Question

The average of 11 numbers is 10.9. If the average of the first six numbers is 10.5 and that of the last six numbers is 11.4, then the middle number is :

• A. 11.5
• B. 11.4
• C. 11.3
• D. 11.0

Answer 1

The Correct Option is A

The problem involves calculating the middle number from a set of 11 numbers when given certain average values. We'll solve it step-by-step to find the middle number.

• The average of all 11 numbers is 10.9. Therefore, the sum of these 11 numbers is:
  Sum₁₁ = 11 × 10.9 = 119.9
• The average of the first six numbers is 10.5, hence their sum is:
  Sum_{first 6} = 6 × 10.5 = 63
• The average of the last six numbers is 11.4, so their sum is:
  Sum_{last 6} = 6 × 11.4 = 68.4
• Notice that the "first six numbers" and the "last six numbers" overlap by one number, which is the middle number, let's denote it by 'x'.
• The sum of all 11 numbers can be expressed with the formula:
  Sum_{first 5} + x + Sum_{last 5} = Sum₁₁
• Sum_{first 6} + Sum_{last 6} - x = Sum₁₁
  63 + 68.4 - x = 119.9
• Combine and solve the equation:
  131.4 - x = 119.9
  x = 131.4 - 119.9
  x = 11.5

Therefore, the middle number is: 11.5