Problem 127 - Elementary Algebra

Question

Lebron loaned his daughter \(\$ 20,000\) to help her buy a condominium. When she sold the condominium four years later, she paid him the \(\$ 20,000,\) plus \(\$ 3,000\) interest. What was the rate of interest?

Step-by-Step Solution

Answer

3.75%

Step 1: Understand the given information

Lebron loaned his daughter \(\$20,000\). She repaid \(\$20,000\) plus \(\$3,000\) interest after four years. Total repayment is \(\$23,000\).

Step 2: Identify the formula

Use the simple interest formula: \[I = P \times r \times t\] where \(I\) is the interest, \(P\) is the principal amount, \(r\) is the rate of interest and \(t\) is the time in years.

Step 3: Substitute the known values

Here, \(I = \$3,000\), \(P = \$20,000\), and \(t = 4\) years. Plug these values into the formula: \[3,000 = 20,000 \times r \times 4\]

Step 4: Solve for the interest rate

Isolate \(r\) by dividing both sides of the equation by \(20,000 \times 4\): \[r = \frac{3,000}{20,000 \times 4} = \frac{3,000}{80,000} = 0.0375\]

Step 5: Convert the decimal to a percentage

Multiply \(0.0375\) by 100 to get the percentage: \[0.0375 \times 100 = 3.75\text{\boldsymbol{\text{\raise.17ex\rlap{\textsuperscript{\textrm{%}}}}}} \]

Key Concepts

principal amount

The principal amount is the initial sum of money that is loaned or invested, excluding any interest or growth over time. In our example, Lebron loaned his daughter a principal amount of \(\$20,000\). This is the base amount used in calculating the interest. Understanding the principal amount is crucial because it anchors all other calculations in the loan repayment process.

If you think of the principal as the starting point, you can more easily grasp how interest accumulates on top of it.

Remember to always identify the principal amount in any problem involving loans or investments—it sets the stage for everything else.

Key points to remember about principal amount:

• It is the initial sum loaned or invested.
• It does not include interest or profits gained over time.
• It is central to calculating interest.

interest rate determination

Determining the interest rate involves some simple algebra using the simple interest formula:

\[ I = P \times r \times t \]

Let’s break this down using the given exercise:

• Interest (I) = \ \$3,000 \
• Principal (P) = \ \$20,000 \
• Time (t) = 4 years \

Now, plug these values into the formula and solve for the rate (r):

\[ 3,000 = 20,000 \times r \times 4 \]

To isolate \ r, \ divide both sides of the equation by \ (20,000 \ cdot 4):

\[ r = \frac{3,000}{20,000 \times 4} = \frac{3,000}{80,000}= 0.0375 \]

Don’t forget to convert the decimal to a percentage by multiplying by 100:

\[ 0.0375 \times 100 = 3.75\% \]

This final number represents the interest rate at which Lebron’s daughter repaid the loan.

Key points to remember about interest rate determination:

• Always use the simple interest formula: \ I = P \times r \times t \
• Substitute the known values and solve for the unknown variable.
• Convert the final value to a percentage.

loan repayment

Loan repayment includes both the principal amount and the interest accrued over the loan period. In this example, Lebron’s daughter repaid \(\$20,000\) (the principal) plus \(\$3,000\) (the interest), summing up to \(\$23,000\). Understanding loan repayment is essential as it impacts your budget and financial planning.

When repaying a loan, you need to consider the total amount you will pay back over time. This helps you avoid surprises.

Key points to remember about loan repayment:

• It includes both the principal and the interest.
• Calculate the total repayment value to manage your finances effectively.
• Understanding repayment terms helps plan for future borrowing or investments.

Regularly reviewing these concepts ensures a solid grasp of financial topics related to loans and investments.