The water level of a lake fell by 1 and one half
 in. during a 3 and three fourths
​-week-long dry spell. Simplify the complex fraction below to find the average rate at which the water level changed every week.
StartFraction negative 1 and one half Over 3 and three fourths EndFraction


Let's solve the problem step by step.

### Step 1: Express the Mixed Numbers as Improper Fractions
First, convert the mixed numbers to improper fractions:

- \( -1 \frac{1}{2} \) becomes \( -\frac{3}{2} \) (since \( 1 \times 2 + 1 = 3 \)).
- \( 3 \frac{3}{4} \) becomes \( \frac{15}{4} \) (since \( 3 \times 4 + 3 = 15 \)).

### Step 2: Set Up the Complex Fraction
The complex fraction is:

\[
\frac{-\frac{3}{2}}{\frac{15}{4}}
\]

### Step 3: Simplify the Complex Fraction
To simplify the complex fraction, multiply the numerator by the reciprocal of the denominator:

\[
\frac{-\frac{3}{2}}{\frac{15}{4}} = -\frac{3}{2} \times \frac{4}{15} 
\]

### Step 4: Multiply the Fractions
Now, multiply the fractions:

\[
-\frac{3 \times 4}{2 \times 15} = -\frac{12}{30}
\]

### Step 5: Simplify the Resulting Fraction
Simplify \( \frac{12}{30} \):

\[
-\frac{12}{30} = -\frac{2}{5}
\]

### Final Answer
The average rate at which the water level changed every week is **-\(\frac{2}{5}\) inches per week**.

Would you like more details, or do you have any questions? Here are some related questions to explore:

1. How do you convert mixed numbers to improper fractions?
2. What is the process for simplifying complex fractions?
3. How does multiplying by the reciprocal help in division of fractions?
4. What are some real-world examples of rates of change?
5. How can you ensure fractions are fully simplified?

**Tip:** When simplifying complex fractions, always check that all fractions are expressed in their simplest form before multiplying or dividing.

Learn how to calculate the average rate at which the water level changes per week during a dry spell. Step-by-step solution to simplify the complex fraction -1 and one half over 3 and three fourths. Suitable for Grades 5-6.

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