Question

A can do a piece of work in 24 days. If \(B\) is \(60 \%\) more efficient than \(A\), then the number of days required by \(B\) to do the same piece of work is : (a) 10 (b) 15 (c) 12 (d) \(9.6\)

Short answer

Answer: 15 days

Step-by-step solution

Understand the given information

We are given that A can complete the work in 24 days. Also, B is 60% more efficient than A.

Calculate the work done in a day by A

If A can complete the work in 24 days, then the work done in a day by A can be calculated as: Work done by A in a day = \(\frac{1}{24}\) of the total work

Calculate the work done in a day by B

Since B is 60% more efficient than A, the work done in a day by B can be calculated as: Work done by B in a day = Work done by A in a day + 60% of work done by A in a day Work done by B in a day = \(\frac{1}{24} + \frac{60}{100} \times \frac{1}{24} = \frac{1}{24} + \frac{3}{8} \times \frac{1}{24} = \frac{1}{24} + \frac{1}{64} = \frac{1}{15}\)

Calculate the number of days required by B to complete the work

Now that we know the work done in a day by B, we can calculate the number of days required by B to complete the same piece of work: Total work = 1 Days required by B = \(\frac{\text{Total work}}{\text{Work done by B in a day}} = \frac{1}{\frac{1}{15}} = 15\) So, B requires 15 days to complete the same piece of work. The correct option is (b) 15.

Key concepts

Time and Work

Understanding the concept of 'time and work' is crucial for grasping the basics of work efficiency problems. In essence, this concept focuses on how long it takes for someone or something to complete a task or job. The relationship is simple: if you have more time, you can typically accomplish more work, and if you work faster, you can complete a task in less time.

In problems that involve 'time and work', we usually define a 'work unit', which could be any task like knitting a sweater, building a wall, or writing a report. We then determine the time it takes for an individual to complete this unit of work. Here's a very handy formula you can use:
\[\text{Work done per day} = \frac{1}{\text{Total days to complete the work}}\] Using this formula helps to simplify complex problems and is an essential tool in the quantitative aptitude toolbox.

Percentage Efficiency

When dealing with work efficiency problems, 'percentage efficiency' is a critical measure. What does it mean to say that one person is a certain percentage more efficient than another? It quantifies how much more work someone can do in the same amount of time compared to another person.

To put it simply, if we say that person B is 60% more efficient than person A, we are stating that for every 100 tasks A can perform, B can do 160 tasks in the same period. This concept is easily expressed in a formula:
\[\text{Efficiency of B} = \text{Efficiency of A} \times (1 + \frac{60}{100})\] By understanding percentage efficiency, students can tackle a variety of problems in the quantitative aptitude domain. It allows for comparisons between different work rates and is vital in planning and optimization challenges.

Quantitative Aptitude

Quantitative aptitude is a vast field involving numerical ability, accuracy, and problem solving. It encompasses various topics, including arithmetic, algebra, geometry, and data interpretation. When tackling problems in this area, it's important to understand the underlying concepts and formulas.

In the context of efficiency problems like our example, quantitative aptitude enables us to translate word problems into mathematical equations. This skill is honed through practice and familiarity with different types of questions, such as ratio and proportion, percentages, and, as in our example, time and work. Enhancing your quantitative aptitude is not just about memorizing formulas; it's also about developing an analytical mind to logically deduce and solve problems.

Problem-Solving Approach

A structured problem-solving approach is key to tackling quantitative aptitude questions effectively. Let's break down the process with the provided exercise as an example:

Step 1: Understand the given information. Always start by identifying and clearly understanding the variables and conditions provided.
Step 2: Translate the problem into equations. This involves converting the word problem into mathematical terms, using formulas as needed.
Step 3: Solve the equations. Apply mathematical rules and operations to find the solution.
Step 4: Check your result. It’s crucial to verify if the solution makes logical sense within the context of the problems.

This approach ensures a systematic method to reach the correct answer and is adaptable to a wide range of topics within mathematical problem solving.