Problem 41 - Precalculus

Question

Diana just graduated from medical school owing \(\$ 80,000\) in student loans. The annual interest rate is \(9 \%\) a. Approximately how many years will it take to pay off her student loan if she makes a monthly payment of \(\$ 750 ?\) b. Approximately how many years will it take to pay off her loan if she makes a monthly payment of \(\$ 1,000 ?\)

Step-by-Step Solution

Answer

a. 15 years with $750 monthly, b. approximately 9.33 years with $1000 monthly.

Step 1: Convert Annual Interest Rate to Monthly

The annual interest rate given is 9%. To use it in monthly loan calculations, convert it into the monthly rate: \[ r = \frac{9\%}{12 \text{ months}} = \frac{0.09}{12} \approx 0.0075 \text{ per month} \]

Step 2: Use Loan Formula to Solve for Time

We'll use the loan payment formula for monthly payments: \[ M = P \frac{r(1+r)^n}{(1+r)^n - 1} \]Where:- \( M \) is the monthly payment- \( P \) is the principal loan amount- \( r \) is the monthly interest rate- \( n \) is the number of months to pay off the loan. Rearrange this formula to solve for \( n \):\[ n = \frac{\log\left(\frac{M}{M - Pr}\right)}{\log(1 + r)} \]

Step 3a: Calculate for Monthly Payment of $750

Substitute \( M = 750 \), \( P = 80000 \), and \( r = 0.0075 \) into the formula:\[ n = \frac{\log\left(\frac{750}{750 - 80000 \times 0.0075}\right)}{\log(1 + 0.0075)} \]Calculate:\[ n \approx \frac{\log\left(\frac{750}{750 - 600}\right)}{\log(1.0075)} \approx \frac{\log(1.8)}{\log(1.0075)} \approx 180 \text{ months} \]Convert months to years:\[ t \approx \frac{180}{12} = 15 \text{ years} \]

Step 3b: Calculate for Monthly Payment of $1000

Substitute \( M = 1000 \), \( P = 80000 \), and \( r = 0.0075 \) into the formula:\[ n = \frac{\log\left(\frac{1000}{1000 - 80000 \times 0.0075}\right)}{\log(1 + 0.0075)} \]Calculate:\[ n \approx \frac{\log\left(\frac{1000}{1000 - 600}\right)}{\log(1.0075)} \approx \frac{\log(2.5)}{\log(1.0075)} \approx 112 \text{ months} \]Convert months to years:\[ t \approx \frac{112}{12} \approx 9.33 \text{ years} \]

Key Concepts

Interest Rate Conversion

When dealing with loans, especially those requiring monthly payments, it's crucial to convert the given annual interest rate to a monthly interest rate. This conversion makes calculations much more accurate and manageable.

Consider an annual interest rate of 9%. To find out the monthly rate, simply divide the annual rate by the number of months in a year.

• Annual interest rate: 9%
• Number of months in a year: 12

The formula becomes: \ \[ r = \frac{9\%}{12} = \frac{0.09}{12} \approx 0.0075\text{ per month} \]

This rate of 0.0075 is what you use in the loan payment formula to determine monthly payments or the loan's lifespan. It's a small step, but converting interest rates this way ensures that your financial calculations are consistent and correct.

Loan Payment Formula

Understanding how to use the loan payment formula is essential for calculating how long it will take to repay a loan. This formula incorporates multiple financial factors: loan principal, monthly interest rate, and the total number of payments needed.

The formula itself looks like this: \ \[ M = P \frac{r(1+r)^n}{(1+r)^n - 1} \]

• \( M \) represents the monthly payment.
• \( P \) is the total loan amount, or principal.
• \( r \) is the monthly interest rate.
• \( n \) stands for the number of payments, typically in months.

This formula helps balance the monthly payments such that the loan principle plus interest is completely paid off over a specified period. Rearranging the formula allows one to find out how many months or payments are needed if other values are known.

Monthly Payment Calculation

Once you have all the necessary variables, calculating the monthly payment becomes straightforward. Simply substitute the values into the loan formula.

For instance, let's calculate it for a monthly payment of \\(750 or \\)1000 using our already converted interest rate. By substituting:

• \( M = 750 \) or \( M = 1000 \)
• \( P = 80000 \)
• \( r = 0.0075 \)

Into this rearranged formula for \( n \): \ \[ n = \frac{\log\left(\frac{M}{M - Pr}\right)}{\log(1 + r)} \]
For \( M = 750 \), \( n \approx 180 \) months, it equates to approximately 15 years to pay off the loan.

For \( M = 1000 \), \( n \approx 112 \) months, roughly translates to about 9 years and 4 months. This highlights the impact monthly payments have on loan duration.

Logarithmic Functions

Logarithmic functions finesse more complex financial equations, such as determining the time required to repay a loan when monthly, not annual, figures are used.

In the context of loan calculations, logarithms help in solving the rearranged loan formula for \( n \), the number of payments: \ \[ n = \frac{\log\left(\frac{M}{M - Pr}\right)}{\log(1 + r)} \]

• \( \log \) is used to solve exponential equations.
• \( \log(\cdot) \) represents the logarithm (often base 10 or natural log).

Logs provide the mechanism for isolating "number of payments" by transforming multiplicative comparisons into additive ones. This utility of logs turns complex calculations into simpler steps, crucial for financial problem-solving. Understanding how they fit into the broader context of a financial formula can markedly simplify many real-world monetary computations.