Question

If the average of 35 numbers is 22, the average of the first 17 numbers is 19, and the average of the last 17 numbers is 20, then the 18 number is:

• A. 133
• B. 107
• C. 132
• D. 108

Answer 1

Answer:

B. 107

• Step 1: Calculate the Total Sum of All Numbers

  • Given that the average of 35 numbers is 22.

  • Using the formula, the Total Sum = Average × Number of Quantities = 22 × 35 = 770.

• Step 2: Calculate the Sum of the First 17 Numbers

  • The average of the first 17 numbers is given as 19.

  • The Sum of First 17 Numbers = Average × Number of Quantities = 19 × 17 = 323.

• Step 3: Calculate the Sum of the Last 17 Numbers

  • The average of the last 17 numbers is given as 20.

  • The Sum of Last 17 Numbers = Average × Number of Quantities = 20 × 17 = 340.

• Step 4: Identify the Overlap/Missing Term Logic

  • Notice that the total number of items is 35. When we consider the first 17 numbers and the last 17 numbers, we are essentially looking at 17 + 17 = 34 numbers.

  • This implies that one number from the original set of 35 is not included in this combined group of 34. This 'missing' number is precisely the 18th number in the sequence.

• Step 5: Determine the Value of the 18th Number

  • The sum of the first 17 numbers and the last 17 numbers combined represents the sum of 34 distinct numbers from the total set of 35.

  • Combined Sum of (First 17 + Last 17) = 323 + 340 = 663.

  • The 18th number is obtained by subtracting this combined sum from the total sum of all 35 numbers.

  • 18th Number = (Total Sum of 35 Numbers) - (Combined Sum of First 17 and Last 17 Numbers)

  • 18th Number = 770 - 663 = 107.