Problem 265 - Elementary Algebra

Question

Sam invested \(\$ 48,000,\) some at \(6 \%\) interest and the rest at \(10 \%\). How much did he invest at each rate if he received \(\$ 4,000\) in interest in one year?

Step-by-Step Solution

Answer

Sam invested \(20,000 at 6% and \)28,000 at 10%.

Step 1: Define the variables

Let the amount invested at 6% be denoted by \( x \) and the amount invested at 10% be denoted by \( 48000 - x \).

Step 2: Write the interest equations

The interest from the amount invested at 6% is \( 0.06x \) and from the 10% investment, it's \( 0.10(48000 - x) \). The total interest is given as \( 4000 \), thus the equation is \: \ \ \[ 0.06x + 0.10(48000 - x) = 4000 \ \ \]

Step 3: Simplify the equation

Expand and simplify the equation: \ \[ 0.06x + 4800 - 0.10x = 4000 \ \ -0.04x + 4800 = 4000 \ -0.04x = 4000 - 4800 \ -0.04x = -800 \ x = 20000 \] Hence, \( x = 20000 \).

Step 4: Calculate the amount invested at 10%

Subtract the amount invested at 6% from the total investment to get the amount invested at 10%: \ \ \[ 48000 - 20000 = 28000 \]

Key Concepts

Algebraic Equations

In simple interest problems, algebraic equations are used to represent different parts of the problem using variables. Here, we defined the amount invested at 6% as \( x \) and the amount invested at 10% as \( 48000 - x \). This allows us to write the relationship using mathematical expressions. The total interest earned from both investments is given by the sum of the interest from each part, which leads to the equation \( 0.06x + 0.10(48000 - x) = 4000 \). By solving this equation, we can find the exact amounts invested at each rate. When dealing with algebraic equations, it's important to:

• Define variables clearly
• Set up expressions based on given information
• Combine terms and simplify
• Solve for the unknowns

Investment Calculations

Investment calculations are a crucial part of financial arithmetic. In our exercise, Sam's total investment needs to be divided into parts with different interest rates. Here, the principal amount is \( 48000 \). Investment at different rates requires calculating the generated interest separately and then combining these to see if they match the conditions given. Here's how it's done:
The interest at 6% is calculated as \( 0.06x \) and at 10% as \( 0.10(48000 - x) \). Combining these gives the total interest, ensuring it matches the specified \( 4000 \). Steps to do investment calculations:

• Divide the principal according to different rates
• Calculate interest for each part using \( \text{Interest} = \text{Principal} \times \text{Rate} \)
• Add them to determine the total interest

This helps in finding the exact distribution required for a given total interest.

Interest Rates

Interest rates determine how much earnings an investment generates over a time period, usually expressed as a percentage. In our problem, Sam invested money at two different rates: 6% and 10%. Interest rates directly affect the total interest earned, calculated using the formula: \[ \text{Interest} = \text{Principal} \times \text{Rate} \times \text{Time} \]
Here, time is 1 year. Calculating interest for each rate separately and adding them helped us find the correct distribution of the investment. Relevant points about interest rates:

• Higher rate yields more interest for the same principal
• It’s directly proportional to the amount and time
• Simple interest does not compound, making calculations straightforward

Understanding interest rates is key to making informed investment decisions.