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Working with ratios
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Direct proportionality

Ratios & fractions

We talked a little bit about the relationship between ratios and fractions in the previous section, but glossed over the differences between them. We will now look at how they are related and how they differ. Let us consider the ratio $3 : 7$. For every $3$ things I get, you get $7$ things. Now the question we ask now is how many things is that? $3$ for me, $7$ for you, so $3 + 7 = 10$ things altogether. So when we ask the question: What is the fraction of all the things that I get out of all of the things there are? I get $3$, the total is $10$, so the fraction is going to be $\frac{3}{10}$.

$\frac{3}{10}$ for me and $\frac{7}{10}$ for you. Now the fallacy people fall into here is the idea that the fraction could be $\frac{3}{7}$ simply by writing the ratio in fraction form: This is wrong, as it makes no logical sense and must be avoided by being prepared in advance to understand that we are talking about a number of things being split up, which, if it were to be reduced down to a single thing would be a ratio of fractions which will add up to 1, like this: $\frac{3}{10}∶ \frac{7}{10}$ and $\frac{3}{10} + \frac{7}{10} = 1$ We will be using this idea that the sum of the parts of a ratio gives us the total number of things being divided up.

The ratio $4 : 9$ can be thought of as $4$ for me and $9$ for you, so $13$ altogether, hence the ratio can be written as $\frac{4}{13}∶ \frac{9}{13}$.

Additionally, we can see that the sum of the fractions in this new ratio, will be $\frac{4}{13} + \frac{9}{13} = \frac{13}{13} = 1$.

This is something which always happens. If you write the ratio in terms of fractions like this, the sum of the fractions in the ratio will always be $1$.