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# How to Calculate Percentages

Chris has a Master's degree in engineering and uses his knowledge to write about a variety of topics from an analytical perspective.

A percentage is simply a ratio between any two numbers. A percentage is basically a fractional number that represents the relative weight of a portion of something compared to its whole portion on a scale of 0 to 100. Understanding how to calculate and use percentages can help you in many ways. For example, when you hear that something is discounted to 50% off, most people will realize that this means that the discounted price is one half of the original price. In other words, 50% is equivalent to the fraction of 50/100 which, when reduced, means 1/2. Other examples of useful percentages include a variety of things such as interest rates, population growth or decline, school grades, and even health/medical related information.

Percentages help standardize the data and information around us making it easier to grasp and explain to others by boiling it down to something that can be related to a value of 100. Learning how to calculate a variety of percentages is relatively easy when you understand the principles behind them as well as the basic steps needed to perform the calculations. In this article, I will explain several common percentage calculations and show you practical examples for their use.

## How to Calculate a Percentage of a Whole Number

Calculating the standard percentage of something is the simplest percent based calculation. Knowing how to calculate percent values is important to be able to calculate things like percent change (increase or decrease), percent discount, percent error, percent yield, and more as explained later in this article

A basic percentage is simply a fraction where you divide the small part of something by the total value of that same thing. For example, if a pizza has 8 slices and I eat two of them, I would have eaten 25% of the pizza. This is because 2 slices divided by 8 total slices is 1/4 or 0.25 which, when multiplied by 100, is 25%. The basic formula is as follows:

[Smaller Number or Portion]/[Total Number or Whole Portion] x 100 = Percent

With this knowledge, you will be able to understand and calculate a variety of types of percentages.

## How to Calculate Percent Off (Discounts)

Retail establishments commonly discount items using a percentage off when they are having a sale. The goal of marking things down in price is often to entice a purchase and/or to move older inventory more quickly. Whatever the case may be, retailers don't often put the actual discounted price on the tag. Emphasis is usually place on the "percent off" and unless you calculate the discount yourself, you may not know the price of the item until you reach the checkout counter. By calculating the discount, and the resulting price, you can learn what you are going to pay for the item before you head to the register.

To calculate the percentage off discount use the following formula:

[Original Price in \$] X [Percent Off in Decimal Form] = [Discount in \$]

Next, to calculate the final price, simply subtract the discount from the original price:

[Original Price in \$] - [Discount in \$] = [Final Price in \$]

Another, simpler way to arrive at the final price more quickly is to subtract the percentage off (in decimal form) from 1 and then multiply the result by the original price. This effectively does the same thing as the above two equations do.

[Original Price in \$] X [1 - Percent Off in Decimal Form] = [Final Price in \$]

Here is an example on how to calculate what the discount and final price is:

Problem:

In a local clothing store you see a shirt that is marked down by 39%. The original price is \$26.95. How much money will you save as compared to paying full price and what will the price be at the register (before taxes)?

Solution:

The discount offered can be found as follows: [\$26.95] X [0.39] = \$10.51. This means that with a 39% discount, you should be saving \$10.51 off of the original price.

Next, to find the final price of item we simply subtract the discount from the original price: \$26.95 - \$10.51 = \$16.44. Therefore, the final price of the item is \$16.44.

For verification, we can use the second method to verify the final price of the item based on the discount percentage: \$26.95 x [1 - 0.39] = \$26.95 x 0.61 = \$16.44

Note that the ratio of the original price to the final price is equal to 1 minus the discount rate (or the percentage of the price that you are actually paying): \$16.44/\$26.95 = 0.61 = 61%.

The ratio of the discount to the original price should equal the discount rate (or the percentage of the price that you are not paying): \$10.51/\$26.95 = 0.39 = 39%.

## How to Calculate Percent Increase

The percentage increase is often used to describe the growth of something over time or between two dates or events. Growth could include anything from populations (example: The Phoenix-Metropolitan area grew 12.98% between 2010 and 2017), to savings accounts and investments (example: my account balance grew 15% in just 36 months), and even things like the number of customers a business sees (example: during the three hottest months of summer Hank's AC Repair Company sees a 45% increase in service calls). As you can see, understanding the percentage increase can be important.

To calculate the percentage increase, you must first calculate the difference (increase) in the two numbers:

[Final Number] - [Original Number] = [Difference (Increase)]

Next, divide the increase by the original number and then multiply it by 100:

{[Difference (increase)]/ [Original Number]} x 100 = [Percent Increase]

When combine into one equation, the formula for percent increase looks like this:

Here's an example of how to calculate the percent increase of something:

Problem:

Last year at this time you had \$10,567.00 invested in high yield, moderate risk mutual funds. Today, your mutual fund is valued at \$16,889.00. What is the percentage increase in the value of the mutual funds over the one year period?

Solution:

First we can easily calculate the difference between the two numbers: \$16,889.00 - \$10,567.00 = \$6,322.

Next, divide this number by the original number and multiply by 100 to calculate the percentage increase: (\$6,322.00 / \$10,567.00) x 100 = 59.83%

Therefore, these mutual funds increased in value by 59.83% in one year.

## How to Calculate Percent Decrease

The percentage decrease is the opposite of the percentage increase and is often used to calculate a variety of things that shrink over time. A great example of calculating the percent decrease is when you want to calculate your percentage of weight loss over time. Other examples of decreasing values could be anything from declining populations (example: the state of Illinois saw a 0.71% decrease in population from 2015 to 2018), to a reduction in product sales (example: sales figures at Mr. Pop's store are down 9.2% compared to last month), and such things like financial losses (example: my investment portfolio decreased by 3.34% in value during the 2008 recession). There are innumerable examples where the percentage decrease could be helpful.

To calculate the percentage decrease first you must calculate the difference (decrease) in the two numbers:

[Final Number] - [Original Number] = [Difference (Decrease)]

Next, divide the decrease by the original number and then multiply it by 100:

{[Difference (Decrease)]/ [Original Number]} x 100 = [Percent Decrease]

In one equation, the formula for percent decrease looks like this:

Here's an example of how to calculate the percent decrease of something:

Problem:

Two years ago our library had an inventory of 3,890,567 books. Due to budget cuts and people failing to return books to the library, the current inventory is now 3,651,119 books. What is the percent decrease in the library's book inventory?

Solution:

First calculate the difference between the two numbers: 3,890,567 - 3,651,119 = 239,448

Next, divide this number by the original number and multiply by 100 to calculate the percentage increase: (239,448 / 3,890,567) x 100 = 6.15%

Therefore, the library's inventory of books decreased by 6.15% in two years.

Here's another example showing how to calculate the percentage of weight lost for a given situation:

Problem:

Sixteen months ago Mike weighed 346 lbs. He decided to lose weight by avoiding sugars, maintaining a calorie deficit, and walking daily for at least 2 hours. He currently weighs 211 lbs. What is Mike's percentage of weight loss over the 16 month period?

Solution:

First calculate the difference between the two numbers: 346 - 211 = 135

Next, divide this number by the original number and multiply by 100 to calculate the percentage increase: (135/ 346) x 100 = 39%

Therefore, Mike lost 39% of his body weight with 16 months of hard work and dedication.

## How to Calculate Percent Error

The percentage error represents how close a particulate value is to that of a true or known value. The percentage error can often be thought of as an accuracy indicator. Calculating this percentage is useful in manufacturing processes (quality control), engineering evaluations and design, as well as statistics.

When calculating the percent error, you must have a known or desired value as well as a measured or observed value for which to compare it too. To calculate the percent error, you must first calculate the difference in the desired value and the measured (observed) value:

[Measured or Observed] - [Actual or Desired Value] = [Difference]

Next, divide the difference by the actual or desired value and then multiply it by 100:

{[Difference]/ [Actual or Desired Value]} x 100 = [Percent Error]

Note that percent error could be positive or negative.

When combined into one equation, the formula for percent error looks like this:

Here's an example of how to calculate the percent error of something:

Problem:

A factory that produces Kevlar bird leashes makes about 4,000 of them per day. The customer's specifications say that each Kevlar bird leash should be 212 centimeters long from end to end. A bird leash selected at random from the day's production is measured at 210.6 centimeters. What is the percent error for this sample?

Solution:

First calculate the difference between the two numbers: 212 - 210.6 = 1.4

Next, divide this number by the desired number (the specification) and multiply it by 100 to calculate the percentage error: (1.4 / 212) x 100 = 0.66%

Therefore, the length of the sampled Kevlar bird leash is out of specification by 0.66%

## How to Calculate Percent Difference

The percentage difference is used to account for the relative difference between two values representing similar things while also taking into account their relative proportions. To properly account for their relative difference, an average value between the two numbers is used to account for the fact that the numbers being compared have no common reference point.

The percentage difference cannot be used to measure the percentage change over time or the percentage change going from one number to another number. Examples where percentage difference can be used include price comparison between similar products (example: the percentage difference in price between shirt A and shirt B is 11.5%) or comparing two similar things (example: the percentage difference between Shaq's height and my height is 22.2%).

To calculate the percent difference, you must first calculate the absolute difference between the numbers being compared. The resulting number is always positive:

|[Number 1] - [Number 2]| = [Difference]

Next, calculate the average of the two numbers:

{[Number 1] + [Number 2]} / 2 = [Average]

Next, divide the calculated difference by the average of the two numbers and then multiply it by 100:

{[Difference]/ [Average]} x 100 = [Percent Difference]

If you combine this into one equation the formula for percent difference looks like this:

Here's an example of how to calculate the percent difference of something:

Problem:

John wears a size 12 sneaker and his best friend wears a size 16. What is the percentage difference between their shoe sizes?

Solution:

First, calculate the absolute difference between the two numbers: 12-16 = -4, and dropping the negative sign, = 4

Next, calculate the average shoe size between the two people: (12+16)/2 = 28/2 = 14

Next, divide the absolute difference by the average and multiply it by 100 to calculate the percentage difference: (4 / 14) x 100 = 28.57%

Therefore, the percentage difference in shoe size between the two men is 28.57%. Note, this is not the same as calculating the percentage larger or smaller one shoe is over the other.

## How to Calculate Percent Yield

The percentage yield is the calculation that shows the ratio between how much something yields or produces as compared to how much it should produce (the theoretical yield). Percentage yield is often used in chemistry and is an indicator of the efficiency of a chemical reaction.

To calculate the percent yield you simply divide the actual (measured) yield by the theoretical yield and then multiply it by 100:

{[Actual Yield]/ [Theoretical Yield]} x 100 = [Percent Yield]

Here's an example of how to calculate the percent yield of a chemical reaction:

Problem:

A sample of pure fat is metabolized into water and carbon dioxide. During the experiment, about 30 grams of water is produced. Given the amount of fat being metabolized, we know that the theoretical yield of water from this reaction should be 33grams. What is the percentage yield?

Solution:

First divide the actual yield by the theoretical yield and then multiple it by 100: (30/33) x 100 = 90.9%

Therefore, the percentage yield of water for this experiment was 90.9%.

## How to Calculate the Percent Tip at Restaurants

Calculating the tip at a restaurant doesn't have to be challenging or stressful. After you determine what percentage you want to offer as a tip, you just multiply that number by the bill amount to determine the tip amount:

[Desired percent tip %)] X [Bill total] = [Tip Total]

You can also just assume a percentage tip and go straight for the total: First, convert the percentage tip to a decimal point, add 1, and then multiply that by the price to get the final price:

{[Desired percent tip (decimal)]+1} X [Bill total] = [Final Total]

One thing that I often do to keep tipping simple is to first calculate what 10% of the bill is. This is easy because you simply take the total bill and move the decimal point one position to the left. The resulting number will be 10% of the total. Now, if want to leave a 20% tip, I double the number I just calculated and then add it to the total. For 30% I multiply that number by 3 instead. In this way, I can quickly calculate a 20% tip in my head.

Here's an example for how to calculate tips:

Problem:

You and a group of friends just ate a delicious meal at a local restaurant. The waitress brings the bill and the total is \$123.89. The waitress was great and she's a struggling college student so you want to give her a 40% tip. What is the tip amount and the final bill?

Solution:

To calculate the tip multiply 40% by the bill: \$123.89 x 40% = \$49.56.

The final bill is the sum of the two numbers: \$123.89 + \$49.56 = \$173.45

...or using the quick mental method:

Using the quick mental method first calculate 10% of the bill by moving the decimal point one position to the left: \$123.89 becomes \$12.39.

Next, multiply this by 4 to get to 40%: \$12.39 x 4 = \$49.56

Now add this tip to get to the original bill to get the final total: \$123.89 + \$49.56 = \$173.45

To make things even easier when calculating tips in your mind be sure to round up to the nearest whole number.