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Problem Analysis:

We are given a logic grid puzzle with the following elements:
- Names: Arnold, Eric, Peter
- Ages: 7 years, 8 years, 9 years
- Birthdays: April, January, September

We need to match each name with their corresponding age and birthday using the clues provided.

Clues:

1. Peter's birthday is in April.
2. Eric is 7.
3. Arnold's birthday is in September.
4. Peter is 8.

Solution Approach:

We will use the clues to fill in the logic grid step by step.

#### Step 1: Analyze Clue 1
Clue 1: Peter's birthday is in April.
- This means we can place a mark in the grid under "Peter" for "April."

#### Step 2: Analyze Clue 2
Clue 2: Eric is 7.
- This means we can place a mark in the grid under "Eric" for "7 years."

#### Step 3: Analyze Clue 3
Clue 3: Arnold's birthday is in September.
- This means we can place a mark in the grid under "Arnold" for "September."

#### Step 4: Analyze Clue 4
Clue 4: Peter is 8.
- This means we can place a mark in the grid under "Peter" for "8 years."

#### Step 5: Deduce Remaining Information
- From Clue 4, Peter is 8 years old, so we mark "8 years" under Peter.
- From Clue 1, Peter's birthday is in April, so we mark "April" under Peter.
- From Clue 2, Eric is 7 years old, so we mark "7 years" under Eric.
- Since Peter is 8 and Eric is 7, Arnold must be 9 years old (the only remaining age).
- From Clue 3, Arnold's birthday is in September, so we mark "September" under Arnold.
- The only remaining month is January, which must belong to Eric.

Filled Logic Grid:

Based on the deductions, the completed grid looks like this:

| Names | Ages | Birthdays |
|---------|---------|-----------|
| Arnold | 9 years | September |
| Eric | 7 years | January |
| Peter | 8 years | April |

Final Answer:

\[
\boxed{
\begin{array}{|c|c|c|}
\hline
\text{Names} & \text{Ages} & \text{Birthdays} \\
\hline
\text{Arnold} & 9 \text{ years} & \text{September} \\
\hline
\text{Eric} & 7 \text{ years} & \text{January} \\
\hline
\text{Peter} & 8 \text{ years} & \text{April} \\
\hline
\end{array}
}
\]