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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 digits is equal to the 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?


A
5
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B
8
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C
1
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D
4
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Solution

The correct option is A $$5$$
$$\Rightarrow$$  Let the 1st, 2nd, 3rd and 4th digits be $$a,\, b,\, c$$ and $$d$$ respectively.
So, four-digit no. will be $$dcba$$

$$\rightarrow$$   According to the given question,
$$\Rightarrow$$  $$a+b = c+d$$        --- ( 1 )
$$\Rightarrow$$  $$a+d = c$$            --- ( 2 )
$$\Rightarrow$$  $$b+d = 2(a+c)$$     ---- ( 3 )

$$\therefore$$   $$a+b=a+2d$$         [ Substituting value of equation ( 2 ) in equation ( 1 ) ]
$$\therefore$$   $$b=2d$$    --- ( 4 )

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

$$\therefore$$   $$d=4a$$ or $$a=\dfrac{d}{4}$$      --- ( 5 )

$$\Rightarrow$$  $$\dfrac{d}{4}+d=c$$   ---  [Substituting ( 5 ) in ( 2 ) ]

$$\therefore$$   $$c=\dfrac{5d}{4}$$ or $$c=\dfrac{5}{4d}$$

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
$$1. a=1, d = 4$$
$$2. a=2, d= 8$$

$$\therefore$$   The value of $$d$$ can be either $$4$$ or $$8$$

$$\Rightarrow$$  When $$d=4$$, then $$c=5$$.
$$\Rightarrow$$  When $$d=8$$, then $$c=10.$$

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

$$\therefore$$  Value of $$c$$ would be $$5$$ which is the third digit of the required number.

Mathematics

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