The correct answer is option (C):
Both the statements together are needed to answer the question
Let's analyze the statements to determine the sufficiency of the information.
Statement 1: A's score in the fourth test was 12 points higher than the average score in the first three tests written.
This statement tells us a relationship: Fourth test score = Average of first three scores + 12. However, we don't know the actual average of the first three tests, so we cannot determine A's score in the fourth test. This statement is insufficient.
Statement 2: A's score on the fourth test raised the average test score from 80 to 85.
This statement describes the change in the average. We know the initial average (80) and the final average (85). Let's use some variables. Let 'x' be the sum of A's scores in the first three tests, and 'y' be A's score in the fourth test. The average of the first three tests is x/3. The overall average after the fourth test is (x+y)/4. The statement implies that the initial average of the first 3 scores is 80 and the final average of 4 scores is 85. So we get x/3 = 80 - incorrect usage of the statement. Correct usage: Let's let the initial average of 3 tests be 'a'. Then (3*a + y) / 4 = 85. Also 3*a / 3 = 80. Then 3*80 + y = 340. 240 + y = 340. Y = 100.
We can determine the fourth test score, which is y. This statement is sufficient. However, we also need to use statement 1.
Using both statements together:
Statement 1 gives us: y = x/3 + 12.
Statement 2 allows us to determine the fourth test score. The initial average is 80 and final is 85.
The average is (x+y)/4=85 and initial average is x/3, so x/3=80. Then x = 240.
Substitute this in first statement, which is y = x/3 + 12. Then y = 240/3 + 12.
So y = 80+12=92
Therefore, both statements are needed together.
Answer: Both the statements together are needed to answer the question