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6. The Percent Proportion and Equation, and Percent of Change
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Chapter 5
6. 

The Percent Proportion and Equation, and Percent of Change

Understanding percent proportion, percent equation, and percent of change is crucial for various real-world applications. These mathematical concepts are essential for calculating discounts in shopping, determining investment growth, and even in scientific research to measure relative error. The lesson delves into these topics, offering insights that can help in making more accurate and informed decisions in both personal and professional settings.
Problem Solving Reasoning and Communication Error Analysis Modeling Using Tools Precision Pattern Recognition
Lesson Settings & Tools
15 Theory slides
12 Exercises - Grade E - A
Each lesson is meant to take 1-2 classroom sessions
Image Credits expand_more
The Percent Proportion and Equation, and Percent of Change
Slide of 15
Life deals plenty of change whether it is doing business or simply noting an increase or decrease in value. This lesson will introduce how to calculate the percent of change between two given amounts. In addition to that, the lesson mentions how to express the amount of error as a percent.

Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.

Challenge

Paying a Loan

Kevin really wants to buy this cherry-red vintage bicycle. However, there is a major issue — he is just shy of the amount he needs! He talks with Uncle Dev, also a cyclist, about the issue. Uncle Dev offers Kevin a loan.

The uncle says: I can give you 84 dollars, and you will pay me 7.70 dollars per month during one year.

What is the percent of change from the price of the bicycle to the total amount that Kevin would pay his uncle?

Discussion

Writing the Percent as a Ratio

There are several ways to solve percent problems. One of them is to write the percent as a ratio.

Concept

Percent Proportion

In a percent proportion, the ratio of a part of a quantity to the whole of that quantity is equal to the percent written as a fraction. a/b=p/100

In this proportion, a is a part of the whole b. Likewise, p is the part - or percent - of the whole 100.
Example

Finding the Percent of Toys

Kevin's uncle Dev works in a toy shop as a sales manager. He is responsible for monitoring the number of products sold. The following graph shows the number of toys sold by the shop in one day.

Toys sold per day are depicted in a bar graph. The vertical axis shows the number of toys sold, and the horizontal axis categorizes the toys as dolls, cars, construction, educational, electronic, and animals.
a What percent of the toys sold are construction toys?

b What percent of the toys sold are electronic toys?

Hint
a Find the total number of toys from the graph and use the percent proportion.
b Use the percent proportion.

Solution
a The given graph categorizes the number of toys sold into six different categories. The task at hand is to figure out the number of construction toys sold as a percent rather than a whole number. Finding the percent proportion can accomplish this.

a/w=p/100 Recall what each variable represents. a &→ Part w &→ Whole p &→ Percent Retrieve from the graph the number of construction toys sold. It is 6, which is the value of a. Now, find the total number of toys sold by adding the corresponding numbers of the six categories from the graph. Dolls=& 8 Cars=& 15 Construction=& 6 Educational=& 4 Electronic=& 15 Animals=& 12 Now calculate the sum of these numbers to find the value of w.

8+15+6+ 4+ 15+ 12
AddTerms

Add terms

60

There are 60 toys sold. This is the value of the whole w. Now that the part and whole are known, the percent can be calculated. Substitute the obtained values into the percent proportion.

a/w=p/100
SubstituteII

a= 6, w= 60

6/60=p/100
▼
Solve for p
ReduceFrac

a/b=.a /6./.b /6.

1/10=p/100
MultEqn

LHS * 100=RHS* 100

100/10=p
CalcQuot

Calculate quotient

10=p
RearrangeEqn

Rearrange equation

p=10

This means that 10% of the total number of toys sold are construction toys.

b Once again, apply the same process from Part A to find the percent that represents the number of electronic toys sold. Start by recalling the percent proportion.

a/w=p/100 The total number of toys sold was already found in Part A. There are 60 toys sold and this is the value of w. On the other hand, 15 electronic toys were sold. This is the value of a. Substitute these values into the percent proportion and solve it for p.

a/w=p/100
SubstituteII

a= 15, w= 60

15/60=p/100
▼
Solve for p
ReduceFrac

a/b=.a /15./.b /15.

1/4=p/100
MultEqn

LHS * 100=RHS* 100

100/4=p
CalcQuot

Calculate quotient

25=p
RearrangeEqn

Rearrange equation

p=25

This means that 25% of the toys sold were electronic toys.

Pop Quiz

Solving a Percent Proportion

Solve the given percent proportion for the missing value. Note that a is part of the whole w and p% or p100 is the percent value.

Finding missing value in a percent proportion
Discussion

Using the Percent in an Equation

Percent problems can also be solved by expressing the given situation as an equation. In that case, it is needed to write the percent as a decimal or as a fraction while solving the equation.

Concept

Percent Equation

In a percent equation, the part is equal to the product of the corresponding percent and the whole. The phrase a is p percent of w is represented as the following equation. a=p % * w For instance, consider the case that the whole is 50 which represents the value of 100%.

As an example, 30% of 50 can be calculated by multiplying these values.

a=30 % * 50
Example

Arranging the Toys According to Age Groups

Kids learn and grow through playing with toys that match their developmental stages. Uncle Dev is organizing toys at the shop according to age groups: infants, toddlers, and preschoolers. By category, the circle graph shows the percents of toddler and infant toys and number of preschool toys.

A pie chart displaying the percentage distribution of toys for toddlers, infants, and preschoolers at a store. The toddler category accounts for 42%, while the infant category accounts for 23%.
a Find the total number of toys.

b Find the number of toys for infants.

Hint
a Express all categories as a percent. Then, use the percent equation to find the value of whole.
b Use the total number of toys in the given three categories. Then use the percent equation to find the part.

Solution
a In the given graph, the numbers of toys are represented with percents or it is just written as a the number.

Toddler & ⇒ 42% Infant & ⇒ 23% Preschooler & ⇒ 140 Recall that the total percents in a circle graph needs to represent 100%. This means that the sum of 42%, 23%, and the missing percent of preschool toys x% will be 100%. Write this situation as an equation and solve for x.

42+23+ x =100
AddTerms

Add terms

65+ x=100
SubEqn

LHS-65=RHS-65

x=35

This means that 140 toys are 35% of the number of toys in all three categories. Since it is asked to find the total number of toys, the following question can be asked. 140 is  35 % of what number? This question can be solved by using a percent equation. a=p% * w Recall that a is the value of part, p is the value of percent, and w is the value of whole. Note that the above question asks for the whole. Now, substitute the obtained values into the equation and then solve it for w. Try to write the percent as a decimal.

a=p% * w
SubstituteII

a= 140, p= 35

140=35% * w
▼
Solve for w
WriteDec

Write as a decimal

140=0.35 * w
DivEqn

.LHS /0.35.=.RHS /0.35.

140/0.35=w
RearrangeEqn

Rearrange equation

w=140/0.35
CalcQuot

Calculate quotient

w=400

This means that there are 400 toys in these three categories.

b The next task for Uncle Dev is to determine the number of toys made for infants. As of right now, he knows it is 23% of all toys. In Part A, he found that there are 400 toys in total.

What number is 23 % of 400? This question can be solved by using a percent equation which states that the product of the percent and whole is equal to the value of part. a=p% * w Notice that 400 is the whole and 23 is the percent value. With this in mind, substitute these values into the percent equation and calculate the result. Remember to write the percent as a decimal.

a=p% * w
SubstituteII

p= 23, w= 400

a= 23% * 400
WriteDec

Write as a decimal

a=0.23 * 400
Multiply

Multiply

92

There are 92 toys for infants in the toy shop.

Pop Quiz

Solving a Percent Equation

In the following applet, there is a percent equation representing the situation in which the part a is p percent of the whole w. Solve the equation for the missing value.

Finding missing value in a percent equation
Discussion

Expressing the Change as a Percent

Sometimes it may be easier to evaluate the amount of change when it is expressed as a percent rather than as a number or a ratio. In that case the percent of change can be used.

Concept

Percent of Change

Percent of change, or percent change, is a percent that expresses an amount of change as a percent of the original amount. It is calculated as the ratio of the change in the amount to the original amount. Percent of Change, p % =Amount of Change/Original Amount If the new amount is greater than the original amount, the percent of change is called a percent of increase. Percent of Increase=New Amount -Original Amount/Original Amount If the new amount is less than the original amount, the percent of change is called a percent of decrease.

Percent of Decrease=Original Amount-New Amount/Original Amount
Discussion

Calculating Percent of Change

The percent change is the change in percent when a quantity has changed. It is calculated by writing a ratio of the amount of change to the original amount as a percent. Percent Change=Amount of Change/Original Amount For example, buying a collector's item for $12 and selling it for $15 results in a profit. What is the percent change? It can be calculated in four steps.

1
Determine if the Change is an Increase or a Decrease
expand_more
Determine if the price of the item increases or decreases by looking at the difference between the new and original amounts.

Increase New amount > Original amount
Decrease Original amount > New amount

The change represents an increase since 15 is greater than 12.

2
Calculate the Amount of Change
expand_more
When the change is an increase, subtract the original amount from the new amount to find the difference. Since the new amount is greater than the original amount, the difference will be positive. 15 - 12 = 3
3
Calculate the Ratio for Percent of Change
expand_more
Now that the amount of increase and the original amounts are known, find the ratio of these amounts.

3/12
ReduceFrac

a/b=.a /3./.b /3.

1/4
CalcQuot

Calculate quotient

0.25

4
Write the Result as a Percent
expand_more
As a final step, write the obtained number as a percent by multiplying 0.25 by 100. 0.25 * 100 =25 The percent of change is 25 %. This means that the price of the collector's item increased by 25 %.
Example

Calculating the Change as a Percent

Uncle Dev now wants to examine the percent of changes between visitors over the last two weeks. He prepares the following table that shows the number of people that visited the toy shop.

Table shows the number of people
a What is the percent of change in the number of people that visit the toy shop on Monday from the first week to the second week?

b What is the percent of change in the number of people that visit the toy shop from Friday to Saturday in the second week?

Hint
a Use the percent of change formula. Is the new amount greater or less than the original amount?
b Use the percent of change formula. Is the new amount greater or less than the original amount?

Solution
a Uncle Dev wants to find the percent of change from the first Monday to the second Monday. Start by recalling the percent of change formula.
Percent of Change Formula

Notice that on the first week's Monday 75 person visited the toy shop but on the second week's Monday 81 people visited the shop. Since it shows an increase, the amount of change will be found by subtracting the original amount from the new amount.

Percent of Increase Formula

Note that the new amount is 81 and the original amount is 75. Percent of Increase=81-75/75 Now calculate this percent of increase!

Percent of Increase=81-75/75
▼
Simplify right-hand side
SubTerms

Subtract terms

Percent of Increase=6/75
ReduceFrac

a/b=.a /3./.b /3.

Percent of Increase=2/25
ExpandFrac

a/b=a * 4/b * 4

Percent of Increase=8/100
WritePercent

Convert to percent

Percent of Increase=8 %

From the first week to the second week, there exists 8% increase in the number of people that visit the toy shop on Monday.

b This time, the percent of change from Friday to Saturday at the second week needs to be calculated. Start by examining the numbers on those days.

Friday=90 Saturday=75 Notice that the number of people decreased. This means that the percent of change will be found by using the percent of decrease formula.

Percent of Decrease Formula

Substitute the original amount as 90 and the new amount as 81 into the formula. Percent of Decrease=90-81/90 Now calculate this percent of decrease!

Percent of Decrease=90-81/90
▼
Simplify right-hand side
SubTerms

Subtract terms

Percent of Decrease=9/90
ReduceFrac

a/b=.a /9./.b /9.

Percent of Decrease=1/10
ExpandFrac

a/b=a * 10/b * 10

Percent of Decrease=10/100
WritePercent

Convert to percent

Percent of Decrease=10 %

The number of people that visit the toys shop is 10% decreased from Friday to Saturday in the second week.

Pop Quiz

Calculating the Percent of Change

In the following applet, the original and new amounts are given in a table. Find the percent of change based on these values. Also determine if this change is an increase or decrease. Round the answer to the nearest hundredth if necessary.

Calculating the percent of change between the given numbers
Discussion

Expressing the Error as a Percent

Sometimes the amount of error can be too big or too small to understand its effect clearly. Expressing the error as a percent is an alternative way to show the amount of the error.

Concept

Relative Error and Percent Error

The relative error is the ratio of the absolute error of a measurement to the exact value. The relative error tells how good a measurement is relative to the size of the object being measured. In other words, the relative error indicates how significant the absolute error is.

Relative Error =Absolute Error/Exact Value

The Relative Error Formula can be rewritten by substituting the Absolute Error Formula.

Relative Error =|Measured Value-Exact Value|/Exact Value

The percent error is the product between the relative error and 100 %. It represents the relative error as a percentage.

Percent Error = Relative Error * 100 %

Consider, for example, a person fishing who expected to catch 480 crayfish. However, the number of crayfish they ultimately caught was 400. The percent error explains the degree of the mistake in the person's estimation.

Absolute Error Relative Error Percent Error
|480- 400| = 80 80/400 = 0.2 0.2* 100 % = 20 %
The absolute error only says that the fisher was wrong by 80 crayfish. In contrast, the percent error says that the estimation was wrong by 20 %, which is more significant information for future predictions.
Example

Estimating a Wrong Price

A customer falls in love with a toy for her daughter. It is called Luchador Teddy! Wait. It is missing a price. Uncle Dev estimates that it costs $12 and writes that. The customer goes to the register to buy it but sees she is about to be charged $13.50!

Uncle Dev apologizes for the misinformation. What is the percent error between the estimated price and actual price? Round the result to the nearest tenth.

Hint
Calculate the amount of error. Then, find the relative error.

Solution
Percent of error problems are kinds of a percent of change problems. With this in mind, start by finding the relative error. Then multiply the relative error by 100% to express it as a percent error. Relative Error=Absolute Error/Exact Value Now, find the absolute error by calculating the difference between the estimated price and the exact price. Absolute Error |$13.50-$12|=$1.50 The absolute error is $1.50. Now, use the relative error formula.

Relative Error=Absolute Error/Exact Value
SubstituteII

Absolute Error= 1.50, Exact Value= 13.50

Relative Error=1.50/13.50
UseCalc

Use a calculator

Relative Error=0.111111 ...

Finally, multiply the obtained number by 100% to express it as a percent.

0.111111 ... * 100%
Multiply

Multiply

11.111111 % ...
RoundDec

Round to 1 decimal place(s)

≈ 11.1 %

The percent error between the Uncle Dev's estimate and the actual price of the toy is about 11.1%. Uncle Dev then decides to offer the customer a discount to make up for the error.

Closure

Calculating the Percent of Change for a Loan

This lesson introduced how to find the percent of change using original and new amounts. The challenge presented at this chapter's start focuses on Kevin's tough decision. If he accepts the loan, he could get his dream bicycle. On the other hand, he could owe too large of an amount!

Kevin is in through about getting 84 dollars, and then paing 7.70 dollars per month for one year.

Help Kevin find the percent of change from the price of the bicycle to the total amount Kevin would have to pay his uncle.

Hint
Find the amount of change between the price of the bicycle and the amount that Kevin would have to pay back. Then, apply the percent of change formula.

Solution
Recall that Kevin's uncle offers to give 84 dollars in total but he wants Kevin to pay him 7.70 dollars every month for one year. Since there are twelve months in a year, start by calculating the total amount that Kevin needs to pay back by multiplying 7.70 and 12. 7.70 * 12 = 92.40 Kevin would need to pay $92.40 back to his Uncle. Now, recall the percent of change formula. Percent of Change=Amount of Change/Original Amount Calculate the difference between the price of the bicycle and the amount Kevin will pay back to find the amount of change. Amount of Change 92.40-84=8.40 Now, substitute the obtained values into the percent of change formula.

Percent of Change=Amount of Change/Original Amount
SubstituteII

Amount of Change= 8.40, Original Amount= 84

Percent of Change=8.40/84
ReduceFrac

a/b=.a /8.4./.b /8.4.

Percent of Change=1/10
ExpandFrac

a/b=a * 10/b * 10

Percent of Change=10/100
WritePercent

Convert to percent

Percent of Change=10 %

Notice that Kevin would pay more money than he receives. This means that 10% represents the percent of increase. Kevin's uncle wants Kevin to pay 10% more than he gives. OMG!

Extra

 Kevin Makes a Counteroffer
The two regather after Kevin found the percent change that his Uncle originally offered.

Kevin makes a counteroffer of 0 percent change

Uncle Dev is impressed, and embarrassed, by Kevin's math skills. Uncle Dev agrees to the counteroffer — realizing the overpayment.

Uncle Dev accepts Kevin's counteroffer.

The two can now take a cruise around town together!

Keving and Uncle Dev go for a ride together.



Level 1
Level 2
Level 3
The Percent Proportion and Equation, and Percent of Change
Exercise 2.1

Mark decides to go on an 8-hour road trip. At the start, his phone's battery charge is at 60%. In normal usage, his battery lasts 15 hours at full charge. Determine if Mark will be able use his phone from the start to the end of the trip.

Note that 15 hours represents 100 % of the battery. Since it is told that Mark's battery charge at 60 % now, we can calculate 60 % of 15 hours to see how many hours are left. Let's write this situation as an equation by using the percent equation. a= p % * w In this equation, a represents the part, p % represents the percent, and w represents the whole. Let's substitute our values into the percent equation. a= 60 % * 15 We will now solve this equation for the value of a.

Since Mark's trip will take 8 hours and his phone's battery will last 9 hours, we can say that yes, he can use his phone during the whole trip.

E
Hint & Answer
S
Solution

A medicine company conducted an experiment on 2500 voluntary people about the effects of a new medicine. The results show that of the 2500 people, 10 had a mild allergic reaction to this new medicine.

scientist and a medicine emojis

Find the percent of the people that have a mild allergic reaction.

We are given that 10 out of the 2500 people that tried a new medicine had a mild allergic reaction. We want to find out what percent of the people had a mild allergic reaction. Let's start by drawing a bar diagram to represent the problem.

Next, we will write a percent proportion to find the percent. a/w=p/100 Note that a is the number of people with a mild allergic reaction and w is the total number of people. With this in mind, now substitute our values into the percent proportion. 10/2500=p/100 Note that we got an equation that we can solve for p. We will multiply both sides of the equation by 100 to get the percent value p alone.

We found that p=0.4. This means that 0.4 % of the people had a mild allergic reaction to the new medicine.

E
Hint & Answer
S
Solution

In the following figure, each grid is the same size.

14x5 rectangle with 19 green squares

What percent of the figure is drawn as green? Round the answer to the nearest integer.

We have been asked to establish the percent of the image is green. In other words, we want to know what percent of all of the squares are green squares. We will represent it by a percent equation. p % * b = a Here, b represents the whole which is the total number of squares and a is the part which is the total number of green squares.

As we can see, there are b=5 * 14 = 70 squares in total and a= 19 green squares in total. Now we can substitute these values into the percent equation and solve it for p.

We can conclude that about 27 % of the figure is green.

E
Hint & Answer
S
Solution
Mark uses a bar graph to display the results of a survey that asks several students to vote on a new candidate who will represent the school in a national dance festival.

school candidate choices

However, Mark forgot to write the scale labels.

 Find the percent of the votes that the most popular candidate received.

If there are 240 votes in total, how many votes does candidate C receive?

Note that the vertical axis is related to votes that each candidate received. However, there are no scale labels, except for the 0 one.

Without the scale we cannot tell how many votes each candidate received. However, we know that the number of lines each bar intersects tells how many portions of votes each candidate received. Let's count how many lines intersect the bars.

Next, let's add the found numbers to find how many overall portions of votes there were. 1+7+3+5+4 = 20 Notice that the most popular candidate, who is B, received 7 votes. Now that we also know the total portion of the votes, we can calculate the percent of the votes for the most popular candidate by using the percent equation. a=p % * w Recall that a represents the part which is the most popular candidate's portion of the votes, p is the percent, and w is the whole which represents the overall portions of the votes. With these in mind, let's substitute the obtained values into the percent equation.

We can conclude that the most popular candidate, who is candidate B, received 35 % of the votes.

Now, we assume that there are 240 votes total. We want to find how many votes are for candidate C. Recall the graph from the previous part in which we counted the number of tick lines each bar intersects.

From Part B we know that overall there are 20 portions of votes and 3 of them are votes for candidate C. Let's find the amount of percent of the votes that candidate C received by using the percent equation. a=p % * w ⇓ 3=p % * 20 Let's solve this equation!

We found that candidate C received the 15 % of the total number of votes. Now, let's calculate 15 % of 240.

This means that the candidate C received 36 votes.

E
Hint & Answer
S
Solution

The table shows the geometry scores of Mark in four tests. Mark needs to get 70 % of the total test points to pass the course.

Test Score Point Value
79 % 200
82.5 % 100
55 % 80
? 50

According to this table, what is the score that Mark needs on the last test to be able to pass the course?

We want to determine the score on the last test for which Mark earns 70 % of the total points on the tests. We can use this information to write the percent equation for the total number of points scored. Total Points Scored= 70 %( Total Point Value) Now, we can consider the given table.

Test Score Point Value
79 % 200
95.5 % 100
55 % 80
? 50

First, we can use the table to calculate the number of points scored on each test. We can do so by multiplying the test score by the point value for each test. Remember that we can represent the score on the last test with an arbitrary variable a.

Test Score Point Value Points Scored
79 % 200 79 %* 200 = 158
82.5 % 100 82.5 %* 100 = 82.5
55 % 80 55 %* 80 = 44
a % 50 a %* 50 = 0.5a

Now, we can calculate the sum of all point values and the sum of points scored in each test.

Test Score Point Value Points Scored
79 % 200 79 %* 200 = 158
82.5 % 100 82.5 %* 100 = 82.5
55 % 80 55 %* 80 = 44
a % 50 a %* 50 = 0.5a
Sum = 430 = 284.5+0.5a

Finally, we can substitute the values in our equation. Total Points Scored= 70 %( Total Point Value) ⇕ 284.5+0.5a= 70 % * 430 Now, we can solve the equation for a.

We found that Mark must get 301 in total from all four tests. Next, we will subtract 284.5 from both sides of the equation.

Finally, we will divide both sides of the equation by 0.5 to get the variable a alone.

Mark needs to score 33 % on the last test to be able to pass the course.

E
Hint & Answer
S
Solution

The Percent Proportion and Equation, and Percent of Change

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