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Chapter 5: Problem 55

Michelle borrows a total of \(\$ 5000\) in student loans from two lenders. One charges \(4.6 \%\) simple interest and the other charges \(6.2 \%\) simple interest. She is not required to pay off the principal or interest for 3 yr. However, at the end of 3 yr, she will owe a total of \(\$ 762\) for the interest from both loans. How much did she borrow from each lender?

Short Answer

Expert verified
Michelle borrowed \( 3500 \) dollars at 4.6% interest and \( 1500 \) dollars at 6.2% interest.

Step by step solution

01

- Define the variables

Let the amount borrowed from the lender with a 4.6% interest rate be denoted as \( x \). Then the amount borrowed from the lender with a 6.2% interest rate will be \( 5000 - x \).
02

- Write the total interest equations

The formula for simple interest is \( I = P \times r \times t \), where \( I \) is the interest, \( P \) is the principal, \( r \) is the interest rate, and \( t \) is the time in years. Using this formula, the interest from the first lender after 3 years is: \( I_1 = x \times 0.046 \times 3 \), and the interest from the second lender after 3 years is \( I_2 = (5000 - x) \times 0.062 \times 3 \).
03

- Set up the total interest equation

The total interest from both loans is given to be \( 762 \). So, we can write the equation: \[ 3 \times 0.046 \times x + 3 \times 0.062 \times (5000 - x) = 762 \].
04

- Simplify the equation

Distribute the constants across the terms: \[ 0.138x + 0.186 \times (5000 - x) = 762 \].
05

- Combine like terms

Expand and combine like terms to simplify: \[ 0.138x + 930 - 0.186x = 762 \]. Combining like terms gives: \[ -0.048x + 930 = 762 \].
06

- Solve for x

Subtract 930 from both sides to isolate \( x \): \[ -0.048x = 762 - 930 \]. This simplifies to: \[ -0.048x = -168 \].
07

- Find the value of x

Divide by -0.048: \[ x = \frac{-168}{-0.048} = 3500 \]. Thus, Michelle borrowed \( 3500 \) dollars from the lender with the 4.6% interest rate.
08

- Find the remaining amount

The remaining amount borrowed from the other lender is: \( 5000 - 3500 = 1500 \). Therefore, Michelle borrowed \( 1500 \) dollars from the lender with the 6.2% interest rate.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simple Interest
Simple interest is a method used to calculate the interest on a loan or investment based on the original principal amount. The formula for simple interest is given by:
\ ( I = P \times r \times t ) \ where:
  • \( I \) is the interest earned or paid.
  • \( P \) is the principal amount, which is the initial amount of money borrowed or invested.
  • \( r \) is the annual interest rate.
  • \( t \) is the time the money is borrowed or invested, in years.
In the context of Michelle's situation, simple interest helps calculate how much extra she needs to repay due to borrowing the money.
Loan Calculations
Loan calculations involve determining how much interest will be paid over the life of a loan, as well as how much principal will be paid off. For Michelle's loans, here is how the calculations work:
  • She borrowed a total of \(5000 from two lenders.
  • One lender charges 4.6% simple interest, and the other 6.2%.
  • She will owe \)762 in interest after 3 years.
Using these details, we create equations to find out how much she borrowed from each lender. Calculations like these are essential in financial planning and help in managing debt effectively.
Algebraic Equations
Algebraic equations help solve problems involving unknown quantities. In this exercise, the algebraic equation was set up to figure out how much Michelle borrowed from each lender.
Here's the breakdown:
  • Define variables: Let \( x \) be the amount borrowed from the lender with a 4.6% interest rate.
  • Thus, \( 5000 - x \) will be from the lender with a 6.2% rate.
  • Write the interest formula: \( 3 \times 0.046 \times x + 3 \times 0.062 \times (5000 - x) = 762 \)
  • Simplify and solve for \( x \).
This algebraic approach helps break down complex financial problems into manageable steps.
Financial Mathematics
Financial mathematics combines mathematical methods with real-world finance problems, helping people understand and manage their finances. In Michelle's case, financial math principles were used to:
  • Calculate the total interest she would owe after 3 years.
  • Determine how much she borrowed from each lender.
  • Create equations to represent her total debt and interest.
Understanding these principles empowers individuals to make informed financial decisions and manage loans effectively. It also helps in financial planning and optimizing borrowing strategies.

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