The difference between a two-digit number and the number obtained by interchanging the positions of its digits is 36. What is the difference between the largest and the smallest of these two digits?
let original number be 10x + y
then the number reversed is 10y + x
difference= 10x+y - (10y+x)
= 9x - 9y
but
9x - 9y= 36
x - y = 4
since x and y must both be integers between 1 and 9
that little equation tells me that the difference between the digits is 4
possible (x,y) are
(9,5), (8,4), (7,3), (6,2) and (5,1)
the difference between the digits for all of them is 4
e.g.
suppose x = 7, y = 3
the first number is 73, and the number reversed is 37
notice 73 - 37 = 36
To find the difference between the largest and the smallest digits, let's work through the problem step by step:
Step 1: Let the two-digit number be represented by "AB", where A represents the tens digit and B represents the units digit.
Step 2: The number obtained by interchanging the positions of the digits AB would be represented as "BA", with B as the tens digit and A as the units digit.
Step 3: The difference between the two-digit number AB and BA is given as 36. We can represent this mathematically as AB - BA = 36.
Step 4: Substituting the values of A and B into the equation, we have 10A + B - (10B + A) = 36.
Step 5: Simplifying the equation, we get 9A - 9B = 36.
Step 6: Factoring out 9 from both terms, the equation becomes 9(A - B) = 36.
Step 7: Dividing both sides of the equation by 9, we have A - B = 4.
Step 8: Based on the equation A - B = 4, we need to find the largest and smallest values for A and B that satisfy this equation.
Step 9: The largest possible value for A is 9, as single-digit numbers cannot be used for a two-digit number. The smallest possible value for A is 5, as choosing a value less than that would result in a negative value for B, which is not possible for a digit.
Step 10: Substituting the values of A into the equation, we get 9 - B = 4 for the largest value of A, and 5 - B = 4 for the smallest value of A.
Step 11: Solving the equations, we find that the largest value for B is 5, and the smallest value for B is 1.
Step 12: Therefore, the difference between the largest and smallest of the digits is 5 - 1 = 4.
So, the difference between the largest and smallest of these two digits is 4.