Question

123 typists typed 984 papers in \(1 / 15\) hour. The number of papers typed per minute by an average typist is : (a) 1 (b) 2 (c) 3 (d) 5

Short answer

Answer: (b) 2 papers per minute.

Step-by-step solution

Convert the given time to minutes

First, we need to convert the given time of \(1 / 15\) hour into minutes. We do this by multiplying the time in hours by 60 since there are 60 minutes in an hour. Time in minutes = \((1 / 15) \text{ hour} \cdot 60 \text{ minutes/hour}\) Time in minutes = \(4 \text{ minutes}\)

Calculate the total papers typed per minute by all typists

Now, we will divide the total number of papers (984) by the total time in minutes (4 minutes) to find the total papers typed per minute by all typists. Total papers typed per minute = \( \frac{984 \text{ papers}}{4 \text{ minutes}}\) Total papers typed per minute = \(246 \text{ papers per minute}\)

Calculate the number of papers typed per minute by an average typist

Finally, we will divide the total papers typed per minute (246) by the number of typists (123) to find the number of papers typed per minute by an average typist. Papers typed per minute by an average typist = \(\frac{246 \text{ papers per minute}}{123 \text{ typists}}\) Papers typed per minute by an average typist = \(2 \text{ papers per minute}\) So the correct answer is (b) 2 papers per minute.

Key concepts

conversion of units

The process of converting units is essential in mathematics, especially when dealing with measurements and time, as these often require standardization. In the original problem, we needed to convert time given in hours to minutes, which is a common conversion since time is regularly calculated in minutes in everyday scenarios.
To convert hours to minutes, multiply the number of hours by 60, because there are 60 minutes in an hour. For example:

• If you have 1 hour, that's equivalent to 60 minutes.
• If you have 2 hours, it's 120 minutes (2 times 60).

In our exercise, we dealt with fractions of an hour. The given time was \(\frac{1}{15}\) hours. By multiplying \(\frac{1}{15}\) by 60, we get 4 minutes. This conversion allows us to use more familiar units to progress further with the problem.
Remember:

• Always check your units before starting a calculation.
• Unit conversions are key for ensuring accuracy.

average rate calculation

Calculating the average rate is essential when you want to understand how much output there is per unit of time or per item. In the context of the problem, the average rate describes how many papers one typist can type every minute.
Here is the approach:

• First, find the total amount of work done within a particular timeframe. For our problem, the work done was 984 papers typed in 4 minutes.
• Next, calculate the work done per minute. We divided the total number of papers by the number of minutes to find out how many papers all typists can complete in one minute: \( \frac{984}{4} = 246 \) papers per minute.

The next step was to find the average rate for one typist. Since 123 typists completed the task together, we divided the rate for all typists by the number of typists:

• \( \frac{246}{123} = 2 \) paper per minute is typed by a single typist, on average.

This calculation helps us understand efficiency and effectiveness when distributed across multiple agents.

division in mathematics

Division is one of the fundamental operations in mathematics, and it is used to distribute a number into smaller equal parts. It plays a crucial role in rate problems and many other mathematical concepts.
The division steps in our exercise were crucial:

• We used division first to determine the number of papers typed per minute by all typists. This is done by dividing the total papers (984) by the number of minutes (4), giving us 246 papers per minute.
• We then needed to find how many papers one typist would type per minute. To do that, we divided the papers per minute by the number of typists: \( \frac{246}{123} = 2 \).

Understanding division can help make sense of rate calculations, proportion problems, and many other situations where you are splitting or sharing quantities evenly. Keep these guidelines in mind:

• Always ensure the numerator (number being divided) and the denominator (number you are dividing by) are in the same units.
• Be clear about what each value represents to avoid mistakes, especially in word problems.

Mathematical operations like division help you break down complex problems into manageable steps.