Question

A notebook is made up of sheets folded in the middle and stapled. Each sheet forms two leaves i.e., four pages. On removing some papers of the first-half and second-half of the book, Joe found the number of the leaves in the first case as odd and in the second-case as even. If the sum of the numbers of the pages on the last leaf of the book is $ 63, $ then what could be the maximum possible sum of the numbers on the pages of leaves that were left in the book?

Answer 1

Hint: Read the given word statements twice and frame the resultant mathematical expressions and from it find the correlation between the given data and the required data and simplify for the resultant value.

Complete step-by-step answer:
Since, we are given that the last two pages sum up to $ 63 $ , and therefore the last two pages will be $ 31 $ and $ 32 $ .
So, the pages of the book in the first half are $ 1 - 16 $ and the pages in the second half are $ 17 - 32 $ .
Now,
Remove leaves with the odd number is the first half being the number one the smallest odd number and the number two being the smallest even number.
Therefore, to minimize the sum of the left pages we should remove page number $ 1 $ and $ 2 $ also, remove pages from the second half $ 17,18,19 $ and $ 20 $
Now, total number of pages subtract the removed pages
 $ = (1 + 32)(16) - (1 + 2 + 17 + 18 + 19 + 20) $
Simplify
 $
   = 528 - 77 \\
   = 451 \;
  $
Hence, the required answer is $ 451 $ maximum sum of the numbers of the pages.
So, the correct answer is “ $ 451 $ ”.

Note: Be good in the difference between the odd and the even numbers. Get the first half and second half of the pages correctly identified since the answer depends on it only. When the number divided by two gives zero as the remainder then that number is even and when any number divided by two gives remainder as the number one then that number is odd.