Problem 36 - Algebra for College Students

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. (a) If some money invested at \(11 \%\) for 2 years earns \(\$ 385\), how much would the same amount earn at \(12 \%\) for 1 year? \(\$ 210\) (b) If some money invested at \(12 \%\) for 3 years earns \(\$ 819\), how much would the same amount earn at \(14 \%\) for 2 years? \(\$ 637\) (c) If some money invested at \(14 \%\) for 4 years earns \(\$ 1960\), how much would the same amount earn at \(15 \%\) for 2 years? \(\$ 1050\)

Step-by-Step Solution

Answer

(a) $210, (b) $637, (c) $1050.

Step 1: Understanding the Relationship

The problem states that simple interest varies jointly with the rate of interest and the time. This means the interest can be modeled by the equation \( I = k \cdot r \cdot t \), where \( I \) is the interest, \( r \) is the rate, \( t \) is the time, and \( k \) is a constant of proportionality depending on the principal amount.

Step 2: Solving Part (a) - Find the Constant

For part (a), we know \( I = 385 \), \( r = 11\% = 0.11 \), and \( t = 2 \). First, solve for \( k \) using these values: \( k = \frac{I}{r \cdot t} = \frac{385}{0.11 \cdot 2} = 1750 \). This \( k \) represents the principal amount.

Step 3: Solving Part (a) - Calculate New Interest

Now, use \( k = 1750 \) to find the interest for a rate \( r = 12\% = 0.12 \) and time \( t = 1 \): \( I = 1750 \cdot 0.12 \cdot 1 = 210 \). Thus, the interest earned is \( \$210 \).

Step 4: Solving Part (b) - Find the Constant

For part (b), we have \( I = 819 \), \( r = 12\% = 0.12 \), and \( t = 3 \). Calculate \( k \): \( k = \frac{819}{0.12 \cdot 3} = 2275 \). This is the principal amount for this scenario.

Step 5: Solving Part (b) - Calculate New Interest

Use \( k = 2275 \) with \( r = 14\% = 0.14 \) and \( t = 2 \): \( I = 2275 \cdot 0.14 \cdot 2 = 637 \). Hence, the interest earned is \( \$637 \).

Step 6: Solving Part (c) - Find the Constant

In part (c), the variables are \( I = 1960 \), \( r = 14\% = 0.14 \), and \( t = 4 \). Solve for \( k \): \( k = \frac{1960}{0.14 \cdot 4} = 3500 \). This \( k \) is the principal amount for this setup.

Step 7: Solving Part (c) - Calculate New Interest

Finally, calculate the new interest with \( k = 3500 \), \( r = 15\% = 0.15 \), and \( t = 2 \): \( I = 3500 \cdot 0.15 \cdot 2 = 1050 \). Hence, the interest earned is \( \$1050 \).

Key Concepts

Joint Variation

Imagine having a direct relation between three variables—this describes joint variation. In the context of simple interest, the total interest earned varies jointly with the interest rate and the time the principal is invested.

What does this mean in simple terms?

• Joint variation indicates that if one of these factors (rate or time) increases, the interest also increases proportionately, assuming the other factor remains unchanged.
• The equation representing this relationship is: \[ I = k \cdot r \cdot t \] Here, \( I \) is the interest, \( r \) is the interest rate, \( t \) is the time in years, and \( k \) is a constant that depends on the amount of money originally invested, known as the principal.

This equation shows that the interest earned varies directly with both the interest rate and the time period. Knowing this makes calculations easier when one or more of these variables change.

Interest Rate

The interest rate is like the price of borrowing money or the reward for investing it. It's expressed as a percentage of the principal for a time period, typically a year. Here, it plays a crucial role in calculating the simple interest using joint variation.

Understanding Interest Rate:

• A higher interest rate means more interest earned or paid over the same period.
• In the equation \( I = k \cdot r \cdot t \), \( r \) is crucial because even a small change in \( r \) can significantly impact \( I \).
• Interest rates can vary based on economic conditions, the lender’s policies, and the borrower’s creditworthiness.

As seen in the examples given in the exercise, changing from an interest rate of 11% to 12% or 14% to 15% results in notable differences in the final interest earned.

Time in Years

Time is the period the principal is invested or borrowed. It's usually measured in years for simplicity in interest calculations, but shorter periods can be calculated by adjusting the time fractionally.

Why Time Matters:

• The longer money is invested, the more interest it accumulates, assuming the interest rate stays constant.
• Time, represented as \( t \) in the equation \( I = k \cdot r \cdot t \), directly influences how much interest is earned.
• When time increases, the product \( r \cdot t \) becomes larger, consequently raising the calculated interest.

In the exercise, you notice different scenarios where time varies from 1 to 4 years. The difference in time impacts the interest calculation significantly.