Question

\(C\) takes twice the number of days to do a piece of work than \(A\) takes. \(A\) and \(B\) together can do it in 6 days while \(B\) and \(C\) can do it in 10 days. In how many days \(A\) alone can do the work? (a) 60 (b) 30 (c) 6 (d) \(7.5\)

Short answer

Answer: 30 days

Step-by-step solution

Find the work rate of A and B together and B and C together

Since A and B together can complete the work in 6 days, their work rate together is (1 work)/(6 days) = 1/6. Similarly, B and C together can complete the work in 10 days, so their work rate together is (1 work)/(10 days) = 1/10.

Define the relationships between the work rates of A, B, and C

Let the work rates of A, B, and C be represented by \(r_A\), \(r_B\), and \(r_C\) respectively. We are given that C takes twice the number of days as A to complete the work, so the work rate of C is half of the work rate of A, or \(r_C = \frac{1}{2} r_A\). According to the definition of work rate, when two or more individuals work together, their combined work rate is the sum of their individual work rates. Therefore, we have: \(r_A + r_B = \frac{1}{6}\) (A and B together) \(r_B + r_C = \frac{1}{10}\) (B and C together)

Use the relationships and given information to find the work rate of A alone

Now substitute \(r_C = \frac{1}{2} r_A\) into the second equation to eliminate \(r_C\): \(r_B + \frac{1}{2} r_A = \frac{1}{10}\) Now you have a system of linear equations: \(r_A + r_B = \frac{1}{6}\) \(r_B + \frac{1}{2} r_A = \frac{1}{10}\) To solve for \(r_A\), first multiply the second equation by 2 to eliminate the fraction: \(2(r_B + \frac{1}{2}r_A) = 2(\frac{1}{10})\) \(r_B + r_A = \frac{1}{5}\) Next, subtract this equation from the first equation to eliminate \(r_B\): \((r_A + r_B) - (r_B + r_A) = \frac{1}{6} - \frac{1}{5}\) \(r_A - r_A = -\frac{1}{30}\) \(r_A = \frac{1}{30}\)

Find how many days A alone can complete the work

Now that we have found the work rate of A alone, we can find how many days it takes A to complete the work by taking the reciprocal of his work rate: Days taken by A alone = \(\frac{1}{r_A} = \frac{1}{\frac{1}{30}} = 30\) So, the correct answer is (b) 30.

Key concepts

Work Rate

The concept of work rate is central to solving problems involving time and work. Work rate refers to the amount of work completed per unit of time. It can be expressed as a fraction, the work done, divided by the time taken. For example, if a work is completed in 6 days, the work rate is \(\frac{1}{6}\). This mathematical tool helps in calculating cooperative work, where multiple parties contribute to completing a task.

When two or more individuals work together, you can add their work rates to find the combined work rate. In our exercise, since A and B together can complete the work in 6 days, their combined work rate is \(\frac{1}{6}\), and similarly, B and C together have a work rate of \(\frac{1}{10}\) as they finish in 10 days. Understanding and calculating work rates allow us to handle intricate problems involving multiple workers and their capabilities.

Linear Equations

In situations like this, where relationships between different variables are given, linear equations come into play. These equations can effectively model such scenarios. By establishing equations based on given data, they help us to solve for unknowns.

In our problem, we represent the work rates of A, B, and C as \(r_A\), \(r_B\), and \(r_C\) respectively. From the problem, we obtain the equations:

• \(r_A + r_B = \frac{1}{6}\)
• \(r_B + \frac{1}{2} r_A = \frac{1}{10}\)

These equations form a system of linear equations. The advantage here is that the solution of these equations gives us the individual work rates, allowing us to determine how many days each worker would need to finish the work alone.

Mathematical Reasoning

Mathematical reasoning is the logic that helps us draw conclusions and solve mathematical problems. In the context of our exercise, reasoning involves understanding the relationships between workers A, B, and C and figuring out how their individual capabilities impact the time it takes to complete a task.

We know from the problem statement that C is twice as slow as A, which means \(r_C = \frac{1}{2} r_A\). This relationship is essential for replacing \(r_C\) in one equation, simplifying the system. This form of reasoning helps uncover valuable connections in the problem, leading us to accurate solutions. It's a critical skill in complex problem-solving, helping analyze each part of the problem logically and derive the necessary steps for the solution.

Problem Solving Strategies

To navigate complex time and work problems, various problem-solving strategies can be applied. Recognizing the nature of the problem often helps in choosing the right strategy. For example, in this problem, defining variables for work rates was the first step.

Next, writing the given relationships as equations is crucial. Then, substitution helped simplify the equations, especially using the relation \(r_C = \frac{1}{2}r_A\). Finally, solving the equations systematically gave us the answer.

• Break down the problem into smaller parts.
• Use substitution to make equations solvable.
• Work systematically through the algebra to find solutions.

Each step of the strategy should be logical and repeatable, ensuring students can handle similar problems with confidence. These strategies build a framework that supports effective problem-solving in real-life scenarios.