Question

If it takes one person 5 hours to load a truck while another person can complete the task in 3 hours, how long should it take then to half fill the truck if they work together at the same rate? (A) 2 hours (B) 3 hours (C) 4 hours (D) 5 hours (E) 6 hours

Short answer

2 hours

Step-by-step solution

- Determine the rates of each worker

First, find the rates at which each person can load the truck. The first person can load the truck in 5 hours, so their rate is \ \(\frac{1}{5} \) of the truck per hour. The second person can load the truck in 3 hours, so their rate is \ \(\frac{1}{3} \) of the truck per hour.

- Calculate the combined rate

To find the combined rate when both individuals are working together, add their rates: \ \(\frac{1}{5} + \frac{1}{3} = \frac{3}{15} + \frac{5}{15} = \frac{8}{15} \) of the truck per hour.

- Determine the time to load half the truck

Since they need to load only half the truck, divide \ \(\frac{1}{2} \) by their combined rate: \ \(\frac{1/2}{8/15} = \frac{1}{2} \times \frac{15}{8} = \frac{15}{16} \) hours.

- Simplify the result

Simplifying \ \(\frac{15}{16} \) hours, which is approximately 1.875 hours. Rounding to the nearest hour, it is closer to 2 hours.

Key concepts

Understanding Work Rate Problems

Work rate problems in the GMAT typically involve calculating how quickly tasks can be completed when individuals or groups perform work over time. The key to these problems is understanding that work rates are additive when multiple workers collaborate.
If one person can load a truck in 5 hours, their work rate is \( \frac{1}{5} \) of the truck per hour. Similarly, if another person can do it in 3 hours, their rate is \( \frac{1}{3} \) of the truck per hour. By combining these rates, you get the total rate for collaborative work.

When solving these problems:

• Identify each worker's individual rate.
• Add the rates to find the combined rate.
• Use this combined rate to calculate the total time needed for the task.

For example, adding the rates \( \frac{1}{5} + \frac{1}{3} \) results in \( \frac{8}{15} \). This sum represents how much of the truck two people can load in one hour together.

Collaborative Work

Collaborative work problems are an extension of individual work rate problems. They involve determining how fast a job can be accomplished when multiple people work together. In these problems, it is crucial to convert the individual rates into a combined rate.

• Calculate each person's rate of completing the job independently—it could be parts of the task done per hour, like \( \frac{1}{5} \) and \( \frac{1}{3} \) of the truck per hour.
• Add these rates to figure out their cumulative rate, which tells you how much of the job is done together in one hour. In our exercise, \( \frac{1}{5} + \frac{1}{3} = \frac{8}{15} \) of the truck per hour.
• Use this cumulative rate to find the total time required for the task or a portion of it by dividing the part of the task by the combined rate.

In our example, to half-fill the truck, divide half the task \( \frac{1}{2} \) by the combined rate \( \frac{8}{15} \): \( \frac{1/2}{8/15} \), which simplifies to \( \frac{15}{16} \) hours. This calculation shows that they would take approximately 1.875 hours, or close to 2 hours, to half-fill the truck.

Time Management in Work Problems

Time management is an essential skill in solving work rate problems. By understanding and managing the time required for each part of a task, you can make more accurate predictions and calculations. These steps can help:

• Break down the task into manageable parts.
• Calculate how long each part will take individually and when combined with others working simultaneously.
• Use proportions to divide the task into smaller parts if necessary.

In the given problem, since half of the truck needs to be filled, we use the combined work rate of \( \frac{8}{15} \). Divide the task (half the truck, which is \( \frac{1}{2} \) of the total work) by this combined rate, yielding \( \frac{15}{16} \) hours.

Effective time management allows you to plan and allocate resources efficiently. This skill is beneficial not just in exams but in practical, everyday situations where collaborative efforts are needed.