# Problems from Percentages

A student has sat an exam in $$8$$ subjects and has obtained $$5$$ passes and $$3$$ failures. Calculate the percentage of every note.

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### Development:

It is a question of applying rules of three to know the percentage of every result.

In the case of the passes: $$\begin{eqnarray} 5 & \rightarrow & 8\\\\ x & \rightarrow & 100 \end{eqnarray}$$

So,he has passed $$5$$ of $$8$$, which represents a $$x$$ per cent.

Expressed in fractions it is:

$$\dfrac{5}{x}=\dfrac{8}{100} \Rightarrow 8x=5\cdot100 \Rightarrow 8x=500 \Rightarrow x=\dfrac{500}{8}=62,5\%$$

In case of the failures: $$\begin{eqnarray} 3 & \rightarrow & 8\\\\ x & \rightarrow & 100 \end{eqnarray}$$

Otherwise:

$$\dfrac{3}{x}=\dfrac{8}{100} \Rightarrow 8x=3\cdot100 \Rightarrow 8x=300 \Rightarrow x=\dfrac{300}{8}=37,5\%$$

It was possible to find the same value by substracting the entire percentage of passes from $$100\%$$:

$$100-62,5=37,5\%$$

### Solution:

The percentage of passes is $$62,5\%$$ and the one of failures is $$37,5\%$$.

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