Problem 34 - Elementary Algebra

Question

Barry can lay a brick driveway by himself in 12 hours. Robert does the same job in 10 hours. How long will it take them to lay the brick driveway working together?

Step-by-Step Solution

Answer

They will take about 5.45 hours to lay the driveway together.

Step 1: Determine Barry's Work Rate

Barry can complete the driveway in 12 hours. Therefore, Barry's work rate is \( \frac{1}{12} \) of the driveway per hour.

Step 2: Determine Robert's Work Rate

Robert can complete the driveway in 10 hours. Therefore, Robert's work rate is \( \frac{1}{10} \) of the driveway per hour.

Step 3: Calculate Combined Work Rate

Add Barry's and Robert's work rates to get their combined work rate. This is \( \frac{1}{12} + \frac{1}{10} = \frac{5}{60} + \frac{6}{60} = \frac{11}{60} \) of the driveway per hour.

Step 4: Find Time to Complete Driveway Together

The time it takes for both to complete the driveway together is the reciprocal of their combined work rate. Thus, they will take \( \frac{60}{11} \) hours to complete the driveway together.

Key Concepts

Combined Work Rate

When two or more people work together on a task, their individual abilities add up to a combined work rate. This is particularly handy in scenarios like Barry's and Robert's problem, where they lay a brick driveway together. The combined work rate is simply the sum of each individual's work rate.

For Barry, his work rate is \( \frac{1}{12} \) because he completes one driveway in 12 hours. Robert's work rate is \( \frac{1}{10} \), as he does the same job in 10 hours.

To find their combined work rate, you add up their individual work rates:

• Barry: \( \frac{1}{12} \)
• Robert: \( \frac{1}{10} \)

By doing the math, Barry's and Robert's combined work rate is \( \frac{11}{60} \). This means that together, they complete \( \frac{11}{60} \) of the driveway in one hour. This combined effort is greater than either working alone, showing how collaboration speeds up such tasks.

Reciprocal of Work Rate

The concept of the reciprocal of work rate is very useful in determining the time it takes to finish a task when working together. Once you have the combined work rate, finding the time required for completion involves taking its reciprocal.

Think of the combined work rate \( \frac{11}{60} \) as a measure of how much of the driveway can be done in one hour when Barry and Robert work together. However, we're interested in the total time to finish the driveway, which is where the reciprocal comes in handy.

The reciprocal is simply flipping the fraction: turn \( \frac{11}{60} \) into \( \frac{60}{11} \). This value gives us a direct answer to the question of how many hours it will take the two working together to finish the driveway. In other words, they require approximately 5.45 hours to complete their task, thanks to their combined efforts.

Time Calculation in Work Rate Problems

Time calculation in work rate problems often involves a few straightforward steps. Once you know the combined work rate, the time required is the reciprocal of this rate. This technique is a powerful tool for solving collaborative tasks and understanding how people or machines can work together efficiently.

In Barry and Robert's case, after computing the combined work rate (\( \frac{11}{60} \)), the main step to find out how long it takes to finish the driveway was to calculate the reciprocal: \( \frac{60}{11} \).

The final result, \( \frac{60}{11} \), simplifies our calculation, showing it takes them just under 5 and a half hours to complete the task. By clearly understanding each person's contribution and leveraging simple arithmetic with fractions and reciprocals, you can quickly solve similar problems in work rate scenarios.