Solving Work and Time Problems with Mixed Work Rates: A Comprehensive Guide

Work and time problems are a common type of mathematical challenge that requires a clear understanding of how individuals or groups can accomplish a task together. This article aims to provide a comprehensive guide to solving such problems, focusing on detailed, step-by-step solutions. We will cover a series of examples that demonstrate how to handle problems with different workers and their respective work rates.

Example 1: A, B, and C Working Together Part 1

Given that A, B, and C can finish a work in 10, 12, and 15 days respectively, we will calculate the total time required to complete the work considering their collaboration and the departure of B and C at different points in time.

First, we find the individual work rates of A, B, and C:

A's work rate: A frac{1}{10} B's work rate: B frac{1}{12} C's work rate: C frac{1}{15}

They work together for the first 4 days, after which B leaves, and 3 days before C leaves, C stops working:

The work completed in the first 4 days:

left(frac{1}{10} frac{1}{12} frac{1}{15}right) times 4 left(frac{6}{60} frac{5}{60} frac{4}{60}right) times 4 frac{15}{60} times 4 frac{1}{4}

C then stops working 3 days before the end, leaving A and B to work for these last days. Let's denote the remaining work by x. The equation for the remaining work is:

left(frac{1}{10} frac{1}{12} frac{1}{15}right) times (t - 4) left(frac{1}{10} frac{1}{15}right) times 3 frac{1}{15} times 3 1

Simplifying further, we get:

left(frac{1}{2} frac{1}{60}right) times (t - 4) frac{1}{10} times 3 frac{3}{15} 1

left(frac{31}{60}right) times (t - 4) frac{1}{10} times 3 frac{1}{5} 1

left(frac{31t - 124 36 72}{60} 1right)

frac{31t - 124 108}{60} 1

31t - 16 60

31t 76

t frac{76}{31} approx 2.45

Therefore, the total time required to complete the work is approximately 8 days, boxed {Eight days total}.

Example 2: A, B, and C Working Together Part 2

In this example, we consider one additional worker, D with a work rate of 1/40. The problem is similar but involves more complex calculations.

The equation for the work completed by A, B, C, and D:

left(frac{1}{10} frac{1}{12} frac{1}{15} frac{1}{40}right) times t - 5 1

After simplifying, we get:

left(frac{120 100 80 60}{1200}right) times t - 5 1

left(frac{360}{1200}right) times t - 5 1

frac{3}{10} times t 6

t 20

Therefore, the total time required to complete the work is 10 frac{10}{11} days, boxed {x 10 frac{10}{11} days}.

Example 3: A, B, and C Working Together Part 3

For the third example, we will include an additional step: more complex time intervals involving multiple workers leaving and joining at different points.

The equation for the work completed by A, B, and C working together for 2 days, then A and C for 3 days, and A working alone for the rest of the time is:

2 times left(frac{1}{10} frac{1}{12} frac{1}{15}right) 3 times left(frac{1}{10} frac{1}{15}right) frac{t - 5}{10} 1

After simplifying, we get:

2 times left(frac{3}{120} frac{5}{120} frac{8}{120}right) 3 times left(frac{3}{120} frac{8}{120}right) frac{t - 5}{10} 1

2 times frac{16}{120} 3 times frac{11}{120} frac{t - 5}{10} 1

frac{32}{120} frac{33}{120} frac{t - 5}{10} 1

frac{65}{120} frac{t - 5}{10} 1

frac{13}{24} frac{t - 5}{10} 1

frac{t - 5}{10} frac{11}{24}

t - 5 frac{110}{24}

t frac{110}{24} 5

t frac{110 120}{24}

t frac{230}{24}

t 9 frac{1}{12}

Therefore, the total time required to complete the work is 7 frac{1}{13} days, boxed {x 7 frac{1}{13} days total}.

Example 4: A, B, and C Working Together Part 4

The final example involves more workers and a different sequence of departures and arrivals. We need to solve for the total time required using similar principles.

Using the work rates of A, B, and C, we set up the following equation:

4 times left(frac{1}{10} frac{1}{12} frac{1}{15}right) 3 times left(frac{1}{10} frac{1}{15}right) 7 times frac{1}{10} 1

After simplifying, we get:

left(frac{1}{2} frac{1}{60}right) times 4 left(frac{1}{5} frac{1}{10}right) times 3 7 times frac{1}{10} 1

left(frac{31}{60}right) times 4 left(frac{3}{10}right) times 3 7 times frac{1}{10} 1

frac{124}{60} frac{9}{10} frac{7}{10} 1

frac{124}{60} frac{16}{10} 1

frac{124}{60} frac{96}{60} 1

frac{220}{60} 1

frac{220}{60} 3.67

Therefore, the total time required to complete the work is 9 days, boxed {9 days total}.

Conclusion

By solving these work and time problems, we not only strengthen our understanding of mathematical principles but also enhance our problem-solving skills. Whether dealing with individual workers or groups, the key to success lies in breaking down the problem into manageable parts and applying the correct work rate calculations. Mastering these techniques can help in various real-world scenarios and in competitive examinations as well.