Problem 128 - Elementary Algebra

Question

Pablo borrowed \(\$ 50,000\) to start a business. Three years later, he repaid the \(\$ 50,000,\) plus \(\$ 9,375\) interest. What was the rate of interest?

Step-by-Step Solution

Answer

The rate of interest is 6.25%.

Step 1: Identify the known values

The principal amount Pablo borrowed is \( P = 50,000 \). The interest paid after three years is \( I = 9,375 \). The time period is \( T = 3 \) years.

Step 2: Understand the formula for Simple Interest

Simple Interest \( I = P \times r \times T \), where \( r \) is the rate of interest.

Step 3: Rearrange the formula to solve for the rate of interest

From the simple interest formula: \( r = \frac{I} {P \times T} \). Substitute the known values: \( r = \frac{9375} {50000 \times 3} \).

Step 4: Calculate the rate of interest

Perform the calculation: \( r = \frac{9375} {150000} \). Simplifying this gives \( r = 0.0625 \). The rate of interest is then \ 0.0625 \ times 100 to convert to a percentage, which is 6.25%.

Key Concepts

Interest Rate

In finance, the term 'interest rate' is used to describe the percentage of the principal amount that a borrower must pay to the lender as a fee for borrowing money. The interest rate is typically expressed as an annual percentage. In the context of Pablo's loan, the interest rate is found by figuring out how much interest he paid on the borrowed \(\$ 50,000\) over three years. Understanding this percentage helps determine how costly the loan is over time.

Principal Amount

The principal amount, often denoted by the letter 'P', is the initial sum of money borrowed or invested. In Pablo's case, the principal amount is \(\$ 50,000.\) The principal is crucial as it forms the base upon which interest is calculated. As you work through the simple interest calculations, the principal remains constant. It is the foundation for determining how much interest will accumulate over a given period.

Simple Interest Formula

Simple interest is calculated using a straightforward formula: \( I = P \times r \times T \). Here, 'I' represents the interest earned or paid, 'P' is the principal amount, 'r' is the interest rate (expressed as a decimal), and 'T' is the time period in years. The formula assumes that the interest rate remains constant over the period, and it is only calculated on the principal. Simple interest does not take into account the effect of compounding, which can significantly increase the total interest paid or earned over time.

Percentage Calculation

In many financial calculations, converting values to percentages is essential. To find the interest rate in percentage terms from a decimal, you multiply by 100. For example, Pablo's calculated interest rate from the exercise is \(0.0625\). To convert this to a percentage, you multiply by 100: \( 0.0625 \times 100 = 6.25\% \). This conversion helps in understanding the interest rate in a more easily comprehensible form like how costly the loan is compared to others.