Solving Work Rate Problems: How Long Does C Take to Complete the Task Alone?

Today, we will go through a problem related to work rate and time. Specifically, we want to determine how long it takes for worker C to complete a piece of work alone, given that A and B together can complete the work in different time frames and with the help of C, they finish it even faster. This problem is a classic example of a time and work problem and can be approached using rate equations.

Introduction to Work Rate Problems

In work rate problems, the key concept is the relationship between work done, the time taken to do the work, and the rate at which the work is done. The basic form of a rate equation is (frac{text{rate} times text{time} }{text{work}}).

The Problem and Initial Setup

A can do a piece of work in 30 days, while B can do the same work in 15 days. With the help of C, they can complete the work in 10 days. The question is: how many days does C take to complete the work alone?

Step 1: Determine Work Rates

First, we need to determine the work rates of A, B, and C. The work rate is the fraction of the work done per day.

Work Rates of A and B

A's rate: (frac{1}{30} text{ work/day})

B's rate: (frac{1}{15} text{ work/day})

Combined Work Rate of A and B

The combined rate is the sum of the individual rates.

[frac{1}{30} frac{1}{15} frac{1}{30} frac{2}{30} frac{3}{30} frac{1}{10} text{ work/day}]

Step 2: Determine the Combined Work Rate with C

When A, B, and C work together, they complete the work in 10 days. Therefore, their combined work rate is:

(frac{1}{10} text{ work/day})

Step 3: Determine C's Work Rate

We can find C's work rate by subtracting the combined rate of A and B from the combined rate of A, B, and C.

(text{Rate of C} frac{1}{10} - frac{1}{10} frac{1}{30} - frac{3}{30} frac{1}{30} - frac{1}{10} frac{1}{15})

Step 4: Calculate the Time for C to Complete the Work Alone

If C's work rate is (frac{1}{15}) work/day, then the time C takes to complete the work alone is the reciprocal of this rate.

(x frac{1}{left(frac{1}{15}right)} 15 text{ days})

Therefore, C can complete the task alone in 15 days.

Conclusion

We have used a step-by-step approach to solve this problem of work rates and time. By determining the individual and combined work rates of A, B, and C, and then subtracting the necessary rates, we derived C's work rate and calculated the time it takes for C to complete the work alone. Such problems are essential in understanding proportional reasoning and work efficiency, which are widely applicable in real-world scenarios, from manufacturing to project management.