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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

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.

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

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Discussion

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

Concept

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.

$ba =100p $

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

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.

a What percent of the toys sold are construction toys?

b What percent of the toys sold are electronic toys?

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

b Use the percent proportion.

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.

$wa =100p $

Recall what each variable represents. $awp →Part→Whole→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=Cars=Construction=Educational=Electronic=Animals= 815641512 $

Now calculate the sum of these numbers to find the value of $w.$
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.
$wa =100p $

SubstituteII

$a=6$, $w=60$

$606 =100p $

▼

Solve for $p$

ReduceFrac

$ba =b/6a/6 $

$101 =100p $

MultEqn

$LHS⋅100=RHS⋅100$

$10100 =p$

CalcQuot

Calculate quotient

$10=p$

RearrangeEqn

Rearrange equation

$p=10$

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.

$wa =100p $

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 $p.$
$wa =100p $

SubstituteII

$a=15$, $w=60$

$6015 =100p $

▼

Solve for $p$

ReduceFrac

$ba =b/15a/15 $

$41 =100p $

MultEqn

$LHS⋅100=RHS⋅100$

$4100 =p$

CalcQuot

Calculate quotient

$25=p$

RearrangeEqn

Rearrange equation

$p=25$

Pop Quiz

Solve the given percent proportion for the missing value. Note that $a$ is part of the whole $w$ and $p%$ or $100p $ is the percent value.

Discussion

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

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

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 Find the total number of toys.

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b Find the number of toys for infants.

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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*.

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

$ToddlerInfantPreschooler ⇒42%⇒23%⇒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 $x.$
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.
$140is35%of what number? $

This question can be solved by using a percent equation.
$a=p%⋅w $

Recall that $a$ is the value of $a=p%⋅w$

SubstituteII

$a=140$, $p=35$

$140=35%⋅w$

▼

Solve for $w$

WriteDec

Write as a decimal

$140=0.35⋅w$

DivEqn

$LHS/0.35=RHS/0.35$

$0.35140 =w$

RearrangeEqn

Rearrange equation

$w=0.35140 $

CalcQuot

Calculate quotient

$w=400$

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 is23%of400? $

This question can be solved by using a percent equation which states that the product of the $a=p%⋅w $

Notice that $400$ is the $a=p%⋅w$

SubstituteII

$p=23$, $w=400$

$a=23%⋅400$

WriteDec

Write as a decimal

$a=0.23⋅400$

Multiply

Multiply

$92$

Pop Quiz

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.

Discussion

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 or *percent change* is the percent that expresses the amount of change as a percent of the original amount. It is calculated as a ratio of the change in the amount to the original amount.
*greater* than the original amount, the percent of change is called a percent of increase.
*less* than the original amount, the percent of change is called a percent of decrease.

$Percent of Change,p%=Original AmountAmount of Change $

If the new amount is $Percent of Increase=Original AmountNew Amount−Original Amount $

If the new amount is $Percent of Decrease=Original AmountOriginal Amount−New Amount $

Discussion

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.
*expand_more*

*expand_more*
*expand_more*
*expand_more*

$Percent Change=Original AmountAmount of Change $

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

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

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

Now that the amount of increase and the original amounts are known, find the ratio of these amounts.

4

Write the Result as a Percent

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

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.

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?

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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?

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?

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.

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.

Note that the new amount is $81$ and the original amount is $75.$$Percent of Increase=7581−75 $

Now calculate this percent of increase!
$Percent of Increase=7581−75 $

▼

Simplify right-hand side

SubTerms

Subtract terms

$Percent of Increase=756 $

ReduceFrac

$ba =b/3a/3 $

$Percent of Increase=252 $

ExpandFrac

$ba =b⋅4a⋅4 $

$Percent of Increase=1008 $

WritePercent

Convert to percent

$Percent of Increase=8%$

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=90Saturday=75 $

Notice that the number of people decreased. This means that the percent of change will be found by using the $Percent of Decrease=9090−81 $

Now calculate this percent of decrease!
$Percent of Decrease=9090−81 $

▼

Simplify right-hand side

SubTerms

Subtract terms

$Percent of Decrease=909 $

ReduceFrac

$ba =b/9a/9 $

$Percent of Decrease=101 $

ExpandFrac

$ba =b⋅10a⋅10 $

$Percent of Decrease=10010 $

WritePercent

Convert to percent

$Percent of Decrease=10%$

Pop Quiz

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.

Discussion

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

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 $=Exact ValueAbsolute Error $

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

Relative Error $=Exact Value∣Measured 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 |
---|---|---|

$∣480−400∣=80$ | $40080 =0.2$ | $0.2⋅100%=20%$ |

Example

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.{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["PI"]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"About","formTextAfter":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.80556em;vertical-align:-0.05556em;\"><\/span><span class=\"mord\">%<\/span><\/span><\/span><\/span>","answer":{"text":["11.1"]}}

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

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

$Relative Error=Exact ValueAbsolute Error $

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=Exact ValueAbsolute Error $

SubstituteII

$Absolute Error=1.50$, $Exact Value=13.50$

$Relative Error=13.501.50 $

UseCalc

Use a calculator

$Relative Error=0.111111…$

Closure

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!

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

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 $12.$
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!

$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=Original AmountAmount of Change $

Calculate the difference between the price of the bicycle and the amount Kevin will pay back to find the amount of change.
$Amount of Change92.40−84=8.40 $

Now, substitute the obtained values into the percent of change formula.
$Percent of Change=Original AmountAmount of Change $

SubstituteII

$Amount of Change=8.40$, $Original Amount=84$

$Percent of Change=848.40 $

ReduceFrac

$ba =b/8.4a/8.4 $

$Percent of Change=101 $

ExpandFrac

$ba =b⋅10a⋅10 $

$Percent of Change=10010 $

WritePercent

Convert to percent

$Percent of Change=10%$

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

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

The two can now take a cruise around town together!

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