# Percentages

We can interpret the percentage as an application of direct proportion, which is only a rule of three in which one of the terms is $100$.

In the IX Session, $125$ women have the deputies' charge in the Congress. Knowing that there are $350$ deputies in total, What is the percentage of women in the Spanish Lower House? And what is the percentage of men?

To solve the exercise it will be necessary to compare the entire number of deputies with $100$ and the number of women with $x$, the percentage that we want to know:

$\begin{eqnarray} 350 & \rightarrow & 100\\ 125 & \rightarrow & x \end{eqnarray}$

So, if $350$ is the whole of the chamber, we are looking at which percentage represents $125$.

The relation converted into fractions will be:

$$\dfrac{350}{125}=\dfrac{100}{x} \Rightarrow 350x=125\cdot100 \Rightarrow 350x=12500 \Rightarrow$$ $$\Rightarrow x=\dfrac{12500}{350}=35,71\%$$

Ormore directly we apply the rule of three for direct proportionality: $x=\dfrac{125\cdot100}{350}=\dfrac{12.500}{350}=\dfrac{1250}{35}=35,71\%$

Rounding, we could say that $36\%$ of the deputies in the Congress are women.

To calculate the percentage of men there are two options. The less direct is to calculate the number of male deputies from the information in the exercise and to raise another rule of three to see what percentage the above mentioned number represents.

$350$ deputies $-125$ women $=225$ men.

$$\dfrac{350}{225}=\dfrac{100}{x} \Rightarrow 350x=225\cdot100 \Rightarrow 350x=22500 \Rightarrow$$ $$\Rightarrow x=\dfrac{22500}{350}=64,29\%$$

The most direct option consists of reducing the percentage of women to $100$, which would represent the whole of the camera:

$100-35,71=64,29\%$

And the fact is that a number expressed in percentage indicates a certain quantity for every $100$. In the previous example, that the percentage of deputy women is $36\%$ represents that from every $100$ deputies of the Spanish Lower House, $36$ are women. In the same way, of those same $100$ deputies, $64$ would be men.

There is another type of problem in which the percentage is known, but another type of fact needs to be computed: a price, for instance.

In an electronics shop we can find a MP4 player at $95$ €, but it is specified that this price is without the VAT (Value Added Tax). If the tax for this type of product is $16\%$: what is the final price of the MP4?

It is necessary to find out how many euros represents $16\%$ of $95$ since this is the amount that we will have to add to the price given to obtain the final price. To do so, we use again a rule of three:

$\begin{eqnarray} 95 \ \mbox{€} & \rightarrow & 100\\ x \ \mbox{€} & \rightarrow & 16 \end{eqnarray}$

So, if $95$ is $100\%$, the number corresponding to $16\%$ will be $x$.

Or what it is the same:

$\dfrac{95}{x}=\dfrac{100}{16} \Rightarrow 100x=95\cdot16 \Rightarrow 100x=1520 \Rightarrow x=\dfrac{1520}{100}=15,20\mbox{€}$

The result we get has to be added to the price offered to obtain the final price of the product:

$95+15,2=110,2$€

In these cases in which there is a percentage of increase in the price of a product, it is possible to calculate directly the final price with the following relation:

$$P_f=P+\Big(P\cdot\dfrac{n}{100}\Big)$$

In which $P_f$ is the final price, $P$ the initial price and $n$ the percentage.

If the relation is applied to the previous problem the final price of the MP4 is obtained by:

$$P_f=95+\Big(95\cdot\dfrac{16}{100}\Big)=95+\Big(\dfrac{1520}{100}\Big)=$$ $$=95+15,2=110,2 \ \mbox{€}$$

It is necessary to highlight that the expression in brackets represents the percentage of the initial number, so that in case of a price increase, as in the present exercise, we will have to add to the initial price, but in case of a reduction we will have to subtract.

A shop of sports clothes offers $25\%$ discount on all the sweaters. How much will a sweater cost if the price before the reduction is $65$ €?

Again, the exercise can be solved by means of a rule of three or applying the relation explained. We will begin with the first case:

$\begin{eqnarray} 65 \ \mbox{€} & \rightarrow & 100\\ x \ \mbox{€} & \rightarrow & 25 \end{eqnarray}$

Otherwise:

$\dfrac{65}{x}=\dfrac{100}{25} \Rightarrow 100x=65\cdot25 \Rightarrow 100x=1625 \Rightarrow x=\dfrac{1625}{100}=16,25\mbox{€}$

Since it is a question of a discount, the obtained number will have to be substracted from the initial price to obtain the discounted final price:

$65-16,25=48,75$€

We get the same result by applying the relation indicated previously, but in this case, instead of adding the expression in brackets, we will have to subtract it since it is a question of a reduction, not of an increase:

$$P_f=P-\Big(P\cdot\dfrac{n}{100}\Big)=65-\Big(65\cdot\dfrac{25}{100}\Big)=65-\Big(\dfrac{1625}{100}\Big)=$$

$$=65-16,25=48,75 \ \mbox{€}$$

Finally, we'd emphasize that it is necessary to be aware of percentages. It is necessary to know what is the total number that it applies to, otherwise we run the risk of making mistakes easily.

A computers shop wants to gain $2.000$ € net for every high class laptop that they sell. As the VAT for this product is $16\%$: what should be the final sale price?

A quite common mistake would be to add $16\%$ to $2.000$ € to obtain the final price. The relation will be applied to verify it:

$$P_f=2000+\Big(2000\cdot\dfrac{16}{100}\Big)=2000+\Big(\dfrac{32000}{100}\Big)=$$ $$=2000+320=2320 \ \mbox{€}$$

But: how much is $2.320$, minus $16\%$ on VAT, the tax that the shop is forced to give back to the Treasury Department?

$$P_f=2320-\Big(2320\cdot\dfrac{16}{100}\Big)=2320-\Big(\dfrac{37120}{100}\Big)=$$ $$=2320-371,2=1948,8 \ \mbox{€}$$

Once the shop has given the $16\%$ of the sale to the Treasury Department, it has $1.948,8$ € net left, not the $2.000$ € expected. What is the problem ?

The computer seller has considered the $2.000$ € as the $100$ %, when in fact this is not the case since only the final price represents the $100$ %. In fact, they should have applied the following rule of three:

$\begin{eqnarray} 2000 \ \mbox{€} & \rightarrow & 84\%\\ x \ \mbox{€} & \rightarrow & 100\% \end{eqnarray}$

So that $2.000$ € represents the $84\%$ of the final price, that is to say, $100\%$ less $16\%$ for the VAT, while $x$ is the desired final price and, therefore, it is $100\%$.

Expressed in fractions we have:

$\dfrac{2000}{x}=\dfrac{84}{100} \Rightarrow 84x=2000\cdot100 \Rightarrow 84x=200.000 \Rightarrow$ $x=\dfrac{200.000}{84}=2380,95\mbox{€}$

Now, once the $16\%$ is removed, the amount we get is:

$$P_f=2380,95-\Big(2380,95\cdot\dfrac{16}{100}\Big)=2380,95-\Big(\dfrac{38095,2}{100}\Big)=$$

$$=2380,95-380,95=2000 \ \mbox{€}$$

Thus obtained is the $2.000$ € net desired by the shop.