Problem 62 - Elementary Algebra

Question

Set up an algebraic equation and then solve. Marty put last year's bonus of \(\$ 2,400\) into two accounts. He invested part in a CD with \(2.5 \%\) annual interest and the rest in a money market fund with \(1.3 \%\) annual interest. His total interest for the year was \(\$ 42.00 .\) How much did he invest in each account?

Step-by-Step Solution

Answer

Marty invested \$900 in the CD and \$1500 in the money market fund.

Step 1: Define Variables

Let's define variables for the amounts invested in each account. Let \( x \) represent the amount invested in the CD with 2.5% interest, and \( y \) represent the amount invested in the money market fund with 1.3% interest.

Step 2: Set Up Equations

We know the total amount invested by Marty is \\(2400, and the total interest earned is \\)42. The first equation based on the total investment is: \[ x + y = 2400 \]The second equation based on the interest earned is: \[ 0.025x + 0.013y = 42 \]

Step 3: Solve for One Variable

Solve the first equation for \( y \): \[ y = 2400 - x \]Substitute this expression for \( y \) in the interest equation.

Step 4: Substitute and Solve

Substitute \( y = 2400 - x \) into the second equation: \[ 0.025x + 0.013(2400 - x) = 42 \] Expand the equation: \[ 0.025x + 31.2 - 0.013x = 42 \]Combine like terms: \[ 0.012x + 31.2 = 42 \]

Step 5: Simplify and Solve for x

Subtract 31.2 from both sides: \[ 0.012x = 10.8 \] Divide both sides by 0.012: \[ x = \frac{10.8}{0.012} = 900 \]Marty invested \$900 in the CD.

Step 6: Solve for y

Use the equation \( y = 2400 - x \) to solve for \( y \): \[ y = 2400 - 900 = 1500 \]Marty invested \$1500 in the money market fund.

Key Concepts

Solving Equations

Solving equations is a method used to find the unknown values that satisfy given conditions in a problem. In this case, Marty has invested in two accounts and earned a total interest. Using this information requires setting up algebraic equations to find the amounts invested in each account.

To solve these equations, we define variables to represent the unknowns. We are given two pieces of information:

• The total money Marty invested: $2,400.
• The total interest he earned: $42.

These details allow us to establish two equations. The first equation is the sum of amounts invested in both accounts. The second relates to the interest earned on each account. By manipulating these equations, we solve for one variable at a time, substituting to find the values of the unknowns.

Investment Problems

Investment problems often involve figuring out how much money is placed in various investment vehicles to achieve certain goals, such as a specific total interest earned. These problems require an understanding of how investments grow over time, affected by interest rates.

In this scenario, Marty split his bonus between two different accounts, each with different interest rates. We need to find how much he invested in each to gain a total of $42 in interest for the year. Such problems exemplify the real-world application of algebra, where investments and returns can be precisely calculated through mathematical equations.

Interest Calculation

Interest calculation is fundamental in investment problems, determining how much will be earned or paid over time based on the principal amount and the rate of interest. Simple interest is used in this exercise, calculated as the principal amount multiplied by the interest rate and time period.

In Marty's example, one account earns interest at 2.5%, and the other at 1.3%. The interest equations reflect these rates:

• For the CD: Interest = \(0.025x\)
• For the money market: Interest = \(0.013y\)

Adding these interests gives us the total interest of $42. Understanding how to apply these rates to different principal amounts allows us to solve for the unknown quantities in the problem.

Variable Definition

Defining variables is an essential first step in solving algebraic equations. In tackling this investment problem, clearly defining what each variable represents is crucial for setting up and solving the appropriate equations.

In the exercise, we defined:

• \( x \) as the amount invested in the CD at 2.5% interest.
• \( y \) as the amount invested in the money market fund at 1.3% interest.

By assigning variables, we translate the written problem into mathematical equations. This step ensures we track each part of the problem correctly and allows us to manipulate the equations logically to find our solution.