1. **State the problem:** We have a data set with numbers 7, 7, 4, 4, 6, 7, and an unknown number $k$. We know the mean of these numbers is 6. We need to find the value of $k$. 2. **Recall the formula for the mean:** $$\text{Mean} = \frac{\text{Sum of all numbers}}{\text{Number of numbers}}$$ 3. **Apply the formula:** There are 7 numbers total (6 known numbers plus $k$). $$6 = \frac{7 + 7 + 4 + 4 + 6 + 7 + k}{7}$$ 4. **Calculate the sum of the known numbers:** $$7 + 7 + 4 + 4 + 6 + 7 = 35$$ 5. **Set up the equation:** $$6 = \frac{35 + k}{7}$$ 6. **Multiply both sides by 7 to eliminate the denominator:** $$6 \times 7 = 35 + k$$ $$\cancel{7} \times 6 = 35 + k$$ 7. **Simplify:** $$42 = 35 + k$$ 8. **Solve for $k$ by subtracting 35 from both sides:** $$42 - 35 = k$$ $$\cancel{35} + (42 - \cancel{35}) = k$$ 9. **Final answer:** $$k = 7$$ So, the value of $k$ that makes the mean 6 is **7**.