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

Introduction / Context:
This question tests proportional reasoning with fractions applied in a real-life context of reading a book. The reader finishes fractional parts in two weeks, and a small known number of pages remains unread. You need to model these fractional readings algebraically and then estimate the total number of pages in the book.

Given Data / Assumptions:

• Let the total number of pages in the book be N.
• Karthik reads 6/13 of the book in the first week.
• In the second week, he reads 5/9 of the remaining pages.
• After the second week, around 100 pages are still unread.
• The question asks for an approximate total number of pages.

Concept / Approach:
We successively remove fractions of the total: first 6/13 of N, then 5/9 of what remains. The unread portion is the fraction that remains after both weeks. We set this remaining amount approximately equal to 100 and solve for N. Finally, we choose the option that gives a leftover closest to 100 pages.

Step-by-Step Solution:
Step 1: After week one, pages read = (6/13)N, so pages remaining = N - (6/13)N = (7/13)N. Step 2: In week two, Karthik reads 5/9 of the remaining pages, so pages read in week two = (5/9) × (7/13)N. Step 3: Remaining after week two = (4/9) × (7/13)N = (28/117)N. Step 4: Set this remaining amount close to 100 pages: (28/117)N ≈ 100. Step 5: Solve for N: N ≈ 100 × (117 / 28) ≈ 417.86 pages. Step 6: Among the options 404, 415, 418, 420, 432, the value 418 is closest to 417.86.

Verification / Alternative check:
For N = 418 pages, remaining pages after both weeks are: (28/117) × 418 ≈ 100.03 pages, which is essentially 100 pages when rounded to the nearest whole number. This confirms that 418 is the best approximate answer.

Why Other Options Are Wrong:
404: Gives remaining pages significantly below 100 (around 96.7 pages).
415: Leaves roughly 99.3 pages, slightly less accurate than 418 in matching 100 pages.
420: Leaves about 100.5 pages, a bit further away from 100 than 418 is.
432: Leaves more than 100 pages, not the closest to the target 100.

Common Pitfalls:
Many learners mistakenly apply the fraction 5/9 to the original total instead of the remaining portion after the first week. Others forget that the problem explicitly asks for an approximate value, so they may be uncomfortable with a non-integer intermediate result. Recognising that real-life quantities like book pages in word problems can sometimes be approximated is important in aptitude tests.

Final Answer:
The total number of pages in the book is approximately 418 pages.