Question

If all the 6 are replaced by 9, then the algebraic sum of all the numbers from 1 to 100 (both inclusive) varies by

• A. 330
• B. 350
• C. 300
• D. 100
• E.

Hint:

To find the algebraic sum after replacing digits, calculate the total change by multiplying the number of replacements with the change in value.

Answer 1

The Correct Option is A

Step 1: Compute the Original Sum The sum of the first 100 natural numbers is determined using the formula: \[ \text{Sum} = \frac{n(n + 1)}{2} \] For \( n = 100 \): \[ \text{Sum} = \frac{100 \times 101}{2} = 5050. \] Thus, the original sum is 5050. Step 2: Identify Numbers Containing the Digit '6' We need to find all numbers from 1 to 100 that contain the digit '6', as these will be modified when '6' is replaced by '9'. \begin{itemize} \item Single-digit number: 6 \item Two-digit numbers: \begin{itemize} \item Numbers with '6' in the tens place: 60, 61, 62, 63, 64, 65, 66, 67, 68, 69 \item Numbers with '6' in the units place: 16, 26, 36, 46, 56, 66, 76, 86, 96 \end{itemize} \end{itemize} Since 66 appears twice in this list, we must avoid counting it twice. Step 3: Compute the Increase for Each Number Replacing '6' with '9' results in the following increases: \begin{itemize} \item Single-digit number: \begin{itemize} \item 6 → 9: Increase by 3 \end{itemize} \item Numbers where the tens digit is 6: \begin{itemize} \item 60 → 90: +30 \item 61 → 91: +30 \item 62 → 92: +30 \item 63 → 93: +30 \item 64 → 94: +30 \item 65 → 95: +30 \item 66 → 99: +33 (both digits change) \item 67 → 97: +30 \item 68 → 98: +30 \item 69 → 99: +30 \end{itemize} \item Numbers where the units digit is 6 (excluding 66, already counted): \begin{itemize} \item 16 → 19: +3 \item 26 → 29: +3 \item 36 → 39: +3 \item 46 → 49: +3 \item 56 → 59: +3 \item 76 → 79: +3 \item 86 → 89: +3 \item 96 → 99: +3 \end{itemize} \end{itemize} Step 4: Calculate the Total Increase Summing up the increases: \begin{itemize} \item Single-digit contribution: \( 3 \) \item Numbers where the tens digit is 6: \[ (30 \times 9) + 33 = 270 + 33 = 303 \] \item Numbers where the units digit is 6 (excluding 66): \[ 3 \times 8 = 24 \] \end{itemize} Total Increase: \[ 3 + 303 + 24 = 330. \] Step 5: Verify with Given Options The total increase is 330. Checking the options: \begin{enumerate}[label=(\Alph*)] \item 330 \item 350 \item 300 \item 100 \end{enumerate} The correct answer is \(\boxed{A}\).