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

Answer 1

The Correct Option is A

Step 1: Understand the Original Sum

The sum of all numbers from 1 to 100 can be calculated using the formula for the sum of the first n natural numbers:

\[ \text{Sum} = \frac{n(n+1)}{2} \]

For \(n = 100\):

\[ \text{Sum} = \frac{100 \times 101}{2} = 5050 \]

Step 2: Identify Where the Digit '6' Appears

We need to find all numbers between 1 and 100 that contain the digit '6'. These numbers will change when '6' is replaced by '9'.

• Single-digit numbers: 6
• Two-digit numbers:
  • Numbers where the tens digit is 6: 60, 61, 62, 63, 64, 65, 66, 67, 68, 69
  • Numbers where the units digit is 6: 16, 26, 36, 46, 56, 66, 76, 86, 96

Note that 66 appears in both lists, so we must be careful not to double-count it.

Step 3: Calculate the Change for Each Number

For each number containing '6', we calculate how much the number increases when '6' is replaced by '9'.

• Single-digit number: 6 → 9: Increase by 3
• Two-digit numbers with tens digit 6:
  • 60 → 90: Increase by 30
  • 61 → 91: Increase by 30
  • 62 → 92: Increase by 30
  • 63 → 93: Increase by 30
  • 64 → 94: Increase by 30
  • 65 → 95: Increase by 30
  • 66 → 99: Increase by 33 (since both digits change)
  • 67 → 97: Increase by 30
  • 68 → 98: Increase by 30
  • 69 → 99: Increase by 30
• Two-digit numbers with units digit 6 (excluding 66, already counted):
  • 16 → 19: Increase by 3
  • 26 → 29: Increase by 3
  • 36 → 39: Increase by 3
  • 46 → 49: Increase by 3
  • 56 → 59: Increase by 3
  • 76 → 79: Increase by 3
  • 86 → 89: Increase by 3
  • 96 → 99: Increase by 3

Step 4: Calculate the Total Increase

Now, we sum up all the increases:

• Single-digit: 6 → 9: +3
• Two-digit numbers with tens digit 6: Total for this group: \( 30 \times 9 + 33 = 270 + 33 = 303 \)
• Two-digit numbers with units digit 6 (excluding 66): Total for this group: \( 3 \times 8 = 24 \)

Overall Total Increase: \( 3 + 303 + 24 = 330 \)

Step 5: Match with the Given Options

The total increase in the sum is 330. Looking at the options:

• (A) 330
• (B) 350
• (C) 300
• (D) 100

The correct answer is A.