1. **State the problem:** We have a data set with entries 2, 4, 6, 8, 8, 8, and 11. We want to add one more number between 1 and 11 so that the median and mode of the new data set are equal. 2. **Recall definitions:** - The **mode** is the number that appears most frequently. - The **median** is the middle value when the data is sorted. 3. **Current mode:** The number 8 appears three times, so the mode is 8. 4. **Add a new number $x$ between 1 and 11:** The new data set has 8 numbers. 5. **Sort the data set including $x$:** The sorted set depends on $x$. 6. **Median for even number of data points:** The median is the average of the 4th and 5th numbers. 7. **Goal:** Median = Mode = 8. 8. **Analyze possible positions of $x$ to keep mode 8:** Adding $x$ should not create a new mode different from 8. 9. **Check median positions:** The 4th and 5th numbers must average to 8. 10. **Try $x=8$:** Sorted set: 2,4,6,8,8,8,8,11 Median = average of 4th and 5th = average of 8 and 8 = 8 Mode = 8 (now appears 4 times) 11. **Conclusion:** Adding $x=8$ makes median and mode both equal to 8. **Final answer:** $\boxed{8}$