Problem 39 - 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. (a) If some money invested at \(7 \%\) for 2 years earns \(\$ 245\), how much would the same amount earn at \(5 \%\) for 1 year? (b) If some money invested at \(4 \%\) for 3 years earns \(\$ 273\), how much would the same amount earn at \(6 \%\) for 2 years? (c) If some money invested at \(6 \%\) for 4 years earns \(\$ 840\), how much would the same amount earn at \(8 \%\) for 2 years?

Step-by-Step Solution

Answer

(a) $87.50; (b) $273; (c) $560.

Step 1: Understanding the formula for simple interest

The simple interest (\( I \)) can be calculated using the formula \( I = P \, r \, t \), where \( P \) is the principal amount, \( r \) is the rate of interest per year, and \( t \) is the time in years. When it says interest varies jointly, it means we can apply only rate and time to see how interest changes because the principal stays constant.

Step 2 (a): Calculating principal using known values

From part (a), we have \( I = 245 \) when \( r = 0.07 \) and \( t = 2 \) years. Using the simple interest formula, we can rearrange to find the principal: \[ P = \frac{I}{r \, t} = \frac{245}{0.07 \, \times \, 2} = \frac{245}{0.14} = 1750. \] Hence, the principal amount \( P \) is \( \$1750 \).

Step 3 (a): Calculating interest for new rate and time

Now we need to calculate the interest for the same \( P = 1750 \) at a new rate \( r = 0.05 \) and \( t = 1 \) year: \[ I = 1750 \, \times \, 0.05 \, \times \, 1 = 87.5. \] So, the interest earned is \( \$87.50 \).

Step 4 (b): Calculating principal for part (b)

For part (b), \( I = 273 \), \( r = 0.04 \), and \( t = 3 \) years. Thus, \( P = \frac{273}{0.04 \, \times \, 3} = \frac{273}{0.12} = 2275. \) So, the principal is \( \$2275 \).

Step 5 (b): Calculating interest for new conditions in part (b)

With \( P = 2275 \), we calculate the interest for new conditions, \( r = 0.06 \) and \( t = 2 \): \[ I = 2275 \, \times \, 0.06 \, \times \, 2 = 273. \] Thus, the interest is \( \$273 \).

Step 6 (c): Calculating principal for part (c)

For part (c), \( I = 840 \), \( r = 0.06 \), and \( t = 4 \) years. Therefore, \( P = \frac{840}{0.06 \, \times \, 4} = \frac{840}{0.24} = 3500. \) Hence, the principal is \( \$3500 \).

Step 7 (c): Calculating interest for new conditions in part (c)

With \( P = 3500 \) for new conditions \( r = 0.08 \) and \( t = 2 \) years: \[ I = 3500 \, \times \, 0.08 \, \times \, 2 = 560. \] Therefore, the interest earned is \( \$560 \).

Key Concepts

Joint Variation

Joint variation occurs when one variable depends on a combination of two or more other variables. In the context of simple interest, it means the interest earned varies based on both the interest rate and the time period.

For simple interest, the formula is:

• Interest \( I \) varies jointly with the rate \( r \) and the time \( t \) when the principal \( P \) is constant.
• This is depicted as \( I = P \cdot r \cdot t \).

So, by altering either the rate or time, while keeping the principal same, you can predict how the interest will change.

Interest Rate

The interest rate is a percentage that determines how much interest you earn on a principal over a period of time. It's a key factor in calculating simple interest.

For example:

• The interest rate of 7% is written as a decimal, 0.07, when using it in calculations.
• This decimal representation is crucial in the formula \( I = P \cdot r \cdot t \).

Understanding how to convert the percentage to decimal form is vital for accurate interest calculations.

Time in Years

Time in years is the duration that money is held in an investment or account to accrue interest. Simple interest calculations assume that time is measured in years.

For instance:

• If you invest for 2 years, \( t \) is 2 in the formula \( I = P \cdot r \cdot t \).
• The longer the time period, the more interest accumulates, assuming rate and principal are constant.

Thus, understanding and accurately using time is crucial for predicting interest outcomes.

Principal Amount

The principal amount is the initial sum of money invested or loaned. This is the basis for calculating earned interest.

• For instance, if \( P = 1750 \) and other variables are known, you can compute the interest.
• The principal remains unchanged in each scenario within the problem, which allows the use of joint variation to calculate interest.

Having a firm grasp on what the principal amount is will help in understanding how the interest is generated.

Interest Calculation

Interest calculation in the context of simple interest involves multiplying the principal, rate, and time. This straightforward formula helps find the interest generated over a period.

Steps:

• Identify the principal \( P \).
• Convert the interest rate from a percentage to a decimal.
• Use the formula \( I = P \cdot r \cdot t \) to find the interest.

This basic calculation is fundamental in everyday financial situations to understand returns or costs over time.

Algebra Applications

Algebra comes into play when solving interest problems, especially when rearranging formulas to find unknown variables.

• For example, to find the principal, rearrange the formula: \( P = \frac{I}{r \cdot t} \).
• This involves understanding inverse operations and ratios, showcasing algebra's practical applications.
• Using algebra in this way allows for problem-solving even when not all variables are immediately known.

This skill enhances financial literacy and can be applied in various real-world situations.