A girl reads two-fifths of a book on the first day, 34 percent of it on the second day, and the remaining pages over the next two days, reading an equal number of pages on each of those days. If she reads 52 pages on the last day, how many pages does the book contain?

Introduction / Context:
This problem combines fractions and percentages of a whole quantity, here the total number of pages in a book. The reading progress is spread over four days with different proportions on different days. The goal is to find the total number of pages using the information about the final day reading. Such questions are frequently used to test comfort with fractional parts and equations.

Given Data / Assumptions:

• The girl reads 2/5 of the book on the first day.
• She reads 34 percent of the book on the second day.
• She reads the remaining pages over the next two days, in equal amounts each day.
• On the fourth (last) day, she reads 52 pages.
• We must find the total number of pages in the book.

Concept / Approach:
We let the total number of pages be N. Fractions of N represent pages read on each day. By adding the fractions for the first two days, we can find the fraction left for the last two days. Since the last two days have equal numbers of pages and the last day is given as 52, we can find the remaining fraction of N and thus compute N. This is a straightforward linear equation in N involving basic percentage and fraction operations.

Step-by-Step Solution:
Step 1: Let total number of pages be N.Step 2: On the first day she reads 2/5 of N, which is 0.4N.Step 3: On the second day she reads 34 percent of N, which is 0.34N.Step 4: Together, pages read on the first two days are 0.4N + 0.34N = 0.74N.Step 5: Remaining fraction of the book after two days is 1 - 0.74 = 0.26. So remaining pages are 0.26N.Step 6: These remaining 0.26N pages are read over the next two days in equal amounts, so each of the last two days has (0.26N) / 2 = 0.13N pages.Step 7: We are told that on the last day she reads 52 pages, so 0.13N = 52.Step 8: Solve for N: N = 52 / 0.13.Step 9: Since 0.13 equals 13 / 100, dividing by 0.13 is multiplying by 100 / 13, giving N = 52 * (100 / 13) = 4 * 100 = 400.

Verification / Alternative check:
Check the distribution with N = 400 pages. On day one, she reads 2/5 of 400, which is 160 pages. On day two, she reads 34 percent of 400, which is 0.34 * 400 = 136 pages. Total after two days is 160 + 136 = 296 pages. Remaining pages are 400 - 296 = 104 pages for the last two days. If she reads equal amounts on days three and four, each day has 104 / 2 = 52 pages, which matches the given last day reading. So N = 400 is correct and consistent.

Why Other Options Are Wrong:
If the book had 376 or 382 or 356 or 420 pages, the calculations for first two days and remaining pages would not produce 52 pages on the last day when split equally across the last two days. For example, with N = 376, the remaining fraction 0.26N would not be exactly 104 pages and half of it would not be exactly 52. Only N = 400 satisfies all the conditions in the problem.

Common Pitfalls:
Some students mistakenly treat 34 percent as 34 pages instead of 34 percent of N. Others may add fractions incorrectly or forget to divide the remaining pages equally between the last two days. It is also easy to make arithmetic mistakes when working with decimals like 0.74 and 0.26. Converting everything carefully and checking each step helps prevent such errors.

Final Answer:
The book contains 400 pages, so the correct option is 400.