1. THE NATION’S NEWSPAPER VOL 2002-08
Math
LESSONS
Student Edition
DATA ANALYSIS:
How Muc h Will
That House Cost?
I N T RO DU C T IO N M AT E R I A L S N EE D E D
Sooner than you think, you will finish u USA TODAY Money (Section B) from
school and be out on your own. Many any Tuesday, Wednesday, Thursday,
young people today want to buy their or Friday edition
own homes, rather than rent. Unless u spreadsheet software (a calculator
you are lucky enough to win the lottery, and a paper-and-pencil table can be
you will probably need to take out a used, if necessary)
loan, often referred to as a mortgage.
How much does it cost per month to
pay back a home loan? How does the
length of time of the loan and the rate of
interest affect your payments? How
much do you actually pay for a house
when you take out a loan? In this lesson,
you will try to answer some of these
questions and see if buying a home may
be in your future.
MATH LESSONS: HOW MUCH WILL THAT HOUSE COST?

2. MATH LESSONS: HOW MUCH WILL THAT HOUSE COST?
P R E PA R AT IO N
Use the information below to help you find monthly payments and total payments for loans of various amounts,
rates, and time periods.
1. On page 4B of USA TODAY (Tuesday through Friday editions), you will find a table labeled Consumer rates.
Look at the bottom of this table for a section labeled Mortgage rates. You will see interest rates labeled 30–yr.
fixed and 15–yr. fixed for various dates. These are the best rates offered for borrowing money for the dates
given. The 30–year rates are for a loan in which you take 30 years to pay the balance and the 15–year rates
are for a 15–year loan.
2. The formula below can be used to find the payment per month for a loan if you know the amount of the loan,
the rate of interest for the loan, and the number of years for which the money will be borrowed. The meaning
of each variable is stated below the formula.
P= Ar
[1 - (1 + r)-t]
a. P is the amount you will pay each month, or the monthly payment.
b. A is the total amount of the loan, or the amount of money you actually borrow.
c. r is the interest rate per month. The interest rate given in the USA TODAY table is the interest rate per year.
You will need to write this percent as a decimal and divide that value by 12 to get the interest rate per month.
For example, a rate of 6.25% is 0.0625 as a decimal and then r in this formula is (0.0625 ÷12). Notice that this
value is used both in the numerator and denominator of the formula.
d. t is the number of time periods for the payments. For a 30–year mortgage, the number of time periods for
payments is 30 • 12 or 360. For a 15–year mortgage, the number of time periods for payments is 15 • 12 or
180.
3. Create a spreadsheet with the following columns to help you analyze the monthly payments and the total
amount paid for 30–year and 15–year loans of various amounts. Use the formula above for Columns C and F.
Be sure the values in your spreadsheet formula reference the correct columns in the spreadsheet. Enter
appropriate formulas to calculate the values for Columns D, G, H, and I.
A B C D E F G H I
Loan 30-year Monthly Total 15-year Monthly Total Ratio of Ratio of
Amount interest payment amount interest payment amount Column Column
rate for paid for rate for paid for F D
30-year a 30- 15-year a 15-year to to
loan year loan loan Column Column
loan C G
MATH LESSONS: HOW MUCH WILL THAT HOUSE COST? Page 2

3. MATH LESSONS: HOW MUCH WILL THAT HOUSE COST?
A N A LY S I S
Here are some questions to help you compare various loans. Use the spreadsheet to do your
calculations.
1. Find the current 15–year and 30–year loan rates in your copy of USA TODAY . Enter 100,000 for a
$100,000 loan into Column A. Enter the two rates from USA TODAY into Columns B and E. Be sure to
enter the rate as a monthly rate. For example, a yearly rate of 7.25% would be entered as 0.0725/12 in
the column.
a. Find the monthly payment for a $100,000 loan at each rate (Columns C and F).
b. Find the value for Column H.
c. Write several sentences comparing the monthly payments.
2. Use the 15–year and 30–year rates you used in Question 1.
a. Find the total payment for a $100,000 loan at each rate (Columns D and G).
b. Find the value for Column I.
c. Write several sentences comparing the total amounts paid over the entire period of the two loans.
3. Enter various loan amounts from $80,000 to $200,000 in Column A of your spreadsheet. Use the two
rates from your copy of USA TODAY in Columns B and E. Fill in the values for all columns in your
spreadsheet.
a. Find the values in Columns H and I of your spreadsheet.
b. Write several sentences describing any patterns that you see.
4. Consider two loans for the same amount of money A. The payments for the loans are calculated for both
a 30–year at 8% per year and a 15–year at 8% per year.
a. Make a conjecture about the ratio of the total amount paid for the 30–year loan to the total amount
paid for the 15–year loan (Column I).
b. Use your spreadsheet to check your conjecture. Describe how you used the spreadsheet to check
your conjecture.
5. Suppose you are offered a 30–year, $100,000 loan on an Internet website. You find that the total amount
of all the payments during the 30–year period is $300,000.
a. To the nearest hundredth of a percent, what is the yearly interest rate of the loan?
b. Describe how you found the interest rate.
MATH LESSONS: HOW MUCH WILL THAT HOUSE COST? Page 3

4. MATH LESSONS: HOW MUCH WILL THAT HOUSE COST?
Q U E S T IO N S
Here are some questions to help you clarify your thinking about loans, interest rates, and
lengths of loans.
6. Suppose the payments for a 15–year and a 30–year loan of $A are calculated using the same
interest rate, r. How will the monthly payments for a 30–year loan compare to the monthly
payments for a 15–year loan?
7. Suppose the payments for a 15–year and a 30–year loan of $A are calculated using the same
interest rate, r. How will the total amount paid for a 30–year loan compare to the total
amount paid for a 15–year loan?
8. Look at your spreadsheet’s calculations for Question 3. Use the formula for calculating the
amount of a monthly payment to show why the ratios in Column H all had the same value.
9. Research mortgages using the Internet or any other source. What are some advantages and
disadvantages of a 15–year loan?
10. Research mortgages using the Internet or any other source. What are some advantages and
disadvantages of a 30–year loan?
C O N C L U S IO N E XT E N S IO N S
u Research the cost of buying a
1. Research starting salaries for a job that you would like to have
home in your area or an area after finishing high school, a training school, or college. Some
in which you would like to lenders suggest that you can pay approximately 25% of your
gross income on a house payment. Estimate the amount of
live. Assume that you can money you would be able to pay per month for a house
make a 20% down payment payment. Use the formula in the Preparation section to
determine how much money you could borrow at the
on a house. Using current current 15–year and 30-year mortgage rates.
interest rates for 15–year and
2. When helping a customer to take out a loan, a bank officer will
30–year loans, find the examine the credit history of the person, such as how
monthly payment for each promptly they pay their bills or how much money the person
type of loan for the amount has in credit card debt. If a person’s credit history is poor, the
loan will probably have a higher rate of interest. Investigate the
you determine you will need effect of various rates of interest on a loan of the same amount.
to borrow. Also, find the total For example, what would be the monthly payments on a
$100,000 30–year loan at 6%, 7%, 8%, and 9%? What would the
amount you will pay for each total amount paid be for the $100,000 30–year loan at 6%, 7%,
loan. Write several paragraphs 8%, and 9%?
explaining which length of 3. Suppose a 15–year loan has an interest rate of 7% and a
loan you would choose. 30–year loan has an interest rate of 8%. Show that the ratio of
the monthly payment for the 15–year loan to the monthly
payment for the 30–year loan is approximately 1.2. [Hint use
the formula in the Preparation section to find the ratio.]
MATH LESSONS: HOW MUCH WILL THAT HOUSE COST? Page 4