https://file.aiquickdraw.com/mathbot/file/1726305539832ub7emnbp.jpg

Let's break down the information from the image for problem 4:

### Problem Statement:
Over a 14-day period, data were collected on the number of passengers traveling on two ferries, A and Z. The results are presented in the table below:

| Ferry A (in 10s) | Ferry Z (in 10s) |
|------------------|------------------|
| 87               | 62               |
| 76               | 42               |
| 65               | 58               |
| 53               | 77               |
| 31               | 50               |

The key explains that:
- 87 represents 530 passengers on Ferry A and 50 passengers on Ferry Z.

#### Questions:
a. **How many more passengers traveled on Ferry Z than on Ferry A?**

b. **The cost of a trip on Ferry A is \$12.50, and the cost of a trip on Ferry Z is \$x. The takings on Ferry Z were \$3.30 less than the takings on Ferry A over this period. Find the value of x.**

c. **Find the least and greatest possible number of days on which the two ferries could have carried exactly the same number of passengers.**

---

### Solving:

#### a. **How many more passengers traveled on Ferry Z than on Ferry A?**

First, convert the data from the table into actual passenger numbers by multiplying by 10:
- Ferry A (in passengers): $ 870, 760, 650, 530, 310 $
- Ferry Z (in passengers): $ 620, 420, 580, 770, 500 $

Next, sum the total number of passengers for both ferries:
- **Total for Ferry A**: $ 870 + 760 + 650 + 530 + 310 = 3120 $
- **Total for Ferry Z**: $ 620 + 420 + 580 + 770 + 500 = 2890 $

Finally, find how many more passengers traveled on Ferry Z than Ferry A:
\[
\text{Difference} = 3120 - 2890 = 230 \text{ more passengers on Ferry A}
\]

It appears the problem asks for the difference, and the result is that **230 more passengers traveled on Ferry A than on Ferry Z**.

---

#### b. **Finding the cost of a trip on Ferry Z ($x$)**

- The total amount taken on Ferry A over this period can be calculated as:
\[
\text{Takings on Ferry A} = 3120 \times 12.50 = 39,000
\]
- The takings on Ferry Z are \$3.30 less:
\[
\text{Takings on Ferry Z} = 39,000 - 3.30 = 38,996.70
\]

The total number of passengers on Ferry Z is 2890. Therefore, to find $x$, the cost of a trip on Ferry Z:
\[
x = \frac{\text{Takings on Ferry Z}}{\text{Total Passengers on Ferry Z}} = \frac{38,996.70}{2890} \approx 13.50
\]

So, the cost of a trip on Ferry Z is approximately **\$13.50**.

---

#### c. **Least and greatest number of days with the same number of passengers**

We need to check when the two ferries carried exactly the same number of passengers, meaning the values in the table for Ferry A and Ferry Z should be the same (when multiplied by 10).

From the table:
- Ferry A (530 passengers) matches Ferry Z (500 passengers), as these two are closest.

This means there are **no days** when both ferries carried exactly the same number of passengers.

---

Would you like further details or have any specific questions about this solution? 

Here are 5 related questions you could explore:
1. How would you calculate the total revenue for both ferries if the costs increased by 10%?
2. How would rounding errors affect the calculations of passenger differences?
3. Can you graph the number of passengers for both ferries over the 14 days to see trends?
4. How would the problem change if additional costs or discounts were introduced?
5. How would the analysis change if the ferry service operated on more than two routes?

**Tip:** When solving problems involving a table of values, always convert and manipulate the raw data first before answering specific questions.

Over a 14-day period, data were collected on the number of passengers traveling on two ferries, A and Z. How many more passengers traveled on Ferry Z than on Ferry A? What is the cost of a trip on Ferry Z if the takings on Ferry Z were $3.30 less than on Ferry A? Find the least and greatest possible number of days when both ferries carried the same number of passengers.

Learn how to solve a multi-step word problem involving ferries A and Z. The problem includes calculating the difference in passengers between two ferries and finding the cost of a trip using arithmetic and proportional reasoning. Suitable for students in Grades 8-10.

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