# Percentages

A **percent** is a number that represents the fractional part out of 100 (*per cent* literally means per one hundred). Thus 94% means \( \frac{94}{100}\). Likewise we can represent a denominator of 100 with decimals by moving the decimal point by 2 places. So \( 94 \times \frac{1}{100} = 0.94 \). Thus \( 94\% = 0.94 = \frac{94}{100} \). The main aim of this wiki is to discuss about percentages.

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## Basic Examples

What percentage of \(50\) is \(25?\)

We have

\[\begin{align} 25 & =\dfrac{25}{50} × 100 \%\\\\ & = \dfrac{1}{2} × 100 \%\\\\ & = 50\%.\ _\square \end{align}\]

What percentage of \(75\) is \(100?\)

We have

\[\begin{align} 100 & =\dfrac{100}{75} × 100 \%\\\\ & = \dfrac{4}{3} × 100 \%\\\\ & = 133.33\%.\ _\square \end{align}\]

What number is \(35\%\) of \(200?\)

We have

\[\begin{align} \text{35% of }200 & = \dfrac {35}{100} × 200\\\\ & = 70.\ _\square \end{align}\]

What number is \(150\%\) of \(50?\)

We have

\[\begin{align} \text{150% of }50 & = \dfrac {150}{100} × 50\\\\ & = 75.\ _\square \end{align}\]

## Adding Percentages

The simplest way to perform arithmetic with percentages is to convert them to their decimal equivalent by moving the decimal two places, then adding the decimals.

If John has \( 37 \% \) of the apples and Sally has \( 42 \% \), what percentage of the apples do they have if they combine their apple piles?

Converting the percentages to decimal and adding, we see that \( 0.37 + 0.42 = 0.79 = 79 \% \). Thus the answer is \( 79 \% \). \( _\square \)

John and Mac were friends. In the election of the class monitor John secured \(30\%\) of the votes and Mac secured \(45\%\) votes. Being friends, they decided to combine their votes and mind the class together. If the total votes polled were \(240\), what is the total number of votes they jointly secured?

We have

\[\begin{align} \text{Total votes secured by John and Mac} & = \text {(30+45)% of 240}\\\\ & = \text{75% of 240}\\\\ & = \dfrac{75}{100} × 240\\\\ & = 180.\ _\square \end{align}\]

## Multiplying Percentages

John has \(10\%\) of \(20\%\) of \(30\%\) of \(1000000\) bananas. How many bananas does John have?

We have

\[\begin{align} \text{Number of bananas John has} & = \dfrac{10}{100} × \dfrac{20}{100} × \dfrac{30}{100} ×1000000\\\\ & = \require{cancel}{\dfrac{10}{\cancel {100}} × \dfrac{20}{\cancel {100}} × \dfrac{30}{\cancel {100}} × \cancel{1000000}}\\\\ & = 6000.\ _\square \end{align}\]

## Percentage Change

Given two values of some variable \(x\) taken at different points of time, percentage change \( \%\Delta\) measures the proportion of the difference between the two values to the original reading of \(x\). If the value of \(\Delta\) is positive, we call it a *percentage increase*; if negative, then *decrease*.

The general formula for

percentage changeis\[\%\Delta = \frac {x_1 - x_0} {x_0} \times 100\% .\]

The price of a commodity changed from \( $20 \) to \( $30 \). Find the percentage change in price of the commodity.

We have

\[\begin{align} \text {Change in the price of the commodity} & = $30 - $20\\ & = $10\\\\ \text {Original price of the commodity} & = $20\\\\ \text{% change in the price of the commodity} & = \dfrac{10}{20} × 100 \%\\ & = 50\%.\ _\square \end{align}\]

Population ChangeImagine a city with 10000 people at the beginning of a year. Given that the population at the end of that year is 10500, compute the percentage change of the population.

Let us assign symbols to the population readings:

\[\begin{align} x_0 &= 10000\\ x_1 &= 10500. \end{align}\]

We apply the formula for pecentage change:

\[\begin{align} \%\Delta &= \frac {x_1 - x_0} {x_0} \times 100\%\\ &= \frac {10500 - 10000} {10000} \times 100\%\\ &= \frac {500} {10000} \times 100\%\\ &= 0.05 \times 100\%\\ &= 5\%. \end{align}\]

We say that the percentage change of the population from the beginning to the end of the year is an

increaseof \(5\%\). This means that population grew 5% in proportion to the initial count. \( _\square \)

The price of a commodity is \($10\). If its price increases by \(10\)%, then what is the new price?

We have

\[\begin{align} \text{10% of 10} & = \dfrac {10×10}{100}\\ & = \dfrac {100}{100}\\ & = 1\\\\ \Rightarrow \text{(New price of the commodity in dollars)} & = 10+1\\ & = 11. \ _\square \end{align}\]

## Percentages - Word Problems

Let's take our knowledge of percentages to a whole new level by practicing word problems on them. Some examples are given below.

## Of \(2384\) students of the school, \(75\%\) attempted the examination, of which \(25\%\) failed. How many students passed the examination?

We have

\[\begin{align} \text{No. of students who attempted the examination} & = \dfrac {75}{100} × 2384\\ & = 1788\\\\ \text {No. of students who failed the examination} & = \text {25% of 1788}\\\\ \text {No. of students who passed the examination} & = \text {(100 - 25)% of 1788}\\ & = \text {75% of 1788}\\ & = \dfrac{75}{100} × 1788\\ & =1341.\ _\square \end{align}\]

Of all the students enrolled at BRCM Public School, \(9\%\) are in the band. The band has \(180\) members. How many students are enrolled at the school?

We know that the \(180 \) band members are equal to \(9\%\) of the total population of the school. We need to find out the total number of students \((100\%)\) that go to BRCM.

We can first divide \(180\) by \(9\) to see that \(20 \) students equal \(1\%\), then multiply by \(100\) to see how many students equal \(100\%\):

\[\begin{align} \text{100 % of students } &=100 \times (\text{1 % of students})\\ &=100 \times (\text{20 students})\\ &=\text{2,000 students}. \end{align}\]

Therefore, there are \(2,000 \) students at the school. \(_\square\)

In Idaho, there are 23 Democratic delegates, to be proportionally distributed to Bernie Sanders and Hillary Clinton based on the popular vote. Bernie Sanders gets 78.8% of the vote. Hillary Clinton gets 21.2% of the vote. How many delegates should be allotted to each candidate?

For Bernie, \(0 .788 \times 23 = 18.124 \), round down to 18 delegates because you can't have a decimal of a delegate.

For Hillary, \( 23- 18 = 5\). \(_\square\)

## Percentages - Problem Solving

You're given 2 gift vouchers for shopping, as shown above. The option A grants you a free second item after purchasing the product in the normal price. On the other hand, the option B gives you the discount of 40% for the first item and allows you to buy the second one at 60% cost of the first one.

Which option will save you more money?