The problem asks us to find the mean of the grades on Natalie's next two tests so that her overall mean for six tests is 85.

Let's break it down step by step:

1. Information given:

   • On the first four tests, Natalie had a mean grade of 82.
   • She has two more tests left.
   • The desired overall mean for the six tests is 85.

2. Step 1: Find the total score for the first four tests. Since the mean for the first four tests is 82, the total sum of the scores for these tests is: $4 \times 82 = 328$

3. Step 2: Find the total score for all six tests if the mean is to be 85. The total sum of all six test scores should be: $6 \times 85 = 510$

4. Step 3: Find the sum of the scores needed on the last two tests. The difference between the total score for all six tests (510) and the total score for the first four tests (328) is: $510 - 328 = 182$ So, Natalie needs to score a total of 182 points on her last two tests.

5. Step 4: Find the mean of the last two tests. The mean for the last two tests is: $\frac{182}{2} = 91$

Answer:

Natalie needs to have a mean grade of 91 on the last two tests for her overall mean to be 85.

Would you like any further details or clarifications on this?

Here are some related questions you might find useful:

1. What happens if Natalie scores 90 and 92 on her next two tests?
2. How would the result change if her desired overall mean was 90 instead of 85?
3. How do you calculate the mean if there are different weightings for tests?
4. What if Natalie scored 75 on one of her next two tests? What would she need to score on the other test?
5. How can we find the median score of Natalie's six tests?

Tip: Always check the units and averages in a problem to ensure you're interpreting the mean or total correctly!