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

Let X's current age = 4m, Y's = 5m. k years ago: (4m–k)/(5m–k) = 1/2, so 8m – 2k = 5m – k, giving 3m = k. So k = 3m. Since k < 10, m can be 1, 2, or 3, giving k = 3, 6, or 9. But ages must be complete years and ages k years ago must be non-negative. For m=1: ages were 3,4 (k=3) ✓. For m=2: ages 6,8 (k=6) ✓. For m=3: ages 9,12 (k=9) ✓. All three values work, so 3 values are possible. Wait, the answer should be (c) 3. Let me re-read the question — the answer is 3.