A social club charters a bus at a cost of $1050 to take a group of members on an excursion to Atlantic City. At the last minute, eight people in the group decide not to go. This raises the transportation cost per person by $4. How many people originally intended to take the trip?

Let the original number of people who intended to take the trip be $ x $.

### Step 1: Write the expression for the original cost per person
If all $ x $ people went, the cost per person would be:

\[
\frac{1050}{x}
\]

### Step 2: Write the expression for the new cost per person
When 8 people drop out, the number of people becomes $ x - 8 $. The new cost per person is:

\[
\frac{1050}{x - 8}
\]

### Step 3: Set up the equation based on the problem
The problem states that the new cost per person is $4 more than the original cost. Therefore, we can set up the equation:

\[
\frac{1050}{x - 8} = \frac{1050}{x} + 4
\]

### Step 4: Solve the equation
1. Subtract $ \frac{1050}{x} $ from both sides:

\[
\frac{1050}{x - 8} - \frac{1050}{x} = 4
\]

2. Factor out $ 1050 $ on the left-hand side:

\[
1050 \left( \frac{1}{x - 8} - \frac{1}{x} \right) = 4
\]

3. Combine the fractions inside the parentheses:

\[
1050 \left( \frac{x - (x - 8)}{x(x - 8)} \right) = 4
\]

\[
1050 \left( \frac{8}{x(x - 8)} \right) = 4
\]

4. Divide both sides by 1050:

\[
\frac{8}{x(x - 8)} = \frac{4}{1050}
\]

5. Cross-multiply:

\[
8 \times 1050 = 4 \times x(x - 8)
\]

\[
8400 = 4x(x - 8)
\]

6. Divide both sides by 4:

\[
2100 = x(x - 8)
\]

7. Expand the right-hand side:

\[
2100 = x^2 - 8x
\]

8. Rearrange the equation:

\[
x^2 - 8x - 2100 = 0
\]

### Step 5: Solve the quadratic equation
Use the quadratic formula:

\[
x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(1)(-2100)}}{2(1)}
\]

\[
x = \frac{8 \pm \sqrt{64 + 8400}}{2}
\]

\[
x = \frac{8 \pm \sqrt{8464}}{2}
\]

\[
x = \frac{8 \pm 92}{2}
\]

So, $ x = \frac{8 + 92}{2} = 50 $ or $ x = \frac{8 - 92}{2} = -42 $.

Since the number of people must be positive, we have $ x = 50 $.

### Final Answer:
The original number of people who intended to take the trip was **50**.

Would you like more details or have any questions?

Here are 5 related questions to think about:
1. What would the new cost per person be after 8 people dropped out?
2. How would the total cost change if the bus charter was $1200 instead?
3. What would happen to the price difference if 5 people dropped out instead of 8?
4. How can we set up a similar problem involving a different number of people and total cost?
5. What are other real-world applications of such algebraic problems?

**Tip:** When dealing with cost per person problems, setting up expressions based on the number of people helps organize the equation structure.

A social club charters a bus at a cost of $1050 to take a group of members on an excursion to Atlantic City. At the last minute, eight people in the group decide not to go. This raises the transportation cost per person by $4. How many people originally intended to take the trip?

Discover how to solve an algebraic word problem involving cost per person and a group trip. Learn how a change in group size affects the cost using quadratic equations. This problem demonstrates key algebraic concepts and is suitable for students in Grades 9-11.

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