Chetan writes the positive integers from \(1\) to \(200\) on a whiteboard. Wassim then erases all the numbers that are multiples of \(9\). Karla then erases all the remaining numbers that contain the digit \(9\). How many numbers are left on the whiteboard?
We first calculate the number of integers that Wassim erases, which is the number of multiples of \(9\) between \(1\) and \(200\). Since \(200 = (22 \times 9) + 2\), there are \(22\) multiples of \(9\) between \(1\) and \(200\). Thus, Wassim erases \(22\) numbers from the whiteboard.
Now let’s figure out how many of the integers from \(1\) to \(200\) contain the digit \(9\). From \(1\) to \(100\), these numbers are \(9, 19, \ldots, 79, 89\) as well as \(90, 91, \ldots , 98, 99\). There are \(19\) of these numbers from \(1\) to \(100\). There are another \(19\) between \(101\) and \(200\), which are obtained by adding \(100\) to each of the numbers from \(1\) to \(100\). Therefore, \(19 + 19 =38\) integers from \(1\) to \(200\) contain the digit \(9\).
However, some of the integers that contain the digit \(9\) are also multiples of \(9\), so were erased by Wassim. There are \(5\) of these numbers between \(1\) and \(200\): \(9\), \(90\), \(99\), \(189\), and \(198\). Thus, Karla erases \(38-5=33\) numbers from the whiteboard.
Hence, the number of numbers left on the whiteboard is \(200 - 22 - 33 = 145\).