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?

a)
5

b)
8

c)
1

d)
4

Correct answer is option 'A'. Can you explain this answer?

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?

a)
5
b)
8
c)
1
d)
4

Correct answer is option 'A'. Can you explain this answer?

Answer

Let the 4 digit no. be xyzw.

According to given conditions we have x + y = z + w, x + w = z, y + w = 2x + 2z.

With help of these equations, we deduce that y = 2w, z = 5x.

Now the minimum value x can take is 1 so z = 5 and the no. is 1854, which satisfies all the conditions. Hence option A.

Answer

Solution:
Let the four-digit number be represented as ABCD, where A, B, C, D are the digits in thousandth, hundredth, tenth and unit place respectively.

Given conditions are:
1. A + B = C + D
2. A + D = C
3. B + D = 2(C + A)

From equation 2, we can write A = C - D
Substituting the value of A in equation 1, we get B = 2D - C

Substituting the values of A and B in equation 3, we get D - C = 2C - 2D + 2(C - D)
Simplifying the above equation, we get 5D = 3C

Since the digits are integers, D must be a multiple of 3 and C must be a multiple of 5.
The only possible values for C are 5 and 0, but C cannot be 0 as it is the first digit of the number.
Therefore, C = 5 and D = 3.

Now, substituting the values of C and D in equations 2 and 1 respectively, we get:
A = C - D = 2
B = 2D - C = 1

Hence, the four-digit number is 2153 and the third digit is 5.

Therefore, the correct answer is option A.

Answer

Solution:
Let the four-digit number be ABCD where A, B, C, and D represent digits in the thousands, hundreds, tens, and ones places, respectively.

Given conditions:
1. A+B = C+D
2. A+D = C
3. B+D = 2(C+A)

Using equation 2, we can express A in terms of C and D:
A = C-D

Substituting this value of A in equation 1, we get:
C+D = C-D+B
2D = B+D
B = D

So, the number can be written as ABAB or CDCD.

Now using equation 3, we can express B in terms of C and D:
B = 2(C+A)-D

Substituting the value of B = D, we get:
D = 2(C+A)-D
2D = 2(C+A)
D = C+A

So, the number can be written as ACDA.

Now using equation 2, we get:
A+D = C
Substituting A = C-D, we get:
C-D+D = C
D = C/2

As D is a digit, C must be even and the only possible value is 4.

Therefore, the third digit of the number is 4.

Hence, the correct answer is option (d) 4.

Answer

Solution:
Let the four-digit number be represented as ABCD,
where A, B, C and D are the digits in the thousandth, hundredth, tenth and unit places respectively.

Given conditions are:

1. A + B = C + D
2. A + D = C
3. B + D = 2(C + A)

Using the first condition, we get:
A - C = D - B

Using the second condition, we get:
D = A + C

Substituting the values of D and A - C in the above equation, we get:
A - C = A + C - B
2C = B

Using the third condition, we get:
B + D = 2(C + A)
Substituting the values of B and D, we get:
2C + A + C = 2(C + A)
3C - A = 0
3C = A

Therefore, we have:
A = 3C
B = 2C
D = 4C

Since the number is a four-digit number, C cannot be 0.
So, C can take values from 1 to 9.

We can now substitute the values of A, B and D in the first condition and simplify it to get:
4C = C + D
3C = D

Thus, the third digit of the number is D = 3C.

Checking the options, we see that only option (A) 5 satisfies the condition as 3C = 15. Therefore, the answer is option (A) 5.

Answer

⇒ 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= 4d --- ( 5 )

⇒ 4d+d=c --- [Substituting ( 5 ) in ( 2 ) ]

c= 4d5

Now d=4a
Since a and d are single digits and we have to form 4 digit no. a<10 and=''>< />< />

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.

Answer

Let the 4 digit no. be xyzw.

According to given conditions we have x + y = z + w, x + w = z, y + w = 2x + 2z.

With help of these equations, we deduce that y = 2w, z = 5x.

Now the minimum value x can take is 1 so z = 5
and the number is 1854 .

Hence option A is the correct one, it satisfies all the conditions given .

Answer

Let the 4 digit no. be xyzw.
According to given conditions we have x + y = z + w, x + w = z, y + w = 2x + 2z.
With help of these equations, we deduce that y = 2w, z = 5x.
Now the minimum value x can take is 1 so z = 5 and the no. is 1854, which satisfies all the conditions. Hence option A.

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