Question

Sum of the digits of a two digit number is 9. If 17 is subtracted from the number, the digits in the number are equal.Find the number.​

Answer 1

one digit-9

let other digit-x

17-x is equal to 9

x is equal to 9+17

x is equal to 26

Answer 2

Since it is a 2 digit number.

Suppose the digit at one’s place is ‘y’ and the digit at ten’s place is ‘x’.

Then the number will be equal to ten times the digit at ten’s place plus the digit at one’s place. i.e ( 10* x) + y

For example: If the number is 15, the digit at its one’s place is 5 and the digit at it’s ten’s place is 1. Then we can write the number as (10*1)+5 = 15

Similarly for digit at one’s place as ‘y’ and the digit at ten’s place as ‘x’ , the number will be (10*x)+y

Now, from question the sum of the digit of the number is 17. Thus,

x + y = 17 —— (i)

Also, it is given in the question that if we subtract 12 from our number, both the digit will become same.

Now here comes the twist part.

Since the sum of digit is 17 ,the one’s digit will must be either 8 or 9. So

If we are subtracting 12 i.e (10*1 +2) from this number , that means the digit at its ten’s place will be reduce by 1. How ?

We will see it by taking some examples:

Suppose we are subtracting 13 i.e (10 *1 + 3) from 58 then we get 58–13=45 , we can see 58 has ten’s place digit as 5 and after subtracting 13 from it , we get 45 which has 4 at its ten’s place. Ten’s place digit reduced by 1.

Now suppose we are subtracting 25 i.e (10*2 +5) from 69 then we get 69–25= 44. So here the digit at ten’s place is reduced by 2 because we subtracted 25 i.e (10*2 +5) .

Now suppose we are subtracting 34 i.e (10*3 +4) from 78 then we get 78–34 = 44. So here the ten’s place digit is reduced by 3 because we subtracted 34 i.e (10*3 +4) .

As you can see, In examples also we are taking the one’s place digit as either 8 or 9.

Now coming back to question, when we subtract 12 from our original number the digit at both ten’s and one’s place of new number become same ( according to question) , suppose they become ‘z’.

Then our new number will be (10*z + z) =11z

i.e original number - 12 = new number

or, (10x + y) - 12 = 11z ——-(ii)

Hence, the digit at ten’s place of original number will be reduced by 1 and that will be equal to the digit at ten’s place of new number, which is ‘z’.

So, z = x - 1 ——-(iii)

Using equation (iii) in equation (ii), we get

10x + y - 12 = 11 * (x - 1)

or, y - x = 1 ——(iv)

And we have x + y = 17 from equation (i)

Now solving equation (i) and equation (iv) ,we get y = 9 , and x = 8.

So the original number is (10*x + y) = 89

We can cross check our answer as, when we subtract 12 from our original number i.e 89 we get 89 - 12 = 77 , both the digits of new number is same. So our answer is correct .
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