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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

Take a look at a container with a total capacity of $60$ 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.

Discussion

Consider a product of a number and a fraction. The numbers have different forms. How can they be multiplied?
## Finding the Percent of a Number

$5⋅73 $

In this case, the number is multiplied by the numerator of the fraction. The denominator remains the same.
$5⋅73 5⋅73 =75⋅3 ⇓=715 $

The last step is to evaluate the quotient.
$715 ≈2.14 $

These calculations are often used when finding the percent of a number. For example, a student scores $84%$ out of $50$ questions and wants to know their number of correct answers. Calculating $84%$ of $50$ can help with this. Method

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.
*expand_more*
*expand_more*
*expand_more*
### Extra

Alternative Method

$40%of86 $

This expression can be evaluated in three steps.
1

Rewrite the Percent as a Fraction

First, recall that a percent is the ratio of a number to $100.$ This fact allows to write a percent as a fraction.
Therefore, $40%$ is equivalent to $52 .$ Use this piece of information to rewrite the task.

$n%=100n $

Apply this formula to $40%.$ Then, simplify the fraction as much as possible.
$40%$

PercentToFrac

$a%=100a $

$10040 $

ReduceFrac

$ba =b/20a/20 $

$100÷2040÷20 $

CalcQuot

Calculate quotient

$52 $

$40%of86⇕52 of86 $

2

Write a Product Expression

Recall that the word

ofis often used to indicate a product in math expressions. This means that $52 $ of $86$ is the same as the product of $52 $ and $86.$

$52 of86⇕52 ×86 $

Multiplying a fraction by a number requires multiplying the numerator of the fraction by the number. The denominator remains the same.
$52 ⋅86=52⋅86 $

3

Evaluate the Expression

Finally, evaluate the expression. Start by performing multiplication in the numerator. Then, calculate the quotient.
Therefore, $40%$ of $86$ is $34.4.$

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 $100$ which leads to moving the decimal point two places to the left.

As shown, $40%$ is equivalent to $0.4.$ The next step is to multiply the decimal number by $86$ in order to calculate $40%$ of $86.$

$0.4⋅86=34.4 $

The calculations show that $40%$ of $86$ is $34.4.$ The same result was found although the process used to get it was different.
Example

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 $850$ fish in the pond. Kevin especially enjoys seeing yellow fish since they are tough to spot.
### Hint

### Solution

Rewrite the product of the percent and the number by using this fraction equivalent to the percent.
Therefore, $18%$ of $850$ is $153.$ This means that there are $153$ yellow fish in the pond.
The next step is to multiply the fraction by $850.$
Therefore, there are $306$ blue fish in the pond.

External credits: @upklyak

a If $18%$ of the fish are yellow fish, how many yellow fish are in the pond?

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b Blue fish represent $36%$ of all the fish. How many blue fish are there?

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a Calculate $18%$ of $850.$ Start by rewriting the percent as a fraction using the definition of a percent.

b Calculate $36%$ of $850.$ Use the definition of a percent to rewrite $36%$ as a fraction. Then, simplify the fraction and multiply by $850.$

a There are $850$ fish in the pond. Yellow fish represent $18%$ of that number. The exact number of yellow fish can be found by calculating $18%$ of $850.$

$18%18% of850⇕×850 $

Start by rewriting the percent as a fraction. Use the fact that a percent is the ratio of a number and $100.$
$n%=100n $

Then, simplify the fraction as much as possible.
$18%$

PercentToFrac

$a%=100a $

$10018 $

ReduceFrac

$ba =b/2a/2 $

$100÷218÷2 $

CalcQuot

Calculate quotient

$509 $

$18%509 ×850⇕×850 $

Multiply the numerator $9$ by $850$ to calculate this product. The denominator remains the same.
$509 ⋅850$

MoveRightFacToNum

$ca ⋅b=ca⋅b $

$509⋅850 $

ReduceFrac

$ba =b/50a/50 $

$19⋅17 $

Multiply

Multiply

$1153 $

DivByOne

$1a =a$

$153$

b This time, $36%$ of $850$ needs to be calculated to find the number of blue fish. Recall that

ofindicates multiplication.

$36%×850 $

To calculate this product, start by rewriting $36%$ as a fraction with the numerator $36$ and the denominator $100.$ Then, simplify the fraction.
$36%$

PercentToFrac

$a%=100a $

$10036 $

ReduceFrac

$ba =b/4a/4 $

$100÷436÷4 $

CalcQuot

Calculate quotient

$259 $

$259 ⋅850$

MoveRightFacToNum

$ca ⋅b=ca⋅b $

$259⋅850 $

ReduceFrac

$ba =b/25a/25 $

$19⋅34 $

Multiply

Multiply

$1306 $

DivByOne

$1a =a$

$306$

Example

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 $15$ centimeters long.
There are also quite large fish like Carp. Their length can reach $420%$ of that of clownfish. Dolphins are even bigger. They can be as long as $260%$ of carp's length.
### Hint

### Solution

This fraction is equivalent to $420%.$
The calculations show that $420%$ of $15$ is $63.$ This means that a carp can grow to a length of up to $63$ centimeters.
Next, multiply the fraction by $63$ by multiplying its numerator by $63.$
Therefore, a dolphin can be $163.8$ centimeters long. This is more than $1.5$ meters, nearly $5$ feet!

External credits: @brgfx

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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a Calculate $420%$ of $15.$ Use the definition of a percent to rewrite it as a fraction.

b Find $260%$ of the carp's length. Start by rewriting the percent as a fraction in simplest form.

a It is given that clownfish can reach a length of $15$ centimeters. A carp is $420%$ of that length. This means that the carp's length is given by $420%$ of $15.$ Recall that

ofindicates multiplication.

$420%420% of15⇕×15 $

First, rewrite the percent as a fraction with the numerator $420$ and the denominator $100.$ Then, simplify the fraction as much as possible.
$420%$

PercentToFrac

$a%=100a $

$100420 $

ReduceFrac

$ba =b/20a/20 $

$100÷20420÷20 $

CalcQuot

Calculate quotient

$521 $

$420%521 ×15⇕×15 $

Next, evaluate this product by multiplying the numerator $21$ by $15$ and leave the denominator as is.
$521 ⋅15$

MoveRightFacToNum

$ca ⋅b=ca⋅b $

$521⋅15 $

ReduceFrac

$ba =b/5a/5 $

$121⋅3 $

Multiply

Multiply

$163 $

DivByOne

$1a =a$

$63$

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 $260%$ of a carp's length. It was found that a carp is $63$ centimeters long.

$260%×63 $

Start by rewriting the percent as a fraction. Remember that a percent is the ratio of a number to $100.$ Again, it is more convenient to work with smaller numbers, so do not forget to simplify the fraction.
$260%$

PercentToFrac

$a%=100a $

$100260 $

ReduceFrac

$ba =b/20a/20 $

$100÷20260÷20 $

CalcQuot

Calculate quotient

$513 $

$513 ⋅63$

MoveRightFacToNum

$ca ⋅b=ca⋅b $

$513⋅63 $

Multiply

Multiply

$5819 $

CalcQuot

Calculate quotient

$163.8$

Discussion

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.
It was determined that $17%$ of $39$ is about $6.$ The exact result is $6.63,$ so the estimation is pretty close.

$17%of39 $

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.
$17is closer to15than to20⇓17%≈15% 39is closer to40than to35⇓39≈40 $

These new rounded numbers can be used to find an estimated result of $17%$ of $40.$ Write a new task.
$15%15% of40⇕×40 $

Start by rewriting the percent as a fraction. Then, simplify the fraction and evaluate the expression.
$15%×40$

▼

Rewrite

PercentToFrac

$a%=100a $

$10015 ×40$

ReduceFrac

$ba =b/5a/5 $

$100÷515÷5 ×40$

CalcQuot

Calculate quotient

$203 ×40$

▼

Evaluate

MoveRightFacToNum

$ca ⋅b=ca⋅b $

$203×40 $

ReduceFrac

$ba =b/20a/20 $

$13×2 $

Multiply

Multiply

$16 $

DivByOne

$1a =a$

$6$

$17%×3917%×39 =6.63≈6 $

Make sure to use this method when there is a need to quickly eyeball or estimate the percent of some number.Example

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!
### Solution

The next step is to calculate the product of this fraction and $20.$
The calculations show that $260%$ of $20$ is $52.$ This result means that it will take Kevin about $52$ days to sail from California to Hawaii.
The exact time of sailing from California to Australia based on the given numbers is $47.16$ days. This number is pretty close to the estimated result of $52$ days. It seems like sailing across the Pacific ocean will be quite a long and adventurous trip!

Kevin learns that a trip from California to Hawaii on a sailboat would take him $18$ days. Additionally, California to Australia is $262%$ 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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a Two key points are given. Firstly, it takes $18$ days to sail from California to Hawaii. Secondly, the distance between California and Australia is $262%$ of the distance between California and Hawaii. This means that $262%$ of $18$ represents the time it takes to sail from California to Australia.

$262%of18 $

It is helpful to estimate this expression by rounding the given numbers.
$262is closer to260than to265⇓262%≈260% 18is closer to20than to15⇓18≈20 $

Next, write $262%$ of $18$ as an expression using the rounded numbers and appropriate operation. Recall that ofoften indicates multiplication.

$260%260% of20⇕×20 $

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 $100.$ Then, simplify the fraction as much as possible.
$260%$

PercentToFrac

$a%=100a $

$100260 $

ReduceFrac

$ba =b/20a/20 $

$100÷20260÷20 $

CalcQuot

Calculate quotient

$513 $

$260%513 ×20⇕×20 $

Multiply the numerator of the fraction by $20$ and leave the denominator as is. Be sure to check if the fraction can be reduced before making any further calculations.
$513 ⋅20$

MoveRightFacToNum

$ca ⋅b=ca⋅b $

$513⋅20 $

ReduceFrac

$ba =b/5a/5 $

$113⋅4 $

Multiply

Multiply

$152 $

DivByOne

$1a =a$

$52$

b From Part A, the exact sailing time from California to Australia is given by $262%$ of $18.$ In other words, the product of $262%$ and $18$ should be evaluated.

$262%262% of18⇕×18 $

Repeat the used procedure one more time. First, rewrite the percent as a fraction in the simplest form.
Next, multiply $18$ by the numerator $131$ to find the product of the fraction and $18.$ Then, reduce the fraction as much as possible and calculate the product and the quotient.
$50131 ×18$

MoveRightFacToNum

$ca ⋅b=ca⋅b $

$50131⋅18 $

ReduceFrac

$ba =b/2a/2 $

$25131⋅9 $

Multiply

Multiply

$251179 $

CalcQuot

Calculate quotient

$47.16$

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