Question

Ethan has $$\$ 60,000$$ to invest. He puts part of the money in a CD that earns \(3 \%\) simple interest per year and the rest in a mutual fund that earns \(8 \%\) simple interest per year. How much did he invest in each if his earned interest the first year was $$\$ 3700$$.

Short answer

Ethan invested \(22,000 in the CD and \)38,000 in the mutual fund.

Step-by-step solution

Define Variables

Let’s define the variables for the amounts invested. Let: - x be the amount of money invested in the CD at 3%. - y be the amount of money invested in the mutual fund at 8%.

Set Up Equations

Set up two equations based on the information given: 1. The total money invested is \(60,000. So, we have: x + y = 60000 2. The total interest earned in one year is \)3,700. So, we have: 0.03x + 0.08y = 3700

Solve the System of Equations - Substitution Method

First, solve the first equation for y: y = 60000 - x Substitute this expression for y in the second equation: 0.03x + 0.08(60000 - x) = 3700

Simplify and Solve for x

Simplify the equation from Step 3 to find x: 0.03x + 4800 - 0.08x = 3700 Combine like terms: -0.05x + 4800 = 3700 Subtract 4800 from both sides: -0.05x = -1100 Divide both sides by -0.05: x = 22000

Find y Using the Value of x

Now that we have x, we can find y: y = 60000 - x Substitute the value of x: y = 60000 - 22000 y = 38000

Verify the Solution

Check if the values satisfy both original equations: 1. x + y = 60000 22000 + 38000 = 60000 60000 = 60000 2. 0.03x + 0.08y = 3700 0.03(22000) + 0.08(38000) = 3700 660 + 3040 = 3700 3700 = 3700 Both equations are satisfied, confirming the solution is correct.

Key concepts

Simple Interest

Simple interest is a way to calculate the interest charge on a loan or investment. It is determined by multiplying the principal amount (the initial amount of money), the interest rate, and the time period for which the money is borrowed or invested.
For example, if Ethan invests part of his $$\text{\textdollar}60,000$$ in a CD with a 3% simple interest rate, the formula to calculate the interest is:
\text { Interest } = \text { Principal } \times \text { Rate } \times \text { Time } \( I = P \times R \times T \)
Where:
- \(I\) is the interest earned
- \(P\) is the principal amount (the amount invested)
- \(R\) is the annual interest rate (expressed as a decimal)
- \(T\) is the time period in years
This formula makes it easy to calculate the interest on investments with fixed rates over a set period.

Systems of Equations

In this problem, we need to solve a system of equations. A system of equations is a set of two or more equations with the same set of unknowns. We are given two key pieces of information:
1. The total amount of money invested is \(\text{\textdollar}60,000\)
2. The total interest earned in one year is \(\text{\textdollar}3,700\)
From this, we can write the following equations:
\begin{align*} x + y &= 60000 \ \text{(total investment)} \ 0.03x + 0.08y &= 3700 \ \text{(total interest)} \ \text {where } x \text{ is the amount invested in the CD, and } y \text{ is the amount invested in the mutual fund.} \ These equations form a system, which we can solve using various methods like substitution or elimination.

Substitution Method

One effective way to solve a system of equations is the substitution method. This method involves solving one of the equations for one variable and then substituting that expression into the other equation.
Let's follow the steps for our problem:
1. Solve the first equation for \(y\) (the amount invested in the mutual fund):
\begin{align*} x + y &= 60000 \ y &= 60000 - x \ \text {2. Substitute this expression for } y \text { into the second equation:} \ 0.03x + 0.08(60000 - x) &= 3700 \ \text {3. Simplify and solve for } x: \ 0.03x + 4800 - 0.08x &= 3700 \ -0.05x + 4800 &= 3700 \ -0.05x &= -1100 \ x &= 22000 \ \text{This tells us that Ethan invested \text{\textdollar}22,000 in the CD.}

Algebraic Problem-Solving

Algebraic problem-solving involves using equations to find the values of unknown variables. In investment problems like this one, we need to understand the relationships between the variables and use algebra to solve them. Here's a quick summary:

- **Define the Variables**: Let \( x \) be the amount invested in the CD, and \( y \) be the amount invested in the mutual fund.
- **Set Up the Equations**: Use the given information to write the equations.
- **Solve the System**: Use methods like substitution or elimination to find the values of the variables.
- **Verify the Solution**: Check if the values satisfy both original equations.

In our case, we used substitution to find \( x = \text{\textdollar}22,000 \), then calculated \( y = 60000 - 22000 = \text{\textdollar}38,000 \). Always verify by plugging the values back into the original equations to ensure they hold true:
\begin{align*} x + y &= 60000 \ (22000 + 38000 = 60000) \ 0.03(22000) + 0.08(38000) &= 3700 \ (660 + 3040 = 3700) \ \text {Both equations are satisfied, confirming the solution is correct.}