Question

A group of workers can complete a job in 120 days. If there were 4 more such workers then the work could be finished in 12 days less. What was the actual strength of workers? (a) 30 workers (b) 40 workers (c) 42 workers (d) 36 workers

Short answer

Answer: 36 workers.

Step-by-step solution

Understand the given information

The problem states that a group of workers can complete a job in 120 days. Let's denote the number of workers as W. We are also given that if there were 4 more workers (W + 4), the job would be completed in 12 fewer days (120 - 12 = 108 days).

Set up the equation using the work formula

We can set up an equation using the work formula, (number of workers) * (number of days) = work. The equation can be written as: (W)(120) = (W + 4)(108)

Solve the equation

Next, we need to solve the equation to find the value of W. Expand the equation: 120W = 108W + 432 Subtract 108W from both sides: 12W = 432 Divide by 12 to find W: W = \frac{432}{12} W = 36 Hence, the actual strength of workers is 36. The correct answer is (d) 36 workers.

Key concepts

Quantitative Aptitude

Quantitative aptitude encompasses the ability to understand and solve mathematical problems that are often presented in work and time-related questions. In this kind of problem, the key is to interpret the text and convert the information into mathematical expressions.

For instance, knowing that 'a group of workers can complete a job in a certain number of days' allows us to understand that the work done is a product of the number of workers and the days they take. As the number of workers changes, the days required for the same amount of work will also change, following the rules of direct or inverse proportion. These problems assess one’s ability to manipulate mathematical equations and use logical reasoning to derive at the correct answer.

Equation Solving

Equation solving is a foundational skill in tackling quantitative problems. In the context of time and work problems, the solution typically involves setting up an equation or system of equations that models the scenario.

Equations are a mathematical statement that assert the equality of two expressions. For this problem, we translate the given scenario into the equation \(W \times 120 = (W+4) \times 108\) by recognizing the work done, as a product of the workers and the time they take, is the same under both conditions. To solve for \(W\), algebraic manipulation such as expanding, grouping like terms, and isolating the variable is necessary. This allows you to determine the number of workers needed to complete the job within the constraints provided.

Number of Workers Calculation

The calculation of the number of workers relates to understanding how labor resources affect the completion of a task. In these problems, you often work backwards from the result—how soon a job is finished—to deduce the number of people who were involved in the work.

In the provided example, we’re solving for the initial number of workers, \(W\), who can complete the job in 120 days. We then compare this to a scenario with an additional 4 workers, finishing the job in 108 days. The calculation involves setting up a proportion between the two scenarios, knowing that adding more workers should result in less time to finish the job, provided the efficiency per worker remains constant.

Direct and Inverse Proportion

Direct and inverse proportions are two fundamental concepts in mathematics that feature prominently in work and time problems. If two quantities are in direct proportion, as one increases, the other also increases at a consistent rate. Conversely, with inverse proportion, as one quantity increases, the other decreases proportionately.

In our problem, the concept of inverse proportion applies: as the number of workers (W) increases, the time (T) required to complete the job decreases. This is shown by the initial set of workers \(W\) taking 120 days, and \(W + 4\) workers taking 108 days. This relationship between workers and time is crucial for setting up the right equation to solve for the unknown number of workers.