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5. Estimation and Percent of a Number
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5. 

Estimation and Percent of a Number

This lesson focuses on teaching the techniques of estimation, particularly when it comes to calculating the percent of a number. It offers practical insights into how these methods can be applied in everyday situations. For example, if you're trying to figure out how much you'll save during a sale, or how much of a certain ingredient you'll need for a recipe, understanding how to estimate percentages can be incredibly useful. The lesson also emphasizes the importance of these skills in various fields, from shopping to scientific research, making it a valuable lesson for both students and professionals.

Student Learning Objectives:
  • Find a percent of a number
  • Estimating the percent of a number
Show more expand_more
9 Theory slides
11 Exercises - Grade E - A
Each lesson is meant to take 1-2 classroom sessions
Image Credits expand_more
Estimation and Percent of a Number
Slide of 9
It is useful to know how to calculate the percent of numbers to understand the actual value represented by the percent. Rounding numbers also comes in handy when we only need an approximate value. In some cases, there is even a trick for quicker calculations! 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

Suppose a container had 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.

A container being filled with a liquid
Discussion

How to Find a Percent of a Number?

The process for finding the percent of a number is very similar to finding the product of a number and a fraction. 5* 3/7&=5* 3/7 &↓ 15/7 &≈ 2.14 For example, suppose a student scores 84 % out of 50 questions and wants to know how many questions they got correct. Calculating 84 % of 50 can help with this.

Method

Finding the Percent of a Number

Percents represent a relative value of something, not the actual value itself. 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.

1
Rewrite the Percent as a Fraction
expand_more
First, recall that a percent is the ratio of a number to 100. This fact allows us to write a percent as a fraction. n %=n/100 Apply this formula to 40 %. Then, simplify the fraction as much as possible.

40 %
PercentToFrac

a %=a/100

40/100
ReduceFrac

a/b=.a /20./.b /20.

40 ÷ 20/100 ÷ 20
CalcQuot

Calculate quotient

2/5

Therefore, 40 % is equivalent to 25. Use this information to rewrite the task. 40 % of 86 ⇕ 2/5 of 86

2
Write a Product Expression
expand_more
Recall that in math, the word of often means to multiply. This means that 25 of 86 is the same as the product of 25 and 86. 2/5 of 86 ⇕ 2/5 * 86 Multiplying a fraction by a number requires multiplying the numerator of the fraction by the number. The denominator stays the same. 2/5 * 86 = 2* 86/5
3
Evaluate the Expression
expand_more
Finally, evaluate the expression. Start by performing the multiplication in the numerator, calculate the quotient.

2* 86/5
Multiply

Multiply

172/5
CalcQuot

Calculate quotient

34.4

Therefore, 40 % of 86 is 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.

40.0 becomes 0.4 after dividing by 100

As we can see, 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. We got the same result even though the process used to get it was different.

Example

Counting Fish in a Pond

A local Fish and Game agency recently found that there are 850 fish in a pond.

Fish swimming in the pond
External credits: @upklyak

a

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

b

If 36 % of the fish are blue, how many blue fish are there?

Hint
a

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

b

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

Solution
a

To find the number of yellow fish in the pond, we multiply 18 % by the total number of fish, 850.

18 % &of 850 &⇕ 18 % &* 850 Let's start by rewriting the percent as a fraction and simplifying it as much as possible.

18 %
PercentToFrac

a %=a/100

18/100
ReduceFrac

a/b=.a /2./.b /2.

18 ÷ 2/100 ÷ 2
CalcQuot

Calculate quotient

9/50

Now we can multiply the fraction 950 by 850 by writing 850 in the numerator of the fraction.

18 % * 850
Substitute

18 %= 9/50

9/50* 850
MoveRightFacToNum

a/c* b = a* b/c

9* 850/50
ReduceFrac

a/b=.a /50./.b /50.

9* 17/1
Multiply

Multiply

153/1
DivByOne

a/1=a

153

There are 153 yellow fish in the pond.

b

This time we want to find 36 % of 850 to find the number of blue fish in the pond.

36 % * 850 To calculate this product, we will use the same process as in Part A. First, we write 36 % as a fraction and simplify it.

36 %
PercentToFrac

a %=a/100

36/100
ReduceFrac

a/b=.a /4./.b /4.

36 ÷ 4/100 ÷ 4
CalcQuot

Calculate quotient

9/25

The next step is to multiply the fraction by 850.

9/25* 850
MoveRightFacToNum

a/c* b = a* b/c

9* 850/25
ReduceFrac

a/b=.a /25./.b /25.

9* 34/1
Multiply

Multiply

306/1
DivByOne

a/1=a

306

There are 306 blue fish in the pond.

Example

Comparing Fish Sizes

Fish come in all sizes. Some clownfish grow to just 15 centimeters long.

Clownfish, carp, and a dolphin
External credits: @brgfx
Carp can reach 420 % of the length of clownfish. Dolphins are even bigger. They can be as long as 260 % of a carp's length.

a

How long can a carp grow to be?

b

What length can a dolphin grow to be?

Hint
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.

Solution
a

To find the length of a carp, we multiply the 15cm a clownfish can grow by 420 %.

420 % & of 15 &⇕ 420 % & * 15 Now we can rewrite the percent as a fraction with a numerator of 420 and a denominator of 100, then simplify it if possible.

420 %
PercentToFrac

a %=a/100

420/100
ReduceFrac

a/b=.a /20./.b /20.

420 ÷ 20/100 ÷ 20
CalcQuot

Calculate quotient

21/5

Next, we find 420 % of 15 by using the equivalent fraction 215.

420 % * 15
WriteAsFrac

Write as a fraction

21/5* 15
MoveRightFacToNum

a/c* b = a* b/c

21* 15/5
ReduceFrac

a/b=.a /5./.b /5.

21* 3/1
Multiply

Multiply

63/1
DivByOne

a/1=a

63

A carp can grow to a length of up to 63 centimeters.

b

In Part A, we found that a carp can grow up to 63 centimeters long, and a dolphin can be 260 % as long as a carp.

260 % * 63 Let's start by rewriting the percent as a fraction.

260 %
PercentToFrac

a %=a/100

260/100
ReduceFrac

a/b=.a /20./.b /20.

260 ÷ 20/100 ÷ 20
CalcQuot

Calculate quotient

13/5

We found that 260 % = 135. Next, we use this equivalent fraction to find the product.

260 % * 63
WriteAsFrac

Write as a fraction

13/5* 63
MoveRightFacToNum

a/c* b = a* b/c

13* 63/5
Multiply

Multiply

819/5
CalcQuot

Calculate quotient

163.8

A dolphin can be 163.8 centimeters long. That is more than 1.5 meters, or about 5.4 feet!

Discussion

Estimating the Percent of a Number

Finding the percent of a number involves quite a few steps. Sometimes, though, an estimation of a percentage is enough. In these cases, rounding the numbers means quicker and easier calculations. Let's consider an example. 17 % of 39 These numbers are not very convenient to work with. We would likely need a calculator. Instead, let's simplify the process and round each number. c|c 17is closer to15 & 39is closer to40 than to20 & than to35 ⇓ & ⇓ 17 % ≈ 15 % & 39≈ 40 We can use these rounded numbers to find an estimated result of 17 % of 40. 15 % &of 40 &⇕ 15 % & * 40 Start by rewriting the percent as a fraction. Then, simplify the fraction and evaluate the expression.

15 % * 40
▼
Rewrite
PercentToFrac

a %=a/100

15/100 * 40
ReduceFrac

a/b=.a /5./.b /5.

15 ÷ 5/100 ÷ 5 * 40
CalcQuot

Calculate quotient

3/20 * 40
MoveRightFacToNum

a/c* b = a* b/c

3* 40/20
ReduceFrac

a/b=.a /20./.b /20.

3* 2/1
Multiply

Multiply

6/1
DivByOne

a/1=a

6
We found that 17 % of 39 is about 6. The exact result is 6.63, so the estimation is pretty close. 17 % * 39 &= 6.63 17 % * 39 &≈ 6

This method is useful when we need to quickly eyeball or estimate the percent of some number.
Example

Traveling Across the Pacific Ocean

A trip from California to Hawaii on a sailboat takes 18 days. California to Australia is 262 % of the distance from California to Hawaii.

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

a

Estimate how much time it would take to sail from California to Australia.

b

Find the exact number of days of the trip would take.

Hint
a

Round 262 and 18 to numbers that end with a 0 or 5 for simpler calculations.

b

Calculate 262 % of 18 by rewriting the percent as a fraction and then multiplying its numerator by 18.

Solution
a

To find the time it takes to sail from California to Australia, we need to find 262 % of 18 days.

262 % * 18 Let's estimate this expression by rounding the given numbers. c|c 262is closer to260 & 18is closer to20 than to265 & than to15 ⇓ & ⇓ 262 % ≈ 260 % & 18≈ 20 Next, let's calculate 260 % of 20. First, we rewrite 260 % as a fraction.

260 %
PercentToFrac

a %=a/100

260/100
ReduceFrac

a/b=.a /20./.b /20.

260 ÷ 20/100 ÷ 20
CalcQuot

Calculate quotient

13/5

Next, we calculate the product of this fraction and 20.

13/5* 20
MoveRightFacToNum

a/c* b = a* b/c

13* 20/5
ReduceFrac

a/b=.a /5./.b /5.

13* 4/1
Multiply

Multiply

52/1
DivByOne

a/1=a

52

It will take about 52 days to sail from California to Hawaii.

b

This time, we want to find the exact sailing time, which is 262 % of 18 days. We use the same method as in Part A, but without rounding the numbers. First, we rewrite 262 % as a fraction and simplify it if possible.

262 % * 18
PercentToFrac

a %=a/100

262/100 * 18
ReduceFrac

a/b=.a /2./.b /2.

131/50 * 18

Next, we can multiply the fraction by the number and find the quotient.

131/50 * 18
MoveRightFacToNum

a/c* b = a* b/c

131* 18/50
ReduceFrac

a/b=.a /2./.b /2.

131* 9/25
Multiply

Multiply

1179/25
CalcQuot

Calculate quotient

47.16

The exact time of sailing from California to Australia is 47.16 days, which is pretty close to the estimated result of 52 days.

Pop Quiz

Finding Percents of Numbers

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

A random generator that asks to calculate some percent of a number
Closure

A Secret for Faster Calculations

There is a trick that can make things much simpler when calculating the percent of a number.

The envelope opens and shows this top secret message: The numbers can be switched and the result will still be the same!
External credits: storyset on Freepik

Let's see this secret in action through the following example. 6 % of 50 It is a little difficult to calculate 6 % of 50 off the top of our head. Let's switch the numbers instead. 50 % of 6 Since 50 % is equivalent to 12, let's multiply 12 by 6. 1/2* 6=3 We found that 50 % of 6 is 3. Let's confirm our trick by calculating 6 % of 50 the usual way.

6 % * 50
▼
Evaluate
PercentToFrac

a %=a/100

6/100* 50
ReduceFrac

a/b=.a /2./.b /2.

3/50* 50
MoveRightFacToNum

a/c* b = a* b/c

3* 50/50
CrossCommonFac

Cross out common factors

3* 50/50
SimpQuot

Simplify quotient

3/1
DivByOne

a/1=a

3
The result is 3. This shows that switching the numbers gives equivalent expressions. n % of m = m % of n This method is especially useful when the new percent is easy to work with. Here are a few examples.

Number Percent Fraction
10 10 % 1/10
20 20 % 1/5
25 25 % 1/4
50 50 % 1/2
75 75 % 3/4
The percent of a number can sometimes be calculated using mental math if we switch the numbers. It is almost like having a super power!



Level 1
Level 2
Level 3
Estimation and Percent of a Number
Exercise 3.1

Describe any real-world problem in which the percent of a number gives a result that is greater than the original number? Then, solve the problem.

Let's consider the following real-life problem.

We need to calculate 150 % of 18 to find the number of pages Emily read that one day. Let's start by rewriting 150 as a decimal number. That can be done by dividing the percent by 100 % by moving the decimal point two places to the left.

Next, multiply 1.5 by 18. 1.5* 18=27 We calculated that 150 % of 18 is 27. This means that Emily read 27 pages instead of the planned 18. In this case, the percent of the number 27 is greater than the number itself, 18. ccc & & Percent of [-0.1cm] Number& & a Number ↓ & & ↓ 18 & < & 27 In general, one number is usually set to represent 100 % in real-world situations. This number is then compared to another number. Calculating any percent that is greater than 100 %, like 136 % or 201 %, will result in a number greater than the original number. Here are a few more examples.

Original Number Percent of the Number Calculation Comparison
The life expectancy in the US was about 77.3 years in 2020. Sarah Knauss born in 1880 lived for 154 % of that number. 154 %* 77.3≈ 119 77.3< 119
The cost of a gallon of milk was $2.78 in 2000. In 2021, the cost of a gallon of milk was 128 % of that price. 128 % * $2.78≈ $3.55 $2.78< $3.55
LaShay earns $18 per hour. After getting a raise, she now earns 115 % of what she earned before. 115 % * $18= $20.7 $18< $20.7
E
Hint & Answer
S
Solution

Order the percents of numbers from least to greatest without finding their exact values.

Let's start by looking at the list of the percents of numbers. 32 % &of 45 70 % &of 28 20 % &of 45 70 % &of 39 We can see that one pair shares the same number 45, while the other pair shares the same percent 70 %. Let's consider each pair one at a time.

Pair With the Same Number

Now, let's take a closer look at the pair that shares the same number. 32 % &of 45 20 % &of 45 Here, we need to calculate 32 % and 20 % of the same number 45. Since 32 is greater than 20, 32 % of 45 is a greater number than 20 % of 45. 20 % of 45 < 32 % of 45

Pair With the Same Percent

We can consider the second pair that shares the same percent in a similar fashion. 70 % &of 28 70 % &of 39 The same percent is calculated of different numbers 28 and 39. Since 28 is less than 39, 70 % of 28 is less than 70 % of 39. 70 % of 28 < 70 % of 39

Final Comparison

We ordered the percents from least to greatest in two pairs. Pair I: 20 % of 45 < 32 % of 45 Pair II: 70 % of 28 < 70 % of 39 Now, let's consider the greater expression from the first pair and the smaller expression from the second pair. 32 % of 45 70 % of 28 Note that we can round both 32 and 28 to 30. Let's do this! ccc 32 % of 45 & ≈ & 30 % of 45 70 % of 28 & ≈ & 70 % of 30 Also, recall that the percent and the number can be interchanged and the result will still be the same. We can use this property to interchange 70 and 30 in the second pair. 70 % & of 30 &⇕ 30 % &of 70 This way we get a pair with the same percent once again. 30 % of 45 30 % of 70 The number 45 is less than 70, so 30 % of 45 is less than 30 % of 70. ccc 30 % of 45 & < & 30 % of 70 ↓ & & ↓ 32 % of 45 & < & 70 % of 28 We found that the greater expression from the first pair is smaller than the smaller expression from the second pair. 20 % of 45 < 32 % of 45 < < 70 % of 28 < 70 % of 39 Finally, we can order the percents of numbers from least to greatest. 20 % of 45 32 % of 45 70 % of 28 70 % of 39

E
Hint & Answer
S
Solution

Estimation and Percent of a Number

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