Correct Answer - Option 3 : 44.5
Given:
Average of twelve numbers = 39
Average of last five numbers = 35
Average of first four numbers = 40
Fifth number = sixth number - 6
Fifth number = seventh number + 5
Formula used:
Average = (Sum of values)/(Number of values)
Calculations:
Let twelve numbers be x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12.
x5 = x6 - 6 ----(i)
x5 = x7 + 5 ----(ii)
Average of first four numbers,
(x1 + x2 + x3 + x4)/4 = 40
⇒ (x1 + x2 + x3 + x4) = 160
Average of last five numbers,
(x8 + x9 + x10 + x11 + x12)/5 = 35
⇒ (x8 + x9 + x10 + x11 + x12) = 175
Average of twelve numbers,
(x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 + x12)/12 = 39
⇒ (x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 + x12) = 468
⇒ (160 + x5 + x6 + x7 + 175) = 468
⇒ x5 + x6 + x7 = 468 - 160 - 175
⇒ x5 + x6 + x7 = 133 ----(iii)
Putting value of x5 from equation (i),
⇒ x6 - 6 + x6 + x7 = 133
⇒ 2x6 + x7 = 139
⇒ x7 = 139 - 2x6 ----(iv)
Putting value of x5 from equation (ii),
x7 + 5 + x6 + x7 = 133
⇒ 2x7 + x6 = 128
⇒ 2(139 - 2x6) + x6 = 128
⇒ 278 - 4x6 + x6 = 128
⇒ 3x6 = 150
⇒ x6 = 50
Putting the value of x6 in equation (iv),
x7 = 139 - 2x6
⇒ x7 = 139 - 2(50)
⇒ x7 = 139 - 100
⇒ x7 = 39
Average of x6 and x7 = (50 + 39)/2
⇒ 89/2
⇒ 44.5
∴ The average of the sixth and seventh numbers is 44.5