Problem 72 - Intermediate Algebra

Question

Investing. Carol receives 67.50 dollars annual income from one investment. John invested 150 dollars more than Carol at an annual rate of \(1 \frac{1}{2} \%\) more. John's annual income is 94.50 dollars. What are the amount and rate of Carol's investment? (Hint: There are two answers.)

Step-by-Step Solution

Answer

Carol invests either 450 dollars at 15% or 300 dollars at 22.5%.

Step 1: Understanding the Variables

Denote Carol's investment amount as \( C \) and the rate of return as \( r \). John's investment can be expressed as \( C + 150 \) dollars, with a rate \( r + 0.015 \). By translating the problem into equations, we can express Carol's annual income as \( C \times r = 67.50 \) and John's income as \((C + 150) \times (r + 0.015) = 94.50\).

Step 2: Formulating Equations

We have two equations based on the given incomes. For Carol, the first equation is \( C \times r = 67.50 \). For John, the second equation is \( (C + 150) \times (r + 0.015) = 94.50 \). These equations will help us find \( C \) and \( r \).

Step 3: Solving for Carol's Investment

First, solve for \( r \) from Carol's equation: \( r = \frac{67.50}{C} \). Substitute this value into John's equation: \((C + 150) \times \left(\frac{67.50}{C} + 0.015\right) = 94.50\). Simplify and solve for \( C \).

Step 4: Substituting and Simplifying

Substitute \( r = \frac{67.50}{C} \) into John's equation: \((C + 150) \times \left(\frac{67.50 + 0.015C}{C}\right) = 94.50\). By expanding and clearing the fraction, obtain the equation \( 67.50 + 0.015C + 150 \times (\frac{67.50}{C} + 0.015) = 94.50\).

Step 5: Solving for C

Further simplification leads to the equation \( 67.50 + 0.015C + \frac{10125}{C} + 2.25 = 94.50\). This can be rearranged to obtain \( 0.015C + \frac{10125}{C} = 24.75\). Multiply by \( C \) to get a quadratic \( 0.015C^2 - 24.75C + 10125 = 0\).

Step 6: Factoring the Quadratic

Solve the quadratic equation \( 0.015C^2 - 24.75C + 10125 = 0 \) using the quadratic formula, \( C = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 0.015 \), \( b = -24.75 \), \( c = 10125 \).

Step 7: Calculating the Roots

Calculate: \( b^2 - 4ac = 24.75^2 - 4(0.015)(10125) = 612.5625 \). The roots are \( C = \frac{24.75 \pm \sqrt{612.5625}}{0.03} \). Calculate the two possible values for \( C \).

Step 8: Determining Carol's Rate

With the values of \( C \) obtained, calculate \( r \) using \( r = \frac{67.50}{C} \). This gives two possible rates for Carol's investment.

Key Concepts

Quadratic Equations

Quadratic equations are an essential part of algebra, commonly taking the form \( ax^2 + bx + c = 0 \). In solving problems related to investment calculations, like Carol and John's scenario, quadratic equations help us uncover unknown variables related to amounts and rates of return. A quadratic equation often contains three terms:

• The quadratic term: \( ax^2 \) where \( a \) is the coefficient of the squared term.
• The linear term: \( bx \) where \( b \) is the coefficient of the term with the variable raised to the first power.
• The constant term: \( c \) which is a constant number.

To solve for the variable \( x \) in such equations, the quadratic formula is frequently used:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula finds the roots, or solutions, of the equation. In Carol's investment case, the quadratic equation \( 0.015C^2 - 24.75C + 10125 = 0 \) was solved using this method to find the possible investment amounts.

Rate of Return

The rate of return is a crucial concept in assessing the performance of an investment. It represents the percentage gain or loss of an investment over a specific period. In Carol's problem, calculating the rate of return involves dividing the annual income by the investment amount and expressing it as a percentage.
For instance, for Carol, her rate of return can be represented as \( r = \frac{67.50}{C} \). This formula helps us to determine how much income is generated per unit of investment.
Knowing the rate of return allows investors to compare different investment opportunities and make informed decisions. A higher rate of return indicates a more profitable investment. In John's situation, his rate of return incorporates a slight increase compared to Carol's, calculated at \( r + 0.015 \), reflecting his additional income based on the amount invested.

Investment Calculations

Investment calculations involve determining values like the amount invested, rate of return, and income generated. These calculations help in effectively managing financial resources and understanding the growth potential of investments.

• Investment Amount: This is the principal sum that is initially placed in an investment. For Carol, this was denoted as \( C \).
• Annual Income: This is the profit generated from the investment annually. Carol's income from her investment was \( 67.50 \) dollars.
• Rate of Return: The percentage income relative to the investment principal, calculated using the formula \( r = \frac{67.50}{C} \) for Carol.

Addressing investment calculations involves solving equations to elucidate unknowns like the investment amount and rate of return. In the exercise, solving the equations developed from given incomes and known variables allowed us to determine the specific amounts and rates of both Carol's and John's investments. When using formulas and calculations in investment analysis, it's important to ensure accuracy and fully understand each variable's role in the overall financial picture.