Problem 95 - Precalculus

Question

Gary and Ginger decide to place \(10,000\) of their savings into investments. They put some in a money market account earning \(3 \%\) interest, some in a mutual fund that has been averaging \(7 \%\) a year, and some in a stock that rose \(10 \%\) last year. If they put \(3,000\) more in the money market than in the mutual fund and the mutual fund and stocks have the same growth in the next year as they did in the previous year, they will carn \(\$ 540\) in a year. How much money did they put in each of the three investments?

Step-by-Step Solution

Answer

Money Market: $5,500, Mutual Fund: $2,500, Stock: $2,000.

Step 1: Identify Variables

Let \( x \) be the amount of money invested in the mutual fund, \( y \) be the amount in the stock, and \( z \) be the amount in the money market account.

Step 2: Set Up Equations

From the problem, we know the total investment is \( x + y + z = 10,000 \). Also, they put \( 3,000 \) more in the money market account than in the mutual fund, so \( z = x + 3,000 \). The interest equation is given by \( 0.07x + 0.10y + 0.03z = 540 \).

Step 3: Substitute and Simplify

Substitute \( z = x + 3,000 \) into the equations. The sum equation becomes \( x + y + (x + 3,000) = 10,000 \), leading to \( 2x + y = 7,000 \). Substitute \( z = x + 3,000 \) into the interest equation: \( 0.07x + 0.10y + 0.03(x + 3,000) = 540 \). Simplify to get \( 0.10x + 0.10y + 90 = 540 \).

Step 4: Solve for One Variable

Solve \( 0.10x + 0.10y + 90 = 540 \) for \( y \) in terms of \( x \). Rearrange to get \( 0.10y = 450 - 0.10x \), leading to \( y = 4,500 - x \).

Step 5: Solve System of Equations

Substitute \( y = 4,500 - x \) into \( 2x + y = 7,000 \): \( 2x + (4,500 - x) = 7,000 \) simplifies to \( x = 2,500 \). From the equation \( y = 4,500 - x \), we get \( y = 2,000 \). Use \( z = x + 3,000 \), resulting in \( z = 5,500 \).

Step 6: Verify Solution

Verify the obtained solution \( x = 2,500 \), \( y = 2,000 \), \( z = 5,500 \) satisfies all the original conditions: total \( 10,000 \), correct interest calculation gives \( 540 \), and correct money allocation in each account.

Key Concepts

Investment Allocation

When dealing with investment allocation, it's important to understand how to distribute funds among different investment options to achieve a desired outcome. Gary and Ginger had a total of $10,000 to invest, which they needed to allocate in a way that straightened with their earning goals. Investment allocation involves more than just dividing money; it's about choosing the right mix of investments to work harmoniously while considering expected rate of returns.

Gary and Ginger decided to place their money into three types of accounts: a money market account, a mutual fund, and stocks. They had specific directions to follow:

• The money market account had to hold $3,000 more than the mutual fund.
• The total allocation had to answer for the entire $10,000 they were working with.
• They wanted to ensure they earn a specific amount of interest over time.

This careful distribution and planning is crucial. It not only helps in reaching financial goals but also minimizes risks by diversifying into various investment vehicles.

Interest Calculation

Interest calculation is a fundamental part of this problem and important for understanding how investments grow over time. To find out how much Gary and Ginger will earn from their investments, we need to pay attention to the interest rates of each investment type.

Each investment earns interest at a different rate:

• The money market account earns 3% annually.
• The mutual fund has an annual return of 7%.
• Stocks have the highest annual return of 10%.

The general formula for interest calculation is:\[\text{Interest} = \text{Principal} \times \text{Rate}\]For multiple investments, you add the interest earned from each. So the total interest is expressed as:\[0.07x + 0.10y + 0.03z = 540\]Here, each term represents the product of the amount invested in each account and its respective interest rate, adding up to the total interest Gary and Ginger aimed to earn the next year.

Variable Substitution

Variable substitution is a mathematical technique used to simplify equations. It makes solving the system of equations easier. In this problem, it helps manage multiple variables by reducing them into a form that is simpler to handle.

Starting with three variables:

• Let \( x \) represent the amount invested in the mutual fund.
• Let \( y \) be the amount invested in stocks.
• Let \( z \) be the money market account investment.

To streamline the equations, we use the relationship \( z = x + 3,000 \) since the money market investment is $3,000 more than the mutual fund. Substituting \( z \) in both the total investment and interest equations allows us to reduce the number of variables, making them easier to solve.

By substituting, the equations become:1. The allocation equation: \[2x + y = 7,000\] 2. The interest equation simplifies to: \[0.10x + 0.10y + 90 = 540\]Through continued substitution and solving these simplified equations, we derive the exact amounts invested in each account, ensuring the conditions are met and the solution is consistent with the problem's requirements.