Problem 13 - Prealgebra

Question

If you invest \(P\) dollars ( \(P\) is for principal) at simple interest rate \(R\) for \(T\) years, the amount of interest you will earn is given by the formula \(I=P \cdot R \cdot T .\) Find \(I\) $$P=\$ 1,000, R=\frac{7}{100}, \text { and } T=2 \text { years }$$

Step-by-Step Solution

Answer

The interest earned is \$140.

Step 1: Identify the Values

First, identify the values provided in the problem. The principal amount \(P\) is \$1,000, the interest rate \(R\) is \(\frac{7}{100}\), and the time \(T\) is 2 years.

Step 2: Use the Interest Formula

Use the simple interest formula given by \(I = P \cdot R \cdot T\). Plug in the values identified: \(I = 1000 \cdot \frac{7}{100} \cdot 2\).

Step 3: Simplify the Expression

Calculate the expression by simplifying step by step: First calculate \(1000 \cdot \frac{7}{100} = 70\). Then multiply the result by 2: \(70 \cdot 2 = 140\).

Step 4: State the Solution

The calculated interest \(I\) is \$140. This is the amount of interest you will earn over the given time period.

Key Concepts

Principal

In finance, the principal is the initial sum of money borrowed or invested, before any interest is added. Imagine you have a piggy bank and decide to put some money into it; that initial sum is your principal.

The principal plays a crucial role because it is the base amount on which all more interest calculations are performed. The simple interest formula, which is often used to calculate the return on an investment, relies heavily on this amount. It's like the starting line in a race:

• If you are loaning money, it represents how much you initially lent out.
• If you're investing, it's the total amount you put into an account or venture at the start.

In the context of the exercise, the principal is $1,000. This is the starting amount on which your earnings or interest is calculated.

Interest Rate

The interest rate is a percentage that tells you how much additional money you will earn or owe based on your principal. Think of it as the fee paid for using someone else's money, or what others pay you for using yours. This rate is typically expressed as a percentage.

There are different types of interest rates, but in this exercise, we focus on simple interest. Simple interest is calculated only on the principal amount. Unlike compound interest, it doesn't accumulate on previously earned interest.

In our exercise, the interest rate is \(\frac{7}{100}\) or 7%. This means for every \(100 you invest, you earn \)7 per year. Here, it's important to remember that:

• A higher interest rate means more earnings or more to pay.
• A lower interest rate means less.

Time Period

The time period reflects how long the money is invested or borrowed. It's a crucial factor in calculating simple interest because it determines how long the principal is subject to earning interest.

In the simple interest formula, the time period is usually measured in years, but it can be in months or days too. The important thing to grasp is that the longer the time period, the more interest you will earn if you are investing, or owe if you are borrowing.

For our specific example in the exercise, the time period is 2 years. This means the principal is accruing interest for 2 full years. When calculating:

• A longer time period results in more interest being generated.
• A shorter period results in less interest.