Problem 96 - Precalculus

Question

Ginger talks Gary into putting less money in the money market and more money in the stock (see Exercise 95 ). They 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 stock than in the mutual fund and the mutual fund and stock have the same growth in the next year as they did in the previous year, they will earn \(\$ 840\) in a year. How much money did they put in each of the three investments?

Step-by-Step Solution

Answer

$3,000 in mutual fund, $6,000 in stock, $1,000 in money market account.

Step 1: Define Variables

Let's define the variables based on the problem statement:- Let \( x \) be the amount of money invested in the mutual fund.- Let \( y \) be the amount of money invested in the stock.- Let \( z \) be the amount of money invested in the money market account.

Step 2: Set Up Equations

From the problem, we know:1. Total investment: \( x + y + z = 10,000 \).2. The stock investment is \( 3,000 \) more than the mutual fund: \( y = x + 3,000 \).3. The total interest earned in one year is \$840, combining the interest from all three investments. This can be expressed as:\[0.07x + 0.10y + 0.03z = 840\]

Step 3: Solve the System of Equations

First, substitute \( y = x + 3000 \) into both the total investment and interest equations:- For the total investment equation: \( x + (x + 3000) + z = 10,000 \) \( 2x + 3000 + z = 10,000 \) \( 2x + z = 7,000 \) - (Equation 1)- For the interest equation: \( 0.07x + 0.10(x + 3000) + 0.03z = 840 \) \( 0.07x + 0.10x + 300 + 0.03z = 840 \) \( 0.17x + 0.03z = 540 \) - (Equation 2)

Step 4: Eliminate and Solve for One Variable

Multiply Equation 1 by 0.03:- \( 0.06x + 0.03z = 210 \)Subtract this from Equation 2:- \( (0.17x + 0.03z) - (0.06x + 0.03z) = 540 - 210 \)- \( 0.11x = 330 \)Solve for \( x \):- \( x = \frac{330}{0.11} = 3,000 \).

Step 5: Solve for the Second Variable

Use the value of \( x = 3,000 \) in Equation 1 to find \( z \):- \( 2(3000) + z = 7,000 \)- \( 6,000 + z = 7,000 \)- \( z = 1,000 \).

Step 6: Solve for the Third Variable

Find \( y \) using \( y = x + 3,000 \):- \( y = 3,000 + 3,000 = 6,000 \).

Step 7: Verify the Solution

Ensure the amounts satisfy all equations:- Total: \( 3,000 + 6,000 + 1,000 = 10,000 \), correct.- Interest: \( 0.07(3,000) + 0.10(6,000) + 0.03(1,000) = 210 + 600 + 30 = 840 \), correct.Thus, the solution is verified.

Key Concepts

Money Market Account

A money market account is a type of savings account that typically comes with higher interest rates than regular savings accounts, but with some constraints. These accounts are advantageous for those who are interested in earning a bit more from their savings while maintaining liquidity.

• They often require a higher minimum balance than regular savings accounts, but they pay interest based on the current interest rates.
• Money market accounts are usually a safe investment option as they are insured by the Federal Deposit Insurance Corporation (FDIC) up to the maximum allowed limits.
• Unlike other savings products, they may offer limited check-writing abilities or debit card access, providing more flexibility when accessing funds.

For example, in our exercise, the money market account is earning a 3% interest rate. This means that for whatever amount Gary and Ginger invest, they will earn 3% of that annually, making it a stable and secure part of their investment portfolio.

Mutual Fund Investment

Mutual fund investments are a way for individuals to pool their money together to invest in a diversified portfolio of stocks, bonds, or other securities. The fund is managed by professional portfolio managers who make investment decisions on behalf of the investors.

• The core advantage of mutual funds is diversification. By investing in a mutual fund, investors can gain exposure to a variety of assets that they might not be able to afford individually.
• They often cater to different investment goals, whether it be aggressive growth, balanced growth, or capital preservation.
• Mutual funds can vary in terms of the risk involved and the type of returns one might expect based on their investment focus.

In our exercise, the mutual fund has been averaging a 7% growth rate per year. This indicates that the fund's historical performance has been strong, potentially providing a higher return compared to more conservative options like money market accounts.

Interest Calculation

Interest calculation is fundamental in understanding how investments grow over time. Calculating interest allows investors to estimate the returns they will receive from their investment. Here's how interest was calculated in our scenario:

• For straightforward cases, the simple interest formula is used: \( I = P \times r \times t \), where:
  • \( I \) is the interest earned,
  • \( P \) is the principal amount (initial investment),
  • \( r \) is the annual interest rate (in decimal form),
  • \( t \) is the time period in years.
• In combined investments, like our exercise, interest from each investment type is summed up: for the money market account, mutual fund, and stock, each has its own rate.
• The total interest earned from all these sources was used to solve for the variables in the problem to ensure the solution matched the given total annual interest of $840.

Understanding and accurately calculating interest helps investors to plan and make informed decisions about their financial goals.