Pete has an average score exactly x points on 4 equally weighted tests. How many points higher than x must Pete score on the 5th equally weighted test to raise his average score after the 5th test to `x + 2` points?

Options

A. `2`
B. `4`
C. `5`
D. `10`

Answer

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

### Step 1: Determine the total score from the first four tests
Pete has an average score of \( x \) points on 4 tests. The total score from these tests can be calculated as follows:
\[ \text{Total score from 4 tests} = \text{Average} \times \text{Number of tests} = x \times 4 = 4x \]

**Hint:** Remember that the total score is the average score multiplied by the number of tests.

### Step 2: Set up the equation for the average after the fifth test
Let \( y \) be the score Pete needs on the fifth test. After the fifth test, the average score should be \( x + 2 \). The total score after 5 tests will be:
\[ \text{Total score after 5 tests} = 4x + y \]
The average after 5 tests can be expressed as:
\[ \text{Average after 5 tests} = \frac{4x + y}{5} \]
Setting this equal to \( x + 2 \):
\[ \frac{4x + y}{5} = x + 2 \]

**Hint:** The average of a set of numbers is the total sum divided by the number of items in the set.

### Step 3: Solve for \( y \)
To eliminate the fraction, multiply both sides of the equation by 5:
\[ 4x + y = 5(x + 2) \]
Expanding the right side:
\[ 4x + y = 5x + 10 \]
Now, isolate \( y \) by subtracting \( 4x \) from both sides:
\[ y = 5x + 10 - 4x \]
This simplifies to:
\[ y = x + 10 \]

**Hint:** When isolating a variable, make sure to perform the same operation on both sides of the equation.

### Step 4: Determine how many points higher than \( x \) Pete must score
To find out how many points higher than \( x \) Pete must score on the fifth test, we compare \( y \) with \( x \):
\[ y - x = (x + 10) - x = 10 \]

Thus, Pete must score **10 points higher than \( x \)** on the fifth test.

**Final Answer:** Pete must score 10 points higher than \( x \).

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