Question

The average of the squares of four consecutive even natural numbers is 126. The average of 8 times of the greatest number and 5 times of the smallest number is:
1. 68
2. 76
3. 74
4. 66

Answer 1

Correct Answer - Option 2 : 76

Formula:

Average = (Sum of elements)/(Number of elements)

Calculation:

Let four consecutive even natural numbers be a, a + 2, a + 4 and a + 6 respectively.

Total sum of squares of four consecutive even natural numbers = 126 × 4 = 504

⇒ a² + (a + 2)² + (a + 4)² + (a + 6)² = 504

⇒ 4a² + 24a - 448 = 0

⇒ a² + 6a - 112 = 0

⇒ a² + 14a - 8a - 112 = 0

⇒ a(a + 14) - 8(a + 14) = 0

⇒ (a + 14)(a - 8) = 0

a = -14 or a = 8

a is a even natural number so a = 8

Greatest number = 8 + 6 = 14

Smallest number = 8

Then,

8 times of greatest number = 14 × 8 = 112

5 times of smallest number = 8 × 5 = 40

∴ average of 8 times of the greatest number and 5 times of the smallest number = (112 + 40)/2 = 76