Problem 104 - Algebra and Trigonometry

Question

Sharing a Job Jack, Kay, and Lynn deliver advertising flyers in a small town. If each person works alone, it takes Jack 4 \(\mathrm{h}\) to deliver all the flyers, and it takes Lynn 1 \(\mathrm{h}\) longer than it takes Kay. Working together, they can deliver all the flyers in 40\(\%\) of the time it takes Kay working alone. How long does it take Kay to deliver all the flyers alone?

Step-by-Step Solution

Answer

Kay takes 3 hours to deliver all the flyers alone.

Step 1: Define Variables

Let Kay's time to deliver all the flyers alone be \( t \) hours. Then, Lynn's time would be \( t + 1 \) hours, and Jack's time is given as 4 hours.

Step 2: Determine Work Rates

Calculate the work rates for each person delivering flyers. Kay's work rate is \( \frac{1}{t} \) of the job per hour, Lynn's work rate is \( \frac{1}{t+1} \), and Jack's work rate is \( \frac{1}{4} \).

Step 3: Combined Work Rate Equation

The combined work rate when they all work together is \( \frac{1}{t} + \frac{1}{t+1} + \frac{1}{4} \). Since they can deliver the flyers together in 40% of the time it takes Kay alone, their combined work rate is \( \frac{1}{0.4t} = \frac{5}{2t} \).

Step 4: Set and Solve the Equation

Set the combined work rate equation: \( \frac{1}{t} + \frac{1}{t+1} + \frac{1}{4} = \frac{5}{2t} \). Multiply through by \( 4t(t+1) \) to clear the fractions, resulting in the equation: \( 4(t+1) + 4t + t(t+1) = 10(t+1) \). Simplify to find \( t^2 - t - 6 = 0 \).

Step 5: Solve Quadratic Equation

Factor the quadratic equation \( t^2 - t - 6 = 0 \) into \( (t - 3)(t + 2) = 0 \). The solutions are \( t = 3 \) and \( t = -2 \), but time cannot be negative, so \( t = 3 \).

Key Concepts

Algebra

Algebra is essential in solving work rate problems like the one involving Jack, Kay, and Lynn. It involves using variables to represent unknown quantities and writing equations that describe relationships between these quantities.
In our exercise, we used algebra to represent the time each person takes to deliver flyers. Kay's time is denoted by the variable \( t \), Lynn's by \( t + 1 \), and Jack's is given as 4 hours.
This approach of defining variables helps us to systematically tackle the problem, by translating a real-world scenario into a mathematical model that we can manipulate and solve.

Quadratic Equations

Quadratic equations are polynomial equations of the form \( ax^2 + bx + c = 0 \). They appeared in our exercise because of the relationship between the work rates and the time taken by the three individuals.
After setting up our equation for the combined work rate, we simplified it to \( t^2 - t - 6 = 0 \). This is a quadratic equation, which we then solved by factoring. Factoring is a method where we express the quadratic as a product of two binomials: \( (t - 3)(t + 2) = 0 \).
This factoring yields potential solutions for \( t \), which tell us how long it takes Kay to deliver all flyers alone. We select only the positive solution, \( t = 3 \), since time cannot be negative in this context.

Collaborative Work Problems

Collaborative work problems are a common type of algebra problem that involves multiple people or machines working together to complete a task. The goal is to find how long it takes them to complete the task alone or jointly.
In the exercise, we calculated individual work rates, which represent the fraction of the work each person can complete in one hour. By adding these work rates, we found the combined work rate when all worked together.
This principle shows how separate efforts can combine to complete a job more quickly, emphasizing the importance of cooperation. Understanding these problems involves recognizing the roles each participant plays and how they contribute to the overall effort.