Sometimes you have to change units before you can
write down a ratio. For instance, say you had to write the ratio between 50
metres and 2 kilometres. Metres and kilometres are different units, *but
they measure the same type of dimension* – they both measure length. So to
write a ratio, I’d need to convert one of the amounts into the same units as
the other. Usually, it’s better to convert the bigger units of measure into
the smaller units of measure. This way can help you avoid decimal numbers in
your ratios.

In this case, the bigger unit of measure is a kilometre, so we’ll convert the amount measured in kilometres into metres. We know there are 1000 metres in a kilometre, so to convert we do:

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Now we can write the ratio between the two distances:

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Using ratios to create models of the Earth and the Sun question |
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The Earth and the Sun are two very huge objects
which are very far apart. Sometimes people create
In our model, we are going to use an orange which has a diameter of 5 cm to represent the Sun. How big should our model Earth be, and how far from our model Sun should we put it? |
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Solution |
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So first, we can summarise what we know about the real world and our model like this:
Part 1 – Working out the size of our model Earth Now, in the real world the Earth is quite a bit smaller than the Sun. How much smaller? Well, we can write a ratio between their two sizes like this: So we have a ratio between the size of the Sun and the size of the Earth. We can use this to work out how big our model Earth should be. First, we can rewrite this ratio as a fraction: This ratio is the ratio of the Sun and Earth’s
sizes in the We want to work out the diameter of our model Earth, so we need to rearrange the equation so it’s the only thing on one side. First we can swap tops and bottoms: Then we multiply both sides by “Model Sun Diameter”:
Now we know how big the model Sun is – 5 cm, so we can write it in, and calculate the model Earth’s diameter: So our model Earth only has to be a fraction of a centimetre in diameter. Next we have to work out how far apart the model Sun and Earth should be. Part 2 – Working out how far apart the model Sun and Earth should be Obviously they’re going to be a lot less than 150 million kilometres apart in our model. But how much less? Well, how much the diameter of the Sun shrinks going from the real world to our model, is the same as how much the distance will shrink going from the real world to our model. The ratio of the Sun’s diameter between the real world and the model is: Whoops! We’ve got different units here. Let’s put everything into centimetres. How many centimetres in a kilometre? Well, there’s 100 centimetres in a metre, and 1000 metres in a kilometre. So: So to change 1,400,000 km to cm, we need to multiply it by 100,000. It becomes 140,000,000,000 cm!!! So our ratio between the Sun’s diameter in the real world and the model is: or in fraction form: So this tells us that measurements in the real
world are 28 billion times larger than in our model. We know that the ratio
between the Sun’s real size and model size is the same as the ratio between
the real distance and the distance in our model. This mean we can write an
equation We want to find out the model distance, so we want to get it all by itself. To do this we need to rearrange the equation, first by swapping tops and bottoms: And then by multiplying both sides by ‘Real Distance’: Now the fraction on the left hand side is the Before, we calculated that: Now, we’ve got: It’s just the same fraction with tops and bottoms swapped over. So the numbers should swap too: Now we can put this into our overall equation: Once we do the calculations we get our answer – the model Sun and model Earth need to be placed 5.4 metres apart. |