Question

Of three workmen, the second and third can complete a job in 10 days. The first and third can do it in 12 days, while the first and second can do it in 15 days. In how many days can each of them do the job alone?

Short answer

Answer: Workman A can complete the job alone in 60 days, workman B in 30 days, and workman C in 20 days.

Step-by-step solution

Assign Variables

Let A, B, and C represent the work rates of the three workmen. The given information can be represented as equations: 1. A + B can do the job in 15 days 2. A + C can do the job in 12 days 3. B + C can do the job in 10 days

Set up equations

The work completed by each workman in a day is a fraction of the entire job. So, we can set up the following equations based on the given information: 1. (A+B) x 15 = 1 2. (A+C) x 12 = 1 3. (B+C) x 10 = 1

Solve the Equations

We can solve the system of linear equations to find the values of A, B, and C. First, we can rewrite the equations, so they are easier to work with: 1. 15A + 15B = 1 2. 12A + 12C = 1 3. 10B + 10C = 1 Now, let's add equations 1 and 2 and subtract equation 3: (15A + 15B) + (12A + 12C) - (10B + 10C) = 1 + 1 - 1 27A + 17B - 10C = 1 Now, let's multiply equation 3 by -1.5 and add it to equation 1: -1.5(10B + 10C) + (15A + 15B) = -1.5 + 1 15A - 1.5C = -0.5 Now, let's multiply equation 3 by -1.2 and add it to equation 2: -1.2(10B + 10C) + (12A + 12C) = -1.2 + 1 12A - 1.2B = -0.2 Finally, let's divide the last equation by 12: A - 0.1B = -0.0167 Now, we can solve for A and B: A = 0.1B - 0.0167 To find the value of B, we can substitute the expanded equation of A into the 27A + 17B - 10C = 1 equation and solve for B. After a few calculations, we get: B = 1/30 Now substitute B back into the equation for A: A = 1/60 Lastly, substitute A and B back into one of the original equations to find C, for example, equation 2: (1/60) + C = 1/12 C = 1/20

Calculate the number of days

Now that we have the work rates (A, B, and C), we can find the number of days it takes each workman to complete the job alone. The number of days is simply the reciprocal of the rate: Workman A: 1/A = 1/(1/60) = 60 days Workman B: 1/B = 1/(1/30) = 30 days Workman C: 1/C = 1/(1/20) = 20 days So, workman A can complete the job alone in 60 days, workman B in 30 days, and workman C in 20 days.

Key concepts

Work Rates

Understanding the concept of work rates is essential when solving problems that involve multiple workers or machines completing a task. A work rate refers to how much of a job can be completed by a worker in a given amount of time, typically expressed as a job per unit of time. In this problem, each workman has a unique work rate, and their combined efforts depend on these rates.

Consider that if Workman A can complete a job in 30 days, his work rate is 1/30 of the job per day. Similarly, if Workman B completes the same job in 20 days, his rate is 1/20 of the job per day. By incorporating this rate-based understanding, we move from just looking at time to focusing on how much job completion happens over time. When multiple workmen collaborate, their work rates add up, allowing us to calculate the total amount of work they can finish together.

In the context of the problem, when Workman A and Workman B work together, their combined work rate is (1/A + 1/B), helping them complete the job quicker than if they worked alone. By identifying and using work rates, you can easily calculate how collaborations affect timeframes and solve such puzzles effectively.

Reciprocal Calculations

Reciprocal calculations help us convert work rates back into timeframes—how long each workman takes to complete the job independently. The reciprocal of a work rate gives the time to finish the entire job if the worker continues at that rate without assistance.

For example, if a worker’s rate is 1/30 of a job per day, taking the reciprocal gives 30 days, showing how long it takes him to finish the job alone. In practical terms, this means if Workman A’s work rate is calculated as 1/60 from the given equations, it tells us that working alone, he will need 60 days to finish the project.

This reciprocal relationship is crucial because it shifts the perspective from how much work is done daily to how long it takes to complete the job. By reciprocating the work rate, you can quickly determine the time required for task completion for not just one worker but across any number of combinations or individual efforts as seen in the problem.

Problem Solving in Mathematics

Solving such mathematical problems involves strategically tackling them step-by-step, breaking them into smaller, more manageable equations. The problem presented is an ideal example of utilizing systems of linear equations to understand and interpret work dynamics among individuals.

Begin by identifying what each equation represents in the context of work performed. For instance, if you know two workers complete a task in a set number of days, this can be framed into an equation about their combined work rates. From there, solve these equations either individually first or by eliminating variables using techniques like substitution or combination.

The whole approach is about simplifying and clarifying complex interactions. In the scenario given, after setting equations with variables representing each workman’s rate (e.g., A, B, C), the task was streamlined by adjusting and solving these equations to derive each work rate. With calculated work rates, the reciprocal calculations are leveraged to obtain the time each workman individually requires, thus perfectly solving the problem.

This systematic, logical method acts as a roadmap for untangling more sophisticated problems in math, using collaborative math skills like equation manipulation, clarity in setting equations, and methodical solving.