Efficiency Calculation: How Long Will A and B Take to Complete a Task Together?

Introduction

This article explores the concept of work rates and how to calculate the combined work rate when two individuals, A and B, collaborate on a task. The problem at hand is to determine how long A and B will take to complete a piece of work when working together, given their individual work rates.

Step 1: Determine Individual Work Rates

First, we need to determine the work rates of A and B individually.

A's Work Rate:

Given that A can complete 1/5 of the work in 12 hours, the work rate of A is:

Rate of A frac{1}{5 times 12} frac{1}{60} text{ of the work per hour}

B's Work Rate:

Given that B can complete 1/6 of the work in 15 hours, the work rate of B is:

Rate of B frac{1}{6 times 15} frac{1}{90} text{ of the work per hour}

Step 2: Combine Work Rates

To find the combined work rate of A and B working together, we add their individual work rates:

Combined Rate Rate of A Rate of B frac{1}{60} frac{1}{90}

First, we need a common denominator to add these fractions. The least common multiple (LCM) of 60 and 90 is 180:

frac{1}{60} frac{3}{180}

frac{1}{90} frac{2}{180}

Now we can add the two rates:

Combined Rate frac{3}{180} frac{2}{180} frac{5}{180} frac{1}{36} text{ of the work per hour}

Step 3: Calculate Time to Complete the Work Together

Finally, to find the time taken to complete the entire work together, we take the reciprocal of the combined rate:

Time frac{1}{frac{1}{36}} 36 text{ hours}

Conclusion

A and B working together can complete the work in 36 hours.

Additional Analysis

To further validate this calculation, let's use the LCM (least common multiple) method. The LCM of 12 and 15 is 60. We can consider 60 hours as the total work:

A's Efficiency:

Efficiency of A frac{60}{12} 5 text{ units per hour}

B's Efficiency:

Efficiency of B frac{60}{15} 4 text{ units per hour}

When A and B work together for 2 hours, they complete:

5 4 9 text{ units per 2 hours}

In 12 hours, they complete:

frac{9 times 6}{12} 45 text{ units}

The remaining work is 15 units, which A can complete in 1 hour and B can finish the remaining unit in 1/4 hour, which is 15 minutes.

Therefore, the total time is 12 1.25 13.25 hours, or 13 hours 15 minutes.

Summary

In conclusion, by calculating individual work rates, combining them, and finding the reciprocal of the combined rate, we determined that A and B working together will complete the work in 36 hours. This method can be applied to similar problems to find the time taken for two entities to complete a task together.