Problem 56 - Algebra for College Students

Question

Solve each investment problem. Mona received a year-end bonus of \(\$ 17,000\) from her company and invested the money in an account paying \(6.5 \%\). How much additional money should she deposit in an account paying \(5 \%\) so that the return on the two investments will be \(6 \% ?\)

Step-by-Step Solution

Answer

Mona should deposit an additional \(\$8500\).

Step 1: Identify Variables

Let's denote the additional money Mona needs to deposit in the 5% account as \(x\).

Step 2: Determine Interest from Existing Investment

Calculate the interest from the initial investment of \(\$17,000\) at an interest rate of 6.5%. This is: \(0.065 \times 17000\).

Step 3: Write the Total Interest Formula

The total interest from both investments should lead to an overall 6% return. The total investment is \(17000 + x\) and the interest should be \(0.06 \times (17000 + x)\). Set up the equality: \(0.065 \times 17000 + 0.05 \times x = 0.06 \times (17000 + x)\).

Step 4: Simplify and Solve for x

Simplify the equation: \(1105 + 0.05x = 1020 + 0.06x\) Subtract \(0.05x\) and 1020 from both sides: \(85 = 0.01x\) Solve for \(x\): \(x = \frac{85}{0.01} = 8500\).

Key Concepts

investment calculation

Investment calculation refers to determining the amount of money placed into different financial assets with the aim of earning a return. These assets can be anything from savings accounts to stocks or bonds. When calculating investments, it's important to consider the initial amount and the expected return. This involves:

• Identifying how much money is invested initially.
• Calculating the returns based on the interest rates and periods.

In the problem, Mona's initial investment is \(17000\), which she puts in a 6.5% interest account. We aim to find out how much additional money, denoted as \(x\), she needs to invest at a 5% rate to achieve an overall 6% return from the two investments. Understanding how to allocate investments and calculate combined returns helps in making informed financial decisions.

interest rate calculation

Interest rate calculation is essential for determining the earnings on investments. Interest can be simple or compound, but here we focus on simple interest, which is calculated as follows:
\[ \text{Interest} = \text{Principal} \times \text{Rate} \times \text{Time} \]Where:

• Principal is the initial amount of money invested.
• Rate is the interest rate.
• Time is the period the money is invested in.

For Mona's initial investment of \17000\ with an interest rate of 6.5%, the interest is \0.065 \times 17000\, totaling \1105\. The additional investment, \(x\), at 5% interest would earn \0.05 \times x\ in interest. By calculating these individual interests, we combine them to meet the overall desired return, vital for planning effective investment strategies.

solving equations

Solving equations is at the heart of investment problems. It involves setting up equations based on known quantities and solving for unknown variables. Steps include:

• Identifying the variables.
• Writing an equation based on the problem's conditions.
• Simplifying and solving the equation.

In this problem, to achieve an overall 6% return, we write an equation using Mona's investments:
\[ 0.065 \times 17000 + 0.05 \times x = 0.06 \times (17000 + x) \] We then simplify and solve for \(x\):
\[ 1105 + 0.05x = 1020 + 0.06x \] Subtracting \0.05x\ and \1020\ from both sides, we get:
\[ 85 = 0.01x \] Finally, solving for \x\:
\[ x = \frac{85}{0.01} = 8500 \] Thus, Mona needs to invest an additional \$8500\ at a 5% interest rate to attain the desired 6% overall return. Understanding and practicing equation solving deepens comprehension of investment problems.