Problem 37 - College Algebra with Modeling and Visualization

Question

A sum of \(\$ 20,000\) is invested in three mutual funds. In one year the first fund grew by \(5 \%,\) the second by \(7 \%,\) and the third by \(10 \% .\) Total earnings for the year were \(\$ 1650\). The amount invested in the third fund was 4 times the amount invested in the first fund. Find the amount invested in each fund.

Step-by-Step Solution

Answer

First: $2500, Second: $7500, Third: $10000.

Step 1: Define Variables

Let the amount invested in the first fund be \(x\). Then the amount invested in the third fund, which is 4 times that of the first, will be \(4x\). If the amounts invested in all three funds add up to \(\$20,000\), then the amount invested in the second fund will be \(20,000 - x - 4x = 20,000 - 5x\).

Step 2: Set Up Equations for Earnings

The earnings from the first fund are \(0.05x\), from the second fund are \(0.07(20,000 - 5x)\), and from the third fund are \(0.10(4x)\). The total earnings from all three funds is given as \(\$1650\). Therefore, the equation for total earnings is:\[0.05x + 0.07(20,000 - 5x) + 0.10(4x) = 1650\]

Step 3: Simplify and Solve the Equation

Distribute the percentage into the brackets:\[0.05x + 0.07 \times 20,000 - 0.35x + 0.40x = 1650\]Calculate \(0.07 \times 20,000\) which is \(1400\):\[0.05x - 0.35x + 0.40x + 1400 = 1650\]Combine like terms:\[0.10x + 1400 = 1650\]Subtract 1400 from both sides:\[0.10x = 250\]Divide by 0.10:\[x = 2500\]

Step 4: Calculate Investments in Each Fund

Now that we have \(x = 2500\):- The first fund amount is \(x = \\(2500\).- The third fund is \(4x = 4 \times 2500 = \\)10000\).- The second fund is \(20,000 - 5x = 20,000 - 12500 = \$7500\).

Key Concepts

Mutual Funds

Mutual funds are a popular investment option where individuals can pool their money to invest collectively in a diversified portfolio of assets. These funds are managed by professional fund managers who allocate the money into various securities such as stocks, bonds, or other investment vehicles.
One of the key benefits of investing in mutual funds is diversification. By holding a mix of various securities, the risk is spread out, reducing the impact of any one asset's poor performance. Additionally, mutual funds offer liquidity, meaning investors can buy or sell their shares on any given business day.
Whether you're new to investing or a seasoned investor, mutual funds offer an accessible way to participate in a managed portfolio. They cater to individuals who may not have the time, expertise, or resources to manage their own investment portfolios. This makes them a convenient option for growing wealth over time.

Simple Interest

Simple interest is a fundamental concept in finance that refers to the calculation of interest on the initial amount (principal) of an investment or loan. It is a straightforward method, where the interest is charged only on the original principal amount, rather than on accumulated interest.
The formula for calculating simple interest can be expressed as:

• Interest = Principal × Rate × Time

Where:

• **Principal** is the original amount of money invested or loaned.
• **Rate** is the annual interest rate as a decimal.
• **Time** is the period the money is invested or borrowed in years.

Understanding simple interest is crucial for making informed investment decisions and evaluating financial contracts. It allows investors to calculate potential earnings or costs, making it easier to compare different investment options.

Algebraic Equations

Algebraic equations are mathematical statements that contain one or more unknown variables represented by symbols, typically letters such as x or y. These equations are used to express relationships between quantities and to find unknown values.
In the context of the original exercise, algebraic equations help us determine the amount invested in each mutual fund. By setting up an equation based on the interest earned and the total investment sum, we can solve for the unknowns like the investments in each fund.
For example, if you know that the total earnings are $1650 and comprise the interest from three different funds, you can set up an equation to distribute and calculate each fund's specific investment. This process involves:

• Defining variables for unknown amounts.
• Writing expressions based on known interest rates.
• Solving the equation by combining like terms and using arithmetic operations.

Understanding how to construct and solve algebraic equations is a valuable skill not only in finance but in numerous problem-solving scenarios across different fields.