In a 4-digit number, the sum of the first two digits is equal to that of last two digits. The sum of the first and last digits is equal to third digit. Finally, the sum of the second and fourth digits is twice the sum of the other two digits. What is the third digit of the number?

Solution(By Examveda Team)

Let the 1^st, 2^nd, 3^rd and 4^th digits be a, b, c and d respectively.
Then,
a + b = c + d ----------(i)
a + d = c ----------(ii )
b + d = 2(a + c) ----------(iii)
from eqn. (i) and (ii),
a + b = a + 2d
→b = 2d
and eqn (iii);
2d + d = 2(a + a + d)
→ 3d = 2(2a + d) → d = 4a
or, a = $$\frac{{\text{d}}}{4};$$
→ Now, from eqn. (ii),
a + d = $$\frac{{\text{d}}}{4}$$ + d = $$\frac{{{\text{5d}}}}{4}$$ = c
Or, c = $$\frac{5}{{4{\text{d}}}}$$
The value of d can be either 4 or 8.
If d = 4, then c = 5
If d = 8, then c = 10
But the value of c should be less than 10
Hence, value of c would be 5