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Let's break down the problem:

### Given Information:
- $ \frac{5}{9} $ of Jesse Black's inventory was destroyed by fire.
- He sold the remaining $ \frac{4}{9} $ of the inventory for $ \frac{3}{5} $ of its value and received \$3700.

We need to find:
1. The value of the destroyed part of the inventory.
2. The value of the inventory before the fire.

### Step-by-Step Solution:

#### (b) What was the value of the inventory before the fire?

Let the total value of the inventory before the fire be $ x $.

- The remaining $ \frac{4}{9} $ of the inventory was sold for $ \frac{3}{5} $ of its value. So, the value of the remaining part is $ \frac{4}{9}x $.
- However, Jesse received $ 3700 $ dollars for it, which was $ \frac{3}{5} $ of the value of the remaining part.

Thus, we set up the equation:
\[
\frac{3}{5} \times \frac{4}{9}x = 3700
\]
Solving for $ x $:

\[
\frac{12}{45}x = 3700
\]
\[
\frac{4}{15}x = 3700
\]
Multiply both sides by 15:
\[
4x = 3700 \times 15
\]
\[
4x = 55500
\]
Now, divide by 4:
\[
x = \frac{55500}{4} = 13,875
\]

Thus, the value of the inventory before the fire was **\$13,875**.

#### (a) What was the value of the destroyed part of the inventory?

Since $ \frac{5}{9} $ of the inventory was destroyed, the value of the destroyed part is:
\[
\text{Destroyed part} = \frac{5}{9} \times 13,875 = 7,708.33
\]

Thus, the value of the destroyed part was **\$7,708.33**.

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Would you like further clarification or more details?

### Follow-up Questions:
1. How would the solution change if the percentage of destroyed inventory was different?
2. What if the amount received for the damaged inventory was less?
3. How can we generalize this type of problem for other similar cases?
4. If Jesse had been insured for the destroyed inventory, how would that affect the calculations?
5. Can we solve this problem using a different method, like proportions?

### Tip:
In problems involving fractions of total values, always break the problem into steps where you solve for the unknowns systematically!

Five-ninths of Jesse Black's inventory was destroyed by fire. He sold the remaining part, which was slightly damaged, for three-fifths of its value and received $3700. (a) What was the value of the destroyed part of the inventory? (b) What was the value of the inventory before the fire?

Learn how to solve an inventory loss problem involving fire damage, where five-ninths of the inventory was destroyed. This problem uses proportions and basic algebra to calculate the value of the damaged and destroyed inventory. Suitable for students in Grades 9-11.

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