Eight consecutive numbers are given. If the average of the two number that appear in the middle is 6 then the sum of the eight given numbers is

Average of 5 boys = 16 yr
sum of age of 5 boys = 16 x 5 = 80 yr
? Average age of 4 boys = 16 - yr 3 month
= 16 yr + 3/12 yr = 16 yr + 1/4 yr
Sum of ages of 4 boys = 4 x (16 +1/4) yr
= 64 +1 = 65 yr
? Age of 5th boy = (80 - 65) yr = 15 yr

Average of 8 numbers = 14
Sum of 8 numbers = 14 x 8 = 112
Similarly, average of 6 numbers = 16
Sum of 6 number = 16 x 6 = 96
? Sum of remaining two number
= Sum of 8 number - sum of 6 numbers
= 112- 96 = 16
Average = (sum of two number)/2 = 16/2 = 8

As per the formula,
Average of the cubes os 1st 'n' natural numbers= n(n+1)²/4
where, n= 5.

? Required average = 5(5+1)²/4
= (5 x 36)/4 = 5 x 9= 45

Even numbers between 11 to 63
= 12,14,16,18,20,..........,62.
Clearly, this is a series of consecutive even number

According to the formula
Average of consecutive even numbers = (First number + Last number)/2
= (12 + 62)/2
= 74/2
= 37

Let a, b, c are the ages of A, B and C respectively

a + b = 2 X 20 = 40 ...(1)
b + c = 2 x 19 = 38 ...(2)
c + a = 2 x 21 = 42 ...(3)

Adding all the 3 equalities
2a + 2b + 2c = 120
a + b + c = 60 --- (4)

From eq. (2) and (4)
a = 22
From eq. (1) and (4)
b = 18
From eq. (4)
c= 20

? Age of A = 22 years
Age of B = 18 years
Age of C = 20 years

Let the four consecutive even number are x, ( x+2), (x+4), (x+6).
Clearly this is a series of consecutive even numbers
According to the formula,
Average of consecutive numbers = (first number + last number)/ 2
? 27 = [x + (x + 6)]/2
? x + 3 = 27
? x =24

? Greatest number = 4th consecutive even number = (x + 6) = 24 + 6 = 30

Total cost of 30 TV sets = ? 300000
Total cost of 15 TV sets = 9000 x 15 = 135000
Total cost of remaining 15 TV sets = 300000 - 135000 = 165000

? Average cost of remaining TV sets = 165000/15 = ? 11000

Total age of 30 girls = 30 x 13 = 390 yr
Total age of 18 girls = 18 x 15 = 270 yr.
? Age of remaining 12 girls = 390 - 270 = 120 yr
? Required average = 120/12 = 10 yr

Let the numbers A, B, C, and D are N , N+1, N + 2 and N + 3, respectively.
According to the question,
[N + (N+1) + (N+2) + (N+3)] / 4 = 49.5
? 4N + 6 = 4 x 49.5
? 4N = 198 - 6 ? N = 192/4
? N = 48

The required product = B x D
= (N + 1) x (N + 3)
= (48 +1) (48 +3 )
= 49 x 51
= 2499

Overall height = (n₁a₁ + n₂a₂) / (n₁ + n₂)
n₁= 20,
a₁ = 5 ft 11 inches = 5 x 12 + 11 = 71 inches
n₂= 18,
a₂ = 6 ft 2 inches = 6 x 12 + 2 = 74 inches

? Overall height = [(20) (71) + (18) (74)] / (20+18)
= (1420 + 1332) / 38 = 72.42 inches