Solving Work Rate Problems: A Comprehensive Guide with Examples
When tackling work rate problems in mathematics, understanding the relationship between the amount of work, the individuals performing the work, and the time they take is crucial. This article will walk you through a detailed example, solving a work rate problem involving three individuals: A, B, and C. We'll break down the process step by step and use algebraic methods to find the solution.
Understanding Work Rate Problems
A time and work problem involves determining how long it takes individuals to complete a job and the impact of their combined efforts. In this guide, we'll focus on a specific problem where A, B, and C can complete a certain piece of work in 6, 8, and 12 days respectively. We'll calculate how long it takes for B to finish the remaining work after A and C leave after working with B for 2 days. Let's dive into the solution.
Step-by-Step Solution
First, we need to determine the work rates of A, B, and C.
Determining Individual Work Rates
Calculation of A's work rate:
[text{Work rate of A} frac{1}{6} text{ work per day}]Calculation of B's work rate:
[text{Work rate of B} frac{1}{8} text{ work per day}]Calculation of C's work rate:
[text{Work rate of C} frac{1}{12} text{ work per day}]Combining the Work Rates of B and C
Now, let's calculate the combined work rate of B and C:
[text{Combined work rate of B and C} frac{1}{8} frac{1}{12} frac{3}{24} frac{2}{24} frac{5}{24} text{ work per day}]Work Done by B and C Together in 2 Days
We need to find the amount of work B and C complete together in 2 days:
[text{Work done by B and C in 2 days} 2 times frac{5}{24} frac{10}{24} frac{5}{12}]Remaining Work
The total work initially is 1 (one whole work). The remaining work after B and C have worked for 2 days is:
[text{Remaining work} 1 - frac{5}{12} frac{7}{12}]Combined Work Rate of A and C
Next, let's calculate the combined work rate of A and C:
[text{Combined work rate of A and C} frac{1}{6} frac{1}{12} frac{2}{12} frac{1}{12} frac{3}{12} frac{1}{4} text{ work per day}]Time Taken by A and C to Finish the Remaining Work
Let (t) be the time taken for A and C to finish the remaining work:
[frac{1}{4} t frac{7}{12} Rightarrow t frac{7}{12} times 4 frac{28}{12} frac{7}{3} text{ days} approx 2.33 text{ days}]Total Time to Complete the Work
The total time taken is the sum of the time B and C worked together and the time A and C worked together:
[text{Total time} 2 text{ days} frac{7}{3} text{ days} frac{6}{3} frac{7}{3} frac{13}{3} text{ days} approx 4.33 text{ days}]Therefore, the total time taken to finish the work is approximately 4.33 days.
Conclusion
Understanding the steps to solve work rate problems is essential for tackling various mathematical challenges. By breaking down the problem and using algebraic methods, we can effectively find the solution. Whether you're in school, preparing for an exam, or looking to improve your problem-solving skills, this guide provides a clear and methodical approach to solving work rate problems.
Related Topics:
Time and Work Problems Algebraic Methods Work Rate ProblemsRelated Keywords:
work rate time and work problem solving algebraic methods