The raw score on a certain standardized test is determined by subtracting `1/4` of the number of incorrect answers from the number of correct answers. IF a student answered 30 question and received a raw score of 20, how many question did the student answer incorrectly?

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Define Variables
Let:
- \( x \) = number of correct answers
- \( y \) = number of incorrect answers

### Step 2: Set Up Equations
From the problem, we know two things:
1. The total number of questions answered is 30:
\[ x + y = 30 \quad \text{(Equation 1)} \]
2. The raw score is calculated as the number of correct answers minus one-fourth of the number of incorrect answers, which equals 20:
\[ x - \frac{1}{4}y = 20 \quad \text{(Equation 2)} \]

### Step 3: Solve the First Equation for One Variable
From Equation 1, we can express \( x \) in terms of \( y \):
\[ x = 30 - y \quad \text{(Substituting into Equation 2)} \]

### Step 4: Substitute into the Second Equation
Now, substitute \( x \) in Equation 2:
\[ (30 - y) - \frac{1}{4}y = 20 \]

### Step 5: Simplify the Equation
Combine like terms:
\[ 30 - y - \frac{1}{4}y = 20 \]
\[ 30 - y - 0.25y = 20 \]
\[ 30 - 1.25y = 20 \]

### Step 6: Isolate the Variable
Subtract 30 from both sides:
\[ -1.25y = 20 - 30 \]
\[ -1.25y = -10 \]

### Step 7: Solve for \( y \)
Divide both sides by -1.25:
\[ y = \frac{-10}{-1.25} = 8 \]

### Step 8: Conclusion
The number of incorrect answers \( y \) is 8.

### Final Answer
The student answered **8 questions incorrectly**.

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