Karthik read 6/13 of a book in the first week and 5/9 of the remaining pages in the second week. If 100 pages were still unread after the second week, approximately how many pages were there in the entire book?

Introduction / Context:
This question combines fraction operations with a word problem involving reading progress. It tests your ability to interpret fractional parts of a whole, handle successive fractions on a remaining quantity, and then work backward from the remaining pages to estimate the total number of pages in the book.

Given Data / Assumptions:

• Let the total number of pages in the book be N.
• In the first week, Karthik reads 6/13 of the book.
• In the second week, he reads 5/9 of the remaining pages.
• After the second week, 100 pages are still unread.
• We assume the fractions are applied exactly, and the final answer is chosen from the nearest integer value among the options.

Concept / Approach:
The key idea is to track what fraction of the book remains after each week. First, subtract the fraction read in the first week to find the remaining part. Then apply the second fraction to that remainder to find how much more is read. The remaining unread portion after two weeks is set equal to 100 pages. This leads to an equation in N, which we solve and then compare to the options, choosing the nearest whole number of pages.

Step-by-Step Solution:
Let total pages be N.Pages read in first week = (6/13)N, so remaining after week one = N - (6/13)N = (7/13)N.In the second week, Karthik reads 5/9 of the remaining pages: (5/9) × (7/13)N = (35/117)N.Remaining after second week = (7/13)N - (35/117)N.Convert 7/13 to a denominator of 117: 7/13 = 63/117.So remaining fraction = (63/117 - 35/117)N = (28/117)N.We are told this remainder equals 100 pages, so (28/117)N = 100.Solve for N: N = 100 × (117/28) ≈ 417.857.The closest whole number among the options is 418 pages.

Verification / Alternative check:
Take N = 418 and recompute the process. After week one, remaining pages = (7/13) × 418 ≈ 225.077, which we interpret as about 225 pages. Five ninths of this is approximately 125 pages, leaving close to 100 pages unread. Because the problem is drawn from an aptitude style context and the options are discrete integers, 418 is the best match, and small rounding adjustments are acceptable in this type of exam question.

Why Other Options Are Wrong:
Taking N as 404, 415, or 420 and repeating the same fraction operations does not produce a remaining count close to exactly 100 pages. The deviations are larger than those obtained with 418, so they are poorer approximations. Option 390 is clearly too small and leads to a remaining count significantly different from 100 pages. Among all choices, 418 fits the derived equation most closely.

Common Pitfalls:
Students often misinterpret the second fraction as 5/9 of the entire book rather than 5/9 of the remaining pages. Another error is adding or subtracting fractions with different denominators incorrectly, which leads to the wrong remaining fraction. Some learners also forget to equate the remaining fraction of the book to 100 pages and instead subtract 100 at an incorrect stage. Careful stepwise tracking of the fractions at each stage avoids these mistakes.

Final Answer:
The total number of pages in the book is best taken as 418 pages based on the given data and available options.