1. **Problem Statement:** Find the mean of the data set $12, 12, b+4, b-5, 17, 10, 9$ given that the mean is 11. Then find the mode of the data after determining $b$. 2. **Formula for Mean:** The mean of $n$ numbers $x_1, x_2, ..., x_n$ is given by: $$\text{Mean} = \frac{x_1 + x_2 + ... + x_n}{n}$$ 3. **Calculate the sum of the data:** Sum $= 12 + 12 + (b+4) + (b-5) + 17 + 10 + 9$ Simplify the sum: $$12 + 12 + b + 4 + b - 5 + 17 + 10 + 9 = (12 + 12 + 4 - 5 + 17 + 10 + 9) + (b + b) = 59 + 2b$$ 4. **Use the mean to find $b$:** Given mean $= 11$ and number of data points $n=7$, $$11 = \frac{59 + 2b}{7}$$ Multiply both sides by 7: $$77 = 59 + 2b$$ Subtract 59 from both sides: $$77 - 59 = 2b$$ $$18 = 2b$$ Divide both sides by 2: $$b = 9$$ 5. **Find the mode:** Substitute $b=9$ into the data: $$12, 12, 9+4=13, 9-5=4, 17, 10, 9$$ The data set is: $$12, 12, 13, 4, 17, 10, 9$$ The mode is the value that appears most frequently, which is $12$. **Final answers:** - $b = 9$ - Mode $= 12$