Collaborative Efforts: How Long Does It Take for A and B to Complete Their Work?

In today's fast-paced world, effective collaboration plays a crucial role in achieving complex tasks efficiently. This article delves into a problem that demonstrates how the combined efforts of two individuals can enhance productivity, leading to faster task completion.

Problem Statement

A and B can complete a specific work in 5 days and 10 days respectively. A starts the work, and after 2 days, B joins to assist. In this article, we will explore how long it will take for A and B to complete the remaining work together.

Analysis

The problem can be approached by determining the individual work rates of A and B, and then combining these rates to find the time required for the remaining work.

Step 1: Calculate Work Rates

Let's start by calculating the work rates for A and B:

Work rate of A: (frac{1}{5}) work per day.

Work rate of B: (frac{1}{10}) work per day.

Step 2: Work Done by A in the First 2 Days

In the first two days, A works alone, and the amount of work completed is:

(2 times frac{1}{5} frac{2}{5})

Step 3: Remaining Work

Considering the total work as 1 unit, after A has worked for 2 days, the remaining work is:

(1 - frac{2}{5} frac{3}{5})

Step 4: Combined Work Rate of A and B

Once B joins, the combined work rate of A and B is:

(frac{1}{5} frac{1}{10} frac{2}{10} frac{1}{10} frac{3}{10}) work per day.

Step 5: Time to Complete Remaining Work

To find out how long it takes for A and B to complete the remaining (frac{3}{5}) of the work, the formula is used:

(text{Time} frac{text{Remaining work}}{text{Combined work rate}} frac{frac{3}{5}}{frac{3}{10}} frac{3}{5} times frac{10}{3} 2 text{ days})

Conclusion

Therefore, after B joins A, it will take (2) days for A and B to complete the remaining work.

Alternative Solutions

For those who prefer alternative methods, two additional approaches are provided:

Solution 1

Let it take x days for A and B to complete the remaining work.

A’s work rate: (frac{1}{5}).

B’s work rate: (frac{1}{10}).

Combined rate: (frac{1}{5} frac{1}{10} frac{2}{10} frac{1}{10} frac{3}{10}).

Equation: (x - frac{2}{5} times frac{x}{6} 1).

Solving: (6x - 2 5x 30), (11x 42), (x frac{42}{11} 3.82 text{ days}).

Solution 2

Let the remaining work be completed in x days.

Combined work rate: (frac{1}{5} frac{1}{10} frac{2}{10} frac{1}{10} frac{3}{10}).

Work done in x days: (frac{x}{5} frac{x}{10} frac{2x x}{10} frac{3x}{10}).

Equation: (1 - frac{3x}{10} 1 - frac{x}{2}).

From B's 2 days work: (frac{1}{6}).

Equation: (frac{1}{6} frac{3x}{10} 1), solving gives (x frac{20}{11} approx 1.81 text{ days}).

These variations demonstrate the robust nature of the problem and the different approaches to a solution.