Question

In a four-digit number, the sum of the first two digits is equal to that of the last two digits. The sum of the first and last is equal to the third digit. Finally, the sum of the second and fourth digit is twice the sum of the other two digits. What is the third digit of the number?
A) 5
B) 8
C) 1
D) 4

Answer 1

Hint: In this question, we have to find the third digit of a number which is a four-digit number by solving the differential equation formed. We can also do it by filling the option given in the equation.

Complete step by step solution: Let us take a four-digit number. Suppose we take dcba as a four-digit number. So, a, b, c, d are in first, second, third and fourth place digit respectively.
Now as given in the question
Firstly, the sum of first two-digit equal to last two digits and the equation formed
\[ \Rightarrow a + b = c + d\] ………… (1)
Secondly, the sum of the first and last is equal to the third digit and the equation formed is
\[ \Rightarrow a + d = c\] ………….(2)
Finally, the sum of the second and last digit is twice the sum of the first and third digit and the equation formed is
\[ \Rightarrow b + d = 2(a + c)\] ………… (3)
Now if we put option A i.e. value of c is 5 than check if this value satisfies all equations or not
Put \[c = 5\] in equation 2 and equation 1, we get
 \[ \Rightarrow a + d = 5\] & \[a + b = 5 + d\]…. (4)
check all the possibility \[a = 1,d = 4;a = 2,d = 3;a = 3,d = 2;a = 4,d = 1\]
\[d \ne 0\] as d is fourth digit place if it is zero than number became 3-digit number.
Only, \[a = 1\] and \[d = 4\] satisfy the equation 4 and 3. So, the number became 4581.
If we put option B, C and D than they don’t satisfy equation 1, 2 and 3.

The correct option is A.

Note: we can also use the conventional method by solving each equation individually and then find the answer but you can get confused in solving them. Please read the statement carefully while making equation. Moreover, some students write equation 3 as \[2(b + d) = a + c\] which is a wrong method.