Applied Math - Percentages - Paying Down Debt

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Introduction

We take out loans (go into debt) to buy things that are necessary, like housing and transportation. We often do not have enough money to buy these things ourselves. If we pay more than our scheduled monthly payments, we can save ourselves some interest and also cut down the number of monthly payments required.

Example:

1.

A person has an auto loan of $30,000. They are making monthly payments of $400 each month. Assume interest is calculated annually at 5% simple interest (real world calculations are likely compounded monthly and require a more complicated calculation).

How much of the person’s first $400 monthly payment goes to interest?

The interest rate is 5% annually. If we make monthly payments, we have to split the 5% into 12 equal pieces before we calculate the interest.

The interest owed on the first monthly payment is

\[\$30,000 \cdot \frac{0.05}{12} \approx\]

\[\$125\]

The person gives their lender $400, but only $400 - $125 = $275 goes to their principal (the amount they owe). They now owe $30,000 - $275 = $29,725.



2.

Suppose a person still owes $29,725 on a loan with 5% interest annually using simple interest. They are making $400 monthly payments. The person gains $3,000 in cash during this month from a project they finished as an independent contractor. Instead of making their usual $400 monthly payment, they choose to pay more and pay $3,000 on the loan. How much of the $3,000 goes to interest and how much goes to the principal?

We first need to calculate how much interest is owed for the month, as we have borrowed money for the month and owe interest for that time. The balance of the loan is $29,725. We have to divide the 5% interest by 12 because 5% interest is an annual rate and we have collected interest for only one month.

\[\$29,725 \cdot \frac{0.05}{12} \approx \$123.85\]

The person paid $3,000 toward their loan, but $123.85 of that money has to go to interest. Their principal is reduced by $3,000 - $123.85 = $2,876.15. They now owe $29,725 - $2,876.15 = $26,848.85.

Practice Problems

1. A person has an auto loan of $25,000. They are making monthly payments of $350 each month. Assume interest is calculated annually at 6% simple interest (real world calculations are likely compounded monthly and likely require a more complicated calculation).

How much of the person’s first $350 monthly payment goes to interest?



2. Suppose a person still owes $35,725 on a loan with 7% interest annually using simple interest. They are making $450 monthly payments. The person gains $5,000 in cash during this month from a project they finished as an independent contractor. Instead of making their usual $450 monthly payment, they choose to pay more and pay $4,000 on the loan. How much of the $4,000 goes to interest and how much goes to the principal? How much do they still owe after this payment?

Theory Questions

1. Is it worth it to make a payment larger than the required monthly payment, like in question 2? Why or why not?