Problem 82 - College Algebra and Calculus: An Applied Approach

Question

For \$2000 to grow to more than \(\$ 2500\) in 2 years, what must the simple interest rate be?

Step-by-Step Solution

Answer

The interest rate must be more than 12.5% to grow \$2000 to more than \$2500 in 2 years. This is calculated by the formula \(R = \frac{I}{PT}\), where \(I=$500\), \(P=$2000\) and \(T=2\) years.

Step 1: Understand the problem

We want to find the simple interest rate required for a principal amount of \$2,000 to grow to more than \$2,500 in 2 years. In other words, the interest earned must be more than \$500 (\$2,500-\$2,000). We will use the simple interest formula \(I=PRT\).

Step 2: Setup the equation

We setup the equation: \(I = PRT\), where \(I=$500\), \(P=$2000\) and \(T=2\) years. We are looking for \(R\).

Step 3: Solve for R

Rearrange the equation for R: \(R = \frac{I}{PT}\), and convert the answer to percentage as interest rate is usually given in percentage.

Step 4: Calculate R

Substitute \(I=$500\), \(P=$2000\) and \(T=2\) years into the equation: \(R = \frac{500}{2000 \times 2}\), calculate the result and convert to percentage.

Step 5: Evaluate the result

The result will be the minimum interest rate which allows the initial amount of \$2000 to grow to more than \$2500 in two years.

Key Concepts

Interest Rate Calculation

To determine the interest rate necessary for a principal amount to grow over time, we use the simple interest formula:

• I stands for the interest earned.
• P is the principal amount or the initial sum of money.
• R represents the interest rate we want to find, expressed as a decimal.
• T is the time duration the money is invested or borrowed for, usually in years.

The basic formula for simple interest is expressed as:\[ I = PRT \]In the original exercise, the aim is to yield an interest of more than \(500 from the original \)2000 in 2 years. We rearrange the formula to solve for the interest rate:\[ R = \frac{I}{PT} \]This formulation tells us how interest depends on the principal, rate, and time. After calculating, it’s important to multiply by 100 to convert from a decimal to a percentage, which is a standard way to state interest rates.

Principal Amount

The principal amount is the original sum of money that is either invested or loaned before interest. In the context of loans, this is the amount borrowed. In investments, it's the starting capital deposit. In our scenario, this is the $2000 you initially start with. It plays a critical role in calculating how much interest you will accumulate over time.

Why the Principal Matters

As the primary basis for all interest calculations, the principal is directly proportional to the total interest earned or paid. Greater principal amounts will yield higher interests for the same interest rate and time period.The formula shows that the interest earned is a product of the initial principal, the interest rate, and the time span:\[ I = PRT \]Therefore, understanding and knowing the principal is crucial for any financial planning or forecasting.

Time in Years

Time, often measured in years, is essential when determining how much interest you'll earn or owe. In simple interest calculations, time is a linear factor, meaning it directly influences the total interest amount.

The Role of Time

In the simple interest formula, \[ I = PRT \]The term \( T \) represents the time period over which the interest is calculated. This can significantly impact your financial returns; the longer the time, the more interest accrues. It is crucial for predicting how investments will grow or loans will increase over years.For the exercise's scenario, the time given was 2 years. Time must be consistent with the interest rate's time unit for accurate calculations; typically, both are in years. Careful consideration of time ensures one correctly estimates the financial outcomes for loans or investments.