To find the total movement of the fault line over the past two years, we need to add the two movements together:
- The movement in the first year: \(1 \frac{1}{2}\) inches
- The movement in the second year: \(7 \frac{2}{5}\) inches
First, let's convert the mixed numbers into improper fractions.
-
For \(1 \frac{1}{2}\): \[ 1 \frac{1}{2} = \frac{2 \times 1 + 1}{2} = \frac{2 + 1}{2} = \frac{3}{2} \]
-
For \(7 \frac{2}{5}\): \[ 7 \frac{2}{5} = \frac{5 \times 7 + 2}{5} = \frac{35 + 2}{5} = \frac{37}{5} \]
Now, we need to add \(\frac{3}{2}\) and \(\frac{37}{5}\). To do this, we need a common denominator. The least common multiple of 2 and 5 is 10.
Next, let's convert both fractions to have a denominator of 10:
-
For \(\frac{3}{2}\): \[ \frac{3}{2} = \frac{3 \times 5}{2 \times 5} = \frac{15}{10} \]
-
For \(\frac{37}{5}\): \[ \frac{37}{5} = \frac{37 \times 2}{5 \times 2} = \frac{74}{10} \]
Now, we can add the two fractions: \[ \frac{15}{10} + \frac{74}{10} = \frac{15 + 74}{10} = \frac{89}{10} \]
Next, we can convert \(\frac{89}{10}\) back to a mixed number: \[ \frac{89}{10} = 8 \frac{9}{10} \]
Thus, the total movement of the fault line over the past two years is: \[ \boxed{8 \frac{9}{10}} \text{ inches} \]