Question

Mr. Modi can copy 40 pages in 10 minutes, Mr Xerox and Mr. Modi both working together can copy 250 in 25 minutes. In how many minutes Mr. Xerox can copy 36 pages? (a) 5 minutes (b) 6 minutes (c) 3 minutes (d) 12 minutes

Short answer

Answer: (b) 6 minutes

Step-by-step solution

Find Mr. Modi's copying rate

We are given that Mr. Modi can copy 40 pages in 10 minutes. To find his rate, we will divide the number of pages copied by the time taken: \(Rate_{Modi} = \frac{40}{10} = 4\) pages per minutes.

Find the combined rate of Mr. Modi and Mr. Xerox

We are given that Mr. Modi and Mr. Xerox together can copy 250 pages in 25 minutes. To find their combined rate, we will divide the number of pages copied by the time taken: \(Rate_{combined} = \frac{250}{25} = 10\) pages per minutes.

Find Mr. Xerox's copying rate

To find Mr. Xerox's copying rate, we will subtract Mr. Modi's rate from their combined rate: \(Rate_{Xerox} = Rate_{combined} - Rate_{Modi} = 10 - 4 = 6\) pages per minutes.

Find the time it takes for Mr. Xerox to copy 36 pages

Now that we have Mr. Xerox's copying rate, we can find out how long it would take him to copy 36 pages using the formula Time = Work ÷ Rate. Time = \(\frac{36}{6} = 6\) minutes. So, it will take Mr. Xerox 6 minutes to copy 36 pages. The correct answer is (b) 6 minutes.

Key concepts

Rate of Work

When solving work and time problems, understanding the rate of work is crucial. The rate of work refers to how much work a person or machine can do in a specific amount of time. For instance, if Mr. Modi can copy 40 pages in 10 minutes, his rate of work is 4 pages per minute.
Understanding rate can be simplified using the formula:
\[ \text{Rate} = \frac{\text{Work Done}}{\text{Time Taken}} \]This formula helps us determine individual rates, as shown in the exercise. Knowing individual rates, we can find combined rates when multiple individuals or machines work together. It's a fundamental step in understanding how collaborative efforts impact total productivity.

Speed and Efficiency

Speed and efficiency are closely related to the rate of work. They describe how fast and how effectively a task can be completed. High efficiency means a worker can complete a task quickly, which typically results in a higher rate of work.

• **Speed**: The faster the rate of work, the less time is needed to complete a task.
• **Efficiency**: Enhanced methods or skills often boost efficiency, enabling individuals to complete more work in the same amount of time.

In the exercise, Mr. Modi’s speed is seen in his ability to copy 4 pages every minute. By understanding efficiency and relative speed, it becomes easier to measure how different workers, like Mr. Modi and Mr. Xerox, contribute to a task both individually and together.
The exercise given shows the calculated efficiency when they work together and demonstrates how shared efforts can lead to greater output than when working individually.

Collaboration in Work

Collaboration can significantly enhance work output by leveraging the combined strengths of multiple workers or machines. In the exercise, Mr. Modi and Mr. Xerox together manage a higher page output in a given time frame compared to when they work separately.
To calculate the effect of collaboration, we simply add their rates of work. If Mr. Modi works at 4 pages per minute and Mr. Xerox at 6 pages per minute, their combined rate when working together becomes:

• \[ \text{Combined Rate} = \text{Rate}_{\text{Modi}} + \text{Rate}_{\text{Xerox}} = 4 + 6 = 10 \text{ pages per minute} \]

This formula reflects the additive nature of collaborative work, enhancing productivity significantly.
Overall, recognizing how collaboration can be used to optimize workload distribution is vital. The key takeaway is that when tasks are shared, the completion time reduces drastically compared to completing them individually. It can lead to more efficient use of resources and time management.