Let the two numbers be 23x and x for some positive value of x. This is based on the given ratio of 23.
According to the problem, if we subtract 2 from the first number and 8 from the second, the new ratio becomes the reciprocal of the original ratio. Therefore, we can set up the equation:
(23x – 2) / (x – 8) = 1/23
Cross-multiplying gives us:
23(23x – 2) = (x – 8)
Expanding both sides:
529x – 46 = x – 8
Now, rearranging the equation:
529x – x = -8 + 46
528x = 38
So, we find that:
x = 38 / 528 = 1/14
Now, substituting back to find the two numbers:
First number = 23x = 23 * (1/14) = 23/14
Second number = x = 1/14
Therefore, the two numbers that satisfy the given conditions are 23/14 and 1/14.