Problem 23 - Prealgebra

Question

Model Trains The size of a model train relative to an actual train is referred to as its scale. Each scale is associated with a ratio as shown in the table. For example, an HO model train has a ratio of 1 to 87, meaning it is \(\frac{1}{87}\) as large as an actual train. Travel Expenses If a car travels 378.9 miles on 50 liters of gas, how many liters of gas will it take to go 692 miles if the car travels at the same rate? (Round to the nearest tenth.)

Step-by-Step Solution

Answer

The car will need approximately 91.3 liters of gas to travel 692 miles.

Step 1: Determine the Car's Mileage

First, calculate how many miles the car can travel per liter. This is done by dividing the total miles traveled by the number of liters used. \[ \text{Mileage per liter} = \frac{378.9 \text{ miles}}{50 \text{ liters}} = 7.578 \text{ miles per liter} \]

Step 2: Setup Proportion for New Distance

Now, set up a proportion to find out how many liters are needed to travel 692 miles, knowing the mileage per liter from the previous step:\[ \text{Liters needed} = \frac{692 \text{ miles}}{7.578 \text{ miles per liter}} \]

Step 3: Calculate Liters Required

Solve the proportion to find out the number of liters needed to travel 692 miles:\[ \text{Liters needed} = \frac{692}{7.578} \approx 91.324 \text{ liters} \]

Step 4: Round to the Nearest Tenth

Finally, round the calculated number of liters to the nearest tenth to get the answer:\[ \text{Liters needed (rounded)} \approx 91.3 \text{ liters} \]

Key Concepts

Scale Models

Scale models are miniature representations of larger objects, retaining accurate proportions to the original. These models can vary in purpose, from hobby crafts to industrial design. A scale is expressed as a ratio, comparing the size of the model to the real-world object. For instance, if a model has a scale of 1:87, like the HO model trains, this means that one unit on the model is equivalent to 87 units on the actual object.

This concept allows for the creation of detailed models, whether it's a train, ship, or any other object, that are much smaller yet accurately proportionate. It is essential to understand the associated ratios to properly build or evaluate these models. In essence, scale models provide a practical way to appreciate and understand structures without needing the space or resources required for their full-scale counterparts.

Unit Conversion

Unit conversion is a fundamental skill in mathematics and science, allowing different units of measurement to be translated into one another. This skill is crucial when solving problems that involve measurements in various units. For example, measuring gas consumption in liters versus calculating distances in miles requires efficient unit conversion to maintain consistency across calculations.

To successfully perform unit conversions:

• Identify the units you have and the units you want to convert to.
• Use a conversion factor, which is a ratio that expresses how many of one unit are equal to another unit.
• Multiply the original measurement by this conversion factor.

This process helps in various applications, such as calculating the fuel efficiency of a vehicle in different countries where units may differ, ensuring accurate and reliable solutions.

Problem Solving Steps

Problem-solving requires a structured approach to break down complex tasks into manageable steps. Let's dissect this through a practical scenario involving travel expenses:

1. **Understand the Problem**: Identifying what is known and what needs to be found. In this case, we know the distance and fuel efficiency, but we need to determine the fuel required for a new distance.

2. **Develop a Plan**: Setting up a proportion based on known relationships, such as miles per liter, helps create an equation to find the unknown quantity.

3. **Execute the Plan**: Perform calculations step by step, ensuring each arithmetic operation follows logically from the last. Calculating the required liters of fuel for a specified distance using the vehicle's mileage per liter, as shown in the example.

4. **Review/Check**: After solving, review the calculations to ensure accuracy. Rounding to the nearest tenth simplifies the answer while maintaining precision.

By following these systematic steps, problem-solving becomes less daunting and more manageable, leading to more accurate results and deeper understanding.