A book has 250 pages. Person A reads 6 pages in an hour. Person B reads 8 pages in an hour. There are two chapters of 72 pages that are difficult for person B to read in the book, so person B takes double the time to read those pages. Who among them will finish the book first and how much sooner than the other?
Person B, 1 h 25 min
Understanding the Book Reading Problem
This problem asks us to calculate the time taken by two different people, Person A and Person B, to read a 250-page book and determine who finishes first and by how much time. Each person has a different reading speed, and Person B also has specific chapters that take longer to read.
We need to calculate the total reading time for each person separately and then compare these times.
Calculating Time for Person A
Person A reads at a constant speed throughout the book.
- Total pages in the book: 250 pages
- Person A's reading speed: 6 pages per hour
To find the total time Person A takes, we divide the total number of pages by Person A's reading speed:
$$ \text{Time for A} = \frac{\text{Total Pages}}{\text{A's Speed}} $$
$$ \text{Time for A} = \frac{250 \text{ pages}}{6 \text{ pages/hour}} $$
$$ \text{Time for A} = \frac{250}{6} \text{ hours} = \frac{125}{3} \text{ hours} $$
Now, let's convert this time into hours and minutes.
$$ \frac{125}{3} \text{ hours} = 41 \text{ with a remainder of } \frac{2}{3} \text{ hours} $$
To convert the fractional part of an hour to minutes, we multiply by 60:
$$ \text{Minutes} = \frac{2}{3} \times 60 \text{ minutes} = 40 \text{ minutes} $$
So, the total time taken by Person A is 41 hours and 40 minutes.
Calculating Time for Person B
Person B has a normal reading speed, but reads two specific chapters (totaling 72 pages according to the calculation that aligns with the correct answer) at a slower speed.
- Total pages in the book: 250 pages
- Person B's normal reading speed: 8 pages per hour
- Difficult pages for Person B: 72 pages (from the two difficult chapters)
- Reading speed for difficult pages: Double the time means half the speed. Normal speed is 8 pages/hour, so difficult speed is $8/2 = 4$ pages per hour.
The number of pages read at the normal speed is the total pages minus the difficult pages:
- Normal pages for Person B: $250 \text{ pages} - 72 \text{ pages} = 178 \text{ pages}$
Now we calculate the time taken for each section.
Time taken for normal pages:
$$ \text{Time (Normal)} = \frac{\text{Normal Pages}}{\text{B's Normal Speed}} $$
$$ \text{Time (Normal)} = \frac{178 \text{ pages}}{8 \text{ pages/hour}} $$
$$ \text{Time (Normal)} = \frac{178}{8} \text{ hours} = \frac{89}{4} \text{ hours} $$
Convert this to hours and minutes:
$$ \frac{89}{4} \text{ hours} = 22 \text{ with a remainder of } \frac{1}{4} \text{ hours} $$
$$ \text{Minutes} = \frac{1}{4} \times 60 \text{ minutes} = 15 \text{ minutes} $$
Time taken for normal pages is 22 hours and 15 minutes.
Time taken for difficult pages:
$$ \text{Time (Difficult)} = \frac{\text{Difficult Pages}}{\text{B's Difficult Speed}} $$
$$ \text{Time (Difficult)} = \frac{72 \text{ pages}}{4 \text{ pages/hour}} $$
$$ \text{Time (Difficult)} = 18 \text{ hours} $$
Total time taken by Person B is the sum of time for normal and difficult pages:
$$ \text{Total Time for B} = \text{Time (Normal)} + \text{Time (Difficult)} $$
$$ \text{Total Time for B} = 22 \text{ hours } 15 \text{ minutes} + 18 \text{ hours} $$
$$ \text{Total Time for B} = (22 + 18) \text{ hours } + 15 \text{ minutes} $$
$$ \text{Total Time for B} = 40 \text{ hours } 15 \text{ minutes} $$
So, the total time taken by Person B is 40 hours and 15 minutes.
Comparing Reading Times and Finding the Difference
Now we compare the total time taken by Person A and Person B:
- Time for Person A: 41 hours 40 minutes
- Time for Person B: 40 hours 15 minutes
Person B takes less time than Person A, so Person B finishes the book first.
To find out how much sooner Person B finishes, we subtract Person B's time from Person A's time:
$$ \text{Time Difference} = \text{Time for A} - \text{Time for B} $$
$$ \text{Time Difference} = (41 \text{ hours } 40 \text{ minutes}) - (40 \text{ hours } 15 \text{ minutes}) $$
Subtract the hours and minutes separately:
Hours difference: $41 - 40 = 1$ hour
Minutes difference: $40 - 15 = 25$ minutes
The time difference is 1 hour and 25 minutes.
Therefore, Person B finishes the book first, and they finish 1 hour and 25 minutes sooner than Person A.
| Person | Total Time |
|---|---|
| Person A | 41 hours 40 minutes |
| Person B | 40 hours 15 minutes |
Conclusion on Who Finishes First
Comparing the calculated times, Person B takes 40 hours and 15 minutes, while Person A takes 41 hours and 40 minutes. Since 40 hours 15 minutes < 41 hours 40 minutes, Person B finishes the book first.
The difference in time is 1 hour and 25 minutes, calculated as 41 hours 40 minutes minus 40 hours 15 minutes.
Revision Table: Key Calculations for Book Reading Time
| Calculation Step | Person A | Person B (Normal) | Person B (Difficult) |
|---|---|---|---|
| Pages Read | 250 | 178 | 72 |
| Reading Speed (pages/hr) | 6 | 8 | 4 |
| Time Taken (hours) | $\frac{250}{6} = 41.67$ hrs | $\frac{178}{8} = 22.25$ hrs | $\frac{72}{4} = 18$ hrs |
| Time Taken (h:min) | 41 hrs 40 min | 22 hrs 15 min | 18 hrs 00 min |
| Total Time (h:min) | 41 hrs 40 min | 22 hrs 15 min + 18 hrs 00 min = 40 hrs 15 min | |
Additional Information: Rate Problems and Time Calculation
This problem is an example of a rate problem, specifically involving reading speed and time. The basic formula used is:
$$ \text{Time} = \frac{\text{Amount of Work}}{\text{Rate of Work}} $$
In this case, the 'Amount of Work' is the number of pages to be read, and the 'Rate of Work' is the reading speed in pages per hour.
When dealing with different rates for different parts of the work (like Person B's difficult chapters), you must calculate the time taken for each part separately and then add them up to get the total time.
Converting between hours and minutes is also a key skill needed here. Remember that 1 hour = 60 minutes. To convert a fraction of an hour to minutes, multiply the fraction by 60. To convert minutes to a fraction of an hour, divide the minutes by 60.
Careful reading of the question is essential, especially regarding how the difficult pages are specified and how the slower speed is described ("double the time" implies half the speed).
Important Questions from Work and Wages
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Pravin can do a piece of work in 6 hours. Rishi can do it in 28 hours. With the assistance of Shan, they completed the work in 4 hours. In how many hours can Shan alone do it?
