Question

The middle digit of a number between 100 and 1000 is zero and the sum of other digit is 13.if the digit. Are reversed,the no.so formed exceed the original no.by 495.find the no.?

Answer 1

$\underline{\underline{\huge{\textbf{Solution:}}}}$

Given,
$\underline{\textit{Statement-1:}}$
The middle digit of a number between 100 and 1000 is zero.

$\underline{\textit{Statement-2:}}$
Sum of the Digits of the Number is 13

$\underline{\textit{Statement-3:}}$
If the digits are reversed, the number so formed exceeds the original Number by 495.

$\underline{\large{\textsf{Step-1:}}}$
Rewrite Statement-1

The Required No. lies in between 100 and 1000
$\implies \sf \textsf{It is a 3-digit Number}$

Middle digit of a three digit Number is it's Ten's place Digit

$\implies \sf \textsf{Ten's digit of 3-digit Number= 0}$

$\underline{\large{\textsf{Step-2:}}}$
Assume the other digits of 3-digit No. as variable

Let Unit's digit of the 3- digit Number be x
Hundred's digit of the 3- digit Number be y
Ten's digit of the 3-digit Number = 0(Given)

$\underline{\large{\textsf{Step-3:}}}$
Rewrite Statement-2

$\implies \sf x+0+y= 13$

$\implies \sf x+y= 13$ ————[1]

$\underline{\large{\textsf{Step-4:}}}$
Express any of the digits, Unit's digit or Hundred's digit, in terms of other

$\implies \sf y= 13-x$

$\underline{\large{\textsf{Step-5:}}}$
Express the Number in terms of digits.

A Number is equal to sum of product of weight of the digit at each place and Face value at that place.

$\textsf{3-digit Number = 1000(Hundred's digit)+10(Ten's digit)+1(Unit's digit)}$

By Substituting,
$\implies \sf 3-digit Number = 1000(y)+10(0)+1(x)$

$\implies \sf 3-digit\: Number = 1000y+x$————[2]

$\underline{\large{\textsf{Step-6:}}}$
Find the Reversed 3-digit Number

If the digits of 3-digits number are reversed,
The Ten's digit remains same, but Unit's digit and Hundred's digit are interchanged.

Unit's Digit of Reversed 3-digit Number = Hundred's Digit of 3-digit Number

Ten's digit of Reversed 3-digit Number = 0

Reversed 3-digit Number = 1000x+0+y

$\implies \sf Reversed 3-digit\: Number = 1000y+x$ ————[3]

$\underline{\large{\textsf{Step-7:}}}$
Rewrite Statement-3

Reversed 3-digit No. = Original 3-digit No. + 495

$\underline{\large{\textsf{Step-8:}}}$
Substituting [2] & [3]

$\implies 100y+x = (100x+y)+495$

$\implies 99y-99x = 495$

$\implies 99(y-x) = 495$

$\implies y-x = 5$ ————[4]

$\underline{\large{\textsf{Step-9:}}}$
Solve equations [1] & [4] to obtain the value of
unknown variables

Do [1]+[4]
$\: \: x+y = 13$
$-x+y = 5$
———————
2y = 18
$\implies y = 9$

Substituting y in [1]

x + 9 = 13
x = 13 - 9
$\implies x = 4$

$\underline{\large{\textsf{Step-10:}}}$
Express the required Number using the variables.

$\therefore$
$\bigstar \textsf{Required 3-digit Number = \underline{\textbf{904}}}$