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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.

Student Learning Objectives:
  • Calculate the percent change between two given amounts
  • Calculate relative and percent error
  • Use a percent proportion to solve an equation
Show more expand_more
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
Values can increase or decrease in many situations. This lesson explains how to find the percent of change between two amounts and how to write an 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 asks his Uncle Dev for money to buy a bike.

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

The following graph shows the number of toys sold by a 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 were construction toys?

b

What percent of the toys sold were 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

To find the percent of toys sold that are construction toys, we can use a percent proportion.

a/b=p/100 Here, a is the part, b is the whole, and p is the percent. The graph shows that 6 construction toys were sold by the store, so a = 6. Next, we find the whole by adding the corresponding values for the six categories shown in the graph.

Category Number
Dolls 8
Cars 15
Construction 6
Educational 4
Electronic 15
Animals 12
Sum 60

Now that we know the part and the whole, we can calculate the percent.

a/b=p/100
SubstituteII

a= 6, b= 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 construction toys made up 10 % of the total number of toys sold.

b

To find the percent of toys sold that were electronic toys, we use the same process as in Part A. We set the ratio of 15 electronic toys to 60 toys in total equal to p100 and solve for p.

a/b=p/100
SubstituteII

a= 15, b= 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 the Percent Proportion

Solve the given percent proportion for the missing 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 such cases, it is usually helpful 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 Suppose the whole of a value is 50. This represents 100 % of the value.

Then, 30 % of 50 can be calculated by multiplying these values.

a=30 % * 50
Example

Grouping Children According to Age Groups

A large daycare center groups its children into three age groups: infants, toddlers, and preschoolers.

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 children at the daycare center.

b

How many infants does the daycare center care for?

Hint
a

Express all categories as percents. Then, use the percent equation to find the value of the whole.

b

Use the percent equation to find the part.

Solution
a

In the given graph, the numbers of children in each age group are represented with percents or integers.

lr Toddlers: & 42 % Infants: & 23 % Preschoolers: & 140 The percentages in a circle graph add up to 100 %. This means the missing percent of preschoolers, x %, is the difference between 100 % and the sum of 42 % and 23 %. Let's write this situation as an equation and solve for x.

x =100 - (42 + 23)
AddTerms

Add terms

x=100-65
SubTerm

Subtract term

x=35

This means the 140 preschoolers make up 35 % of the total number of children in the daycare center. Now we can find the total number of children using a percent equation. a= p % * w We want to find the whole w. Let's substitute the values into the equation and solve for w.

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

There are 400 children in total at the daycare center.

b

To find the number of infants, we can use our results from Part A. We know that there are 400 children in total, so we have to find 23 % of 400, or 23 % of the whole 400. Let's substitute these values into the percent equation and simplify.

a=p % * w
SubstituteII

p= 23, w= 400

a= 23 % * 400
WriteDec

Write as a decimal

a=0.23 * 400
Multiply

Multiply

92

The daycare center cares for 92 infants.

Pop Quiz

Solving the Percent Equation

Solve the given percent 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 describe a change as a percent rather than as a number or ratio. We can do this using percent of change.

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 increase. Percent 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 decrease.

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

Calculating Percent of Change

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. The percent change 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. 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
Write the 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 Change as a Percent

The table shows the number of people that visited a small nature center each day over the last two weeks.

Table shows the number of people
a

What is the percent of change in the number of people that visit the nature center 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 nature center 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

On the first Monday, 75 people visited the nature center, but on the second Monday, 81 people visited. Since this is an increase, we use the formula for percent increase.

p %=New Amount - Original Amount/Original Amount Let's substitute the values into the formula and simplify.

p %=81- 75/75
▼
Simplify right-hand side
SubTerms

Subtract terms

p %=6/75
ReduceFrac

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

p %=2/25
ExpandFrac

a/b=a * 4/b * 4

p %=8/100

a/100=a %

p %=8 %

From the first week to the second, there was an 8 % increase in the number of people who visited the nature center on Monday.

b

In the second week, 90 visitors came to the nature center on Friday and 75 on Saturday. Since this is a decrease, we use the formula for percent decrease.

p %=Original Amount - New Amount/Original Amount Let's substitute the values and calculate the percent decrease.

p %=90- 81/90
▼
Simplify right-hand side
SubTerms

Subtract terms

p %=9/90
ReduceFrac

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

p %=1/10
ExpandFrac

a/b=a * 10/b * 10

p %=10/100

a/100=a %

p %=10 %

In the second week, the number of people who visited the nature center decreased by 10 % from Friday to Saturday.

Pop Quiz

Calculating the Percent of Change

Find the percent of change based on the values in the table. Then 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 Error as a Percent

Percent of change compares how much a value changes from its original amount. We can use a similar idea to compare an estimated or measured value with its actual value. The difference between these values is the error. When the error is written as a percent of the actual value, it is called the percent error.

Concept

Relative Error and Percent Error

Relative error is the ratio of the absolute error of a measurement to the exact value. It shows how large the error is relative to the size of the value being measured. In other words, 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

Percent error is the product of 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. Suppose he only caught 400 crayfish. The percent error explains the degree of the mistake in the fisherman'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. The percent error says that the estimation was wrong by 20 %, which is more significant information for future predictions.
Example

How Close Was the Estimate?

Emily found a cute teddy bear. She estimates that the bear will cost $12. The actual cost is $13.50.

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
We will start by finding the relative error, then multiply it by 100 % to express it as a percent error. Relative Error=Absolute Error/Exact Value The absolute error is the absolute value of the difference between the estimated price and the exact price. | $13.50- $12|= $1.50 Now we can use the relative error formula by substituting the absolute error of 1.50 and the exact value of 13.50.

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, we multiply this number by 100 % to express it as a percent.

0.111111 ... * 100 %
Multiply

Multiply

11.1111 % ...
RoundDec

Round to 1 decimal place(s)

≈ 11.1 %

The percent error between Emily's estimate and the actual price of the toy is about 11.1 %.

Closure

Calculating the Percent of Change for a Loan

The challenge presented at the beginning of the lesson focuses on Kevin's tough decision.

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
We start by finding the total amount Kevin has to repay his uncle by multiplying the 12 months in a year by the $7.70 he repays each month. $ 7.70 * 12 = $ 92.40 Next, we find how much more Kevin will repay than the amount he borrowed by subtracting the borrowed amount from the total repayment. $ 92.40- $ 84= $ 8.40 Finally, we use the percent of change formula to find what percent more Kevin will repay.

Percent of Change=Amount of Change/Original Amount
SubstituteII

Original Amount= 84, Amount of Change= 8.40

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 %

Kevin's uncle wants Kevin to pay back 10 % more than he would borrow.



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

What percent of 120 is 24?

In the percent proportion, the ratio of the part a to the whole b is equal to the percent p written as a fraction. a/b=p/100 Given any two of these three variables, we can find the third one. In this exercise, we are given the values a= 24 and b= 120, and we are looking for the percent p.

This means that 24 is 20 % of 120.

E
Hint & Answer
S
Solution

What is 32% of 50?

In the percent proportion, the ratio of the part a to the whole b is equal to the percent p written as a fraction. a/b=p/100 Given any two of these three variables, we can find the third one. In this exercise, we are given the values p= 32 and b= 50, and we are looking for the value of a.

This means that 16 is 32 % of 50.

E
Hint & Answer
S
Solution

0.4% of what number is 80?

In the percent proportion, the ratio of the part a to the whole b is equal to the percent p written as a fraction. a/b=p/100 Given any two of these three variables, we can find the third one. In this exercise, we are given the values p= 0.4 and a= 80, and we are looking for the value of b.

This means that 80 is 0.4 % of 20 000.

E
Hint & Answer
S
Solution

Find the percent of change between the following values. Original Amount: &366 New Amount: &153 Round the answer to the nearest tenth.

In order to determine a percent of change p, we can use the following general formula. p % =Amount of Increase or Decrease/Original Amount From the given values, we can see the original amount was 366 and the new amount is 153, which shows a decrease. 366-153= 213 The difference between the new amount is 213 and the original amount is 366. Now we can use the formula.

The percent decrease is about 58.2 %.

E
Hint & Answer
S
Solution

Mark estimated that a baby cat weighs 12 pounds. He used a scale and measured the actual weight of the baby cat — it is 15 pounds. What is the percent error in this case?

Mark's estimate about the baby cat's weight is 12 pounds, but the actual weight is 15 pounds. We want to find the percent error of this estimate. Let's remember the formula for it. Percent Error = Amount of Error/Actual Amount Let's first find the amount of error. Mark's estimate of 12 pounds is less than the actual amount of 15 pounds. We will subtract the Mark's estimate from the actual amount to get the amount of error. Amount of Error = 15-12 = 3 Now we will divide the amount of error by the actual amount to get the percent error. Percent Error = Amount of Error/Actual Amount ⇓ Percent Error = 3/15 Finally, let's simplify the obtained ratio and express it as a percent.

The percent error in this case is 20 %.

E
Hint & Answer
S
Solution

The Percent Proportion and Equation, and Percent of Change

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