## What is a Percentage?

A percentage is a number indicated as a fraction of 100. It is a dimensionless number often expressed using the symbol %.

## What Kind of Word is Percent?

The word "percent" comes from the Latin word "cent," which means hundred. Thus, "percent" means per hundred.

For example, 20% is read as "20 percent." We know that percent means per 100, so "20%" is like saying 20 marks out of 100 marks. This can also be expressed as a fraction, 20/100, with 20 as the numerator and 100 as the denominator. The word "per" here means we are dividing one number by the other.

Where are percentages used in real life? We come across percentages in everyday life when finding profits or losses, calculating discounts when shopping, paying interest, assessing our grades in classes, and in many other areas.

But how do percentages work? One might ask you the percentage you scored on your exam or the discounted price when purchasing a product. But how do we calculate that?

## Converting Percent to Fractions

Taking the fraction $\frac{20}{100}$, let us think about how to solve the percentage. First, we can reduce the fraction by dividing 20 by 100 and making it into a simple term.

$\begin{array}{c}20\%=\frac{20}{100}\\ =\frac{1}{5}\end{array}$

Here, $\frac{20}{100}$ is reduced to $\frac{1}{5}$ . We cannot reduce it further as a fraction.If you have a big cake and you divide it into 5 equal pieces for 5 friends, each will get one-fifth of the cake. Mathematically, we can show this as $\frac{1}{5}$. This is equivalent to 20 percent since $\frac{20}{100}=\frac{1}{5}$.

To convert a percentage into a fraction, remove the percent sign, write the number left as a fraction over 100, and reduce it to its lowest form.

## Converting Numbers to Percentages

Suppose that you are going shopping and find a shirt with a $15 discount. The cost of the shirt is $150, but you do not have to pay $150 because of the $15 discount. How do you calculate the percentage of the discount?

Here, the total is not 100 but 150. Can we still find the percentage? Yes.

## What is the Formula for Figuring Out Percentage?

**Step 1: **We have to divide the discount by the total and convert to a decimal.

$\frac{15}{150}=\frac{1}{10}$

($\frac{1}{10}$ = 0.1 when converting to decimal value)

**Step 2:** Multiply the result by 100 to write in terms of percentage.

$\frac{1}{10}\times 100=10$

(Multiplying 0.1 with 100 gives 10)

The shirt has a 10% discount.

But how to calculate the percentage of a number like $\frac{74}{50}$?

**Hint:** Multiply the number by 100 and simplify.

## How to Calculate the Percentage of a Number

If there are 200 mangoes in a mango tree at your house, and your neighbor asks for 25% of the total mangoes, how many mangoes should you give him?

25% of 200 can be mathematically written as $\frac{25}{100}\times 200$ . Upon simplifying, we get the value 50.

$\begin{array}{c}\frac{25}{100}\times 200=\frac{1}{4}\times 200\\ =50\end{array}$

## How to Write Ratios as Percentage

To make chocolate ice cream, we need 3 portions of milk, 1 portion of sugar, and 2 portions of cocoa powder. It is of the ratio 3:1:2. The total portion is 3+1+2=6. After mixing it all together, the ice cream has $\frac{3}{6}$ portion of milk mixed with $\frac{1}{6}$ portion of sugar, and $\frac{2}{6}$ portion of cocoa powder. What percentage of milk, sugar, and cocoa powder is there in this mixture?

The percentage of milk is 60%.

$\begin{array}{c}\frac{3}{6}\times 100=\frac{300}{6}\\ =50\%\end{array}$.

The percentage of sugar is 16.67%.

$\begin{array}{c}\frac{1}{6}\times 100=\frac{100}{6}\\ =16.67\%\end{array}$

The percentage of milk is 33.33%.

$\begin{array}{c}\frac{2}{6}\times 100=\frac{200}{6}\\ =33.33\%\end{array}$

You are going to the market with a friend and your friend buys 5 oranges and 2 apples. You have also bought 1 orange and 2 apples. What is the percentage of apples and oranges you both have in total?

**Hint:** Calculate the total apples and oranges and the percent of apples and oranges.

## Practice Problems

**1.** **A chocolate has a total length 30 centimeters. Some percentage of the chocolate, x, is 12 centimeters in length. What is x?**

Solution: x% of 30=12

$\begin{array}{l}\frac{x}{100}\times 30=12\\ x\times 30=12\times 100\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\text{divideby30onbothsides}\right)\\ x=40\end{array}$

**Answer: **The 12 centimeters of chocolate is 40% of the total 30-centimeter chocolate.

**2. When a long stick is broken by 25%, it becomes 120 meters long. What is the total length of the stick?**

Solution: We know the actual stick is lengthier than the broken portion of the stick. But we know the value of the broken stick and not the actual stick. To find its original length, it takes a series of steps. Let the length of the actual stick be x. 25% of x is written as $\frac{25}{100}\times x$. From the information given, we can write the following step.

$x-\frac{25x}{100}=120$

Simplify and calculate the value of x.

Multiplying both the sides by 100, we get,

$\left(x-\frac{25}{100}\times x\right)\times 100=120\times 100$

Take 100 inside the bracket and multiply with each term.

$\left(x\times 100\right)-\left(\frac{25}{100}\times x\times 100\right)=120\times 100$

$100x-25x=12000$

$75x=12000$

Divide both sides by 75 to get the value of x.

$\begin{array}{c}x=\frac{12000}{75}\\ x=160\end{array}$

**Answer: **The actual length of the stick is 160 meters.

**3. The price of a notebook increased by 20% and costs $96. What was the original price of the notebook?**

Solution: The original price of the notebook is unknown. Let us call it x. The price is increased by 20%. The new cost of the notebook is 96. We have to find x. From the given information, adding 20% of the cost of the notebook to the original price gives the new price. Mathematically, it can be written as $x+\frac{20x}{100}=96$.

Calculate the value of x from this like we did earlier.

**Answer: **The value of x is $80.

**4. If your uncle hands you $1,000 and asks you to take 40% of it and give the remaining to your sibling, how much money will you get?**

Solution: Calculate the amount you get by taking 40% of 1000.

$\frac{40}{100}\times 1000=400$

You will get 400 bucks. Now calculate how much amount your sibling would get.

(Hint: Subtract 40 from 100 and calculate the percentage of it).

**5. Suppose that a man got a dress for $2,000 and sold it for $2,200. What is the percentage of profit?**

Solution: If the man sold a dress which costs $2,000 for $2,200, the profit he gets from selling is $2,200 – $2,000 = $200. The profit percentage is calculated by dividing the gain by the sale price and then multiplying by 100.

$\frac{200}{2200}\times 100=9.09\%$

**Answer: **Therefore, he gets a profit of 9.09%.

**6. If a man buys one kilogram of grapes for $80 and sells it for $60, is it a loss or a profit? Find the percent of loss/profit.**

Solution: This is a loss because on subtracting the selling price from the cost price, if the value is positive, it would be a profit. But 60 – 80 = -20. So, the value is negative. After knowing that it is a loss, calculate the loss by subtracting the actual cost from the selling price so that the value becomes positive.

To calculate the loss percentage, divide the loss by the cost price and then multiply by 100.

$\frac{20}{60}\times 100=33.33\%$

**Answer: **He had a 33.33% loss.

## Formulas

- The formulas for profit and loss are:

Loss = Cost Price - Selling Price

Profit = Selling Price - Cost Price

- The formula to obtain the percentage of a number is:

$\frac{\text{Part}}{\text{Whole}}\times 100$

- The formula to convert a percentage into a fraction is:

$x\%=\frac{x}{100}$

## Context and Applications

This topic is significant in professional exams for both undergraduate and graduate courses, especially for:

- B.S. in Mathematics
- M.S. in Mathematics

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