below.

Therefore, the answer to part a) falls between the squares for 5.2 and 5.4 pints. It looks to be halfway between the two, so the answer is 5.3 pints.

The answer to part b) falls between the squares for 2 and 2.2 litres. It is past halfway and is closer to 2.2 than 2, so the answer is 2.15 litres.

a) We already have the first point we will plot: (1, 2.54). Now, for the other points, observe that the ratio of “inches : cm” is 1 : 2.54. Scaling up this ratio by multiply both values by 3, we get that

$1 : 2.54 = 3 : 7.62$

So, the 2nd point we’ll plot is (3, 7.62). Scaling up the ratio again, multiplying both terms by 2, we get

$3 : 7.62 = 6 : 15:24$

So, the third point we’ll plot will be (6, 15:24). Plotting these three points and joining them with a straight line, we get the graph shown below.

b) To use this graph to convert 10cm to inches, we must draw a horizontal line from 10cm on the y-axis to the graph, and then from the point on the graph draw a vertical line down to the x-axis. This should look like:

Reading off the x-axis, we see that the answer lies halfway between 3.8 and 4, so we get the answer to be

$10\text{ cm }\equiv 3.95 \text{ inches}$