To solve the problem, let's break it down step by step:

Known scores:

Thao's current scores on six exams are:

• 78, 83, 44, 89, 78, 89

The score of 44 will be replaced by the score on a "make-up" exam, and Thao wants to achieve an overall average (mean) of 80.

Step 1: Calculate the current total of the other 5 scores.

First, sum the scores excluding the one Thao will replace (44): $78 + 83 + 89 + 78 + 89 = 417$

Step 2: Set up the equation.

Let $x$ be the score that Thao must get on the make-up exam to achieve an overall average of 80. Since the new total score will include $x$ instead of 44, we can set up the equation as follows:

$\text{New total score} = 417 + x$ The total number of exams is still 6, so the equation for the average is: $\frac{417 + x}{6} = 80$

Step 3: Solve for $x$.

Multiply both sides by 6: $417 + x = 480$ Now, subtract 417 from both sides: $x = 480 - 417$ $x = 63$

Conclusion:

Thao must score 63 on the make-up exam to get an overall average of 80.

Do you have any questions or need further explanation? Here are some related questions to consider:

1. How do you calculate the mean for a set of numbers?
2. What would happen if Thao scored higher or lower than 63 on the make-up exam?
3. How is the mean affected by replacing just one score in a set of numbers?
4. What if Thao needed an average higher than 80? How would the solution change?
5. What is the impact of a very low score (like 44) on the overall average?

Tip: When working with averages, replacing an outlier score can significantly change the result, especially in smaller data sets.