the sum of digits of a two digit number is 11. if the digits are interchanged and the new number is subtracted from the original, we get 9. what can be the original number?
Answers
This 'number theory' question as I see it has 3 parts or criteria.
First is a number is made up of 2 digits which limits the answer to between 10 and 99.
Second is that the sum of the two digits that make up this number is 11, which limits the answer to between 2 and 9 - 2,3,4,5,6,7,8,9.
The third part is that if the digits in the number are interchanged, the number would be increased by 9.
So, the answer is 56.
There are several digits whose sum is 11 - 2 & 9 / 3 & 8 / 4 & 7 / 5 & 6.
There are also the reverse iterations of these digits whose sum is also 11 -
6 & 5 / 7 & 4 / 8 & 3 /9 & 2.
This 'solves' the second part .
These digits also make up several numbers as follows:
29
38
47
56
65
74
83
92
This 'solves' the first part'
You can start at the top or bottom of the list above and add 9 as follows:
29 + 9 = 38
38 + 9 = 47
47 + 9 = 56
56 + 9 = 65
65 + 9 = 74
74 + 9 = 83
83 + 9 = 92
Then take the first set of numbers above and interchange or reverse them to see if you can solve the question as follows:
29 - 92 difference of 63 FAILS
38 - 83 difference of 45 FAILS
47 - 74 difference of 27 FAILS
56 - 65 difference of 9 CORRECT
65 - 56 difference of -9 FAILS
74 - 47 difference of -27 FAILS
83 - 38 difference of -45 FAILS
92 - 29 difference of - 63 FAILS
Let the digit in tens place be x and digit in ones place be y.
So, the original number be 10x + y.
Given sum of its digits is 11
⇒ x + y = 11
If digits are interchanged i.e digit in tens place be y and digit in ones place be x.
Now, the number becomes 10y + x
Again given that original number - new number = 9
⇒ 10x + y - (10y + x) = 9
10x + y - 10y - x = 9
9x - 9y = 9
x - y = 1
Adding equations x + y = 11 and x - y = 1,
we get
2x = 12
x = 6
x + y =11
so, y = 5
The original number is 10x + y = 10*6 + 5 = 65
Hope this helps you.