Problem 57 - Algebra for College Students

Question

For Problems \(57-67\), solve each problem by setting up and solving an appropriate inequality. Suppose that Lance has \(\$ 500\) to invest. If he invests \(\$ 300\) at \(9 \%\) interest, at what rate must he invest the remaining \$200 so that the two investments yield more than \(\$ 47\) in yearly interest?

Step-by-Step Solution

Answer

The rate must be more than 10%.

Step 1: Understand the problem

Lance has a total of $500 to invest, with $300 already being invested at a 9% interest rate. We need to determine the interest rate for the remaining $200 such that the total interest from both investments exceeds $47.

Step 2: Set up an equation for each investment

The interest from the $300 investment can be calculated by multiplying the principal ($300) by the interest rate (9%), which gives us $27. For the $200 investment, we express the interest as $200 times an unknown rate (let's call it r).

Step 3: Establish the inequality

We know the sum of the interests from both investments must be greater than $47. Hence, our inequality is: \[ 27 + 200r > 47 \].

Step 4: Simplify and solve the inequality

Subtract 27 from both sides to isolate the term with r, resulting in \[ 200r > 20 \]. Then, divide both sides by 200 to solve for r, leading to \[ r > 0.1 \].

Step 5: Convert the solution into interest rate terms

Since r represents the interest rate in decimal form, multiplying by 100 converts it to a percentage. Thus, \[ r > 0.1 \] translates to an interest rate greater than 10%.

Key Concepts

Investment Problems

Investment problems often involve finding ways to maximize returns based on certain constraints. In various scenarios, you may want to know how to allocate a sum of money across different investment options to achieve a specific financial goal. For example, Lance's situation involves splitting an investment between two options.

• He partially invests in a known interest rate.
• He aims to find the minimum required interest rate for the remaining investment to achieve a certain income goal.

By understanding how to break down these problems into simpler parts, setting up and solving for unknowns becomes more manageable. Always start by identifying the total amount available for investment, the amount already allocated, and the interest or return required from the overall investment.

Interest Rates

Interest rates represent the percentage of the principal that will be earned or paid over a specific period, typically a year. In investments, they are crucial for calculating the potential return. Lance wants to ensure that his total returns exceed a particular threshold.

• The $300 is invested at a known fixed rate of 9%, generating a calculable amount of interest.
• The rest, $200 in this case, requires finding an interest rate to meet the returns goal.

To solve these problems, understand that fixed interest rates allow for exact calculation of returns, while unknown rates require algebraic methods to isolate and solve for them. Be mindful of converting rates into decimal form for calculations, then back to percentages for the final answer.

Solving Inequalities

Inequalities help us understand what values satisfy conditions where a simple equation may not suffice, especially in optimization problems like investment returns. In Lance's case, we needed to determine when his total interest exceeded $47, leading us to set up an inequality.

• The known interest is subtracted from the target to isolate interest from the unknown rate.
• Once isolated, the inequality is solved algebraically to find the minimum rate needed.

Once solved, inequalities often provide a range of solutions rather than a single value, offering flexibility in decision-making. Remember to convert any rate from decimal form into a percentage to maintain consistency with general financial practices.