Kanaka took 8 tests for her social studies class. Each test has a maximum score of 100 and a minimum score of 0. On the 8 tests, Kanaka's mean score was 90. More than a quarter of her tests have scores less than 85. IF the average of the remaining tests is x, which of the following must be true?

Options

A. `x le 85`
B. `85 lt x lt 90`
C. `x=90`
D. `xgt90`

Answer

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

### Step 1: Calculate the Total Score
We know that Kanaka took 8 tests and her mean score was 90. The mean score is calculated as the total score divided by the number of tests.

\[ \text{Total Score} = \text{Mean Score} \times \text{Number of Tests} = 90 \times 8 = 720 \]

**Hint:** Remember that the mean is the total divided by the number of items.

### Step 2: Determine the Minimum Score for Tests Below 85
We are told that more than a quarter of her tests have scores less than 85. Since there are 8 tests, a quarter would be 2 tests. Therefore, more than 2 tests (at least 3) must have scores below 85.

**Hint:** Calculate one-fourth of the total number of tests to find the minimum count of tests below a certain score.

### Step 3: Calculate the Maximum Possible Score for Tests Below 85
Assuming the minimum case where 3 tests scored below 85, we can calculate the maximum possible score for these tests. If we assume the highest score for these tests is 84 (since they are below 85):

\[ \text{Maximum Score for 3 Tests} = 3 \times 84 = 252 \]

**Hint:** Use the highest possible score below the threshold to find the total score for those tests.

### Step 4: Calculate the Total Score for the Remaining Tests
Now, we can find the total score for the remaining tests (which are 5 tests, since 8 - 3 = 5):

\[ \text{Total Score for Remaining Tests} = \text{Total Score} - \text{Score for Tests Below 85} \]
\[ \text{Total Score for Remaining Tests} = 720 - 252 = 468 \]

**Hint:** Subtract the total score of the lower-scoring tests from the overall total to find the score of the remaining tests.

### Step 5: Calculate the Average Score for the Remaining Tests
Now, we can find the average score (x) for the remaining 5 tests:

\[ x = \frac{\text{Total Score for Remaining Tests}}{\text{Number of Remaining Tests}} = \frac{468}{5} = 93.6 \]

**Hint:** To find the average, divide the total score of the remaining tests by the number of those tests.

### Conclusion
Since we calculated that \( x = 93.6 \) and we know that more than a quarter of her tests scored below 85, we can conclude that the average score of the remaining tests must be greater than 90.

Thus, the statement that must be true is:
**X is greater than 90.**

### Final Answer
The correct option is **D: X > 90.**