Question

The middle digit of a number between 100 and 1000 is zero and the sum of the other digit is 13. If the digits are reversed, the number so formed exceeds the original number by 495. Find the number.

Answer 1

Hint: We will first form the number by assuming the digit in ones, tens and hundredth place. Here, being the middle digit 0, no digit else than 0 will be there. So, we can write the number as: 100 y + x, where y is the digit at hundredth place and x is the digit at ones since the digit at tens place is 0. The reversed number will be 100 x + y. Then, we will solve both of the obtained equations for a relation between x and y using the condition that the reversed number exceeds the original number by 495. Also, we will use the initial condition that the sum of the digits at ones and hundredth place is 13.

Complete step-by-step answer:
We are given a number which is between 100 and 1000.
The middle digit is 0 and since it is a number between 100 and 1000 therefore, the digit at tens place is 0.
Let x be the digit at ones and y be the digit at hundredth place.
We can form the number as 100 y + x.
Now, we are said that the sum of digits other than tens (i.e., 0) is 13.
$ \Rightarrow $x + y = 13 equation (1)
If the digits are reversed, then the number formed will be 100 x + y.
We are told that the reversed number is greater than the original number by 495.
$ \Rightarrow $100 x + y – (100 y + x) = 495
$ \Rightarrow $100 x + y – 100 y – x = 495
$ \Rightarrow $99 x – 99 y = 495
Dividing both sides by 99, we get
$ \Rightarrow $x – y = 5
Or we can write this equation as: x = 5 + y equation (2)
Now using equation (1), we can write
$ \Rightarrow $5 + y + y = 13
$ \Rightarrow $ 2y = 8
$ \Rightarrow $y = 4
Using equation (2), we get
$ \Rightarrow $x = 5 + y
$ \Rightarrow $x = 5 + 4
$ \Rightarrow $x = 9
Therefore, the number will be 100 y + x $ \Rightarrow $100 (4) + 9
$ \Rightarrow $the original number = 409
Hence, the number is 409.


Note: In such questions, you may get confused in the formation of the number which lies between 100 and 1000. You may get wrong while deducing the relation from the word problem and because of that, you may not be able to reach the desired 3 – digit number. You can also solve this equation by putting the value from equation (1) into equation (2) and solve it for the values of x and y.