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In daily life, percents are often related to certain numbers. It is useful to know how to calculate the percent of such numbers to understand the actual value represented by the percent. Rounding numbers also comes in handy when only an approximation is needed. In some cases, a trick for quicker calculations even exists! This lesson covers it all.

### Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

## Connection Between Percents and Numbers

Take a look at a container with a total capacity of liters. Move the slider to fill it with a liquid. See how the number of liters of the liquid relates to the percent of the occupied container. ## How to Find a Percent of a Number?

Consider a product of a number and a fraction. The numbers have different forms. How can they be multiplied?
In this case, the number is multiplied by the numerator of the fraction. The denominator remains the same.
The last step is to evaluate the quotient.
These calculations are often used when finding the percent of a number. For example, a student scores out of questions and wants to know their number of correct answers. Calculating of can help with this.

## Finding the Percent of a Number

Percents are commonly used in real-life situations. They represent a relative value of something, not the actual value. The percent of a number should be calculated to find the actual value. Consider the following example.
This expression can be evaluated in three steps.
1
Rewrite the Percent as a Fraction
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First, recall that a percent is the ratio of a number to This fact allows to write a percent as a fraction.
Apply this formula to Then, simplify the fraction as much as possible.
Therefore, is equivalent to Use this piece of information to rewrite the task.
2
Write a Product Expression
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Recall that the word of is often used to indicate a product in math expressions. This means that of is the same as the product of and
Multiplying a fraction by a number requires multiplying the numerator of the fraction by the number. The denominator remains the same.
3
Evaluate the Expression
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Finally, evaluate the expression. Start by performing multiplication in the numerator. Then, calculate the quotient.
Therefore, of is

### Extra

Alternative Method
Another method used to calculate the percent of a number is to rewrite the percent as a decimal number. This is done by dividing the percent by which leads to moving the decimal point two places to the left. As shown, is equivalent to The next step is to multiply the decimal number by in order to calculate of
The calculations show that of is The same result was found although the process used to get it was different.

## Counting Fish in a Pond

Kevin is thrilled to spend time at a pond near his house. The local Fish and Game agency recently did a study finding that there are fish in the pond. Kevin especially enjoys seeing yellow fish since they are tough to spot. External credits: @upklyak
a If of the fish are yellow fish, how many yellow fish are in the pond?
b Blue fish represent of all the fish. How many blue fish are there?

### Hint

a Calculate of Start by rewriting the percent as a fraction using the definition of a percent.
b Calculate of Use the definition of a percent to rewrite as a fraction. Then, simplify the fraction and multiply by

### Solution

a There are fish in the pond. Yellow fish represent of that number. The exact number of yellow fish can be found by calculating of
Start by rewriting the percent as a fraction. Use the fact that a percent is the ratio of a number and
Then, simplify the fraction as much as possible.
Rewrite the product of the percent and the number by using this fraction equivalent to the percent.
Multiply the numerator by to calculate this product. The denominator remains the same. Therefore, of is This means that there are yellow fish in the pond.
b This time, of needs to be calculated to find the number of blue fish. Recall that of indicates multiplication.
To calculate this product, start by rewriting as a fraction with the numerator and the denominator Then, simplify the fraction.
The next step is to multiply the fraction by Therefore, there are blue fish in the pond.

## Comparing Fish Sizes

After enjoying his time at the pond, Kevin heads home and opens his favorite website FishFrenzy.com. He is fascinated by the vastness of the underwater world. There are quite small fish, like clownfish, which can grow to just centimeters long. External credits: @brgfx, @brgfx, @brgfx
There are also quite large fish like Carp. Their length can reach of that of clownfish. Dolphins are even bigger. They can be as long as of carp's length.
a What length can a carp grow to be?
b What length can a dolphin grow to be?

### Hint

a Calculate of Use the definition of a percent to rewrite it as a fraction.
b Find of the carp's length. Start by rewriting the percent as a fraction in simplest form.

### Solution

a It is given that clownfish can reach a length of centimeters. A carp is of that length. This means that the carp's length is given by of Recall that of indicates multiplication.
First, rewrite the percent as a fraction with the numerator and the denominator Then, simplify the fraction as much as possible.
This fraction is equivalent to
Next, evaluate this product by multiplying the numerator by and leave the denominator as is. The calculations show that of is This means that a carp can grow to a length of up to centimeters.
b Now that the length of a carp is known, the length of a dolphin can be found. Dolphins are said to be as long as of a carp's length. It was found that a carp is centimeters long.
Start by rewriting the percent as a fraction. Remember that a percent is the ratio of a number to Again, it is more convenient to work with smaller numbers, so do not forget to simplify the fraction.
Next, multiply the fraction by by multiplying its numerator by
Therefore, a dolphin can be centimeters long. This is more than meters, nearly feet!

## Estimating the Percent of a Number

It was shown that finding the percent of a number involves quite a few steps. The good news is that sometimes, the exact result is not necessary and an estimation is absolutely enough. In those cases, rounding the numbers means quicker and easier calculations. Consider an example.
These numbers are not very convenient to work with. It is likely that a calculator would be needed. Instead, simplify the process and round each number.
These new rounded numbers can be used to find an estimated result of of Write a new task.
Start by rewriting the percent as a fraction. Then, simplify the fraction and evaluate the expression.
Rewrite
Evaluate
It was determined that of is about The exact result is so the estimation is pretty close.
Make sure to use this method when there is a need to quickly eyeball or estimate the percent of some number.

## Traveling Across the Pacific Ocean

Loving aquatic life as much as he does, Kevin dreams of sailing across the Pacific Ocean someday. He might even get to witness a sea lion belching or a whale sprouting! Kevin learns that a trip from California to Hawaii on a sailboat would take him days. Additionally, California to Australia is of the distance from California to Hawaii. Kevin now wants to estimate the time it would take to sail from California to Australia.
a Use the method of rounding the given numbers and estimate the travel time from California to Australia.
b Find the exact number of days of the trip and compare it with the estimated result.

### Hint

a The sailing time from California to Australia can be found by calculating of Round and to numbers that end with a or for simpler calculations.
b Calculate of by first rewriting the percent as a fraction and then multiplying its numerator by

### Solution

a Two key points are given. Firstly, it takes days to sail from California to Hawaii. Secondly, the distance between California and Australia is of the distance between California and Hawaii. This means that of represents the time it takes to sail from California to Australia.
It is helpful to estimate this expression by rounding the given numbers.
Next, write of as an expression using the rounded numbers and appropriate operation. Recall that of often indicates multiplication.
Now, calculate this new expression. Start by rewriting the percent as a fraction. Use the fact that a percent is the ratio of a number to Then, simplify the fraction as much as possible.
The next step is to calculate the product of this fraction and
Multiply the numerator of the fraction by and leave the denominator as is. Be sure to check if the fraction can be reduced before making any further calculations. The calculations show that of is This result means that it will take Kevin about days to sail from California to Hawaii.
b From Part A, the exact sailing time from California to Australia is given by of In other words, the product of and should be evaluated.
Repeat the used procedure one more time. First, rewrite the percent as a fraction in the simplest form. Next, multiply by the numerator to find the product of the fraction and Then, reduce the fraction as much as possible and calculate the product and the quotient.
The exact time of sailing from California to Australia based on the given numbers is days. This number is pretty close to the estimated result of days. It seems like sailing across the Pacific ocean will be quite a long and adventurous trip!

## Finding Percents of Numbers

Calculate the given percent of a number. Round the answer to one decimal place if needed. ## A Secret for Faster Calculations

There is trick that can make things much simpler when calculating the percent of a number. Ready for it? External credits: storyset on Freepik
Keep this top secret method of calculations at hand. See it in action through the following example.
There are typically quite a few calculations to evaluate of The good news is that interchanging the numbers makes the task much simpler.
The fraction equivalent of is This means that calculating of comes down to multiplying by That is the same as dividing by
The calculations show that of is This trick can be double checked by calculating of using the known method.
Evaluate
The result is, indeed, This proves that interchanging the numbers creates equivalent expressions.
Remember to use this trick when calculating the percent of a number, which in percent form corresponds to a simple known fraction. Here are a few examples.
Number Percent Fraction
This way the percent of a number can be calculated using mental math. It is like having a super power!

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