Question

In a four-digit number, the sum of the first 2 digits is equal to that of the last 2 digits. The sum of the first and last digits is equal to the third digit. Finally, the sum of the second and fourth digits is twice the sum of the other 2 digits. What is the third digit of the number?

Answer 1

Let the 1st, 2nd, 3rd and 4th digits be a, b, b, c and d respectively.

Then,

a+b = c+d;

a+d = c;

b+d = 2(a+c);

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 = d/4; → Now, from eqn. (ii), a+d = (d/4)+d = 5d/4 = c

Or, c = 5/4d.

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.

Answer 2

Answer:

5

Step-by-step explanation:

Let the 1st, 2nd, 3rd and 4th digits be a,b,c and d respectively.

So, four-digit no. will be dcba

→   According to the given question,

⇒  a+b=c+d        --- ( 1 )

⇒  a+d=c            --- ( 2 )

⇒  b+d=2(a+c)     ---- ( 3 )

∴   a+b=a+2d         [ Substituting value of equation ( 2 ) in equation ( 1 ) ]

∴   b=2d    --- ( 4 )

⇒  2d+d=2(a+a+d)        [ Substituting eq ( 4 ) and ( 2 ) in eq. ( 3 ) ]

∴   d=4a or a=

4

d

​

      --- ( 5 )

⇒  

4

d

​

+d=c   ---  [Substituting ( 5 ) in ( 2 ) ]

∴   c=

4

5d

​

 or c=

4d

5

​

Now d=4a

Since a and d are single digits and we have to form 4 digit no. a<10 and 0<d<10

So, the possible integer values of a and d which satisfy d=4a are

1.a=1,d=4

2.a=2,d=8

∴   The value of d can be either 4 or 8

⇒  When d=4, then c=5.

⇒  When d=8, then c=10.

⇒  But the value of c should be less than 10 as it is a single digit.

∴  Value of c would be 5 which is the third digit of the required number.