Question

The ratio of working efficiency of \(A\) and \(B\) is \(5: 3\) and the ratio of efficiency of \(B\) and \(C\) is \(5: 8\). Who is the most efficient? (a) \(A\) (b) \(B\) (c) \(C\) (d) can't be determined

Short answer

Answer: A

Step-by-step solution

Find a common ratio for A, B, and C's efficiencies

We are given two ratios. The first one is the ratio of A's efficiency to B's efficiency, which is \(5 : 3\). The second one is the ratio of B's efficiency to C's efficiency, which is \(5 : 8\). To compare all three efficiencies, we'll need to find a ratio that includes A, B, and C. First, we need to find a common value for B's efficiency in both ratios. We can do this by multiplying both ratios by necessary values to obtain the same value for B's efficiency in both ratios.

Finding the common value for B

We can see that the LCM of 3 and 5 is 15. So, we can multiply both ratios by the necessary values to get a common efficiency for B: For the first ratio \(5 : 3\), we can multiply it by 5 to get a new ratio \(25 : 15\). This represents that A has an efficiency of 25 units and B has an efficiency of 15 units. For the second ratio \(5 : 8\), we can multiply it by 3 to get a new ratio \(15 : 24\). This represents that B has an efficiency of 15 units and C has an efficiency of 24 units.

Combining the two ratios

Now that we have a common efficiency value for B in both ratios, we can combine the ratios to get the efficiency ratio of A : B : C which is \(25 : 15 : 24\).

Determine the most efficient person

Comparing the efficiency values in the ratio \(25 : 15 : 24\), we can see that A has the highest efficiency with a value of 25 units. So, A is the most efficient. The answer is (a) \(A\).

Key concepts

Ratio Analysis

Ratio analysis is a useful method for comparing different quantities or values. In this problem, we're looking at the working efficiencies of three individuals: A, B, and C. By analyzing the given ratios of their efficiencies, we can determine how they compare to one another.

- The ratio of A's efficiency to B's is 5:3, which means for every 5 units of work A does, B does 3 units.
- The ratio of B to C is 5:8, indicating that for every 5 units B accomplishes, C completes 8 units.

Understanding ratios is critical because it allows us to determine relationships and proportions between different quantities. In this exercise, the goal was to find a common basis for comparison by finding an equivalent ratio, which included all three individuals. By calculating the least common multiple for B's efficiency in both original ratios, we were able to align them on a common scale (15 units for B) and draw a clear comparison across A, B, and C.

Work Efficiency

Work efficiency refers to the ability of a worker to accomplish a task in a set amount of time. The higher the work efficiency, the more work the person can complete in the same duration, or the lesser time they need to accomplish the same amount of work.

In this problem, work efficiency was key to understanding who among A, B, and C could produce the most output.

• A's efficiency was found to be 25 units when normalized to a common measure.
• B's efficiency remains as 15 units in this common framework.
• Similarly, C's efficiency equated to 24 units when scaled with the same measure.

Efficient workers complete tasks more quickly and effectively than less efficient ones. Thus, knowing A completed 25 units illustrates a higher efficiency compared to B and C, helping us conclude that A is the most efficient.

Problem Solving

Problem-solving is the process of identifying a problem, developing strategies to solve it, and implementing those strategies. In context with our efficiency ratio problem, effective problem-solving involved a few key steps.

First, understanding the concept of ratios was essential to set the foundation. Then, identifying how to align the different given ratios was a crucial step. Using concepts like least common multiple helped us achieve a unified framework for comparison. This step-by-step method of solving ensured that all information was used correctly, and the end goal was clear.

Finally, the solution involved comparing the new ratios to determine the person with the highest work efficiency. Problem-solving requires not just analysis but logical application and reasoning through each step. This problem demonstrated how methodical approaches enlighten solutions effectively.