What are Percentages?

Percentages are a mathematical concept used to express a number as a fraction of 100. They are a way to represent a part-to-whole relationship or a relative comparison between two numbers and are usually are denoted by the symbol “%” or “pct“.

For example, if you score 45 out of 50 on a test, you can express your score as a percentage. By calculating \(\frac{45}{50}\times 100\), you find that your score is 90%.

Wile the term percent is often attributed to Latin per centum (“per hundred”), it actually comes from the French “pour cent.”. The whole idea of thinking in the terms of hundredths comes from ancient Greece.

Percentages are commonly used in many contexts, such as calculating discounts, tips, taxes, changes in stock prices, and economic indicators. They can also be used every day to represent information.

For example, if you are told that your monthly utility bill is high due to excess infiltration, or leaking of air into your home, you may wish to know what percent of this leakage comes from a specific area. This way, you may take action to reduce infiltration, energy usage, and your utility bill.

How to Calculate Percentages

Calculating percentages involves determining what portion of a whole a particular number represents. For instance, if you want to find out what percentage 30 is of 150, you would use the formula:

\[\frac{numerator}{denominator} \times 100\]

Mathematical problems involving percentages usually fall into one of three categories. Each will be described in turn. First, read over these reminders

• All problems involving the finding of percentages, or the actual amount represented by a percentage, are either multiplication or division problems.
• All percentages are parts of 100 that can also be expressed as a fraction with denominator 100.
• All percents are parts of 100 that can also be written as decimals to the hundredths place.

What is X percent of Y?

To calculate “What is X percent of Y”, use the following formula:

\[(X \times Y) \div 100\]

For example, the dress, shoes or pants you have been dreaming about are $75, and they are on sale at 30% off this weekend. In this example, \(X=30\) and \(Y=$75\). Let us now plug these values into the formula:

\[(30 \times $75) \div 100 = $22.50\]

So you save $22.50 on the item.

X is what percent of Y?

To calculate “X is what percent of Y”, use the following formula:

\[\frac{X}{Y} \times 100\]

For example, you are shopping for a car and the car salesman gave you $1000 off the car you bought for $30,000. In this example, \(X=1000\) and \(Y=30000\). Plugging these values into the formula:

\[\frac{1000}{30000} \times 100 = 3.33%\]

So the salesman gave you a 3.33% discount, which is not a very good deal.

 X is  Y% of what?

This equation is asking to find the base value that X represents as a percentage of Y. Here, we use the following formula:

\[X = \frac{Y%}{100} \cdot \text{base}\]

Where:

\(X\) is the given value
\(Y%\) is the percentage that \(X\) represents
\(\text{base}\) is the unknown value that \(X\) is a percentage of

For example, if the equation is “12 is 15% of what?”, we can express this as:

\[12 = \frac{15%}{100} \cdot \text{base}\]

Solving for the base value, we get:

\[\text{base} = \frac{12}{\frac{15%}{100}} = 80\]

So the equation is asking to find the base value that 12 represents as 15% of.

Percentage Change

Percentage change refers to a measure of the relative change in a value over time. It shows the amount of increase or decrease in a value compared to the original or starting value. This metric is useful for analyzing trends, evaluating performance, and making comparisons over time in various fields, such as finance, economics, and business.

A positive percentage change indicates an increase, while a negative percentage change indicates a decrease. Both percentage increase and percentage decrease can be used to express how much a value has risen or fallen as a percentage of its original value. Here’s how to calculate both:

Percentage Increase:

Percentage increase tells us how much a number increased. It occurs when the new value is greater than the original value.

The formula is:

\[\text{Percentage increase} = \frac{x_2 – x_1}{x_1} \times 100%\]

Where:

• \(x_1\) is the initial value
• \(x_2\) is the final value

Real world example: last week a bag of oranges cost five dollars and now it’s six. What’s the percentage increase? Let’s start with just the increase. It was 5 and is now 6. It’s 1 dollar more. How many percents does that equal? Let’s see. The formula for percentage is:

Okay, let’s work through this real-world example of the percentage increase for the price of oranges.

Given:

• Last week, a bag of oranges cost $5.
• This week, the bag of oranges costs $6.

To calculate the percentage increase:

Calculate the increase in price:

The initial price was $5.

The final price is $6.

The increase in price is $6 – $5 = $1.

\[\text{Percentage increase} = \frac{6 – 5}{5} \times 100% = \frac{1}{5} \times 100% = 20%\]

Therefore, the percentage increase in the price of a bag of oranges from $5 to $6 is 20%.

Percentage Decrease:

A percentage decrease, on the other hand, tells us how much a number dropped. The formula for calculating percentage decrease is:

\[\text{Percentage decrease} = \frac{x_1 – x_2}{x_1} \times 100%\]

Where:

• \(x_2\) is the final value
• \(x_1\) is the initial value

Let’s learn it on an example:

12 kids in our class did not know how to calculate percentages. Now that we have practiced it a bit, only 9 kids cannot do it. What’s the percentage decrease?

Let’s break down the calculation step-by-step:

1. The original number of kids who couldn’t calculate percentages was 12.
2. The new number of kids who can’t calculate percentages is 9.
3. The decrease is 12 – 9 = 3 kids.

Now, to calculate the percentage decrease:

\[\text{Percentage Decrease} = \frac{3}{12} \times 100% = 0.25 \times 100% = 25%\]

Therefore, the percentage decrease in the number of kids who cannot calculate percentages is 25%.

Percentage point change: If you need to show the change in percentage points, you can simply subtract the two percentages. For example, if the initial percentage was 40% and the final percentage is 60%, the change in percentage points would be:

\[60\% – 40\% = 20\text{ percentage points}\]

Percentage Discount (% off)

Imagine a jacket originally priced at $200 is on sale for 15% off. To find the discount amount:

Discount Amount=(15×200100)=30Discount Amount=(10015×200​)=30

The final price of the jacket would be:

Final Price=200−30=170Final Price=200−30=170

Percentage as a Decimal

Decimals and percentages are closely related and can be easily converted from one to the other. They both provide ways to represent parts of a whole and can be used interchangeably to compare the relative magnitudes of different quantities. For example, 0.25 < 0.5 and 25% < 50%.

Converting a Decimal to a Percentage

To convert a decimal to a percentage, multiply the decimal by 100 and add the percent symbol (%).
For example: Convert 0.83 to a percent.

• Multiply the decimal by 100 : \(0.83 \times 100 = 830.83×100=83\)
• Add a percent sign after the answer:\(\%83%\)

Thus, 0.83 as a percentage is 83%

Converting a Percent to a Decimal

To convert a percentage to a decimal, divide the percentage by 100.

For example: Convert 75% to a decimal.

• \[75\% \div 100 = 0.7575%÷100=0.75\]

So, 75% as a decimal is 0.75.

Fraction as a Percentage

A percentage is essentially a fraction with a denominator of 100. Infact, any fraction can be converted to a percentage by dividing the numerator by the denominator and then multiplying by 100. For example, the fraction 3/4 can be converted to the percentage 75% (3/4 = 0.75 = 75%).

Conversely, any percentage can be converted to a fraction by dropping the % sign and writing the number over 100. For example, 40% can be written as the fraction 40/100.

Converting a Fraction to a Percent

These examples illustrate how percentages can be applied in various scenarios, enhancing your ability to make informed decisions and perform accurate calculations.

Fractions, decimals, and percentages are all different ways to represent the same underlying quantity. They are interchangeable – any number that can be written as a fraction can also be written as a decimal or a percentage, and vice versa.

Here is a table that shows the relationship between the most commonly searched fractions, decimals, and percentages:

Fraction	Decimal	Percentage
1/16	0.0625	6.25%
1/17	0.0588	5.88%
2/20	0.10	10%
1/10	0.10	10%
2/40	0.05	5%
1/8	0.125	12.5%
2/16	0.125	12.5%
1/6	0.1667	16.67%
3/16	0.1875	18.75%
1/5	0.20	20%
2/12	0.1667	16.67%
3/7	0.4286	42.86%
5/25	0.20	20%
7/12	0.5833	58.33%
10/16	0.625	62.5%
12/20	0.60	60%
13/20	0.65	65%
14/20	0.70	70%
16/20	0.80	80%
15/20	0.75	75%
15/18	0.8333	83.33%
18/20	0.90	90%
19/20	0.95	95%
15/25	0.60	60%
17/25	0.68	68%
18/25	0.72	72%
19/25	0.76	76%
21/25	0.84	84%
22/25	0.88	88%
20/25	0.80	80%
9/15	0.60	60%
14/18	0.7778	77.78%
1/6	0.1667	16.67%
28/40	0.70	70%
7/12	0.5833	58.33%
10/15	0.6667	66.67%
17/18	0.9444	94.44%
16/24	0.6667	66.67%
11/16	0.6875	68.75%
8/12	0.6667	66.67%
14/25	0.56	56%
11/14	0.7857	78.57%
10/12	0.8333	83.33%
15/24	0.625	62.5%
12/17	0.7059	70.59%
11/17	0.6471	64.71%
40/50	0.80	80%
50/80	0.625	62.5%
29/40	0.725	72.5%
9/12	0.75	75%
21/24	0.875	87.5%
15/16	0.9375	93.75%

FAQs

Is it “Percent” or “per cent”?

It depends on your diet, really. If you eat hamburgers for majority of your meals, it is percent. If you prefer fish and chips, it is per cent. If you spray your fish-smelling chips with vinegar, then it is per cent, mate (as opposed to burger eaters’ percent, dude).

When it comes to percentage, both sides of the pond are in agreement: it should be a single word. Still confused? Americans say percent, British use per cent. Something tells me American English is more popular nowadays, so this website uses a single-word form.

What is per mille?

Per mille, per mil, or per mill all refer to a unit that represents one part in one thousand(1/1000 or 0.001), or 0.1% (one-tenth of a percent). It is usually denoted by the symbol “‰”

The advantage of using a per mille calculation instead of a percentage is that it allows for more precise measurements of very small amounts relative to a larger whole. This can be particularly useful in financial, scientific, or statistical applications where small fractions need to be quantified accurately.

What are Percentage points?

Percentage points are a unit of measurement used to express the absolute change in a percentage value. We can express this concept as follows:

\[\text{Percentage Point Change} = \text{New Percentage} – \text{Old Percentage}\]

We often use percentage points when we talk about a change between two percentages, especially when the starting and ending values are far apart. They provide a more intuitive way to understand the magnitude of the change.

For example:

• If a test score increases from 80% to 85%, the increase is 5 percentage points.
• If the inflation rate goes from 3% to 5%, the increase is 2 percentage points.
• If a market share drops from 40% to 35%, the decrease is 5 percentage points.