Learn Estimation Techniques for Calculating the Percent of a Number

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.

Percent
Rate
Converting a Percent Into a Fraction
Converting a Fraction Into a Percent

Explore

Connection Between Percents and Numbers

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

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?

5 \cdot \frac{3}{7}

In this case, the number is multiplied by the numerator of the fraction. The denominator remains the same.

5 \cdot \frac{3}{7} 5 \cdot \frac{3}{7} = \frac{5 \cdot 3}{7} ⇓ = \frac{1 5}{7}

The last step is to evaluate the quotient.

\frac{1 5}{7} \approx 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 %

50

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.

40 % of 86

This expression can be evaluated in three steps.

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.

n % = \frac{n}{1 0 0}

Apply this formula to

40 % .

Then, simplify the fraction as much as possible.

40 %

PercentToFrac

$a % = \frac{a}{1 0 0}$

\frac{4 0}{1 0 0}

ReduceFrac

$\frac{a}{b} = \frac{a / 2 0}{b / 2 0}$

\frac{4 0 \div 2 0}{1 0 0 \div 2 0}

CalcQuot

Calculate quotient

\frac{2}{5}

Therefore,

40 %

is equivalent to

\frac{2}{5} .

Use this piece of information to rewrite the task.

40 % of 86 ⇕ \frac{2}{5} of 86

Write a Product Expression

Recall that the word of is often used to indicate a product in math expressions. This means that

\frac{2}{5}

86

is the same as the product of

\frac{2}{5}

and

86 .

\frac{2}{5} of 86 ⇕ \frac{2}{5} \times 86

Multiplying a fraction by a number requires multiplying the numerator of the fraction by the number. The denominator remains the same.

\frac{2}{5} \cdot 86 = \frac{2 \cdot 8 6}{5}

Evaluate the Expression

Finally, evaluate the expression. Start by performing multiplication in the numerator. Then, calculate the quotient.

\frac{2 \cdot 8 6}{5}

Multiply

\frac{1 7 2}{5}

CalcQuot

Calculate quotient

34.4

Therefore,

40 %

86

34.4 .

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

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 %

86 .

0.4 \cdot 86 = 34.4

The calculations show that

40 %

86

34.4 .

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

850

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

External credits: @upklyak

a If

18 %

of the fish are yellow fish, how many yellow fish are in the pond?

b Blue fish represent

36 %

of all the fish. How many blue fish are there?

Hint

a Calculate

18 %

850 .

Start by rewriting the percent as a fraction using the definition of a percent.

b Calculate

36 %

850 .

Use the definition of a percent to rewrite

36 %

as a fraction. Then, simplify the fraction and multiply by

850 .

Solution

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 %

850 .

18 % 18 % of 850 ⇕ \times 850

Start by rewriting the percent as a fraction. Use the fact that a percent is the ratio of a number and

100 .

n % = \frac{n}{1 0 0}

Then, simplify the fraction as much as possible.

18 %

PercentToFrac

$a % = \frac{a}{1 0 0}$

\frac{1 8}{1 0 0}

ReduceFrac

$\frac{a}{b} = \frac{a / 2}{b / 2}$

\frac{1 8 \div 2}{1 0 0 \div 2}

CalcQuot

Calculate quotient

\frac{9}{5 0}

Rewrite the product of the percent and the number by using this fraction equivalent to the percent.

18 % \frac{9}{5 0} \times 850 ⇕ \times 850

Multiply the numerator

9

850

to calculate this product. The denominator remains the same.

\frac{9}{5 0} \cdot 850

MoveRightFacToNum

$\frac{a}{c} \cdot b = \frac{a \cdot b}{c}$

\frac{9 \cdot 8 5 0}{5 0}

ReduceFrac

$\frac{a}{b} = \frac{a / 5 0}{b / 5 0}$

\frac{9 \cdot 1 7}{1}

Multiply

\frac{1 5 3}{1}

DivByOne

$\frac{a}{1} = a$

153

Therefore,

18 %

850

153 .

This means that there are

153

yellow fish in the pond.

b This time,

36 %

850

needs to be calculated to find the number of blue fish. Recall that of indicates multiplication.

36 % \times 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

$a % = \frac{a}{1 0 0}$

\frac{3 6}{1 0 0}

ReduceFrac

$\frac{a}{b} = \frac{a / 4}{b / 4}$

\frac{3 6 \div 4}{1 0 0 \div 4}

CalcQuot

Calculate quotient

\frac{9}{2 5}

The next step is to multiply the fraction by

850 .

\frac{9}{2 5} \cdot 850

MoveRightFacToNum

$\frac{a}{c} \cdot b = \frac{a \cdot b}{c}$

\frac{9 \cdot 8 5 0}{2 5}

ReduceFrac

$\frac{a}{b} = \frac{a / 2 5}{b / 2 5}$

\frac{9 \cdot 3 4}{1}

Multiply

\frac{3 0 6}{1}

DivByOne

$\frac{a}{1} = a$

306

Therefore, there are

306

blue fish in the pond.

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

15

centimeters long.

External credits: @brgfx

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.

a What length can a carp grow to be?

b What length can a dolphin grow to be?

Hint

a Calculate

420 %

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.

Solution

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 %

15 .

Recall that of indicates multiplication.

420 % 420 % of 15 ⇕ \times 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

$a % = \frac{a}{1 0 0}$

\frac{4 2 0}{1 0 0}

ReduceFrac

$\frac{a}{b} = \frac{a / 2 0}{b / 2 0}$

\frac{4 2 0 \div 2 0}{1 0 0 \div 2 0}

CalcQuot

Calculate quotient

\frac{2 1}{5}

This fraction is equivalent to

420 % .

420 % \frac{2 1}{5} \times 15 ⇕ \times 15

Next, evaluate this product by multiplying the numerator

21

15

and leave the denominator as is.

\frac{2 1}{5} \cdot 15

MoveRightFacToNum

$\frac{a}{c} \cdot b = \frac{a \cdot b}{c}$

\frac{2 1 \cdot 1 5}{5}

ReduceFrac

$\frac{a}{b} = \frac{a / 5}{b / 5}$

\frac{2 1 \cdot 3}{1}

Multiply

\frac{6 3}{1}

DivByOne

$\frac{a}{1} = a$

63

The calculations show that

420 %

15

63 .

This means that a carp can grow to a length of up to

63

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

260 %

of a carp's length. It was found that a carp is

63

centimeters long.

260 % \times 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

$a % = \frac{a}{1 0 0}$

\frac{2 6 0}{1 0 0}

ReduceFrac

$\frac{a}{b} = \frac{a / 2 0}{b / 2 0}$

\frac{2 6 0 \div 2 0}{1 0 0 \div 2 0}

CalcQuot

Calculate quotient

\frac{1 3}{5}

Next, multiply the fraction by

63

by multiplying its numerator by

63 .

\frac{1 3}{5} \cdot 63

MoveRightFacToNum

$\frac{a}{c} \cdot b = \frac{a \cdot b}{c}$

\frac{1 3 \cdot 6 3}{5}

Multiply

\frac{8 1 9}{5}

CalcQuot

Calculate quotient

163.8

Therefore, a dolphin can be

163.8

centimeters long. This is more than

1.5

meters, nearly

5

feet!

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.

17 % of 39

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.

17 is closer to 15 than to 20 ⇓ 17 % \approx 15 % 39 is closer to 40 than to 35 ⇓ 39 \approx 40

These new rounded numbers can be used to find an estimated result of

17 %

40 .

Write a new task.

15 % 15 % of 40 ⇕ \times 40

Start by rewriting the percent as a fraction. Then, simplify the fraction and evaluate the expression.

15 % \times 40

▼

Rewrite

PercentToFrac

$a % = \frac{a}{1 0 0}$

\frac{1 5}{1 0 0} \times 40

ReduceFrac

$\frac{a}{b} = \frac{a / 5}{b / 5}$

\frac{1 5 \div 5}{1 0 0 \div 5} \times 40

CalcQuot

Calculate quotient

\frac{3}{2 0} \times 40

▼

Evaluate

MoveRightFacToNum

$\frac{a}{c} \cdot b = \frac{a \cdot b}{c}$

\frac{3 \times 4 0}{2 0}

ReduceFrac

$\frac{a}{b} = \frac{a / 2 0}{b / 2 0}$

\frac{3 \times 2}{1}

Multiply

\frac{6}{1}

DivByOne

$\frac{a}{1} = a$

6

It was determined that

17 %

39

is about

6 .

The exact result is

6.63,

so the estimation is pretty close.

17 % \times 39 17 % \times 39 = 6.63 \approx 6

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

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!

A sailing boat, an island with a seal and a sprouting whale

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.

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

262 %

18 .

Round

262

and

18

to numbers that end with a

0

5

for simpler calculations.

b Calculate

262 %

18

by first rewriting the percent as a fraction and then multiplying its numerator by

18 .

Solution

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 %

18

represents the time it takes to sail from California to Australia.

262 % of 18

It is helpful to estimate this expression by rounding the given numbers.

262 is closer to 260 than to 265 ⇓ 262 % \approx 260 % 18 is closer to 20 than to 15 ⇓ 18 \approx 20

Next, write

262 %

18

as an expression using the rounded numbers and appropriate operation. Recall that of often indicates multiplication.

260 % 260 % of 20 ⇕ \times 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

$a % = \frac{a}{1 0 0}$

\frac{2 6 0}{1 0 0}

ReduceFrac

$\frac{a}{b} = \frac{a / 2 0}{b / 2 0}$

\frac{2 6 0 \div 2 0}{1 0 0 \div 2 0}

CalcQuot

Calculate quotient

\frac{1 3}{5}

The next step is to calculate the product of this fraction and

20 .

260 % \frac{1 3}{5} \times 20 ⇕ \times 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.

\frac{1 3}{5} \cdot 20

MoveRightFacToNum

$\frac{a}{c} \cdot b = \frac{a \cdot b}{c}$

\frac{1 3 \cdot 2 0}{5}

ReduceFrac

$\frac{a}{b} = \frac{a / 5}{b / 5}$

\frac{1 3 \cdot 4}{1}

Multiply

\frac{5 2}{1}

DivByOne

$\frac{a}{1} = a$

52

The calculations show that

260 %

20

52 .

This result means that it will take Kevin about

52

days to sail from California to Hawaii.

b From Part A, the exact sailing time from California to Australia is given by

262 %

18 .

In other words, the product of

262 %

and

18

should be evaluated.

262 % 262 % of 18 ⇕ \times 18

Repeat the used procedure one more time. First, rewrite the percent as a fraction in the simplest form.

262 % \times 18

PercentToFrac

$a % = \frac{a}{1 0 0}$

\frac{2 6 2}{1 0 0} \times 18

ReduceFrac

$\frac{a}{b} = \frac{a / 2}{b / 2}$

\frac{1 3 1}{5 0} \times 18

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.

\frac{1 3 1}{5 0} \times 18

MoveRightFacToNum

$\frac{a}{c} \cdot b = \frac{a \cdot b}{c}$

\frac{1 3 1 \cdot 1 8}{5 0}

ReduceFrac

$\frac{a}{b} = \frac{a / 2}{b / 2}$

\frac{1 3 1 \cdot 9}{2 5}

Multiply

\frac{1 1 7 9}{2 5}

CalcQuot

Calculate quotient

47.16

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!

Pop Quiz

Finding Percents of Numbers

Calculate the given percent of a number. Round the answer to one decimal place if needed.

	{{ 'ml-lesson-number-slides' \| message : article.intro.bblockCount }}
	{{ 'ml-lesson-number-exercises' \| message : article.intro.exerciseCount }}
	{{ 'ml-lesson-time-estimation' \| message }}

{{ article.displayTitle }}

Catch-Up and Review

Extra

Hint

Solution

Hint

Solution

Hint

Solution