1. **State the problem:** Lisa took 4 chemistry tests with an arithmetic mean of 82. She then scores 97 on her 5th test. We need to find the new arithmetic mean for all five exams. 2. **Recall the formula for arithmetic mean:** $$\text{Mean} = \frac{\text{Sum of all scores}}{\text{Number of tests}}$$ 3. **Calculate the total sum of the first 4 tests:** Since the mean of the first 4 tests is 82, $$\text{Sum of first 4 tests} = 4 \times 82 = 328$$ 4. **Add the 5th test score:** $$\text{New total sum} = 328 + 97 = 425$$ 5. **Calculate the new mean for all 5 tests:** $$\text{New mean} = \frac{425}{5} = 85$$ 6. **Answer:** The arithmetic mean for all five exams is **85**.