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.

What percentage of $50$ is $25?$

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We have

$$\begin{aligned} 25 & =\dfrac{25}{50} × 100 \%\\\\ & = \dfrac{1}{2} × 100 \%\\\\ & = 50\%.\ _\square \end{aligned}$$

What percentage of $75$ is $100?$

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We have

$$\begin{aligned} 100 & =\dfrac{100}{75} × 100 \%\\\\ & = \dfrac{4}{3} × 100 \%\\\\ & = 133.33\%.\ _\square \end{aligned}$$

What number is $35\%$ of $200?$

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We have

$$\begin{aligned} \text{35\% of }200 & = \dfrac {35}{100} × 200\\\\ & = 70.\ _\square \end{aligned}$$

What number is $150\%$ of $50?$

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We have

$$\begin{aligned} \text{150\% of }50 & = \dfrac {150}{100} × 50\\\\ & = 75.\ _\square \end{aligned}$$

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?

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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?

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We have

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

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

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We have

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

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 change is

$$\%\Delta = \frac {x_1 - x_0} {x_0} \times 100\% .$$

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

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We have

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

Population Change

Imagine 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.

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Let us assign symbols to the population readings:

$$\begin{aligned} x_0 &= 10000\\ x_1 &= 10500. \end{aligned}$$

We apply the formula for pecentage change:

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

We say that the percentage change of the population from the beginning to the end of the year is an increase of $5\%$. 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?

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We have

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

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?

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We have

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

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?

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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 $100\%$:

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

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?

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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, $23- 18 = 5$. $_\square$