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Let's break this problem down step by step.

### Given Information:
- Three-fourths of Jesse Black's inventory was destroyed by fire.
- The remaining part (one-fourth) was slightly damaged and sold for **one-seventh** of its value, which amounted to **$3900**.

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

### Part (b): Value of the inventory before the fire
- The remaining part of the inventory (after the fire) is **one-fourth** of the original value.
- This remaining part was sold for **one-seventh** of its value, and the amount received was **$3900**.

Thus, the value of the remaining inventory (before the sale) is:
\[
\frac{1}{7} \times \frac{x}{4} = 3900
\]
Solving for $ x $:
\[
\frac{x}{28} = 3900
\]
Multiplying both sides by 28:
\[
x = 3900 \times 28
\]
\[
x = 109200
\]
So, the total value of the inventory before the fire was **$109,200**.

### Part (a): Value of the destroyed part of the inventory
The destroyed part of the inventory was three-fourths of the total inventory:
\[
\text{Destroyed value} = \frac{3}{4} \times 109200 = 81900
\]
So, the value of the destroyed part of the inventory was **$81,900**.

### Summary:
- (a) The value of the destroyed part of the inventory was **$81,900**.
- (b) The value of the inventory before the fire was **$109,200**.

Would you like further details or have any questions?

Here are 5 related questions:
1. How would the result change if Jesse had received one-sixth of the remaining inventory's value?
2. What if the remaining part of the inventory was sold for one-fifth of its value instead?
3. How would the calculation change if 60% of the inventory was destroyed instead of three-fourths?
4. What if Jesse had received $4500 instead of $3900 for the damaged part?
5. How would the total inventory value change if Jesse sold the remaining part at half its value?

**Tip:** When solving problems involving fractions of a total, it's useful to represent the total as a variable to simplify the calculations.

Three-fourths of Jesse Black's inventory was destroyed by fire. He sold the remaining part, which was slightly damaged, for one-seventh of its value and received $3900.
(a) What was the value of the destroyed part of the inventory?
(b) What was the value of the inventory before the fire?

Solve a problem involving inventory loss after a fire. Jesse Black lost three-fourths of his inventory and sold the remaining part for $3900, representing one-seventh of its value. Learn how to calculate the value of the destroyed part and the total inventory value before the fire.

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