Problem 17 - 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. Length of a Boxcar How long is an actual boxcar that has an HO scale model 5 inches long? Give your answer in inches, then divide by 12 to give the answer in feet.

Step-by-Step Solution

Answer

The actual boxcar is 435 inches or 36.25 feet long.

Step 1: Understand the Problem

We have a model train boxcar using the HO scale, which means the ratio of the model to the actual boxcar is 1:87. The length of the model is 5 inches. We need to find the actual length of the boxcar.

Step 2: Set Up the Proportion

Since the scale is 1:87, the model to actual dimension ratio is \( \frac{1}{87} \). Set up the proportion based on the scale, which is \( \frac{5}{x} = \frac{1}{87} \), where \( x \) is the actual length of the boxcar in inches.

Step 3: Solve the Proportion

To find \( x \), cross-multiply: \( 5 \times 87 = 1 \times x \). This simplifies to \( x = 5 \times 87 \). Calculate \( x \).

Step 4: Calculate the Actual Length

Multiply 5 by 87 to find the actual length of the boxcar in inches: \( x = 435 \). The actual boxcar is 435 inches long.

Step 5: Convert Inches to Feet

To convert inches to feet, divide the actual boxcar length by 12: \( \frac{435}{12} \approx 36.25 \). The actual boxcar is approximately 36.25 feet long.

Key Concepts

Ratios

Ratios are a way to compare two quantities, showing the relationship between them. When you think of ratios, imagine them as a fraction or a comparison between two numbers. This idea is used in many areas, such as cooking, mathematics, and even in building things like model trains.
For example, in the model train exercise, the ratio of 1:87 indicates a miniature version's proportion compared to the real-life item. This means for every 1 unit (like an inch) of the model train, the actual train measures 87 units. Ratios are vital in scaling because they help us easily understand how much larger or smaller one thing is compared to another.

• A ratio of 1:87 means the model is 1/87th the size of the real object.
• Ratios can also be written as fractions, like \( \frac{1}{87} \).
• They help in converting measurements proportionally from one scale to another.

Understanding ratios allows you to measure accurately in real-world applications, ensuring the scale of models like those in model trains is consistent and true to life.

Model Train Scales

Model train scales depict the ratio between the size of the model and the real-world object it's based on. These scales are standardized, so anyone building or collecting model trains can ensure accuracy in size.
In this context, the HO scale is a common scale used in model railroading, noted for its ratio of 1:87. This means the model is 1/87th the size of the real train. Scales are significant because they bring uniformity across the modeling industry, allowing for interchangeable components and parts that fit well together.

• HO scale is popular due to its manageable size and high detail level.
• Understanding model scales helps in constructing realistic train sets.
• Different scales cater to different space requirements and detail preferences.

Whether a hobbyist or a serious collector, knowing the train scale ensures you can create scenes that look both cohesive and authentic.

Length Conversion

Length conversion involves changing measurements from one unit to another. In our problem, converting from inches to feet is an example of this, and it's an important skill in making sure measurements are easy to relate to real-world sizes.
For the boxcar problem, once the actual length in inches is found, converting it to feet provides a clearer perspective on its size using a more commonly used unit. Since there are 12 inches in a foot, this conversion is straightforward:

• To convert inches to feet, divide the inches by 12.
• For example, \(\frac{435}{12} \approx 36.25\), meaning the boxcar is roughly 36.25 feet long.
• Conversion helps in visualizing and communicating dimensions efficiently.

Mastery of length conversions eases the transition between different forms of measurement, facilitating better understanding and communication of size-related information.

Problem Solving Steps

Approaching a problem with a clear set of steps makes finding a solution much easier. For mathematical problems, following an organized procedure helps in avoiding mistakes and providing clear answers.
For the model train boxcar problem, the five-step process outlined ensures clarity:

• Understand the Problem: Identify what you know and what you need to find out.
• Set Up the Proportion: Use the ratio to create an equation expressing the relationship between model and real-life size.
• Solve the Proportion: Cross-multiply and solve for the unknown variable.
• Calculate the Actual Length: Compute the results and check for accuracy.
• Convert to Required Units: Change the measurements to a desired unit, like from inches to feet.

This systematic approach not only applies to model train problems but is also a solid framework for tackling many other mathematical challenges, helping you solve them efficiently and correctly.