Question

Out of 7 numbers the average of first 4 is 4 and the last 4 is also 4 and the average of all 7 is 3 find the 4th number ?

• A. 8
• B. 11
• C. 7
• D. 10

Answer 1

Answer:

B. 11

Deconstructing the Problem: Overlapping Averages

• This type of problem involves overlapping sets, where one or more numbers are common to multiple groups whose averages are given. Identifying this overlap is crucial for solving such problems quickly in exams.

• Let the 7 numbers be N1, N2, N3, N4, N5, N6, N7.

Step 1: Calculate the Sum of the First Four Numbers

• Given: Average of first 4 numbers = 4

• Number of quantities = 4

• Sum of first 4 numbers (N1 + N2 + N3 + N4) = Average × Number of Quantities = 4 × 4 = 16.

Step 2: Calculate the Sum of the Last Four Numbers

• Given: Average of last 4 numbers = 4

• Number of quantities = 4

• Sum of last 4 numbers (N4 + N5 + N6 + N7) = Average × Number of Quantities = 4 × 4 = 16.

Step 3: Calculate the Sum of All Seven Numbers

• Given: Average of all 7 numbers = 3

• Number of quantities = 7

• Sum of all 7 numbers (N1 + N2 + N3 + N4 + N5 + N6 + N7) = Average × Number of Quantities = 3 × 7 = 21.

Step 4: Identify the Overlap and Formulate the Equation

• When you add the sum of the first 4 numbers and the sum of the last 4 numbers, the 4th number (N4) is included twice. This is the key insight for solving overlapping average problems.

• The total of the two partial sums will exceed the overall sum of all numbers by the value of the overlapping element.

• Therefore, (Sum of First 4) + (Sum of Last 4) = (Sum of All 7) + (The Overlapping Number)

Step 5: Solve for the Overlapping Number (4th number)

• Substitute the calculated sums into the equation:

• 16 (from Step 1) + 16 (from Step 2) = 21 (from Step 3) + N4

• 32 = 21 + N4

• N4 = 32 - 21 = 11