Problem 1: Priscilla's Monthly Payment Calculation

Priscilla has a bill of $5,077, and she wants to pay it off over 4 years (48 months) with an interest rate of 23.4%. We will assume this is a monthly compounded loan, so we can use the following formula to calculate the monthly payment:

$M = P \cdot \frac{r(1+r)^n}{(1+r)^n - 1}$

Where:

• $M$ is the monthly payment
• $P$ is the principal (amount borrowed, $5,077)
• $r$ is the monthly interest rate (annual interest rate divided by 12)
• $n$ is the total number of payments (in months)

Step 1: Identify the given values

• $P = 5,077$
• Annual interest rate = 23.4%, so the monthly interest rate $r = \frac{23.4\%}{12} = 0.234 / 12 = 0.0195$
• $n = 4 \text{ years} \times 12 \text{ months/year} = 48 \text{ months}$

Step 2: Substitute the values into the formula

$M = 5077 \cdot \frac{0.0195(1 + 0.0195)^{48}}{(1 + 0.0195)^{48} - 1}$

Step 3: Simplify the formula

First, calculate $(1 + 0.0195)^{48}$:

$(1 + 0.0195)^{48} \approx 1.0195^{48} \approx 2.5475$

Now, substitute this back into the formula:

$M = 5077 \cdot \frac{0.0195 \cdot 2.5475}{2.5475 - 1}$

$M = 5077 \cdot \frac{0.04965}{1.5475}$

$M \approx 5077 \cdot 0.03208$

$M \approx 162.91$

So, Priscilla would have to pay $162.91 per month.

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Problem 2: Diego's Loan Payoff Time

Diego has a student loan and we need to determine how long it will take him to pay it off. Let’s use the formula for the loan balance over time:

$B = \frac{M \cdot \left(1 - (1 + r)^{-n}\right)}{r}$

Where:

• $B$ is the loan balance (the amount Diego owes)
• $M$ is the monthly payment
• $r$ is the monthly interest rate (annual interest rate divided by 12)
• $n$ is the number of months (which we need to find)

To clarify, the missing information from the problem includes:

• The amount Diego owes (loan balance $B$) and the monthly payment $M$. Please provide those details, so I can continue.

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Would you like me to walk through the math for the second problem once you provide the missing details?

Related questions:

1. How would Priscilla’s monthly payment change if the interest rate were lower?
2. What is the total amount of interest Priscilla will pay over the 4 years?
3. What formula is used to calculate monthly payments for loans with fixed payments?
4. How would the length of the loan term affect Priscilla’s monthly payment?
5. How could Diego lower the total interest he pays on his loan over time?

Tip: Always double-check your interest rates and time periods when calculating loan payments to ensure accuracy.