A group of friends decided to divide the $800 cost of a trip equally among themselves. When two of the friends decided not to go on the trip, those remaining still divided the $800 cost equally, but each friend’s share of the cost increased by $20. How many friends were in the group originally?

Explanation

To solve the problem, we need to determine the original number of friends in the group who planned to go on the trip. Let's denote the original number of friends as \( x \).

### Step-by-Step Solution:

1. **Initial Cost Per Friend**:
The total cost of the trip is $800. If there are \( x \) friends, the cost per friend initially is:
\[ \text{Initial Cost Per Friend} = \frac{800}{x} \]

2. **Cost Per Friend After Two Friends Back Out**:
When two friends decide not to go, the number of friends becomes \( x - 2 \). The new cost per friend is:
\[ \text{New Cost Per Friend} = \frac{800}{x - 2} \]

3. **Setting Up the Equation**:
According to the problem, the new cost per friend is $20 more than the initial cost per friend. Therefore, we can set up the equation:
\[ \frac{800}{x - 2} = \frac{800}{x} + 20 \]

4. **Clearing the Fractions**:
To eliminate the fractions, multiply through by \( x(x - 2) \):
\[ 800x = 800(x - 2) + 20x(x - 2) \]

5. **Expanding and Simplifying**:
Expanding both sides gives:
\[ 800x = 800x - 1600 + 20x^2 - 40x \]
Simplifying this, we get:
\[ 0 = 20x^2 - 40x - 1600 \]

6. **Dividing the Equation**:
To simplify, divide the entire equation by 20:
\[ 0 = x^2 - 2x - 80 \]

7. **Factoring the Quadratic**:
We can factor the quadratic equation:
\[ (x - 10)(x + 8) = 0 \]

8. **Finding the Roots**:
Setting each factor to zero gives:
\[ x - 10 = 0 \quad \Rightarrow \quad x = 10 \]
\[ x + 8 = 0 \quad \Rightarrow \quad x = -8 \quad (\text{not valid since } x \text{ must be positive}) \]

9. **Conclusion**:
Therefore, the original number of friends in the group is:
\[ \boxed{10} \]