Problem 100 - Algebra and Trigonometry

Question

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 1 hour to deliver all the flyers alone.

Step 1: Understand the problem

We need to find out the time it takes for Kay to deliver all the flyers alone. We know the times it takes for Jack and Lyn to complete the task individually and that all three working together take 40% of Kay's time.

Step 2: Define variables

Let Kay's time to deliver the flyers be \( x \) hours. Since Lynn takes 1 hour longer than Kay, we can define Lynn's time as \( x + 1 \) hours.

Step 3: Write individual rates

Express the rate of each person in terms of work done per hour. Jack can deliver all flyers in 4 hours, so his rate is \( \frac{1}{4} \) flyers per hour. Kay's rate is \( \frac{1}{x} \) flyers per hour, and Lynn's rate is \( \frac{1}{x + 1} \) flyers per hour.

Step 4: Express the combined rate

When working together, the combined rate is the sum of individual rates: \[ \frac{1}{4} + \frac{1}{x} + \frac{1}{x+1}. \] Given that they complete the work in 40% of the time it takes Kay, their combined rate is \( \frac{1}{0.4x} = \frac{5}{2x} \).

Step 5: Set up the equation

Equate the sum of the individual rates to the combined rate: \[ \frac{1}{4} + \frac{1}{x} + \frac{1}{x+1} = \frac{5}{2x}. \]

Step 6: Clear fractions and solve for \( x \)

Cross-multiply and clear denominators to solve for \( x \):1. Multiply through by \( 4x(x+1) \) to eliminate fractions.2. Simplify to get \( x(x+1) + 4(x+1) + 4x = 10. \)3. Solve \( x^2 + 5x + 4 = 10 \).4. Rearrange to \( x^2 + 5x - 6 = 0. \)5. Factor to \( (x-1)(x+6) = 0. \)6. Solve for \( x \): \( x = 1 \) or \( x = -6 \). Since negative time doesn't make sense, \( x = 1 \).

Step 7: Verify the solution

Check the solution by ensuring it satisfies all conditions. With Kay taking 1 hour, Lynn would take 2 hours, requiring verification of total combined rate which aligns with original data.

Key Concepts

rates of work

Understanding rates of work is essential when solving real-world word problems. Here, the rate of work refers to how much work gets done in a specific period. It's typically expressible as a fraction, where the numerator is the "total work" (in this case, delivering all the flyers) and the denominator is the "time taken."

Let's consider Jack's rate. He finishes the task in 4 hours, so his rate is \( \frac{1}{4} \) of the work per hour. If Kay takes \( x \) hours, his rate is \( \frac{1}{x} \) per hour, while Lynn's is \( \frac{1}{x+1} \) because she needs an additional hour compared to Kay.

Combining individual rates helps calculate how quickly multiple people complete a task together. When Jack, Kay, and Lynn work as a team, their rates add up. Understanding both individual and combined rates helps us set up equations to determine unknown time variables efficiently.

equation solving

Equation solving plays a crucial role in finding unknown values in algebra word problems. Once you establish the equation from the word problem data, solving it involves isolation of the variable you need to find.

In this exercise, we derive the equation from the known individual rates of work and the combined rate of work. The equation \[ \frac{1}{4} + \frac{1}{x} + \frac{1}{x+1} = \frac{5}{2x} \]comes from summarizing both individual rates and their combined completion rate (40% of Kay's time).

The next step is to clear any fractions by multiplying through by the least common multiple of the denominators. You rearrange the equation into a simpler quadratic form, solving directly or by factoring. Successfully solving such equations involves understanding both algebraic manipulation and the logic behind each step, ensuring the result is meaningful in context.

problem solving steps

Breaking down a problem into clear, manageable steps guides successful problem-solving. Begin by understanding the problem context and determining what needs to be solved.

Setting up equations from the problem description involves:

• Identifying variables
• Expressing rates of work as per given conditions
• Establishing equations through rate expressions and given time percentages

After writing the equation, proceed to solve it.

Finally, always verify your solution. Check that it logically satisfies the conditions described in the problem, such as using the derived time to confirm the combined working rate.

Following structured steps ensures a logical flow from understanding to solving, and eventually confirming the solution, creating a clear pathway to answer complex word problems.