Problem 36 - Intermediate Algebra

Question

The simple interest earned by a certain amount of money varies jointly as the rate of interest and the time (in years) that the money is invested. If the money is invested at \(8 \%\) for 2 years, \(\$ 80\) is earned. How much is earned if the money is invested at \(6 \%\) for 3 years?

Step-by-Step Solution

Answer

The interest earned is $90.

Step 1: Understanding the Problem

We are given that the simple interest varies jointly as the rate of interest and the time. This tells us that the relationship is given by \(I = k imes R imes T\), where \(I\) is the interest, \(R\) is the rate, \(T\) is the time, and \(k\) is the constant of variation.

Step 2: Finding the Constant of Variation (k)

We know \(I = 80\), \(R = 0.08\), and \(T = 2\). Plug these values into the equation: \(80 = k \times 0.08 \times 2\). Simplify to find \(k\): \(80 = k \times 0.16\), thus \(k = \frac{80}{0.16}\). Calculating gives \(k = 500\).

Step 3: Calculate Interest for New Rate and Time

Now use \(k = 500\), with \(R = 0.06\) and \(T = 3\) to find the new interest. Substitute values into the formula \(I = k \times R \times T\): \(I = 500 \times 0.06 \times 3\).

Step 4: Simplify the Expression

Simplify the equation: \(I = 500 \times 0.18\). Calculate \(I = 90\). This is the amount of simple interest earned at 6% for 3 years.

Key Concepts

Joint Variation

Joint variation is a concept where one quantity varies with two or more other quantities simultaneously. It’s like a dance where each partner affects the others. In this context, the interest earned (\( I \) varies according to both the rate (\( R \) and the time (\( T \). Mathematically, this is represented as \( I = k \cdot R \cdot T \), where \( k \) is a constant term known as the constant of variation.

This means if you change the rate or the time, the interest changes predictably. If the rate goes up or you leave the money in longer, more interest is earned.

• **Proportional Change:** Changing the rate or time will proportionally change interest.
• **Constant Proportion:** \( k \) remains the same unless the initial conditions change.

Understanding joint variation helps in predicting outcomes like how much interest is earned with different rates and times.

Rate of Interest

The rate of interest is a crucial element in calculating simple interest. It's essentially a percentage that determines how much money your investment earns over a period. In our scenario, the rate of interest (\( R \) affects how quickly the initial amount grows.

A higher rate means more reward for your investment during the same time, while a lower rate results in slower growth. In our case, we calculated interest using two different rates: \( 8\% \) (0.08) and \( 6\% \) (0.06).

Here are some key points:

• **Percentage Format:** Rates are usually expressed as a percentage and need to be converted into decimals for calculations.
• **Impact on Growth:** A small change in rate can significantly impact the total interest earned.

Changing the rate of interest is like adjusting the speed of your profit's growth.

Time (Years)

Time, often measured in years, is another factor that jointly affects simple interest. The time (\( T \) that the money is invested is directly proportional to the interest earned.

The longer you invest your money, the more interest it generates. Time hence plays a critical role and can lead to significant differences in the interest accrued.

Consider the features of how time impacts interest:

• **Linear Relationship:** Longer time leads to more interest in simple interest calculations.
• **Independence of Rate:** While the rate determines the pace, time sets the track length for interest accumulation.

By understanding this, you can plan how long to invest to reach your financial goals.