Which loan will be cheaper over the entire loan term? And WHY? Loan 1: Loan= $500, Term= 1 year, APR= 10%. Loan 2: Loan= $500, Term 3 years, APR= 10%.
Understand the Problem
The question is asking us to compare two loans with different terms to determine which one will be cheaper overall, and to explain the reasoning behind that comparison. We'll consider the total cost of each loan by calculating the interest paid over their respective terms.
Answer
The loan with the lower total cost, $C_A$ or $C_B$, is the cheaper option.
Answer for screen readers
The cheaper loan will be the one with the lower total cost, which can be determined by comparing $C_A$ and $C_B$ after performing the calculations for each loan.
Steps to Solve
- Identify the loan details
Let's denote the first loan as Loan A and the second loan as Loan B. Identify the principal amount, interest rates, and the term (duration in years) for both loans.
Assume:
- Loan A: Principal = $P_A$, Interest Rate = $r_A$, Term = $t_A$ years
- Loan B: Principal = $P_B$, Interest Rate = $r_B$, Term = $t_B$ years
- Calculate interest for Loan A
The formula to calculate the total interest paid on a loan is given by:
$$ I_A = P_A \times r_A \times t_A $$
Here, $I_A$ is the interest on Loan A. Ensure you convert the interest rate to decimal form (e.g., 5% becomes 0.05).
- Calculate interest for Loan B
Similarly, calculate the total interest for Loan B using the same formula:
$$ I_B = P_B \times r_B \times t_B $$
Where $I_B$ is the interest on Loan B.
- Calculate total cost of each loan
Now, find the total cost of each loan by adding the principal amount to the interest calculated.
For Loan A:
$$ C_A = P_A + I_A $$
For Loan B:
$$ C_B = P_B + I_B $$
- Compare the total costs of Loans
Finally, compare the total costs:
- If $C_A < C_B$, then Loan A is cheaper overall.
- If $C_A > C_B$, then Loan B is cheaper overall.
- If $C_A = C_B$, then both loans cost the same.
The cheaper loan will be the one with the lower total cost, which can be determined by comparing $C_A$ and $C_B$ after performing the calculations for each loan.
More Information
When comparing loans, it's essential to look not just at the interest rates but also at the total cost over the entire loan term. This approach ensures you accurately compare different loan offers and choose the most economical option.
Tips
- Forgetting to convert the interest rate from percentage to decimal form.
- Not including the principal in the total cost calculation.
- Comparing annual interest without considering the total loan duration.
