Question

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

• A. $5$
• B. $8$
• C. $1$
• D. $4$

Answer 1

The correct option is A $5$

Let the 1st , 2nd , 3rd and 4th digits be, a, b, c and d respectively
Then
$a+b=c+d$ .....eq1
$a+d=c$ .........eq2
$b+d=2(a+c)$ ....eq3
From eq 1 and 2
$a+b=a+2d$
$b=2d$ .....eq4
Now use eq2 and eq4 in eq3
$2d+d=2(a+a+d)\Rightarrow 3d=2(2a+d)\Rightarrow d=4a$
or, $a=\frac{d}{4}$
Now , from eq2 $a+d=\frac{d}{4}+d=\frac{5d}{4}=c$
Or, $c=\frac{5}{4d}$
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 $5$.