5. Estimation and Percent of a Number
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Chapter 2
5.
Estimation and Percent of a Number
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Why
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.
Lesson Settings & Tools
| | 9 Theory slides |
| | 11 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Estimation and Percent of a Number
Slide of 9
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.
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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.
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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? 5* 37 In this case, the number is multiplied by the numerator of the fraction. The denominator remains the same. 5* 37&=5* 3/7 & ⇓ 5* 37&=15/7 The last step is to evaluate the quotient. 15/7≈ 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
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.
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 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 piece of information to rewrite the task. 40 % of 86 ⇕ 2/5 of 86
2
Write a Product Expression
expand_more Recall that the word
ofis often used to indicate a product in math expressions. 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 remains the same. 2/5 * 86 = 2* 86/5
3
Evaluate the Expression
expand_more Finally, evaluate the expression. Start by performing multiplication in the numerator. Then, calculate the quotient.
Therefore, 40 % of 86 is 34.4.
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 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.
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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?
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b Blue fish represent 36 % of all the fish. How many blue fish are there?
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Hint
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.
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 % of 850.
18 % &of 850 &⇕ 18 % &* 850 Start by rewriting the percent as a fraction. Use the fact that a percent is the ratio of a number and 100. n %=n/100 Then, simplify the fraction 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
Rewrite the product of the percent and the number by using this fraction equivalent to the percent. 18 % &* 850 &⇕ 9/50 &* 850 Multiply the numerator 9 by 850 to calculate this product. The denominator remains the same.
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
Therefore, 18 % of 850 is 153. This means that there are 153 yellow fish in the pond.
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 %=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
Therefore, there are 306 blue fish in the pond.
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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.
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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 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 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 % & of 15 &⇕ 420 % & * 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 %=a/100
420/100
ReduceFrac
a/b=.a /20./.b /20.
420 ÷ 20/100 ÷ 20
CalcQuot
Calculate quotient
21/5
This fraction is equivalent to 420 %. 420 % &* 15 &⇕ 21/5 &* 15 Next, evaluate this product by multiplying the numerator 21 by 15 and leave the denominator as is.
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
The calculations show that 420 % of 15 is 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 % * 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 %=a/100
260/100
ReduceFrac
a/b=.a /20./.b /20.
260 ÷ 20/100 ÷ 20
CalcQuot
Calculate quotient
13/5
Next, multiply the fraction by 63 by multiplying its numerator by 63.
13/5* 63
MoveRightFacToNum
a/c* b = a* b/c
13* 63/5
Multiply
Multiply
819/5
CalcQuot
Calculate quotient
163.8
Therefore, a dolphin can be 163.8 centimeters long. This is more than 1.5 meters, nearly 5 feet!
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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. c|c 17is closer to15 & 39is closer to40 than to20 & than to35 ⇓ & ⇓ 17 % ≈ 15 % & 39≈ 40 These new rounded numbers can be used to find an estimated result of 17 % of 40. Write a new task. 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
▼
Evaluate
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
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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 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 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.
This means that 262 % of 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.
c|c
262is closer to260 & 18is closer to20 than to265 & than to15 ⇓ & ⇓ 262 % ≈ 260 % & 18≈ 20
Next, write 262 % of 18 as an expression using the rounded numbers and appropriate operation. Recall that of
often indicates multiplication.
260 % &of 20 &⇕ 260 % &* 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 %=a/100
260/100
ReduceFrac
a/b=.a /20./.b /20.
260 ÷ 20/100 ÷ 20
CalcQuot
Calculate quotient
13/5
The next step is to calculate the product of this fraction and 20. 260 % &* 20 &⇕ 13/5&* 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.
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
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.
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 % &of18 &⇕ 262 % &* 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.
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 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!
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Pop Quiz
Finding Percents of Numbers
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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?
External credits: storyset on Freepik
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
| Number | Percent | Fraction |
|---|---|---|
| 10 | 10 % | 1/10 |
| 20 | 20 % | 1/5 |
| 25 | 25 % | 1/4 |
| 50 | 50 % | 1/2 |
| 75 | 75 % | 3/4 |
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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.
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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 |
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Solution
Order the percents of numbers from least to greatest without finding their exact values.
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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
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Estimation and Percent of a Number
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