Question

There is a certain number consisting of 3 digits which is equal to 25 times the sum of the digits, and if 198 be added to the number, the digits will reversed, also the sum of the extreme digits exceeds the middle digit by unity. Find the number

• A. 753
• B. 345
• C. 375
• D. 573

Answer 1

The correct option is C 375

Let the numbers in the 100's place be x, 10's place be y and1's place be z.

H T O
x y z
So,
$100x+10y+z=25(x+y+z)$
As, the number is 25 times the sum of the digits of the number.

$100x+10y+z=25x+25y+25z$
$75x-15y-24z=0$
$25x-5y-8z=0....\left(1\right)$

Again,

100x + 10y + z + 198 = 100z + 10y + x

As, the number will reverse if 198 is added to it
$99x-997=-198$
$x-z=-\mathrm{2.............}\left(2\right)$

Now,
$x+z=y+1$
As, the sum of extreme two digits is one more than the middle term
$x-y+z=1.......\left(3\right)$

$\Rightarrow eqn\left(1\right)-5\times eqn\left(3\right)$

$25x-5y-8z=0$
$-5x+5y-5z=-5$
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$20x-13z=-\mathrm{5.....................}\left(4\right)$

$\Rightarrow eqn\left(2\right)\times 13-eqn\left(4\right)$

$13x-13z=-26$
$-20x+13z=+5$
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$-7x=-21$

$x=\mathrm{3.....................}\left(5\right)$

Now, substituting eqn(5) in eqn(2), we get,
$x-z=-2$
$3-z=-2$
$z=\mathrm{5.....................}\left(6\right)$

Again, substituting eqn(5) and eqn(6) and z in eqn(3), we get

$x-y+z=1$
$3-y+5=1$
$-y=-7$
$y=7$

Hence the number is 375.

Check: 375 + 198 = 573 (digits are reversed)