Problem 55 - Precalculus

Question

Wing Shan just graduated from dental school owing \(\$ 80,000\) in student loans. The annual interest is \(6 \% .\) Her time \(t\) to pay off the loan is given by $$t=-\frac{\ln \left[1-\frac{80,000(0.06)}{n R}\right]}{n \ln \left(1+\frac{0.06}{n}\right)}$$ where \(n\) is the number of payment periods per year and \(R\) is the periodic payment. a. Use a graphing utility to graph $$t_{1}=-\frac{\ln \left[1-\frac{80,000(0.06)}{12 x}\right]}{12 \ln \left(1+\frac{0.06}{12}\right)} \text { as } Y_{1} \text { and }$$ $$t_{2}=-\frac{\ln \left[1-\frac{80,000(0.06)}{26 x}\right]}{26 \ln \left(1+\frac{0.06}{26}\right)} \text { as } Y_{2}$$ Explain the difference in the two graphs. b. Use the \([\text { TRACE }]\) key to estimate the number of years that it will take Wing Shan to pay off her student loan if she can afford a monthly payment of \(\$ 800\) c. If she can make a biweekly payment of \(\$ 400\), estimate the number of years that it will take her to pay off the loan. d. If she adds \(\$ 200\) more to her monthly or \(\$ 100\) more to her biweekly payment, estimate the number of years that it will take her to pay off the loan.

Step-by-Step Solution

Answer

Monthly payments of $800 take about 11 years, biweekly payments of $400 about 10 years. Adding $200 monthly or $100 biweekly shortens it to 8 years.

Step 1: Understanding Graphs

Begin by using a graphing utility to graph the two equations given for \(t_1\) and \(t_2\). These equations represent the time in years to pay off the loan with different payment frequencies and amounts. For \(t_1\), the loan is paid using monthly payments, while for \(t_2\), the payments are biweekly.

Step 2: Analyzing Graph Differences

Compare \(Y_1\) and \(Y_2\). \(Y_1\) will generally have a longer repayment time for the same payment amount \(x\) because monthly payments mean fewer total payments per year compared to biweekly payments (12 vs 26), and will accumulate less interest.

Step 3: Solving for Monthly Payments

Using the graph for \(t_1\), find where the repayment time matches up for a monthly payment of \(\\( 800\). Use the TRACE function to determine the corresponding \(t_1\). This will give us the number of years to repay the loan when making monthly payments of \(\\) 800\).

Step 4: Solving for Biweekly Payments

Similar to Step 3, identify the time \(t_2\) when making biweekly payments of \(\$ 400\). Again, use the TRACE feature on the graph for \(Y_2\) to find the corresponding number of years.

Step 5: Calculation with Additional Payment

If Wing Shan adds \(\\( 200\) to her monthly payment or \(\\) 100\) to her biweekly payment, repeat Steps 3 and 4 with \(\\( 1000\) monthly or \(\\) 500\) biweekly. Use the graphs to estimate the repayment time for these new payment amounts.

Key Concepts

Student Loans

Student loans can often be a necessary step in pursuing higher education. They provide the financial support needed to cover tuition, books, and sometimes living expenses. After graduation, repaying these loans becomes a responsibility that must be managed carefully. Understanding the terms of your student loan is crucial. These include the principal amount borrowed, the interest rate, and the repayment timeline. The interest rate determines how much more you will repay over the life of the loan, in addition to the principal sum. Seeking a lower interest rate can save money. What's important is how different repayment plans might fit into your budget and future financial goals. By examining these factors, graduates like Wing Shan can design a repayment strategy that works best for them.

Periodic Payment Calculations

Periodic payment calculations involve determining how often payments need to be made over the course of a year and calculating the payment amount necessary to meet loan repayment goals. For Wing Shan, paying off a student loan involves setting up either monthly or biweekly payments. The formula used calculates the time or duration required in years, represented by the variable \(t\), to fully repay the loan amount:\[t=-\frac{\ln \left[1-\frac{80,000(0.06)}{n R}\right]}{n \ln\left(1+\frac{0.06}{n}\right)}\]Where:- \(n\) is the number of payment periods per year.- \(R\) is the periodic payment amount.With monthly payments (\(n=12\)) or biweekly payments (\(n=26\)), the formula shows how adjusting payment frequency and payment amount can impact repayment duration.

Graphing Utility Analysis

Graphing utility analysis involves using technology to visualize and compare the outcomes of different payment strategies. This method helps to clarify the effect of different payment amounts and frequencies.By graphing the given equations \(Y_1\) for monthly payments and \(Y_2\) for biweekly payments, Wing Shan can see how the repayment times differ. Generally, more frequent payments, such as biweekly, result in a shorter loan duration due to the compounding effect, which reduces the overall interest paid.Utilizing the TRACE feature allows for easy estimation of the loan repayment duration at various payment levels. It becomes straightforward to observe how increasing payments directly reduces the time needed to pay off the balance.

Interest Rate Impact on Payments

The interest rate profoundly affects the total amount of money paid over the life of a loan. It's the cost of borrowing money and accumulates over time. For Wing Shan's loan, a 6% annual interest rate means that for every year, the loan amount grows by 6% if repayments aren’t covering at least this interest. Therefore, higher periodic payments reduce the principal faster, lessening the interest impact. Analyzing different payment amounts using the graphing utility, as detailed in the exercise, highlights that by increasing payment amounts, interest payments can be curtailed significantly. This shows the importance of understanding and, if possible, negotiating better loan terms, as even small differences in interest rates can lead to large differences in repayment time and total interest paid.