Average daily balance method math

Unit II Discussion Board Math

MAT 1301, Liberal Arts Math 1

Course Learning Outcomes for Unit II Upon completion of this unit, students should be able to:

1. Apply mathematical principles used in real-world situations. 1.1 Apply percent concepts and equations to solve problems. 1.2 Utilize the average daily balance method to compute credit card charges. 1.3 Utilize the simple and compound interest formula to solve applied problems. 1.4 Calculate the annual percentage rate. 1.5 Compute finance charges on a credit card using the unpaid balance method.

Reading Assignment Chapter 8: Consumer Mathematics: The Mathematics of Everyday Life

 Section 8.1: Percents, Taxes, and Inflation, pp. 379-387

 Section 8.2: Interest, pp. 388-398

 Section 8.3: Consumer Loans, pp. 399-407

 Section 8.6: Looking Deeper: Annual Percentage Rate, pp. 424-430

Unit Lesson Chapter 8: Consumer Mathematics Assume that you are buying a car. Before paying for it, you research different loans and calculate the amount the interest you will owe if you chose a particular loan. Once a loan is chosen, you write a check for the down payment and balance your checkbook accordingly. Although this scenario may not be a common occurrence, the use of mathematics it portrays is common. The basic math skills required to manage money is called consumer mathematics. Consumer mathematics is frequently used every day. The information shared in this lesson will help you understand the basis behind taxes, raises, interest payments, loans, and other financial situations that arise. 8.1 Percents, Taxes, and Inflation: In this section, we will learn how to use percentages to calculate taxes and percent of inflation. Percent Many monetary calculations use the notion of percentages. Percentages are an easy way to describe the amount of something. For example, you may be required to pay off 10% of your $1,000 loan each month. This means that you will be paying $100 each month. We say the percentage instead of the actual amount for simplicity and because the amount of the loan may vary. Writing Percent Using Decimals The word “percent” means “per hundred”. Therefore, one percent is one part to a hundred parts. We write a percent as a fraction where one hundred is the denominator.

UNIT II STUDY GUIDE

Consumer Mathematics

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Any fraction with a denominator of 100 can easily be changed to a decimal by dividing by hundred or moving the decimal two places to the left. This is shown in the example below:

Example: Convert each percent to a decimal:

a) 78% b) 3% c) 27.35% d) 0.08%

Solution:

a) Convert 78% to a decimal.

We can think of 78% as seventy-eight hundredths or 78

100 .

Therefore, we write 78 and move the decimal to the left two places.

The decimal form of seventy-eight hundredths is 0.78.

b) Convert 3% to a decimal.

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We can think of 3% as three-hundredths. Therefore, we write 3.and move the decimal to the left two places.

3% = 0.03

The decimal form of three-hundredths is 0.03.

c) Convert 27.35% to a decimal.

We can write 27.35% as

Since we are dividing by 100, we will move the decimal two places to the left.

The decimal form of 27.35% is 0.2735.

d) Convert 0.08% to a decimal.

We can write 0.08% as

Since we are dividing by 100, we will move the decimal two places to the left.

0.08% = 0.0008

The decimal form of 0.08% is 0.0008. Converting a Decimal to a Percent We have learned to move the decimal of a number two places to the left when converting a percent to a decimal. We can do the opposite when converting a decimal to a percent. Example: Write 0.0035 as a percent. Solution: We can write 0.0035 as a percent by multiplying the decimal by 100 and adding a % symbol at the end of the number. When we multiply by 100, we are really moving the decimal two places to the right.

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Converting a Fraction to a Percent Fractions can be converted to percentages by dividing the denominator into the numerator and then multiplying the decimal by 100. For example:

Example: Convert each fraction to a percent:

a) 3

4

b) 9

25

c) 11

8

Solution:

a) Convert 3

4 to a percent.

We must first divide the denominator into the numerator and then convert the decimal form into a percent.

3 ÷ 4 = 0.75

We can now multiply 0.75 by 100 and add the % symbol.

b) Convert 9

25 to a percent.

We must first divide the denominator into the numerator and then convert the decimal form into a percent.

9 ÷ 25 = 0.36

We can now multiply 0.36 by 100 and add the % symbol.

c) Convert 11

8 to a percent.

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We must first divide the denominator into the numerator and then convert the decimal form into a percent.

11 ÷ 8 = 1.375

We can now multiply 1.375 by 100 and add the % symbol.

Percent of Change The percent of change refers to an increase or decrease in an amount. For example, assume that there are 50 students in a course. It is projected that the percent of change will increase by 50% next year. This means that an additional 25 students will take the course next year making the total number of students to be 75. We can use the following formula when calculating a percent of change:

The base amount is the amount that we started with. It is sometimes referred to as our original or initial amount. Example: Dealer markup on a car: If a dealer buys a car from the manufacturer for $19,875 and then sells it for $21,065, what is his markup? Solution: For this problem, we will use the following formula:

We substitute our given information and solve, converting the final decimal answer to a percent:

This tells us that the markup (percent of change) is 6%. The Percent Equation The percent equation is used by taking some percent of a base quantity and setting it equal to an amount. The percent equation is:

We will learn how to use this equation by doing the following examples. Example:

a) 12 is what percent of 80? b) 77 is 22% of what number?

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c) What is 23% of 140? Solution: For each problem, we will use the percent equation.

The first step in solving these problems is to identify what information we are given and what information we are asked to find.

a) 12 is what percent of 80?

We will use the formula:

b) 77 is 22% of what number?

We will use the formula:

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First, convert 22% into a decimal by dividing by 100 or by moving the decimal two places to the left.

a) What is 23% of 140?

We will use the formula;

First, convert 23% into a decimal by dividing by 100 or by moving the decimal two places to the left.

Example: In a certain fiscal quarter, 40% of McDonald’s profits came from Happy Meals. If Happy Meals’ profits were $508 million, how much profit came from selling other items? Solution: We will use the formula:

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In this problem, $508 million accounted for 40% of McDonald’s profits. Therefore, the base is not known. First, convert 40% to a decimal, then substitute the given information into the formula and solve for the base.

Therefore, $1,270 million were total profits realized by McDonald’s that quarter. To find the profits that came from items other than Happy Meals we will need to subtract the Happy Meal profits from the total base profits.

$1,270 - $508 = $762, so $762 million in profits came from other items. Taxes Taxes are usually given in the form of a percent. We will use the same concepts learned above to solve tax- related problems. Example: Use the following table to calculate the federal income tax due for a taxable income of $148,000:

If your taxable income is over—

But not over— The tax is Of the amount over—

Line 1 $0 7,550 ………………10% 0$

Line 2 7,550 30,650 $755.00 + 15% 7,550

Line 3 30,650 74,200 $4,220.00 + 25% 30,650

Line 4 74,200 154,800 $15,107.50 + 28% 74,200

Line 5 154,800 336,550 $37,675.50 + 33% 154,800

Line 6 336,550 ……………. $97,653.00 + 38% 336,550 Federal income taxes due for a single person.

Solution: We will use Line 4 because the taxable income is greater than $74,200 but less than $154,800. Line 4 states that the tax is $15,107.50 + 28% for any amount over 74,200. Therefore, we will subtract 74,200 from our amount (148,000) to find the amount that will be taxed.

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Therefore, $35,771.50 is owed. 8.2 Interest: Interest is the amount of money that is accumulated over time from a base amount. In today’s world, people can earn interest from a savings account or can pay interest for a new car, house, or loan. In this section, we will learn how to use simple interest and compound interest formulas. We will need to know some key terms before calculating interest. The chart below contains some terms that will be used in both simple interest and compound interest formulas.

Letter Used in Equation

Term Definition

P Principal The amount of money that is deposited. This is sometimes called the present value.

r Interest rate The percentage earned on the deposit. The rate is given as a percent.

t Time (in years) The length of time that the deposit remains in the bank. The time is usually stated in years.

I Interest earned The amount of money earned on your deposit.

A Future value The amount that will be in your account at some point in the future.

Simple Interest

The most basic way to compute interest is by using the simple interest formula. This formula calculates the amount of interest earned from depositing a principal (P) amount over a certain length of time (t) for a defined interest rate (r). Formula for Computing Simple Interest Rate – We calculate simple interest using the following formula:

Where I is the interest earned, P is the principal, r is the annual interest rate, and t is the time in years. As shown by the formula, the interest earned (I) is found by multiplying the principal (P), the annual interest rate (r), and the time (t) together. Note:

 The annual interest rate (r) must be converted to a decimal before it is substituted into the formula.

 The time (t) must be converted to years before it is substituted into the formula. For example, 6 months equals 0.5 years.

Example:

Use the simple interest formula, 𝐼 = 𝑃𝑟𝑡 and elementary algebra to find the missing quantity: I = ?, P = $1,000, r = 8%, t = 3 years

Solution: First, convert the interest rate (r) into decimal form. To do this, divide 8 by 100 or move the decimal two places to the left.

r = 8% = .08

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Next, substitute the given values into the simple interest formula and solve.

Example: Use the simple interest formula 𝐼 = 𝑃𝑟𝑡 and elementary algebra to find the missing quantity:

I = $700, P = $3,500, r = ?, t = 4 years Solution: We are asked to find the interest rate (r). We will substitute the values given for I, P, and t and solve for r. 𝐼 = 𝑃𝑟𝑡 700 = 3,500 ∙ 𝑟 ∙ 4 Multiply: 3500 × 4 = 14,000 700 = 14,000 ∙ 𝑟

700

14,000 = 𝑟 Divide: 700 ÷ 14,000 = 0.05

0.05 = 𝑟 Next, convert r = 0.05 to a percent by multiplying by 100 or by moving the decimal two places to the right. Therefore, r = 5%. The simple interest formula allowed us to calculate the interest earned from our deposit. The next formula will allow us to calculate the future amount of our deposit over a certain number of years. Computing the Future Value using Simple Interest –To find the future value of an account that pays simple interest, use the formula:

Where A is the future value, P is the principal, r is the annual interest, and t is the time in years. Example:

Use the future value formula and elementary algebra to find the missing quantity.

A = $1,770, P = ?, r = 6%, t = 3 years Solution: First, we need to convert the rate into decimal form. To do this, divide 6 by 100 or move the decimal two places to the left.

r = 6% = .06

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Now, substitute and solve.

Therefore, P = $1,500. Example: Use the future value formula and elementary algebra to find the missing quantity.

A = $966, P = $840, r = 5%, t =? Years Solution: First, we need to convert the rate into decimal form. To do this, divide 6 by 100 or move the decimal two places to the left.

r = 5% = .05 Now, substitute and solve.

Therefore, t = 3 years Compound Interest Compounding interest means the bank is paying interest on the previously earned interest. For example, assume that we deposited $100 at 10% interest. Simple interest only applies the 10% to the $100. Compound interest will apply the 10% to the 100 plus any additional money that was earned through the interest.

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Compound Interest Formula – Assume that an account with principal P is paying an annual interest r and compounding is being done m times per year. If the money remains in the account for n time periods, then the future value, A, of the account is given by the formula:

This formula introduces two new variables:

 m = number of times compounding is done per year (compounding period)

 n = number of compounding periods

Example: Use the formula for computing future value using compound interest to determine the value of the account at the end of the specified time period.

Solution: First, convert 7% to a decimal. To do this, divide by 100 or move the decimal two places to the left.

7% = .07

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Example: Use the formula for computing future value using compound interest to determine the value of the account at the end of the specified time period.

Solution: First, convert 4% to a decimal. To do this, divide by 100 or move the decimal two places to the left.

4% = .04 Next, substitute and solve: Example:

For this exercise, you are given an annual interest rate and the compounding period for two investments. Decide which the better investment is.

4.75% compounded monthly; 4.70% compounded daily Solution:

You must compare (1 + 𝑟

𝑚 )𝑛 for each case where m is the number of times the money compounds in a single

year.

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Solving for both we get,

Therefore, we can see that 4.75% compounded monthly is the better choice. Solving for Unknowns in the Compound Interest Formula We will be asked to solve for unknown variables using the compound interest formula. This formula contains exponential expressions. Therefore, we need to learn the properties of exponents and how to use these properties to solve for an unknown variable in the exponential position The diagram below lists the basic components of an exponential equation. As shown, the base is the number that is being multiplied, the exponent is the small number attached to the base which tells us how many times the base is being multiplied to itself, and the answer is the solution after the exponent is applied to the base.

Note: The exponent is sometimes referred to as a power. The common logarithmic function is used to solve for an exponent. The logarithmic function has a base of 10. This function is located on your calculator and is usually represented by the word “log.” Try practicing with your calculator by pressing the “log” key and then entering 100,000. You should receive an answer of 5. This means that 10 raised to the 5th power is 100,000.

The exponential problem, 105 = 100,000 is related to the following logarithmic problem:

Now that we know some key terms, we will introduce a property that will enable us to solve for exponents: Exponent Property of the Log Function –

This property enables us to solve for the exponent by taking it out of its original position.

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Example: Solve:

Solution: X is in the exponent position. Therefore, we will use the exponent property of the log function to solve for x. Steps:

Example: Solve:

(1.15)𝑥 = 3 Solution: X is in the exponent position. Therefore, we will use the exponent property of the log function to solve for x. Steps:

We use the exponent property of log functions to solve for the time variable in compound interest problems.

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Example: You have $1,500 to invest. Find the time in years that it takes your investment to double with annual compounding if your annual interest rate is 3.5%. Solution:

We will use the compound interest formula,𝐴 = 𝑃 (1 + 𝑟

𝑚 )

𝑛

, to solve. The problem states that we want to find

the number of years it will take to double our investment. Therefore, our investment ($1,500) is our principle (P) and our future value is double our investment (2 ∙ 1500 = $3,000). The problem also states that compounding is done once a year, so m=1. Our annual interest rate is 3.5%. The variables are as follows:

A = 2 x $1,500 = $3,000 P = $1,500

r = 3.5% = .035 m = 1

Substituting we have:

Example: Use the compound interest formula 𝐴 = 𝑃(1 + 𝑟)𝑡 and the given information to solve for t.

A = $1,500 P = $1,000

r = 4% = .04

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Solution: Substituting we have:

8.3 Consumer Loans: A loan is sum of money that is borrowed and then paid back within a specified period. Loans that have a fixed number of payments are called installment loans. Each payment is called an installment. For example, suppose you borrowed $1,000 and were required to pay $100 a month for 10 months. This is an installment loan with an installment of $100. Loans are usually borrowed with interest. This means the amount of the loan will be charged with an interest rate determined by the person you are borrowing money from. The interest rate on a loan is often called a finance charge. This charge will determine the payments of each installment. The Add-On Interest Method The add-on interest payment method is used to determine the monthly payment of an installment loan. This method uses the simple interest formula to calculate the interest of an installment loan. Once the interest is calculated, we will add the interest to the purchase or base price and divide by the number of monthly payments. The formula for calculating the monthly payment using the add-on method is:

Where 𝑃 is the amount of the loan, 𝐼 is the amount of interest due on the loan, and 𝑛 is the number of monthly payments.

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Example: Compute the monthly payments for the add-on interest loan. The amount of the loan, the annual interest rate, and the term of the loan are given.

Solution: We will use the add-on method to solve. First, use the simple interest formula to solve for 𝐼:

Next, find the number of monthly payments (n): The problem states that we will pay off the loan in t=3 years. Convert years to months by multiplying by 12 because there are 12 months in a year.

Lastly, determine the monthly payments by substituting into the formula.

Therefore, the monthly payments will be $30.34. Example: Angela’s bank gave her a 4-year add-on interest loan for $6,480 to pay for new equipment for her antiques restoration business. The annual interest rate is 11.65%. How much interest will she pay? What are her monthly payments? Solution: From the problem, we know the following:

P = $6,480 r = 11.65% = 0.1165

t = 4 years First, use the simple interest formula, 𝐼 = 𝑃𝑟𝑡 to solve for 𝐼:

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Therefore, Angela will pay $3,019.68 in interest. Next, find the number of monthly payments (n): The problem states that we will pay off the loan in 4 years. Convert years to months by multiplying by 12 because there are 12 months in a year.

Lastly, determine the monthly payments by substituting into the formula 𝑀𝑜𝑛𝑡ℎ𝑙𝑦 𝑃𝑎𝑦𝑚𝑒𝑛𝑡 = 𝑃+𝐼

𝑛 .

Therefore, Angela will pay $197.91 a month. The Unpaid Balance Method The unpaid balance method is used for loans where the initial borrowed amount and the monthly payment varies from month to month. This is called an open credit loan. Credit cards provide an open credit loan because the interest charged depends on the amount of purchases that were charged to the card. Unpaid Balance Method for Computing the Finance Charge on a Credit Card –

Where

P = previous month’s balance + finance charge + purchases made – returns – payments r = annual interest rate

t = 1

12

Example: Use the unpaid balance method to find the finance charge on the credit card account. Last month’s balance, the payment, the annual interest rate, and any other transactions are given.

Last month’s balance = $475 Payment = $225

Interest rate = 18% = 0.18 Bought ski jacket = $180 Returned camera = $145

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Solution: First, calculate the interest rate (finance charge) for last month’s balance using the simple interest formula:

Next, calculate the current principle: P = previous month’s balance + finance charge + purchases made – returns –- payments

Lastly, use the simple interest formula to find your new finance charge:

The Average Daily Balance Method The average daily balance method is another method that calculates the monthly payment for open-ended credit or credit card loan. Use the following steps to compute the finance charge on a credit card loan (open ended credit) –

1. Add the outstanding balance for your account for each day of the month.

2. Divide by the total found in step 1 by the number of days in the month to find the average daily balance.

3. Find the finance charge by using the simple interest formula , 𝐼 = 𝑃𝑟𝑡 , where P is the average daily

balance found in step 2, r is the annual interest rate, and t is the number of days in the month divided by 365.

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Example: Use the average daily balance method to compute the finance charge on the credit card account for the previous month. The starting balance and transactions on the account for the month are given. Assume an annual interest rate of 21%. Month: October (31 days); previous month’s balance: $190

Date Transaction

October 9 Charged $35 for a book

October 11 Charged $20 for gasoline

October 20 Made payment of $110

October 26 Charged $13 for lunch Solution:

1. Add the outstanding balance for your account for each day of the month. We will construct a table:

Transaction Number of days that passed until next activity

Balance Number of days ∙ Balance

Balance on October 1st

Oct. 1st, 2nd, 3rd, 4th, 5th, 6th, 7th, 8th = 8 days

$190 8∙190 = $1520

Charged $35 for a book on Oct. 9th

Oct. 9th, 10th = 2 days 190+35 = $225

2∙225 = $450

Charged $20 for gasoline Oct. 11th

Oct. 11th , 12th, 13th, 14th, 15th, 16th, 17th, 18th, 19th = 9 days

225+20 = $245

9∙245 = $2205

Made payment of $110 on Oct. 20th

Oct. 20th, 21st, 22nd, 23rd, 24th, 25th = 6 days

245-110= $135

6∙135 = $810

Charged $13 for lunch on Oct. 26th

Oct. 26th, 27th, 28th, 29th, 30th, 31st = 6 days

135+13 = $148

6∙148 = $888

Total = $5,873 2. Divide by the total found in step 1 by the number of days in the month to find the average daily

balance.

𝑃 = 5873

31 = 189.451613 ≈ $𝟏𝟖𝟗. 𝟒𝟓

3. Find the finance charge by using the simple interest formula,

𝐼 = 𝑃𝑟𝑡 , where P is the average daily balance found in step 2, r is the annual interest rate, and t is the number of days in the month divided by 365.

Recall that P = $189.45, r = 21% = 0.21, t = 31

365

𝐼 = 𝑃𝑟𝑡

𝐼 = (189.45)(0.21) ( 31

365 )

𝐼 = 39.7848 ( 31

365 )

𝐼 = 1233.33

365

𝐼 = 3.38 Therefore, your finance charge will be $3.38.

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8.6 Annual Percentage Rate: The interest rate methods introduced in section 8.3 discussed monthly interest rates. In this section, we will discuss a method for finding an annual interest rates. This is called the annual percentage rate or APR. Previously, we are given monthly interest rates. However, as payments are made and our principle becomes smaller, our initial interest rate will not truly reflect the monthly payments. For example, assume that you had a loan of $1000 with a fixed payment of $100 over 10 months. This would be a 10% interest rate for the first month. Our remaining balance is $900. Our monthly payment is $100 for the second month; however, our

interest rate is no longer 10%. Our interest rate for the second month is 100

900 = 0.11 or 11%. If we continue, we

will find that our interest rate increases each month. Therefore, stating that our interest rate 10% each month is not a true statement. The “true” interest rate is represented by the annual interest rate. We will first calculate the APR using the simple interest formula and elementary algebra. Example: Find the APR of the loan by using the given information.

Loan amount = $8,000 Yearly payments = 4 Rate = 12% = 0.12

Solution: First, solve for the total interest paid:

Next, we can divide $8,000 by 4 yearly payments to give us a re-payment of $2,000 per year. Note t = 1 because the payments are made once a year. Therefore, we can use the interest formula 𝐼 = 𝑃𝑟𝑡 re-written as:

Combine like terms and solve for r:

The APR is 19.2%.

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Finding the APR using Tables Standard financial tables have been constructed to find the APR. The table that will be used throughout this section is located on page 426 of your textbook and is shown below. It provides the annual interest for per $100 on a loan.

Steps for finding the APR using the table on page 426 –

1. Find the finance charge on the loan if it is not already given.

2. Determine the finance charge per $100 on the loan.

3. Use the line of the table that corresponds to the number of payments to find the number closest to the amount found in step 2.

4. The column heading for the column containing the number found in step 3 is the APR.

Example: Use the following table to find the APR to the nearest whole percent for the given situation. Assume that all interest rates are annual rates.

Jonathan has agreed to pay off a $4,500 loan by making 24 monthly payments. The total finance charge on his loan is $600. Solution:

1. Find the finance charge on the loan if it is not already given. The finance charge is $600.

2. Determine the finance charge per $100 on the loan.

Annual Interest Table, (Pirnot, 2014, p. 426)

Annual Interest Table, (Pirnot, 2014, p. 426)

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3. Use the line of the table that corresponds to the number of payments to find the number closest to the amount found in step 2.

We are making 24 monthly payments. Therefore, we will use line that corresponds with 24 in the table.

4. The column heading for the column containing the number found in step 3 is the APR.

From the table, we can see that for a 24-payment loan, this corresponds to an APR of roughly 12%. Example: Use the following table to find the APR to the nearest whole percent for the given situation. Assume that all interest rates are annual rates.

Megan pays a finance charge of $310 on a 12-month, $5,000 loan. Solution:

1. Find the finance charge on the loan if it is not already given. The finance charge is $310.

2. Determine the finance charge per $100 on the loan.

3. Use the line of the table that corresponds to the number of payments to find the number closest to the amount found in step 2.

We are making 12 monthly payments. Therefore, we will use line that corresponds with 12 in the table.

4. The column heading for the column containing the number found in step 3 is the APR.

From the table, we can see that for a 12-payment loan, this corresponds to an APR of roughly 11%. Estimating the APR The APR can be calculated by using a formula for add-on interest loans. This estimate does not require the use of a table as the previous examples did.

Annual Interest Table, (Pirnot, 2014, p. 426)

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Formula to Approximate the APR for the Add-On Interest Loan–

Where 𝑟 is the annual interest rate and 𝑛 is the number of payments. Example: Estimate the annual percentage rate for the add-on loan using the given number of payments and annual interest rate.

n = 36 r = 6.4% = 0.064

Solution: Use the following formula:

Reference Pirnot, T. L. (2014). Mathematics all around (5th ed.). Boston, MA: Pearson.