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Let's define the number of sweatshirts, T-shirts, and caps as $ s $, $ t $, and $ c $ respectively. We'll set up a system of equations based on the information provided:

1. **Total number of items**: 
   \[ s + t + c = 186 \]

2. **Total revenue if all items are sold**:
   \[ 26s + 14t + 8c = 3132 \]

3. **Items sold**:
   - Three-fourths of the sweatshirts were sold: $ \frac{3}{4}s $
   - Half of the T-shirts were sold: $ \frac{1}{2}t $
   - One-fourth of the caps were sold: $ \frac{1}{4}c $
   - Total items sold were 99: 
   \[ \frac{3}{4}s + \frac{1}{2}t + \frac{1}{4}c = 99 \]

Now, we'll solve this system of equations.

### Equation 1:
\[ s + t + c = 186 \]

### Equation 2:
\[ 26s + 14t + 8c = 3132 \]

### Equation 3:
To eliminate fractions, multiply the third equation by 4:
\[ 3s + 2t + c = 396 \]

Now, we will solve these equations step-by-step.

#### From Equation 1 and Equation 3:

First, isolate $ c $ from Equation 1:
\[ c = 186 - s - t \]

Substitute $ c $ in Equation 3:
\[ 3s + 2t + (186 - s - t) = 396 \]
\[ 3s + 2t + 186 - s - t = 396 \]
\[ 2s + t + 186 = 396 \]
\[ 2s + t = 210 \]

Now, we have:
\[ c = 186 - s - t \]
\[ 2s + t = 210 \]
\[ 26s + 14t + 8c = 3132 \]

#### Solving for $ s $ and $ t $:

From $ 2s + t = 210 $, isolate $ t $:
\[ t = 210 - 2s \]

Substitute $ t $ and $ c $ in Equation 2:
\[ 26s + 14(210 - 2s) + 8(186 - s - (210 - 2s)) = 3132 \]

Let's solve this equation step-by-step:

1. **Calculate terms:**
   - $ 14(210 - 2s) = 2940 - 28s $
   - $ 8(186 - s - 210 + 2s) = 8(-24 + s) = -192 + 8s $

2. **Substitute and simplify:**
   \[ 26s + 2940 - 28s - 192 + 8s = 3132 \]
   \[ 6s + 2748 = 3132 \]

3. **Solve for $ s $:**
   \[ 6s = 3132 - 2748 \]
   \[ 6s = 384 \]
   \[ s = 64 \]

#### Finding $ t $ and $ c $:

- $ t = 210 - 2(64) $
- $ t = 210 - 128 $
- $ t = 82 $

- $ c = 186 - 64 - 82 $
- $ c = 40 $

### Solution:
- **Sweatshirts (s)**: 64
- **T-shirts (t)**: 82
- **Caps (c)**: 40

Would you like further details or have any questions?

Here are some related questions for further practice:
1. What is the total revenue generated by selling all sweatshirts alone?
2. How much revenue was generated from selling three-fourths of the sweatshirts?
3. If the price of caps were increased by $2, what would be the new total revenue?
4. How many more items would they need to sell to reach a revenue of $3500?
5. What percentage of the total items were caps?
6. How much more revenue would they have if they sold all items at the prices mentioned?
7. What is the average price per item if all items were sold?
8. How would the total revenue change if they sold half of the caps instead of one-fourth?

**Tip:** Always double-check your calculations, especially when dealing with systems of equations, to ensure accuracy.

Learn how to solve a system of equations problem involving sweatshirts, T-shirts, and caps. This problem includes calculating the number of each item sold and their prices to find the total revenue. Suitable for high school students learning about systems of equations and fractional equations.

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