Problem 37 - Elementary Algebra

Question

Fawn invested a certain amount of money at \(3 \%\) interest and she invested \(\$ 1250\) more than that amount at 5\%. Her total yearly interest was \(\$ 134.50\). How much did she invest at each rate?

Step-by-Step Solution

Answer

Fawn invested $900 at 3% and $2150 at 5%.

Step 1: Define Variables

Let \( x \) be the amount Fawn invested at 3%. Therefore, the amount she invested at 5% will be \( x + 1250 \).

Step 2: Write Interest Equation

The interest from the 3% investment is \( 0.03x \) and from the 5% investment is \( 0.05(x + 1250) \). The total interest is given as \( 134.50 \). Thus, the equation for total interest is: \[ 0.03x + 0.05(x + 1250) = 134.50 \]

Step 3: Simplify the Equation

Simplify the equation: \[ 0.03x + 0.05x + 62.50 = 134.50 \] \[ 0.08x + 62.50 = 134.50 \]

Step 4: Solve for \( x \)

Subtract 62.50 from both sides: \[ 0.08x = 72 \] Divide both sides by 0.08: \[ x = 900 \]

Step 5: Calculate Amounts Invested

Fawn invested \( x = 900 \) at 3% and \( x + 1250 = 900 + 1250 = 2150 \) at 5%.

Key Concepts

Interest Calculation

Interest calculation is a fundamental concept in understanding how investments grow over time. Whenever you deposit money into an interest-bearing account or investment, it accrues interest over periods like months or years. The amount of interest you earn depends on two main components:

• Principal: This is the initial amount of money invested or borrowed.
• Interest Rate: This is the percentage at which the principal amount grows over a period of time.

For example, in the problem provided, Fawn invested a certain amount at 3% interest. This means for every dollar invested, 3% of it is added as interest each year. Therefore, if her investment is represented as \(x\), the interest from the 3% portion is calculated using the formula:\[ 0.03x \]Similarly, knowing she invested an additional \(1250\) dollars at 5% interest, the interest for this portion is:\[ 0.05(x + 1250) \] Understanding interest calculation is crucial for both personal savings and investment strategies as it helps in predicting the growth of money over time.

Equation Simplification

Equation simplification is a critical mathematical process in solving any algebra problem. It involves reducing complex equations to simpler forms, making it easier to find the solution. In algebra, this often involves:

• Combining like terms
• Using basic arithmetic operations such as addition, subtraction, multiplication, or division

In the exercise, after setting up the interest equation:\[ 0.03x + 0.05(x + 1250) = 134.50 \] We simplified it by first expanding and then combining like terms:\[ 0.03x + 0.05x + 62.50 = 134.50 \] This turns into:\[ 0.08x + 62.50 = 134.50 \] The simplification process reduces the equation into a more manageable form where you can easily isolate \(x\) by performing inverse operations, ultimately solving it. Mastering equation simplification allows you to tackle more complex mathematical problems efficiently.

Investment Problems

Investment problems are common in algebra and involve calculating how money grows over time when placed in different investment scenarios. These problems often require setting up equations based on given scenarios and then finding unknown values that represent real-world quantities like initial investment amounts or expected returns.The core idea in solving investment problems is to:

• Define variables to represent unknown quantities.
• Set up a mathematical equation based on the relationships stated in the problem.
• Simplify and solve the equation to find the value of the unknowns.

In Fawn's example, we defined \(x\) as the amount invested at 3%, while \(x + 1250\) represented the amount invested at 5%. We used the total interest value of \(134.50\) to establish a relationship between these investments. By solving the simplified equation:\[ 0.08x + 62.50 = 134.50 \]We found the exact amounts Fawn invested at each rate. Investment problems help in understanding the impact of different interest rates on investments, providing important financial insights.