Problem 16 - Elementary and Intermediate Algebra

Question

Investments. A business class divided an imaginary investment of \(\$ 80,000\) among three mutual funds. The first fund grew by \(10 \%,\) the second by \(6 \%,\) and the third by \(15 \% .\) Total earnings were \(\$ 8850 .\) The earnings from the first fund were \(\$ 750\) more than the earnings from the third. How much was invested in each fund?

Step-by-Step Solution

Answer

Invested amounts were \$ 45,000\ in the first fund, \$ 10,000\ in the second fund, and \$ 25,000\ in the third fund.

Step 1: Define Variables

Let the amounts invested in the three mutual funds be denoted as follows: Let the amount invested in the first fund be \(a\), in the second fund be \(b\), and in the third fund be \(c\).

Step 2: Set Up Equations Based on Investment Totals

Given that the total investment is \$ 80,000, we can write the equation: \[a + b + c = 80,000.\]

Step 3: Set Up Equations Based on Earnings

Knowing the earnings percentages, the total earnings equation is: \[0.1a + 0.06b + 0.15c = 8,850.\]

Step 4: Use Earnings Relationship

The problem states that the earnings from the first fund were \$ 750\ more than the earnings from the third fund. So, \[0.1a = 0.15c + 750.\]

Step 5: Solve the Equations

Now solve the system of equations: \[1. a + b + c = 80,000\] \[2. 0.1a + 0.06b + 0.15c = 8,850\] \[3. 0.1a = 0.15c + 750\]. First, solve equation (3) for \(a\): \[0.1a = 0.15c + 750 \rightarrow a = 1.5c + 7,500\].Substitute \(a = 1.5c + 7,500\) into equations (1) and (2).

Step 6: Substitute and Simplify

Substituting in equation (1): \[1.5c + 7500 + b + c = 80,000 \rightarrow b = 80,000 - 2.5c - 7,500 \rightarrow b = 72,500 - 2.5c\].Substituting in equation (2): \[0.1(1.5c + 7,500) + 0.06(72,500 - 2.5c) + 0.15c = 8,850\].

Step 7: Solve for \(c\)

Expand and simplify the combined earnings equation: \[0.15c + 750 + 4,350 - 0.15c + 0.15c = 8,850 \rightarrow 0.15c + 5,100 = 8,850 \rightarrow 0.15c = 3,750 \rightarrow c = 25,000\].

Step 8: Solve for other Variables

Substitute \(c = 25,000\) back into the equations for \(a\) and \(b\): \[a = 1.5(25,000) + 7,500 = 37,500 + 7,500 = 45,000\], \[b = 72,500 - 2.5(25,000) = 72,500 - 62,500 = 10,000\].

Step 9: Verify the Solution

Check if all the constraints are satisfied with the values: \[a + b + c = 45,000 + 10,000 + 25,000 = 80,000\], \[0.1(45,000) + 0.06(10,000) + 0.15(25,000) = 4,500 + 600 + 3,750 = 8,850\], \[0.1(45,000) = 4,500, 0.15(25,000) + 750 = 3,750 + 750 = 4,500 (which matches).\]. All equations are verified.

Key Concepts

System of Equations

A system of equations is a collection of two or more equations with the same set of unknowns. In this exercise, we have three equations involving the investments in three mutual funds. The key to solving a system of equations is to find values for the unknowns that simultaneously satisfy all the given equations.

To start, we define three variables: \(a\), \(b\), and \(c\), representing the amounts invested in the first, second, and third mutual funds, respectively.

From the exercise, we create three equations:
1) \[a + b + c = 80,000\]
2) \[0.1a + 0.06b + 0.15c = 8,850\]
3) \[0.1a = 0.15c + 750\]

These equations form our system, and our goal is to solve for \(a\), \(b\), and \(c\).

Percentage Growth

Percentage growth refers to the increase in a value expressed as a percentage of the original amount. In this exercise, the growth rates for the three mutual funds are 10%, 6%, and 15%.

To find the earnings from each fund, we use the growth percentages:

• The first fund grows by 10%, so its earnings are \(0.1a\).
• The second fund grows by 6%, so its earnings are \(0.06b\).
• The third fund grows by 15%, so its earnings are \(0.15c\).

Using these growth rates, we formed the second equation: \[0.1a + 0.06b + 0.15c = 8,850\]

This equation represents the total combined earnings of the three funds.

Variable Substitution

Variable substitution is a method used to simplify and solve a system of equations. It involves solving one of the equations for one variable and substituting this expression into the other equations.

In this problem, we solve the third equation for \(a\): \[0.1a = 0.15c + 750 \rightarrow a = 1.5c + 7,500\]

Next, we substitute \(a = 1.5c + 7,500\) into the first and second equations:
1) \[1.5c + 7,500 + b + c = 80,000 \rightarrow b = 72,500 - 2.5c\]
2) \[0.1(1.5c + 7,500) + 0.06(72,500 - 2.5c) + 0.15c = 8,850\]

This substitution reduces the complexity of the equations, making it easier to solve for the remaining variables.

Mutual Funds Investment

Investing in mutual funds involves pooling money from many investors to purchase a diversified portfolio of securities. Each mutual fund focuses on specific types of investments and targets different growth rates.

In this scenario, we have three different mutual funds, each with its own growth rate:

• The first fund grows by 10%.
• The second fund grows by 6%.
• The third fund grows by 15%.

The aim is to determine how much money was invested in each fund, given the total investment amount and the growth rates.

After setting up and solving our system of equations, the amounts invested in the first, second, and third mutual funds are found to be:
\(a = 45,000\), \(b = 10,000\), and \(c = 25,000\).

These calculated amounts ensure the total investment and earnings percentages match the given conditions.