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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.

Challenge

## Leaving a Tip

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

External credits: studiogstock

How much tip did they leave?

Discussion

## Interest

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.

Concept

## Simple Interest

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 simple interest per year on a deposit of
This means that the simple interest earned on in one year is 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 simple interest each year.
Years  Amount of Simple Interest  Balance
Notice that the balance at the end of each period is calculated by adding the principal and simple interest earned during that period.
Example

## Calculating the Amount of Interest

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 in this account as principal.

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

a If the bank offers simple interest per year, what will be the balance after years?
b If an account earns in years, what is the annual interest rate that the bank offers?

### Hint

a Use the simple interest formula.
b Substitute the given values into the simple interest formula and solve the equation for

### Solution

a Notice that the balance is the final amount of money in the account after adding the interest. The amount of principal is and the bank offers 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.
In this formula, is the amount of interest, is the principal, is the annual interest rate, and is the time in years. Since Vincenzo's money will stay in this account for years, substitute the values and into the formula to calculate the total amount of interest.
The account will earn in years. As a final step, add this amount to the principal to calculate the final balance in years.
The balance of the account will be after years.
b This time the amount of interest earned in 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.
Substitute these values into the formula and solve for the annual interest rate
Solve for
The bank offers a annual interest rate.
Example

## Who is Correct?

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

However, the salesperson says that to find the selling price, the price of the coat needs to be multiplied by Who is correct?

### Hint

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

### Solution

It is given that the sales tax rate in Miami is This means that to purchase an item, an additional of its original price must be paid as sales and use tax.
Calculate the tax for a coat in two ways to determine who is correct, Vincenzo or the salesperson.

### Vincenzo's Way

Vincenzo prefers to calculate the total cost in two steps, by finding of 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.
In this equation, represents the amount of tax,  is the percent, and is the original price of the item. Substitute the values into the percent equation and calculate  of Remember to write the percent as a decimal number.
The amount of the tax for a  coat is Now add this amount to the original price of the coat to find the total cost.
Vincenzo can buy this coat for Next, calculate the total cost following the salseperson's method.

### The salesperson's Way

The salesperson calculates the total cost by multiplying by 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

### Conclusion

Notice that both methods gave the same result. This is because multiplying by or and adding the result to is the same as multiplying by
Both of the ways are correct and give the same total cost. This means that both Vincenzo and the salesperson are correct.
Example

## Calculating the Markup and Markdown

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.

a Vincenzo received a discount on the jeans he purchased. What percent discount did he receive on them? Round the answer to the nearest integer.
b A few days later, Vincenzo saw the same shoes on an online shopping platform. He noticed that the price in-store had a markup compared to the price online. What is the price of the sneakers online? Round the answer to two decimal places.

### Hint

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.

### Solution

a Vincenzo received a discount on the jeans, but the percent of that discount is unknown. Start by examining the receipt to find this amount as a percent.
According to the receipt, the jeans cost and Vincenzo received a discount of Now, recall the formula for the percent equation.
In the percent equation, the part of the whole is the percent multiplied by the whole Here, the discount of dollars is the part the percent is and the whole is the original price, dollars.
Solve for
The discount is equal to about of the original price.
b Vincenzo noticed that the price of the sneakers in store includes a markup compared to the price online. To find the price online, start by adding to
The in-store price of the sneakers is of the price online. Now, recall the formula for the percent equation once again.
According to the receipt, the in-store price of the sneakers is This represents the part The percent is and the whole is the price online. Now, solve the equation for
Solve for
The price of the sneakers is about on the online platform.
Example

## Calculating a Bonus

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 of the contract's total value.

If Vincenzo's aunt makes a month, how much money will she make as a bonus?

### Hint

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

### Solution

Start by calculating the total value of the contract. Since Vincenzo's aunt earns per month, multiply this amount by to find the total amount of money that she earns in two years.
She earns dollars in two years. Per her contract, of this amount will be given as a bonus. The percent equation can be used to calculate of
Recall that the part is equal to the product of the percent and the whole Substitute and into the percent equation to find the value of
Vincenzo's aunt will get as a bonus after working for the company for two years.
Example

## Paying Back a Loan

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

a Help Vincenzo's uncle calculate how much money he will pay in interest.
b How much will he pay back altogether?

### Hint

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.

### Solution

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.
Vincenzo's uncle wants to know how much money he will pay in interest. Start by using the percent equation to find the amount of interest he will pay in one year.
In this case, the of the loan is and the is Let the be represented by For simplicity, ignore the units for now.
Next, solve the equation for 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 Finally, multiply the interest for one year by to calculate the total amount of interest that Vincenzo's uncle will pay over years.
Vincenzo's uncle will pay 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, is the amount of interest, is the principal, is the interest rate, and is the time in years.
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 in total in interest over the four years of the loan.
This means that he will pay altogether when repaying the loan.
Closure

## Calculating the Amount of Tip

The challenge presented at the beginning of the lesson was to find of 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.
Note that the percent is and the total cost of the meal represents the whole, which is Now substitute these values into the percent equation.
Evaluate the right-hand side of the equation to find the value of Remember to rewrite the percent as a fraction.
They left a tip.