Question

A number consists of three digits, the right hand being zero. If the left hand and the middle digits are interchanged the number is diminished by 180. If the left hand digit is halved and the middle and right hand digits are interchanged the number is diminished by 454. Find the number.

Answer 1

Hint: Let the three-digit number be \[100x + 10y + 0\] , then, by first condition which is ; if x and y are interchanged i.e. \[100y + 10x\], the number is reduced by 180.
  \[\therefore \left( {100x + 10y} \right) - \left( {100y + 10x} \right) = 180\] \[ \Rightarrow x - y = 2\]
And by the second condition which is , if x is halved i.e. 50x and y& z are interchanged i.e. \[10 \times 0 + y\left( {\because z = 0} \right)\] , the number gets reduced by 454.
 \[\therefore \left( {100x + 10y} \right) - \left( {50x + y} \right) = 454\] \[ \Rightarrow 50x + 9y = 454\]
Solving the above two equations will give the answer.

Complete step-by-step answer:
Lets us consider a three-digit number xyz, where x, y and z are some digits.
It is given that; right hand side digit is 0 i.e. digit at unit place is 0.
 \[\therefore z = 0\]
We can also write the number xyz as \[100x + 10y + z\] , but \[z = 0\] , so \[100x + 10y + 0\] is the number.
It is given that; if x and y are interchanged i.e. \[100y + 10x\], the number is reduced by 180.
\[\therefore \]\[\left( {100x + 10y} \right) - \left( {100y + 10x} \right) = 180\]
 \[
  \therefore 90x - 90y = 180 \\
  \therefore 90\left( {x - y} \right) = 180 \\
  \therefore x - y = \dfrac{{180}}{{90}} \\
 \]
 \[\therefore x - y = 2\] … (1)
Now, if x is halved i.e. 50x and y & z are interchanged i.e. \[10 \times 0 + y\left( {\because z = 0} \right)\] , the number gets reduced by 454.
 \[\therefore \left( {100x + 10y} \right) - \left( {50x + y} \right) = 454\]
 \[\therefore 50x + 9y = 454\] … (2)
From equation (1) we get
 \[x = y + 2\]
Now, substitute \[x = y + 2\] in equation (2)
 \[\therefore 50\left( {y + 2} \right) + 9y = 454\]
 \[
  \therefore 50y + 100 + 9y = 454 \\
  \therefore 59y = 454 - 100 \\
  \therefore 59y = 354 \\
  \therefore y = \dfrac{{354}}{{59}} \\
  \therefore y = 6 \\
 \]
Now, substitute \[y = 6\] in equation (1) to get the value of x.
 \[
  \therefore x - 6 = 2 \\
  \therefore x = 2 + 6 \\
  \therefore x = 8 \\
 \]
We got \[x = 8\] , \[y = 6\] and \[z = 0\] .
So, the number is \[\left( {100 \times 8} \right) + \left( {10 \times 6} \right) + 0 = 800 + 60 + 0 = 860\] .

Note: Any number, for example 357, can also be written as \[\left( {100 \times 3} \right) + \left( {10 \times 5} \right) + 7\] . The numbers in ascending order give you the smallest number, and the numbers in reverse order. i.e. descending order is your biggest number.