Problem 329 - Elementary Algebra

Question

In the following exercises, solve. Five student government officers want to go to the state convention. It will cost them \(\$ 110\) for registration, \(\quad \$ 375\) for transportation and food, and \(\$ 42\) per person for the hotel. There is \(\$ 450\) budgeted for the convention in the student government savings account. They can earn the rest of the money they need by having a car wash. If they charge \(\$ 5\) per car, how many cars must they wash in order to have enough money to pay for the trip?

Step-by-Step Solution

Answer

The students need to wash 49 cars.

Step 1: Calculate Total Registration, Transportation, and Food Costs

First, find the combined cost of registration and transportation plus food. Add together the registration cost of \$110\ and the transportation and food cost of \$375\. \[ 110 + 375 = 485 \]

Step 2: Calculate Total Hotel Cost

Next, find the total cost for the hotel. There are 5 students, and it costs \$42\ per student for the hotel. Multiply \$42\ by 5: \[ 42 \times 5 = 210 \]

Step 3: Calculate Overall Total Cost

Now, add the combined cost of registration and transportation/food to the total hotel cost to get the overall total cost for the trip. \[ 485 + 210 = 695 \]

Step 4: Determine How Much Money is Needed

Subtract the amount already budgeted from the overall total cost to find out how much more money is needed. The budgeted amount is \$450\. \[ 695 - 450 = 245 \]

Step 5: Calculate Number of Cars Needed to be Washed

Finally, divide the extra amount needed by the charge per car to find out how many cars the students need to wash. \[ \frac{245}{5} = 49 \]

Key Concepts

linear equations

Understanding linear equations is essential for solving many algebraic word problems. In this exercise, a linear equation helps to calculate the number of cars that must be washed to cover the trip expenses. Linear equations are mathematical expressions representing a line in the form of \ y = mx + c \.

In your problem, the linear equation will help you understand the relationship between the money required and the number of cars washed.
Let's breakdown the relationship:
- The total cost of the trip is calculated to be \(695.
- The student government already has \)450, so the remaining amount needed is \(245.
- The cost per car wash is \)5.
The equation to solve would be: \550 - 450 = 245 \

cost calculation

Cost calculation is a key part of budgeting and planning. In this problem, there are multiple costs to consider, including registration, transportation, food, and hotel.
Here are the steps:

• First, combine the registration cost (\(110) and transportation plus food cost (\)375). This gives a subtotal of \(485.
• Next, calculate the total hotel cost. For 5 students in hotels costing \)42 each, the total hotel expense is \(210.
• Finally, sum the registration and transportation/food subtotal with the hotel total to get the overall trip cost (\)485 + \(210 = \)695).

budget analysis

Budget analysis involves comparing your available resources to your expenses to determine if you have enough to meet your needs. In the exercise, budget analysis helps to determine how much additional money is required for the trip.
To analyze the budget:
- Determine the overall cost of the trip (\(695).
- Identify the amount currently budgeted (\)450).
- Subtract the budgeted amount from the total cost (\(695 - \)450 = \(245).
The result shows that the students need an extra \)245 to cover the trip.

problem-solving steps

Effective problem-solving involves breaking down the problem into smaller, manageable steps. This makes it easier to follow and ensure accuracy. Here's a step-by-step approach for the given problem:

• Step 1: Calculate the combined registration, transportation, and food costs by adding \(110 (registration) and \)375 (transportation and food). This gives \(485.
• Step 2: Calculate the total hotel cost by multiplying the number of students (5) by the hotel cost per student (\)42), resulting in \(210.
• Step 3: Add the combined cost (\)485) to the total hotel cost (\(210) to get the overall trip cost (\)695).
• Step 4: Subtract the budgeted amount (\(450) from the overall cost to find the amount still needed (\)695 - \(450 = \)245).
• Step 5: Divide the additional amount needed (\(245) by the charge per car wash (\)5) to find the number of cars they need to wash (49 cars).