Problem 46 - Algebra for College Students

Question

Solve each equation. If \(\$ 1500\) is invested at \(6 \%\) interest, how much money must be invested at \(9 \%\) so that the total return for both investments is \(\$ 301.50\) ?

Step-by-Step Solution

Answer

Invest $2350 at 9%.

Step 1: Understand the Problem

We need to find out how much money should be invested at 9% interest rate so that the total interest earned from two investments is $301.50. One investment is already made with $1500 at a 6% interest rate.

Step 2: Calculate Interest from the First Investment

First, calculate the interest gained from the $1500 investment at 6% interest rate using the formula \( I = P \times r \). So, \( I = 1500 \times 0.06 \).

Step 3: Compute the Interest: Part 1

When we calculate \( I = 1500 \times 0.06 \), the interest from the first investment is \$90.

Step 4: Determine Remaining Interest Needed

Subtract the interest from the first investment from the total desired interest: \( 301.50 - 90 = 211.50 \). This is the interest we need from the second investment.

Step 5: Calculate the Required Second Investment

We need the investment at 9% that results in \$211.50. Use the formula \( I = P \times r \) and rearrange it to find \( P = \frac{I}{r} = \frac{211.50}{0.09} \).

Step 6: Solve for the Second Investment Amount

Calculate \( \frac{211.50}{0.09} \) to find that the necessary amount to invest at 9% is \$2350.

Key Concepts

Interest Calculation

Understanding how interest works is essential when dealing with investments. Interest can be seen as the cost of borrowing money or the earnings on invested funds. In simple terms, interest can be calculated using the formula: \[ I = P \times r \]where:

• \( I \) is the interest earned or paid,
• \( P \) is the principal amount (initial investment or loan),
• \( r \) is the interest rate, expressed as a decimal.

For example, if you invest \\(1500 at a 6% interest rate, you use the formula to find the interest: \[ I = 1500 \times 0.06 \]This gives you \\)90 as the interest earned from the first investment. Calcualting interest is crucial in evaluating the return from various investments, helping make informed financial decisions.In the context of our problem, this calculation allows us to determine how much profit is initially made, forcing us to calculate how much is left and what further investment is required to meet a goal.

Investment Equations

Investment equations are crucial tools in financial algebra, allowing one to calculate how much should be invested to achieve a certain financial return. The general form of an investment equation stems from the interest formula, rearranged to solve for various unknowns, often the principal.In our example, after calculating the interest from the first investment, we need to find the principal for a second investment. For this, the formula can be rearranged as:\[ P = \frac{I}{r} \]Given that our remaining needed interest is \\(211.50, and the second investment has a 9% interest rate, we plug these values into the formula:\[ P = \frac{211.50}{0.09} \]This rearrangement and computation show that \\)2350 must be invested at a 9% rate to achieve the remaining interest necessary for our total desired return. These calculations underscore the power of algebra in solving practical financial problems.

Percentage Problems

Percentage problems are common in everyday math, helping convert figures into more understandable and relatable terms. They are particularly important in financial calculations involving interest rates. Percentages are always calculated out of 100, simplifying comparisons and conversions. For instance, an interest rate of 6% is mathematically translated as 0.06, making it easier to compute with in equations. In our problem, both interest rates need conversion before plugging them into equations:

• Convert 6% to 0.06,
• Convert 9% to 0.09.

Such simple conversions make complex problems more manageable, transforming interest calculations from abstract to applicable. When dealing with "returns" or "profits," it's important to convert percentage gains into decimals to use in multiplication with principal amounts. This helps anticipate the actual dollar amounts when planning investments, budgets or other financial scenarios.