UGC-NET-Paper1

Let the four consecutive odd numbers be:
a = n (first odd number), b = n + 2, c = n + 4, d = n + 6.

The sum L of these numbers is:
L = a + b + c + d = n + (n + 2) + (n + 4) + (n + 6) = 4n + 12.

The average M of these numbers is:
M = L / 4 = (4n + 12) / 4 = n + 3.

Let the three consecutive even numbers be:
x = m (first even number), y = m + 2, z = m + 4.

The sum P of these numbers is:
P = x + y + z = m + (m + 2) + (m + 4) = 3m + 6.

The average Q of these numbers is:
Q = P / 3 = (3m + 6) / 3 = m + 2.

We have two equations based on the problem statement:

M = Q - 6
P = L - 16
Substituting the expressions for M and Q into the first equation:
n + 3 = (m + 2) - 6.
n + 3 = m - 4.
n = m - 7 (Equation 1).

Substitute in the second equation Now, substitute L and P into the second equation:
3m + 6 = (4n + 12) - 16.
3m + 6 = 4n - 4.
3m + 10 = 4n (Equation 2).

Substituting n = m - 7 from Equation 1 into Equation 2:
3m + 10 = 4(m - 7).
3m + 10 = 4m - 28.
10 + 28 = 4m - 3m.
Thus, 38 = m.

m = 38 back into Equation 1 to find n:
n = 38 - 7 = 31.
M = n + 3 = 31 + 3 = 34.