How Many Days Can B Complete a Work Alone? A Comprehensive Guide to Understanding Work Rate Problems

When dealing with work rate problems, such as determining how long it will take for a single entity to complete a task given the combined effort, it's crucial to understand the underlying principles. This article will walk you through a detailed step-by-step solution to the problem of how many days B alone can complete a piece of work, given that A and B together can complete it in 4 days, and A alone can complete it in 12 days.

Understanding Work Rates

The first step in solving work rate problems is to understand the concept of work rate. Work rate is the measure of how quickly a task can be completed, expressed as the amount of work done per unit of time. In this problem, we need to find B's work rate to determine how long B can complete the work alone.

Step-by-Step Solution

Calculating A's Work Rate

A can complete the work in 12 days. Therefore, A's work rate is:

Work rate of A frac{1}{12} text{ work per day}

Calculating the Combined Work Rate of A and B

A and B together can complete the work in 4 days. Thus, their combined work rate is:

Combined work rate of A and B frac{1}{4} text{ work per day}

Finding B's Work Rate

Let B's work rate be (frac{1}{b}), where b is the number of days B takes to complete the work alone. The equation for their combined work rate is:

Work rate of A Work rate of B Combined work rate

Substituting the known values, we get:

frac{1}{12} frac{1}{b} frac{1}{4}

To solve for b, we rearrange the equation:

frac{1}{b} frac{1}{4} - frac{1}{12}

We find a common denominator for the fractions, which is 12:

frac{1}{4} frac{3}{12}

So:

frac{1}{b} frac{3}{12} - frac{1}{12} frac{2}{12} frac{1}{6}

Therefore, b 6. This means B alone can complete the work in 6 days.

Stop and Think: Additional Problem-Solving Techniques

While the above method is effective, it's also useful to explore other techniques that lead to the same solution, such as using algebraic equations and solving quadratic equations.

Algebraic Equation Method

In another approach, we can set up the equation based on the given information. Using the equation:

frac{1}{A} frac{1}{B} frac{3}{40}

and considering that B works 6 days less:

frac{1}{B-6} frac{1}{B} frac{3}{40}

Multiplying through by the common denominator, we get:

40(B) 40(B-6) 3B(B-6)

Expanding and simplifying:

80B - 240 3B^2 - 18B

Bringing all terms to one side:

3B^2 - 98B 240 0

Solving this quadratic equation using the quadratic formula:

B frac{98 pm sqrt{98^2 - 4times 3times 240}}{2times 3}

B frac{98 pm sqrt{9604 - 2880}}{6}

B frac{98 pm sqrt{6724}}{6}

B frac{98 pm 82}{6}

Considering the positive root:

B frac{98 82}{6} frac{180}{6} 30

Hence, B alone can complete the work in 30 days.

Alternative Methods

In a similar situation, if A's efficiency is twice B's efficiency, we can equate:

frac{40}{A} frac{40}{B} 3

Given that B A - 6, substituting this into the equation, we get:

frac{40}{A} frac{40}{A-6} 3

Using the least common multiple, which is 40, and solving for A, we find:

A 24

Thus, B 24 - 6 18 days.

However, a more straightforward method involves calculating the efficiency of A and B:

A: 3 units/day, B: 1 unit/day

Together, they complete 4 units/day, so the time required is:

12 units / 4 units/day 3 days

This method simplifies the problem into a basic arithmetic calculation.

For an alternative, you can also solve it by:

1/A 1/B 3/40

1/B - 6 1/B 3/40

B - 6 12/B 3/40

B^2 - 5B - 12B 30

B^2 - 5B - 12 0

Solving the quadratic equation:B 10 days

Thus, B can complete the work in 10 days.

Conclusion

Understanding work rate problems requires a step-by-step approach and multiple methods to ensure accuracy. Whether using basic algebra, quadratic equations, or efficiency calculations, the solutions to these problems help in various real-world scenarios, such as project management, resource allocation, and workforce planning.