Question

Ajit is 3 times as efficient as Bablu, then the ratio of numberof days required by each to work alone, completely? (a) \(3: 1\) (b) \(1: 3\) (c) \(6: 3\) (d) \(3: 6\)

Short answer

Answer: (b) 1 : 3

Step-by-step solution

Represent Efficiencies

Let \(A\) and \(B\) represent the amount of work done by Ajit and Bablu in one day, respectively. According to the problem, Ajit is 3 times as efficient as Bablu, so we can write the relationship as: \(A = 3B\).

Find the Relationship Between Number of Days

Let \(D_A\) and \(D_B\) represent the number of days required by Ajit and Bablu to work alone, respectively. Since they are working alone, we can represent the total work done by each person as their efficiency multiplied by the number of days they work: \(A \times D_A = B \times D_B\).

Substitute Efficiency Values

Now we can substitute the relationship between the efficiencies from Step 1 into the equation from Step 2. We get \((3B) \times D_A = B \times D_B\).

Simplify and Solve for Ratio

Divide both sides of the equation by \(B\) to get rid of it from the equation. This gives us \(3D_A = D_B\). The ratio of number of days required by Ajit to Bablu is \(D_A : D_B = 1 : 3\).

Choose the Correct Option

From the given options, the correct answer is (b) \(1 : 3\).

Key concepts

Ratio and Proportion

Understanding the concept of ratio and proportion is crucial when dealing with work and efficiency problems. A ratio is a way to compare two quantities by division, highlighting the relative size of one quantity in comparison to another. For instance, if Ajit's efficiency is denoted as 3 and Bablu's as 1, the ratio of their efficiencies is expressed as 3:1. This means Ajit is three times as efficient as Bablu.

Proportion, on the other hand, indicates that two ratios are equivalent. When we look at the relationship between their efficiencies and the time taken to complete the same amount of work, we set up a proportion. In the exercise, if Ajit takes one day to complete a task, Bablu would take three days for the same task because he is only a third as efficient. This inverse relationship—where higher efficiency implies lesser time—is the basis for the proportion we see in time and work problems.

Quantitative Aptitude

Quantitative aptitude is all about the ability to handle numerical and logical reasoning. It's a skill set that's particularly important when solving time and work problems. These require quick thinking, logical structuring of equations, and the manipulation of numbers. Students need to grasp the efficiency ratios likely seen in the problem and then convert these ratios into actionable equations.

In the given exercise, quick quantitative analysis was necessary to establish the relationship between Ajit and Bablu's work rates and to use that to find the ratio of days required to complete the task. Mastering quantitative aptitude means being comfortable with representing real-world situations as mathematical models and then manipulating these models to find a solution.

Time and Work

The concept of 'time and work' in quantitative problems is grounded in understanding how time and work are reciprocal to each other, especially when calculating efficiency. If one person or entity is more efficient than another, it means that they can complete more work in the same amount of time, or the same work in less time.

This inversely proportional relationship is pivotal when solving problems where individuals or groups have different efficiencies. In our exercise example, Ajit's heightened efficiency translates to a decreased need for time to complete the same amount of work as Bablu. Thus, when discussing time and work, it's vital to remember that increasing the work speed or efficiency results in a reduced time requirement, which is always expressed as an inverse ratio of the time taken by each party.