A **combined proportion** occurs when two or more different proportional relationships are involved in a situation. For example, let's consider a situation where the total cost of a trip is influenced by both the distance traveled and the number of people attending.

### Example:

A bus rental company charges based on both:
1. The **distance** traveled (direct proportion): The cost increases as the distance increases.
2. The **number of people** sharing the bus (inverse proportion): The cost per person decreases as the number of people increases.

Let's say:
- The total cost of renting the bus is directly proportional to the distance traveled, $d$, and inversely proportional to the number of people, $n$, sharing the bus.

This can be expressed as:

\[
C = k \cdot \frac{d}{n}
\]

Where:
- $C$ is the cost per person,
- $k$ is the constant of proportionality,
- $d$ is the distance,
- $n$ is the number of people.

### Example Values:

Suppose the bus company charges $400 for a 200-mile trip with 10 people. Find the cost per person.

1. First, determine the constant $k$:

\[
C = k \cdot \frac{d}{n} \quad \Rightarrow \quad 40 = k \cdot \frac{200}{10}
\]

\[
40 = 20k \quad \Rightarrow \quad k = 2
\]

2. Now, to find the cost for a different trip, say 300 miles with 15 people:

\[
C = 2 \cdot \frac{300}{15} = 2 \cdot 20 = 40
\]

Thus, the cost per person for a 300-mile trip with 15 people would be **$40**.

This is an example of **combined proportion**, where cost is both directly proportional to distance and inversely proportional to the number of people.

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Do you need further details or have any questions?

Here are five related questions for you to explore:
1. How does the cost change if only the distance increases but the number of people remains the same?
2. What happens to the cost per person if the distance is halved and the number of people doubles?
3. Can you find the total cost for a trip without knowing the cost per person?
4. How does the constant of proportionality affect the overall calculation?
5. Can you create a formula for a situation where the cost is directly proportional to both distance and time?

**Tip:** Always identify which variables are in direct proportion and which are in inverse proportion to set up the equation properly.

Explore an example of combined proportion where the total cost of a trip is influenced by both the distance traveled and the number of people sharing the cost. Learn how direct and inverse proportions interact using the formula C = k * (d / n) to calculate cost per person. This example covers concepts in proportional relationships, suitable for students in Grades 9-11.

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