Question

\(A\) is twice efficient as \(B\) and together they do the same workin as much time as \(C\) and \(D\) together. If \(C\) and \(D\) can complere the work in 20 and 30 days respectively, working alone, ther in how many days \(A\) can complete the work individually: (a) 12 days (b) 18 days (c) 24 days (d) 30 days

Short answer

Based on the given information, it takes A 18 days to complete the job individually.

Step-by-step solution

Determine the work done by individuals

To express their work in equations, we'll use the following notation: - The amount of work done by A in one day: W_A - The amount of work done by B in one day: W_B - The amount of work done by C in one day: W_C - The amount of work done by D in one day: W_D According to the problem, A is twice as efficient as B. This means: W_A = 2 * W_B Additionally, we are given that C and D can complete the work in 20 and 30 days, respectively. So: W_C = 1/20 W_D = 1/30

Find the combined work of A and B, and C and D

To find the combined work of A and B, we'll add their work done in one day (W_A + W_B). And for C and D, we'll add their work done in one day (W_C + W_D): W_A+B = W_A + W_B W_C+D = W_C + W_D = 1/20 + 1/30 = 1/12 According to the problem statement, A and B together can finish the job in the same time as C and D together. So: W_A+B = W_C+D = 1/12

Substitute the relationship between A and B

We know that W_A = 2*W_B, so we can replace W_A in the equation for W_A+B: 1/12 = 2*W_B + W_B

Solve for W_B and find W_A

Combine the terms and solve for W_B: 1/12 = 3*W_B W_B = 1/36 Now find W_A using the relationship W_A = 2*W_B: W_A = 2 * (1/36) W_A = 1/18

Find the number of days it takes for A to complete the job

Since W_A represents the fraction of work A completes in one day, to find the number of days it takes for A to complete the job, we take the reciprocal of W_A: Days taken by A = 1 / (1/18) Days taken by A = 18 So, it takes A 18 days to complete the job individually. The correct answer is (b) 18 days.

Key concepts

Work and Time Problems

Work and Time problems in quantitative aptitude are crucial concepts that test an individual's problem-solving abilities. They involve understanding the relationship between the amount of work to be done and the time taken to complete it by one or more entities, often people or machines. The fundamental rule for these problems is that if an entity is more efficient, it will take less time to do the same amount of work, and conversely, if it's less efficient, it will take more time.

To tackle these problems, you usually assign a work unit to the entire task, like '1 work unit'. If a person or machine completes the work in 'd' days, we say their work rate is '1/d' work units per day. By adding or comparing these rates, we can determine how long it would take for different combinations of people or machines to complete the same task. Remember, the key to solving these problems is to establish the work rates and then to use them to calculate time or work, depending on what the question asks for.

Efficiency in Work Problems

Efficiency in work problems is all about comparisons. It's a measure of work done per time unit and is a relative metric. In quantitative aptitude exercises, efficiency helps us compare different workers' (or machines') abilities to perform work. Generally, if one worker is 'n' times more efficient than another, they would complete a set amount of work 'n' times faster.

For example, if worker A is twice as efficient as worker B, A would do in 1 hour what B does in 2 hours. Mathematically, when considering efficiency, you can set up a direct proportion to compare the efficiency of two entities: If the efficiency of A is twice that of B, then A's work (which we can denote as W_A) is 2 times B's work (W_B). As part of solving these problems, understanding and establishing these proportions correctly is crucial to calculate either the amount of work done or the time taken to do it, depending on the available data.

Quantitative Comparison

Quantitative comparison problems involve analyzing and comparing quantities to determine their relationship. These are common in standardized tests and can range from simple arithmetic comparisons to complex algebraic expressions. The strategy is to either manipulate the given information to directly compare the quantities or convert them into a common unit of measure.

In the context of work and time problems, this could translate to comparing the amount of work done by different entities in the same amount of time or the time taken by different entities to do the same amount of work. Effective strategies include equalizing one side of the comparison by finding a common denominator, or cross-multiplication to compare ratios. A clear understanding of ratios, proportional thinking, and the ability to simplify expressions are instrumental in efficiently solving these types of problems.