A ferry boat holds 800 people. There are 260 people already on the boat. The owner of the boat wants to load several tour groups on. The tour groups come in sizes of 40 people or 50 people. The owner would like an equal number of each size of tour group on the boat. What is the greatest number of tour groups that can be loaded onto the boat? Use the verbal model to write an inequality. Then solve the inequality. What does this solution mean in this equation?

Number of people the boat can hold is greater than or equal to The Number of people already on the boat + Number of 40 people tour groups x 40 people + Number of 50 people tour groups x 50 people..

Please explain how to solve this in step by step detail

To solve this problem, let's break it down step by step:

Step 1: Let's assign variables to the unknown quantities in the problem.
- Let x represent the number of 40 people tour groups.
- Let y represent the number of 50 people tour groups.

Step 2: Write the inequality based on the given information.
The number of people the boat can hold is greater than or equal to the number of people already on the boat plus the number of 40 people tour groups multiplied by 40 people plus the number of 50 people tour groups multiplied by 50 people.

800 ≥ 260 + 40x + 50y

Step 3: Simplify the inequality.
Combine like terms on the right side of the inequality:
800 ≥ 260 + 40x + 50y

Step 4: Rearrange the inequality.
To make it easier to solve, let's subtract 260 from both sides of the inequality:
800 - 260 ≥ 40x + 50y

Simplifying further:
540 ≥ 40x + 50y

Step 5: Find the greatest number of tour groups that can be loaded onto the boat.
To find the greatest number of tour groups, we need to maximize either the number of 40 people tour groups (x) or the number of 50 people tour groups (y) within the given constraint.

For simplicity, let's first consider that no 40 people tour groups are loaded (x = 0). In this case, the inequality becomes:
540 ≥ 50y

To find the maximum number of 50 people tour groups, divide both sides of the inequality by 50:
10.8 ≥ y

Since the number of groups must be a whole number, the maximum number of 50 people tour groups is 10.

Now let's consider the case when no 50 people tour groups are loaded (y = 0). The inequality becomes:
540 ≥ 40x

Divide both sides of the inequality by 40:
13.5 ≥ x

Again, since the number of groups must be a whole number, the maximum number of 40 people tour groups is 13.

Step 6: Determine the common number of tour groups that can be loaded.
To satisfy the equal number of each size of tour group, we need to find the greatest common number between the maximum number of 40 people tour groups (13) and the maximum number of 50 people tour groups (10).

The greatest common number is 10.

Therefore, the greatest number of tour groups that can be loaded onto the boat is 10, with 10 groups of 40 people and 10 groups of 50 people.

In summary, the solution to the inequality is:
x = 10 (number of 40 people tour groups)
y = 10 (number of 50 people tour groups)

Step 1: Let's define the variables:

Let x be the number of 40 people tour groups.
Let y be the number of 50 people tour groups.

Step 2: Write the inequality:
The number of people the boat can hold is greater than or equal to the number of people already on the boat plus the number of 40 people tour groups times 40 plus the number of 50 people tour groups times 50.
800 ≥ 260 + 40x + 50y

Step 3: Simplify the inequality:
Rearrange the inequality to get all the terms on one side:
40x + 50y ≤ 800 - 260
40x + 50y ≤ 540

Step 4: Find the greatest number of tour groups:
Since we want an equal number of each size of tour group, we need to find the greatest possible value for both x and y. Both x and y can be either 0 or any positive integer.

Step 5: Find the greatest number of tour groups for x:
To find the greatest number of tour groups for x, we need to find the largest value of x that satisfies the inequality.
Let x = 0:
40(0) + 50y ≤ 540
0 + 50y ≤ 540
50y ≤ 540
y ≤ 10.8

The greatest integer value of y that satisfies the inequality when x = 0 is 10.

Step 6: Find the greatest number of tour groups for y:
To find the greatest number of tour groups for y, we need to find the largest value of y that satisfies the inequality.
Let y = 0:
40x + 50(0) ≤ 540
40x + 0 ≤ 540
40x ≤ 540
x ≤ 13.5

The greatest integer value of x that satisfies the inequality when y = 0 is 13.

Step 7: Determine the solution:
Both x and y can be either 0 or any positive integer up to the respective greatest values found.

The greatest number of tour groups that can be loaded onto the boat is when:
x = 13 (13 groups of 40 people)
y = 10 (10 groups of 50 people)

This means the owner can load 13 tour groups of 40 people and 10 tour groups of 50 people onto the boat.

To solve this problem, we'll start by setting up an inequality using the given information. Let's denote the number of 40 people tour groups as 'x' and the number of 50 people tour groups as 'y'.

According to the verbal model, the number of people the boat can hold (800) must be greater than or equal to the number of people already on the boat (260) plus the number of 40 people tour groups (x) multiplied by 40 people plus the number of 50 people tour groups (y) multiplied by 50 people.

So the inequality becomes:

800 ≥ 260 + 40x + 50y

To solve this inequality, we'll need to reduce it to a more simplified form. Start by subtracting 260 from both sides:

800 - 260 ≥ 260 + 40x + 50y - 260
540 ≥ 40x + 50y

Now, let's analyze the inequality. Since we want an equal number of each size of tour group, we need to find the greatest number of tour groups that can be loaded onto the boat. Therefore, we want to maximize the number of both 40 people and 50 people tour groups, which means finding the maximum value of 'x' and 'y'.

To do this, we'll consider the constraints of the problem. Each tour group can have a maximum of 40 people or 50 people, so 'x' and 'y' cannot be negative. Additionally, we can assume the variables are integers since we're dealing with whole groups.

Now, let's analyze the inequality further. To maximize the number of tour groups, we want to divide the 540 people (the excess capacity of the boat) equally between the 40 people and 50 people tour groups.

Let's assign 'x' and 'y' the same value, which we'll call 'n'. This means that there will be 'n' 40 people tour groups and 'n' 50 people tour groups. Therefore, we can rewrite the inequality as:

540 ≥ 40n + 50n

To simplify, combine like terms on the right side:

540 ≥ 90n

Divide both sides of the inequality by 90:

540/90 ≥ 90n/90
6 ≥ n

This tells us that 'n' must be less than or equal to 6 in order to satisfy the inequality.

So, the greatest number of tour groups that can be loaded onto the boat is 6, assuming the owner wants an equal number of each size of tour group.

In conclusion, setting up the inequality, solving it, and finding that 'n' must be less than or equal to 6 means that the owner can load a maximum of 6 tour groups onto the boat, with an equal number of 40 people and 50 people tour groups.