Loan Basics
What is the total amount of money you borrow called?
Principal
Arya takes out a loan of $5,000 at a 4% fixed interest rate for 2 years. How much simple interest will Arya pay?
$400
What are the labels of an axis when showing the interest accrued on a loan?
X - Time (in years), Y - Interest (in dollars)
Diego takes on a loan at $5000 at a 4% fixed rate for 4 years. Thiago takes on a loan for $5000 with a variable rate. The interest rate changes as follows:
Year 1: 3%
Year 2: 5%
Year 3: 4%
Year 4: 4%
Who pays more?
Neither; both of them pay the same amount!
Why does a loan with a fixed interest rate create a straight-line graph of total interest over time?
Because the amount of interest added each year is the same, so the rate of change is constant.
____________ is the "cost of borrowing". It is how banks and other organizations that lend money make a profit.
Interest
Rio borrows $8000 at 5% interest for 3 years. How much interest will he pay in total?
$1,200
Analyze the total interest paid on a loan by year: Is this a fixed or variable rate loan?
Year 0: 0
Year 1: $250
Year 2: $500
Year 3: 750
Year 4: 1000
Fixed
Ella takes on a loan of $8000 with a 5% interest rate for 6 years. Nadia takes on a variable rate loan with a principal of $8000 and interest rates as follows:
Year 1: 4% 320
Year 2: 5% 400
Year 3: 5% 400
Year 4: 4% 320
Year 5: 6% 480
Year 6: 2% 160
Who pays more? By how much?
Ella pays more, by $320.
“Variable interest loans are always worse because the rate changes.” Do you agree or disagree? Explain using an example or reasoning.
Student Answer
Sasha is taking on an $8,000 loan to buy a new car. She will be paying monthly payments of at least $184 for four years. What is the term of the loan?
4 years
Yuno takes out a loan of $6,000, with a variable interest rate.
Year 1: 3%
Year 2: 5%
Year 3: 4%
Year 4: 6%
How much total interest is paid after 4 years?
$1080
A graph shows total interest over time. The points on the graph are as follows: (0,0), (1, 180), (2, 360), (3, 540), (4, 720). Is this a fixed or variable interest rate loan?
Fixed
Vidhy takes on a loan of $6000 at 4% fixed rate for 4 years. Erick takes on a variable rate loan with the following interest rates: 3%, 6%, 4%, 5%. Who pays less? BY how much?
Vidhy, $120 less
Two loans have the same principal and term. One is fixed, one is variable. The variable loan ends up costing less total interest.
Explain how this is possible even though the rate changes each year.
Because some years have lower interest rates that reduce the total amount added, and those lower rates can outweigh higher-rate years.
What is the difference between the interest and the interest rate?
The interest is the actual amount of additional money that you must pay back; the interest rate is the percentage used to calculate this sum.
Bernat takes out a loan for $10,000 with a 4% interest rate over 4 years. How much does Bernat pay in total?
$11,600
Rany takes on a loan for $9,000, with an interest rate of 6% over 5 years. Graph the interest on the loan over time.
Graphs!
Ian takes on a $12,000 fixed rate loan at 5% for 4 years. Josue takes on a $12,000 variable rate loan with the following interest rates: 4%, 6%, 5%, 5%. Whose loan is cheaper, and by how much?
Both Ian and Josue pay the same amount.
A student looks at two graphs of total interest over time:
The student says:
“Graph A must be the better option because it is predictable.”
Do you agree or disagree? Defend your answer using math reasoning.
Students Answer
Dishi takes on a variable rate loan of $7,500. The interest rates for the loan are as follows:
Year 1: 2%
Year 2: 4%
Year 3: 5%
Year 4: 3%
Meanwhile, Amaya takes on a loan that's fixed at 4% for 4 years. Who ends up paying less? By how much?
Dishi ends up paying less, by $150.
Neneyo takes on a variable rate loan for $6500.
Year 1: 2%
Year 2: 6%
Year 3: 3%
Year 4: 7%
Graph the interest rate on the loan.
Graphs!
Kristelle takes on a $7,500 at 4.5% interest rate for 4 years. Sophia takes on a $7,500 variable rate loan with the following interest rates: 5%, 4%, 4%, 6%. Whose loan is cheaper?
Kristelle's loan is cheaper by $75.
Two loans have the same principal and term. A student says:
“If two loans have the same total interest after 4 years, their graphs must look identical.”
Do you agree or disagree? Explain carefully.
Disagree. Fixed vs. Variable Rate graphs.