Problem 71 - Intermediate Algebra

Question

Investing. Grant receives 225 dollars annual income from one investment. Jeff invested 500 dollars more than Grant, but at an annual rate of \(1 \%\) less. Jeff's annual income is 240 dollars. What are the amount and rate of Grant's investment?

Step-by-Step Solution

Answer

Grant invested $4500 at a rate of 5%.

Step 1: Define Variables for Grant's Investment

Let \( P \) be the principal amount of Grant's investment and \( r \) be the rate of return on Grant's investment. Thus, Grant's annual income can be expressed as \( P \cdot \frac{r}{100} = 225 \).

Step 2: Define Jeff's Investment Variables

Since Jeff invested \( 500 \) dollars more than Grant, his principal amount is \( P + 500 \). If Grant's rate is \( r \), then Jeff's rate would be \( r - 1 \% \) and his annual income is given by \( (P+500) \cdot \frac{r-1}{100} = 240 \).

Step 3: Solve for Grant's Principal Amount

From the equation \( P \cdot \frac{r}{100} = 225 \), we can express \( P \) as \( P = \frac{225 \cdot 100}{r} \). We will use this expression in the equation obtained from Jeff's income.

Step 4: Set Up Equation for Jeff's Income

Use the expression for \( P \) and substitute in Jeff's income equation: \[ \left(\frac{225 \cdot 100}{r} + 500\right) \cdot \frac{r-1}{100} = 240 \] Simplifying this equation will allow us to find \( r \).

Step 5: Simplify the Equation and Solve for \( r \)

Simplify the equation: \( \left(\frac{22500}{r} + 500\right) \cdot \frac{r-1}{100} = 240 \). By further simplifying we get: \( 22500 \cdot \frac{r-1}{r} + 500 \cdot (r-1) = 24000 \). Solving faithfully for \( r \), we find that \( r = 5 \).

Step 6: Calculate Grant's Investment Amount

With \( r = 5 \), use the relationship \( P = \frac{225 \cdot 100}{r} \) to find \( P = \frac{22500}{5} \). Hence, the principal amount \( P = 4500 \) dollars.

Key Concepts

Investment Problems

Investment problems often involve understanding how money grows over time. These problems require calculating returns based on the amount invested and the interest rates involved. In this scenario, determining how much Grant and Jeff have invested involves:

• Identifying the principal amount, or the initial investment they each made.
• Understanding the rate of return, which affects how much income they earn from their investments annually.
• Using these elements to set up equations that describe their financial situations.

Investment problems like this one are often posed as algebraic puzzles where you need to find unknowns such as principal sums or rates of interest. To do this, you form relationships using known quantities, usually incomes, and unknown quantities, like rates or amounts.

Rate of Return

The rate of return represents how much an investment generates relative to its initial amount, usually expressed as a percentage. It is pivotal in assessing the profitability of an investment:

• A higher rate of return means earning more income from the same amount invested.
• In this example, Grant's rate of return is critical to solving how much he originally invested.

This rate can be found by rearranging the formula for annual income: \[ \text{Income} = \text{Principal} \times \frac{\text{Rate of Return}}{100} \]By substituting known values, the formula allows us to solve for the unknown rate, which tells us Grant's investment was more profitable despite having a lower income than Jeff.

Algebraic Equations

Algebraic equations form the backbone of solving investment problems. They help you express relationships between different financial elements clearly and solve for unknown variables.

• Initially, you identify and define what each variable represents – like Grant's investment or rate.
• Next, write equations based on the problem's conditions; for example, relating known incomes to the rate and amount invested.

For Grant and Jeff:- Grant's equation is formed as: \[ P \cdot \frac{r}{100} = 225 \]- Jeff's, accounting for his additional $500 investment at a lower rate, is:\[ (P + 500) \cdot \frac{r - 1}{100} = 240 \]These equations are solved simultaneously, allowing us to find answers for both variables, illustrating how algebra simplifies complex financial scenarios into solvable mathematical problems.