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This lesson focuses on the several ways to manage money and explores some related concepts. Concepts such as sales tax, markup, discount, markdown, principal, and simple interest will be introduced through examples. By the end of the lesson, a general understanding of these concepts and confidence in managing financial matters will be increased.
### Catch-Up and Review

**Here are a few recommended readings before getting started with this lesson.**

Vincenzo went out for dinner together with his family. The total cost of the meal was $$124.$ Because they really enjoyed the food and the service, they left a $20%$ tip.

External credits: studiogstock

How much tip did they leave?

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*Interest* is the cost of borrowing money from a bank or additional money earned from keeping money in a bank account. There are several types of interest, but only *simple interest* will be introduced in this lesson.

The interest that is applied only to an initial amount of money is called simple interest. The initial amount is known as the principal. Simple interest is calculated as a product of principal, annual interest rate, and the time in years.

An interest rate is a percent used to calculate the interest on the principal. It may be easier to write it in decimal form to make the calculations easier. For instance, assume that a savings account earns $3%$ simple interest per year on a deposit of $$1000.$$1000⋅0.03⋅1=$30 $

This means that the simple interest earned on $$1000$ in one year is $$30.$ The final amount of money in the account is called the balance. The following table shows the balance over five years of an account that earns $3%$ simple interest each year. Years | Amount of Simple Interest | Balance |
---|---|---|

$1$ | $1000⋅0.03⋅1=$30$ | $1000+30=$1030$ |

$2$ | $1000⋅0.03⋅2=$60$ | $1000+60=$1060$ |

$3$ | $1000⋅0.03⋅3=$90$ | $1000+90=$1090$ |

$4$ | $1000⋅0.03⋅4=$120$ | $1000+120=$1120$ |

$5$ | $1000⋅0.03⋅5=$150$ | $1000+150=$1150$ |

Vincenzo is a very thrifty child and saves some of his allowance every month. His father thinks that opening a savings account for Vincenzo's savings would be great for him. He wants to support Vincenzo and plans to deposit $$5000$ in this account as principal.

After talking with several bankers, he notices that banks offer different interest rates.

a If the bank offers $4%$ simple interest per year, what will be the balance after $10$ years?

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b If an account earns $$100$ in $4$ years, what is the annual interest rate that the bank offers?

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a Use the simple interest formula.

b Substitute the given values into the simple interest formula and solve the equation for $r.$

a Notice that the balance is the final amount of money in the account after adding the interest. The amount of principal is $$5000$ and the bank offers $4%$ simple interest for this amount. Using this information, the interest for Vincenzo's savings account can be calculated. Recall the simple interest formula to find the amount of interest.

$I=Prt $

In this formula, $I$ is the amount of interest, $P$ is the principal, $r$ is the annual interest rate, and $t$ is the time in years. Since Vincenzo's money will stay in this account for $10$ years, substitute the values $P=5000,$ $r=4%,$ and $t=10$ into the formula to calculate the total amount of interest.
$I=Prt$

SubstituteValues

Substitute values

$I=5000⋅4%⋅10$

WriteDec

Write as a decimal

$I=5000⋅0.04⋅10$

Multiply

Multiply

$I=2000$

$5000+2000=7000 $

The balance of the account will be $$7000$ after $10$ years.
b This time the amount of interest earned in $4$ years is given. Since the values of principal, time, and the amount of interest are known, the formula for simple interest can be applied again.

$I=Prt $

Substitute these values into the formula and solve for the annual interest rate $r.$
$I=Prt$

SubstituteValues

Substitute values

$100=5000⋅r⋅4$

Solve for $r$

Multiply

Multiply

$100=20000r$

DivEqn

$LHS/20000=RHS/20000$

$20000100 =r$

ReduceFrac

$ba =b/100a/100 $

$2001 =r$

WriteDec

Write as a decimal

$0.005=r$

WritePercent

Convert to percent

$0.5%=r$

RearrangeEqn

Rearrange equation

$r=0.5%$

Vincenzo goes to Miami, Florida, to visit his aunt and uncle. The city's sales tax rate is $7%.$ Vincenzo wants to buy a new coat. The price of the coat before tax is $$120.$ Vincenzo says that the total cost can be calculated by finding $7%$ of $$120$ and adding this value to $$120.$

However, the salesperson says that to find the selling price, the price of the coat $$120$ needs to be multiplied by $1.07.$ Who is correct?{"type":"choice","form":{"alts":["Vincenzo","The salesperson","Both of them","Neither of them"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":2}

Calculate the total cost using the methods described by Vincenzo and the salesperson. Use the percent equation to calculate the amount of the tax.

It is given that the sales tax rate in Miami is $7%.$ This means that to purchase an item, an additional $7%$ of its original price must be paid as sales and use tax.
### Vincenzo's Way

Vincenzo prefers to calculate the total cost in two steps, by finding $7%$ of $$120$ and then adding this amount to the original price of the coat. Recall that a percent of a number can be calculated as a product of the percent and that number. The percent equation can be helpful to do that.
The amount of the tax for a $$120$ coat is $$8.40.$ Now add this amount to the original price of the coat to find the total cost.
### The salesperson's Way

The salesperson calculates the total cost by multiplying $$120$ by $1.07.$ Check the result to see if it is the same as Vincenzo's solution.
According to the salesperson's calculations, the total cost of the coat is $$128.40.$ ### Conclusion

Notice that both methods gave the same result. This is because multiplying $120$ by $7%,$ or $1007 ,$ and adding the result to $120$ is the same as multiplying $120$ by $1.07.$
**both** Vincenzo and the salesperson are correct.

$Selling Price=Original Price+Tax $

Calculate the tax for a $$120$ coat in two ways to determine who is correct, Vincenzo or the salesperson. $a=p%⋅w $

In this equation, $a$ represents the amount of tax, $p$ is the percent, and $w$ is the original price of the item. Substitute the values into the percent equation and calculate $7%$ of $$120.$ Remember to write the percent as a decimal number.
$a=p%⋅w$

SubstituteII

$p=7$, $w=120$

$a=7%⋅120$

PercentToFrac

$a%=100a $

$a=1007 ⋅120$

MoveRightFacToNum

$ca ⋅b=ca⋅b $

$a=100840 $

CalcQuot

Calculate quotient

$a=8.40$

$120+8.40=128.40 $

Vincenzo can buy this coat for $$128.40.$ Next, calculate the total cost following the salseperson's method. $120⋅1.07$

WriteSum

Write as a sum

$120⋅(1+0.07)$

Distr

Distribute $120$

$120⋅1+120⋅0.07$

Multiply

Multiply

$120+8.40$

AddTerms

Add terms

$128.40$

$120⋅1007 +120⇔120⋅(1.07) $

Both of the ways are correct and give the same total cost. This means that Vincenzo visits several more shopping centers in Miami. He buys a pair of sneakers and some jeans at a department store. Below is his receipt.

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b A few days later, Vincenzo saw the same shoes on an online shopping platform. He noticed that the price in-store had a $20%$ markup compared to the price online. What is the price of the sneakers online? Round the answer to two decimal places.

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a *Discount* or *markdown* refer to a reduction in the price of an item.

b *Markup* refers here to the difference between the price in-store and the price online.

a Vincenzo received a $$10$ discount on the jeans, but the percent of that discount is unknown. Start by examining the receipt to find this amount as a percent.

$a=p%⋅w $

In the percent equation, the $a=p%⋅w$

SubstituteII

$a=10.00$, $w=34.00$

$10.00=p%⋅34.00$

Solve for $p$

RearrangeEqn

Rearrange equation

$p%⋅34.00=10.00$

DivEqn

$LHS/34.00=RHS/34.00$

$p%=34.0010.00 $

UseCalc

Use a calculator

$p%=0.294117…$

WritePercent

Convert to percent

$p%=29.4117…%$

RoundInt

Round to nearest integer

$p%≈29%$

b Vincenzo noticed that the price of the sneakers in store includes a $20%$ markup compared to the price online. To find the price online, start by adding $20%$ to $100%.$

$100%+20%=120% $

The in-store price of the sneakers is $120%$ of the price online. Now, recall the formula for the percent equation once again.
$a=p%⋅w $

According to the receipt, the in-store price of the sneakers is $$44.00.$ This represents the $a=p%⋅w$

SubstituteII

$a=44.00$, $p%=120%$

$44.00=120%⋅w$

Solve for $w$

PercentToFrac

$a%=100a $

$44.00=100120 ⋅w$

MoveRightFacToNum

$ca ⋅b=ca⋅b $

$44.00=100120w $

MultEqn

$LHS⋅100=RHS⋅100$

$4400=120w$

DivEqn

$LHS/120=RHS/120$

$1204400 =w$

CalcQuot

Calculate quotient

$36.666666…=w$

RoundDec

Round to $2$ decimal place(s)

$36.67≈w$

RearrangeEqn

Rearrange equation

$w≈36.67$

Vincenzo's aunt signed an agreement when she joined a company two years ago. If she is still working with the company after two years, she will receive a bonus equivalent to $8%$ of the contract's total value.

If Vincenzo's aunt makes $$1200$ a month, how much money will she make as a bonus?{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<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>","formTextAfter":null,"answer":{"text":["2304"]}}

Calculate the total amount of money that she earns in twenty-four months, then calculate $8%$ of that amount.

Start by calculating the total value of the contract. Since Vincenzo's aunt earns $$1200$ per month, multiply this amount by $24$ to find the total amount of money that she earns in two years.
*part* $a$ is equal to the product of the percent $p$ and the *whole* $w.$ Substitute $p%=8%$ and $w=28800$ into the percent equation to find the value of $a.$
Vincenzo's aunt will get $$2304$ as a bonus after working for the company for two years.

$1200⋅24=28800 $

She earns $28800$ dollars in two years. Per her contract, $8%$ of this amount will be given as a bonus. The percent equation can be used to calculate $8%$ of $28800.$
$a=p%⋅w $

Recall that the $a=p%⋅w$

SubstituteII

$p=8$, $w=28800$

$a=8%⋅28800$

PercentToFrac

$a%=100a $

$a=1008 ⋅28800$

MoveRightFacToNum

$ca ⋅b=ca⋅b $

$a=1008⋅28800 $

SplitIntoFactors

Split into factors

$a=1008⋅288⋅100 $

CancelCommonFac

Cancel out common factors

$a=1008⋅288⋅100 $

SimpQuot

Simplify quotient

$a=8⋅288$

Multiply

Multiply

$a=2304$

Vincenzo's uncle borrows $$2400$ to repair his car. He will pay off the loan over $4$ years by paying back the principal plus $3.5%$ simple interest for each year.

a Help Vincenzo's uncle calculate how much money he will pay in interest.

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b How much will he pay back altogether?

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a Start by using the percent equation to find the amount of interest Vincenzo's uncle will pay over one year.

b Add the amount of the loan to the total amount of interest found in Part A.

a Remember that *simple interest* represents a proportional relationship between the annual interest and the principal, or initial amount, of a certain amount of money. The ratio of annual interest to the principal is the interest rate.

$part=percent×whole↓interest amount=interest rate×principal $

In this case, the $principal$ of the loan is $$2400$ and the $interest$ $rate$ is $3.5%.$ Let the $interest$ $amount$ be represented by $s.$ For simplicity, ignore the units for now.
$interest amount=interest rate×principal↓s=3.5%×2400 $

Next, solve the equation for $s.$ Remember that percents can be written as decimals by moving the decimal point two places to the left.
This means that the simple interest for one year is equal to $$84.$ Finally, multiply the interest for one year by $4$ to calculate the total amount of interest that Vincenzo's uncle will pay over $4$ years.
$$84⋅4=$336 $

Vincenzo's uncle will pay $$336$ in total in interest. It is worth keeping in mind that this amount can be found directly by using the simple interest formula. In this formula, $I$ is the amount of interest, $P$ is the principal, $r$ is the interest rate, and $t$ is the time in years.
$Prt=I⇓2400⋅3.5%⋅4=336 $

b To find out how much Vincenzo's uncle will pay in total when repaying the loan, add the principal to the total amount of interest. Remember that it was found in Part A that Vincenzo's uncle will pay $$336$ in total in interest over the four years of the loan.

$$2400+$336=$2736 $

This means that he will pay $$2736$ altogether when repaying the loan.
The challenge presented at the beginning of the lesson was to find $20%$ of $$124$ in order to find the amount to tip. As done in the previous examples, use the percent equation to calculate a percent of the whole.
*percent* $p$ is $20%$ and the total cost of the meal represents the *whole,* which is $$124.$ Now substitute these values into the percent equation.
They left a $$24.80$ tip.

$a=p%⋅w $

Note that the $a=20%⋅$124 $

Evaluate the right-hand side of the equation to find the value of $a.$ Remember to rewrite the percent as a fraction.
$a=20%⋅124$

PercentToFrac

$a%=100a $

$a=10020 ⋅124$

MoveRightFacToNum

$ca ⋅b=ca⋅b $

$a=10020⋅124 $

Multiply

Multiply

$a=1002480 $

CalcQuot

Calculate quotient

$a=24.80$