Solving Work Rate Problems: A and B's Job Completion Time Analysis

In the context of algebra and work rate problems, the scenario of two workers, A and B, completing a job together and individually can be a fascinating application of mathematical principles. This article delves into the step-by-step solution to determining the individual completion times of workers A and B based on the given conditions.

Introduction to Work Rate Problems

Work rate problems often involve determining the time it takes for individuals to complete a task independently or in collaboration. These problems can be solved using various methods, including algebraic equations. By establishing the work rates of each individual, we can effectively find the time taken when they work together and separately.

Understanding the Problem

The problem states that workers A and B together can complete a job in 4 days. Additionally, it is mentioned that worker A spends twice as much time as worker B to complete the job alone.

Setting Up Equations

Let us denote the time taken by worker B to complete the job alone as b days. Consequently, worker A, taking twice as long, would take 2b days to complete the job alone.

The work rate of each worker can be defined as follows:

A's work rate: 1/2b jobs per day B's work rate: 1/b jobs per day

Combining Work Rates

When A and B work together, their combined work rate would be the sum of their individual work rates:

1/2b 1/b 3/2b

According to the problem, the combined work rate of A and B is 1/4 jobs per day. Therefore, we can set up the following equation:

3/2b 1/4

Cross-multiplying to solve for b, we get:

3 × 4 2b

b 6

Hence, B would take 6 days to complete the job alone.

Since A takes twice as long as B, A would take:

2 × 6 12 days

Alternative Methods and Additional Examples

Using Substitution

If we let B take b days, then A would take 2b days. This can be represented algebraically as:

A's work rate: 1/2b jobs per day B's work rate: 1/b jobs per day

Combining the work rates:

1/2b 1/b 3/2b

Setting this equal to 1/4:

3/2b 1/4

Solving for b:

b 6

Therefore, for B: b 6 days, and for A: 2 × 6 12 days.

Additional Example

Consider another example where if B takes b days, A takes 3b days, and C takes 2c days. We know:

3b 2c, hence b 2c/3

The individual work rates for A, B, and C are:

A's work rate: 1/3b B's work rate: 1/b C's work rate: 1/c

Combining work rates and setting equal to 1/4:

1/3b 1/b 1/c 1/4

Substituting b 2c/3:

1/2c 3/2c 1/c 1/4

4/2c 1/c 1/4

Multiplying by 4c to clear the denominators:

2 4 4 c

c 12, hence b 8 and a 24

Hence A takes 24 days, B takes 8 days, and C takes 12 days each working alone.

Conclusion

By following the steps outlined above, we can solve work rate problems effectively. Understanding the relationships between work rates, individual completion times, and combined work rates provides a robust framework for tackling similar problems in algebra and higher mathematics. Whether using straightforward substitution or complex algebraic manipulations, these techniques offer valuable insights into problem-solving in the realm of work rate problems.

References

For further reading and practice, consider exploring additional resources on Math Warehouse: Work Rate Problems and other educational platforms.