The ratio of the present age of X to that of Y is 4 : 5. k (less than 10) years ago, this ratio was 1 : 2. How many values of k are possible if the ages are expressed in complete years?

Let present ages be X = 4t and Y = 5t. k years ago: (4t - k)/(5t - k) = 1/2, giving 2(4t - k) = 5t - k, so 8t - 2k = 5t - k, hence 3t = k, i.e., k = 3t. Since k < 10 and k must be a positive integer (and ages must be complete years), t can be 1, 2, or 3, giving k = 3, 6, 9. So 3 values are possible.