Solving the Problem of A, B, and C Working Together and Alone

Pagination is a common scenario in work rate problems, where we need to determine how long a task will take when different individuals work either together or separately. The problem at hand involves determining the time required for each individual, A, B, and C, to complete a project alone based on their work ratios and the information given about their combined effort.

A, B, and C's Work Ratios

The problem states that A, B, and C work in the ratio 5:6:7. This means for every 2 parts of work A completes, B completes 3 parts, and C completes 4 parts. Given that together they can finish the project in 30 hours, we can determine the individual times required for A, B, and C to complete the project alone.

Let's first solve the problem step-by-step using the provided ratios and the given total working time of 30 hours.

Step-by-Step Solution

To solve this, we first need to find the combined work rate of A, B, and C. If together they can finish the project in 30 hours, and their combined work rate is 1/30 of the project per hour.

Using Ratios and Combined Work Rate

If A, B, and C's work rates are in the ratio 5:6:7, let's assume their work rates are 5x, 6x, and 7x respectively. The total work done per hour by A, B, and C together is 5x 6x 7x 18x. Since this equals 1/30 of the project, we can set up the equation:

18x 1/30

Solving for x:

x 1/540

Now we can find the individual work rates:

A's work rate: 5x 5/540 1/108 B's work rate: 6x 6/540 1/90 C's work rate: 7x 7/540 7/540 1/77.14

Since the work rate is 1/(time to complete the project alone), the time taken by A is 108 hours, by B is 90 hours, and by C is approximately 77.14 hours.

Checking the Solution

To verify the solution, we need to ensure that the combined work rate equals 1/30 when A, B, and C work together:

1/108 1/90 1/77.14 1/30

This confirms our solution is correct. Therefore, the time required for B to complete the project alone is 90 hours.

Alternative Methods

There are also alternative methods to solve this problem. For instance, using the combined work time directly, we can determine the time each person takes by setting up the equations based on their individual work rates.

Using Combined Work Time Equations

If 5y the number of hours for A alone, then 6y for B and 7y for C, and they together finish the project in 30 hours, we can set up the following equation:

30/5y 30/6y 30/7y 1

Solving for y:

y 107/7

Therefore, the time for B to complete the project alone is:

6y 6 * (107/7) 642/7 ≈ 91.7 hours

Alternatively, if we assume the efficiency ratio is 5:6:7, the time ratio will be 7:6:5. Thus, B will take:

30 * (6/6) 90 hours

Conclusion

The problem provides various ways to determine the time required for each individual to complete a project alone. By understanding work rate and ratios, we can accurately solve such problems. The correct solution, based on the information provided, is that B will take approximately 90 hours to complete the project alone.