Question

Working together, Kendra, Latasha, and Melanie can complete a certain task in 4 hours. If Kendra alone could complete the task in 8 hours and Latasha could complete the task in half the time it would take Melanie, how long would it take Latasha to complete the task by herself?

Short answer

Latasha would take 12 hours to complete the task by herself.

Step-by-step solution

- Assign Variables

Let the time it takes Latasha to complete the task alone be denoted as \(L\) hours, and the time it takes Melanie to complete the task alone be \(M\) hours.

- Relate Latasha and Melanie's Times

According to the problem, Latasha can complete the task in half the time it would take Melanie, so we can write: \(L = \frac{M}{2}\).

- Write Individual Rates

Kendra's work rate is \(\frac{1}{8}\) of the task per hour, Latasha's work rate is \(\frac{1}{L}\) of the task per hour, and Melanie's work rate is \(\frac{1}{M}\) of the task per hour.

- Write Combined Work Rate

Given they can complete the task together in 4 hours, their combined rate is: \(\frac{1}{4}\) of the task per hour. Therefore, the combined rate equation is: \(\frac{1}{8} + \frac{1}{L} + \frac{1}{M} = \frac{1}{4}\).

- Substitute Latasha's Rate

Using \(L = \frac{M}{2}\), substitute \(\frac{1}{L}\) with \(\frac{2}{M}\) into the combined work rate equation: \(\frac{1}{8} + \frac{2}{M} + \frac{1}{M} = \frac{1}{4}\).

- Simplify the Equation

Combine the terms with \(M\) in the denominator: \(\frac{1}{8} + \frac{3}{M} = \frac{1}{4}\).

- Solve for \(M\)

Subtract \(\frac{1}{8}\) from both sides: \(\frac{3}{M} = \frac{1}{4} - \frac{1}{8}\). This simplifies to: \(\frac{3}{M} = \frac{1}{8}\). Hence, \(M = 24\).

- Find Latasha's Time \(L\)

Using \(L = \frac{M}{2}\), compute Latasha's time: \(L = \frac{24}{2} = 12\). Thus, Latasha takes 12 hours to complete the task alone.

Key concepts

Algebra

To solve work rate problems, we need to use algebra to set up and solve equations. In this exercise, we use algebra to express the work rates of individuals and combine them into an equation. For instance, Kendra’s work rate is \(\frac{1}{8}\) since she can complete the job in 8 hours. By expressing Latasha's time as \(\frac{M}{2}\), we incorporated Melanie's work rate into the equation. Algebra helps in isolating variables, such as using substitution \(L = \frac{M}{2}\), making it easier to combine rates into a single equation and solve for unknowns. This systematic approach makes it simple to find each individual’s contribution and solve for the time taken.

Collaborative Work

When tackling collaborative work problems, it's important to understand how multiple entities can combine their efforts to complete a task more efficiently. The key is to sum up their individual work rates. Here, Kendra, Latasha, and Melanie have respective work rates, which combined give us the collaborative rate. We find these rates and then sum them up:

• Kendra's rate: \(\frac{1}{8}\)
• Latasha's rate: \(\frac{1}{L}\)
• Melanie's rate: \(\frac{1}{M}\)

Adding these rates gives the total work rate when they work together. Understanding collaborative work helps you see how different parts contribute to a whole.

Mathematical Reasoning

Mathematical reasoning lets us solve complex problems with logic and structured thinking. In this exercise, we applied reasoning to set up relationships between the rates. By recognizing that Latasha's time is half of Melanie's, we express Latasha's rate in terms of Melanie's rate \(L = \frac{M}{2}\). This let us simplify the problem into a single variable and follow logical steps
to find a solution. The reasoning step of isolating variables (solving for \(M)\) and then using those values to find other unknowns (\(L\)) is fundamental. This structured approach ensures accurate and reliable results when solving work rate problems.

GRE Preparation

Work rate problems are common in the GRE quantitative section. Preparing for them involves practicing similar questions to understand various types of collaborative work scenarios. Here are some tips:

• Familiarize yourself with the concept of individual and combined work rates.
• Practice breaking down complex problems into simpler algebraic equations.
• Work on substitution and solving linear equations efficiently.
• Make sure to review your steps for logical consistency.

Mastering these techniques will enhance your confidence and accuracy in the GRE exam, especially when dealing with time and work-related questions.