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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.
Explore
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.
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Discussion
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.
Method
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
expand_more 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
expand_more Recall that the word
Multiplying a fraction by a number requires multiplying the numerator of the fraction by the number. The denominator remains the same.
ofis often used to indicate a product in math expressions. This means that of is the same as the product of and
3
Evaluate the Expression
expand_more Finally, evaluate the expression. Start by performing multiplication in the numerator. Then, calculate the quotient.
Therefore, of is
Extra
Alternative MethodAnother 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.
Example
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.
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a If of the fish are yellow fish, how many yellow fish are in the pond?
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b Blue fish represent of all the fish. How many blue fish are there?
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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
b This time, of needs to be calculated to find the number of blue fish. Recall that
ofindicates multiplication.
Example
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.
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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?
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b What length can a dolphin grow to be?
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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
ofindicates multiplication.
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.
Discussion
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.
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.
▼
Rewrite
▼
Evaluate
Example
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!
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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.
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b Find the exact number of days of the trip and compare it with the estimated result.
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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.
This means that of represents the time it takes to sail from California to Australia. ofoften indicates multiplication.
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.
Pop Quiz
Finding Percents of Numbers
Closure
A Secret for Faster Calculations
There is trick that can make things much simpler when calculating the percent of a number. Ready for it?
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Keep this top secret method of calculations at hand. See it in action through the following example.
| Number | Percent | Fraction |
|---|---|---|
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