In how many days will 5 men and 4 women together complete the work?\newline\textbf{(I).} 15 men and 12 women can complete the same work in 12 days.\newline\textbf{(II).} 10 men and 8 women can complete the same work in 16 days.

Let m and w be the work done by 1 man and 1 woman per day. From statement I: 15m + 12w = 1/12, which simplifies to 5m + 4w = 1/36. This means 5 men and 4 women complete 1/36 of the work daily, so they need 36 days total. From statement II: 10m + 8w = 1/16, which simplifies to 5m + 4w = 1/32, giving 32 days. Each statement alone gives us the value of 5m + 4w, which is exactly what we need to find the time for 5 men and 4 women.