There are a certain number of pages in a book. Arjun tore a certain page out of the book and later found that the average of the remaining page numbers is `46(10)/(13)` . Which of the following were the page number of the page that Arjun had torn ?

Options

A. 57 and 58
B. 59 and 60
C. 45 and 46
D. 47 and 48

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the Problem
Arjun tore out a certain page from a book, and the average of the remaining page numbers is given as \(46 \frac{10}{13}\). We need to find out which page number he tore out.

### Step 2: Convert the Average to an Improper Fraction
First, convert the mixed number \(46 \frac{10}{13}\) into an improper fraction:
\[ 46 \frac{10}{13} = \frac{46 \times 13 + 10}{13} = \frac{598 + 10}{13} = \frac{608}{13} \]

### Step 3: Set Up the Equation for the Average
Let \(N\) be the total number of pages in the book. Since Arjun tore out one page, the number of remaining pages is \(N - 1\). The average of the remaining pages can be expressed as:
\[ \text{Average} = \frac{\text{Sum of remaining pages}}{\text{Number of remaining pages}} = \frac{S}{N - 1} \]
Where \(S\) is the sum of the remaining page numbers.

### Step 4: Calculate the Total Sum of Pages
The sum of the first \(N\) pages is given by the formula:
\[ S = \frac{N(N + 1)}{2} \]
After tearing out one page, the sum of the remaining pages becomes:
\[ S - m \]
Where \(m\) is the page number that was torn out.

### Step 5: Set Up the Average Equation
From the average we have:
\[ \frac{S - m}{N - 1} = \frac{608}{13} \]
Cross-multiplying gives:
\[ 13(S - m) = 608(N - 1) \]

### Step 6: Substitute for \(S\)
Substituting \(S = \frac{N(N + 1)}{2}\) into the equation:
\[ 13\left(\frac{N(N + 1)}{2} - m\right) = 608(N - 1) \]
This simplifies to:
\[ \frac{13N(N + 1)}{2} - 13m = 608N - 608 \]

### Step 7: Rearranging the Equation
Rearranging gives:
\[ 13m = \frac{13N(N + 1)}{2} - 608N + 608 \]
\[ m = \frac{\frac{13N(N + 1)}{2} - 608N + 608}{13} \]

### Step 8: Finding Possible Values for \(N\)
To find \(N\), we need \(N - 1\) to be a multiple of 13. Testing values close to the average gives us \(N = 93\) (as it is close to \(91\) which is a multiple of \(13\)).

### Step 9: Calculate \(m\)
Substituting \(N = 93\):
\[ S = \frac{93 \times 94}{2} = 4371 \]
Now substituting back into the equation:
\[ m = \frac{4371 - 608(N - 1)}{13} \]
Calculating \(N - 1 = 92\):
\[ m = \frac{4371 - 608 \times 92}{13} \]
Calculating gives:
\[ m = \frac{4371 - 55856}{13} = \frac{115}{13} = 57 \]
Thus, the two pages that were torn out are \(57\) and \(58\).

### Step 10: Conclusion
The page numbers that Arjun tore out are \(57\) and \(58\).

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