5. Estimation and Percent of a Number
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Chapter 2
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.
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| | 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
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.
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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.
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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
ofoften 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.
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 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.
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Example
Counting Fish in a Pond
A local Fish and Game agency recently found that there are 850 fish in a pond.
External credits: @upklyak
a
If 18 % of the fish are yellow, how many yellow fish are in the pond?
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b
If 36 % of the fish are blue, how many blue fish are there?
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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
There are 850 fish in the pond. Yellow fish represent 18 % of that number. We can find the exact number of yellow fish by calculating 18 % of 850.
18 % &of 850 &⇕ 18 % &* 850 Start by rewriting the percent as a fraction. We know that a percent is the ratio of a number and 100. n %=n/100 Now let's 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 using this fraction. 18 % &* 850 &⇕ 9/50 &* 850 Multiply the numerator 9 by 850 to calculate the product. The denominator stays 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 we want to find 36 % of 850 to find the number of blue fish in the pond. Recall that of
indicates 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
Fish come in all sizes. Some clownfish grow to just 15 centimeters long.
External credits: @brgfx
a
How long 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
A clownfish can grow up to 15 centimeters. A carp can be 420 % as long. Let's find 420 % of 15 to determine the carp's length. Recall that of
indicates 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, multiply 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 we know the length of a carp, we can find the length of a dolphin. Dolphins can be 260 % as long as a carp. In Part A we found that a carp can grow up to 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. Since it is more convenient to work with smaller numbers, 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
We found that a dolphin can be 163.8 centimeters long. That is is more than 1.5 meters, nearly 5 feet!
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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
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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
Estimate how much time it would take to sail from California to Australia.
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b
Find the exact number of days of the trip would take.
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Solution
a
We are given two key points of information. 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 % of 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, 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
To calculate this new expression, we start by rewriting the percent as a fraction. Use the fact that a percent is the ratio of a number to 100. Then, we 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.
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 about 52 days to sail from California to Hawaii.
b
The exact sailing time from California to Australia is given as 262 % of 18. In other words, we need to find the product of 262 % and 18.
262 % &of18 &⇕ 262 % &* 18 First, let's rewrite the percent as a fraction in 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.
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Pop Quiz
Finding Percents of Numbers
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Closure
A Secret for Faster Calculations
There is a trick that can make things much simpler when calculating the percent of a number.
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. We can confirm our trick by calculating 6 % of 50 using the known method.
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 2.1
Consider the diagram showing how much time Zosia spent on different activities during a typical day.
How many more hours did Zosia spend sleeping than doing homework?
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We want to find about how many more hours Zosia spent sleeping than doing homework. Zosia spent 33 % of her day sleeping and 13 % of her day doing homework according to the diagram. Let's start by calculating the difference between these percents. 33 % - 13 % = 20 % This means that we need to find 20 % of 24 hours. We can do this by multiplying 20 % by 24 hours. 20 %of 24h = 20 %* 24h Let's rewrite the percent as a decimal number by dividing it by 100. To do so, we will move the decimal point two places to the left.
Next, multiply 0.2 by 24 to evaluate 20 % of 24 hours. 0.2* 24=4.8 We found that Zosia spent 4.8 hours more sleeping than doing homework.
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Solution
Dominika wants to buy a purse regularly priced at $52. It is on sale for 20 % off. Dominika estimates that she will save 15 of $ 50 or $10.
Will the actual amount Dominika saves be more or less than $10? Explain.
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Let's start by writing 20 % as a fraction. To write a percent as a fraction, we write the number part as a numerator over 100. 20 % = 20/100 Let's simplify this fraction by dividing the numerator and the denominator by 20.
We found that Dominika will save 15 of $ 52. She estimates that she will save 15 of $ 50. This means that she rounded $ 52 to $ 50. Since 52 is greater than 50, we conclude that 15 of $ 52 is greater than 15 of 50. 52 > 50 ⇔ 1/5 * 52 > 1/5* 50 Therefore, the actual amount of saved money is greater than the amount Dominika estimated.
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The SkyRise Airline records snack orders made by their passengers. Last year, 28 % of passengers ordered a sandwich and a soda. Today, there are 396 passengers on the flight to Seattle, Washington.
About how many passengers could the airline expect to order a sandwich and a soda on this flight?
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We need to estimate 28 % of 396 to find the approximate number of passengers who will order a sandwich and a soda on the flight to Seattle. Let's start by rounding both numbers. c|c 28is closer to30 & 396is closer to400 than to20 & than to350 ⇓ & ⇓ 28 % ≈ 30 % & 396≈ 400 Now, we calculate 30 % of 400. We can do this by multiplying 30 % by 400. Let's also rewrite the percent as a fraction.
We found that 28 % of 396 is about 120. This means that the airline can expect about 120 passengers on the flight to Seattle to order a sandwich and a soda. Note that 396 can also be rounded to 395, which results in a different estimate 118.5. 30 % of 395=118.5 Since this number represents the number of people, it should be an integer. Let's round 118.5 to 119. That means 119 is another estimate of the number of passengers on the flight to Seattle who will order a sandwich and a soda.
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Kevin is getting ready for a snowboard race that has three rounds. After the first round, 25 % of the participants with the best results will compete in the second round. Then, 15 % of the fastest snowboarders will compete in the final third round.
If 528 people are in the first round of the competition, how many competitors will there be in the third round? Round the answer to the nearest integer.
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We know that 528 snowboarders take part in the first round of the competition. Only 25 % best snowboarders will compete in the second round. Let's find the number of competitors in the second round by calculating 25 % of 528. 25 % &of 528 &⇕ 25 % &* 528 We can start by rewriting the percent as a fraction and simplifying it as much as possible.
We found that 132 snowboarders will compete in the second round. We also know that 15 % of the snowboarders with the best results will move on to the third round. Let's find how many people this is by calculating 15 % of 132. This time, we can rewrite the percent as a decimal number by dividing it by 100.
We get that 15 % is equivalent to 0.15. Next, multiply the decimal number 0.15 by 132. 0.15* 132=19.8 The product is 19.8, which we round to 20. Therefore, about 20 competitors will take part in the final third round of the competition.
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Estimation and Percent of a Number
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