Problem 40 - Elementary Algebra

Question

A sum of \(\$ 6000\) is invested, part of it at \(5 \%\) interest and the remainder at \(7 \%\). If the interest earned by the \(5 \%\) investment is \(\$ 160\) less than the interest earned by the \(7 \%\) investment, find the amount invested at each rate.

Step-by-Step Solution

Answer

Invest \$2166.67 at 5% and \$3833.33 at 7%.

Step 1: Define Variables

Let the amount invested at 5% be \( x \). Then the amount invested at 7% will be \( 6000 - x \) since the total investment is \$6000.

Step 2: Write Interest Equations

The interest from the 5% investment is \( 0.05x \). The interest from the 7% investment is \( 0.07(6000 - x) \).

Step 3: Set Up the Equation Based on Given Conditions

According to the problem, the interest from the 5% investment is \$160 less than the interest from the 7% investment. This gives us the equation: \( 0.05x + 160 = 0.07(6000 - x) \).

Step 4: Distribute and Simplify Equation

Distribute the 0.07 to terms inside the parentheses: \( 0.05x + 160 = 420 - 0.07x \).

Step 5: Solve for \( x \)

Combine like terms: \( 0.05x + 0.07x + 160 = 420 \), which simplifies to \( 0.12x + 160 = 420 \). Subtract 160 from both sides: \( 0.12x = 260 \). Divide by 0.12 to find \( x \): \( x = \frac{260}{0.12} = 2166.67 \).

Step 6: Calculate the Amount at 7%

Since \( x \) is the amount invested at 5%, then \( 6000 - 2166.67 = 3833.33 \).

Key Concepts

Understanding Simple Interest

Simple interest is a way to calculate the interest you earn or owe on a certain amount of money, often referred to as the principal. It is different from compound interest as it is calculated on the principal alone, without taking into account any previously accrued interest. It is straightforward, making it a basic concept in financial mathematics.

Simple interest is calculated using the formula:

• The formula is: \( I = P \times r \times t \)
  Where:
  • \( I \) is the interest earned or paid
  • \( P \) is the principal amount
  • \( r \) is the rate of interest per period as a decimal
  • \( t \) is the time period the money is invested or borrowed for, in years

To visualize simple interest using the problem's context: if you invest part of your money at 5% and part at 7%, each investment will earn interest separately based on the percentage rate applied to the amount invested in each section. The equation \( 0.05x \) represents the interest from the 5% investment, while \( 0.07(6000 - x) \) represents the interest from the 7% portion.

Solving with Algebraic Equations

Algebraic equations are equations involving variables, coefficients, and constants. They can be used to find unknown values when certain conditions are known, as demonstrated in this investment problem. By using variables to represent unknown quantities, we can set up equations based on relationships and conditions provided in a problem.

In this specific investment problem, we'll use algebra to solve for the unknown amounts invested at two different interest rates. The variable \( x \) is defined to represent the amount invested at the 5% interest rate, and consequently, \( 6000 - x \) represents the amount invested at the 7% interest rate. Setting this up in equation form enables solving for \( x \):

• The equation is formulated from the information: the interest from the 5% investment is $160 less than the interest from 7% investment.
  

This leads to the equation \( 0.05x + 160 = 0.07(6000 - x) \), which is an orchestration of setting up our known quantities and conditions into a solvable format. Simplifying and solving this equation involves expanding terms and combining like terms to isolate \( x \) and eventually find its value.

Effective Problem Solving Strategies

Problem-solving in mathematics often involves a series of structured steps to make analysis and calculations clear and attainable. Here’s how this is exemplified within our investment scenario:

Start by defining variables to represent the unknown quantities. This is crucial, as it turns a word problem scenario into a mathematical model that can be calculated. Next, write out equations that represent the situation: in this problem, you create an equation based on the relationship between different amounts of interest.

Simplifying is the next essential step: distribute terms, combine like terms, and reorganize the equation to isolate the variable. This process is not only systematic but also helps in breaking down complex information into manageable parts.

Finally, solving the algebraic equation gives you the precise values of unknowns, achieving the problem goal. Once you have found \( x \), substitute back to find other needed values, confirming both amounts invested at each interest rate. This structured method ensures each step logically leads to the next, honing problem-solving skills applicable in broader contexts.