Problem 23 - Prealgebra

Question

Montego and Luisa are rewriting the scale 1 inch \(=2\) feet in \(a: b\) form. Montego wrote 1: 2 and Luisa wrote \(1: 24 .\) Who is correct? Explain your reasoning.

Step-by-Step Solution

Answer

Luisa is correct; the correct scale as a ratio in terms of inches is 1:24.

Step 1: Understand the Initial Scale

The problem provides a scale where 1 inch equals 2 feet. Here, 1 unit in inches corresponds to 2 units in feet for the given distance. This means that if you have a model or drawing using these dimensions, every inch represents 2 feet in real life.

Step 2: Convert Feet to Inches

Since the scale involves feet and we need to express the ratio in the same units, convert 2 feet to inches. There are 12 inches in 1 foot, so 2 feet is equal to 2 \(\times\) 12 = 24 inches.

Step 3: Write the Ratio in a:b Form

With the conversion, the problem becomes comparing 1 inch to 24 inches (because 2 feet is 24 inches). Therefore, the ratio in inches is 1:24.

Step 4: Compare the Ratios

Montego wrote the ratio as 1:2, meaning he understood the scale as is without converting feet to inches. Luisa wrote 1:24, accurately converting the feet into inches first. Thus, Luisa's ratio correctly represents the scale in terms of inches.

Key Concepts

Understanding Scale Conversion

Imagine you have a blueprint or model, and you need to relate it to real-world dimensions. This connection between the model and reality is known as a "scale." To convert scales correctly, you need to ensure the units match up perfectly.

In the problem, the given scale was 1 inch equating to 2 feet. When converting scales, maintaining consistent units is critical. Using the same type of measurement like inches or feet ensures the scale accurately represents the desired measurement in both real life and the model.

Luisa, in the exercise, converted the 2 feet into inches before expressing the ratio, thus ensuring the scale conversion was accurate. Remember, consistent units in scale conversion make it simpler to grasp and apply.

Mastering Measurement Units

Measurement units are fundamental in understanding and working with scales or any quantitative problem. Knowing the conversion between units like inches and feet is essential for solving problems involving measurements.

Let's break down the conversion used in the exercise:

• 1 foot = 12 inches
• Thus, 2 feet = 2 \(\times\) 12 inches = 24 inches

Converting all units to be the same type, here into inches, helps in stating the ratio clearly and effectively. Using a consistent unit across comparisons avoids confusion and ensures accuracy.

Whenever faced with different measurement units, always aim for consistency by converting them to the same standard. This practice simplifies understanding and solving any measurement-related problems.

Effective Problem-Solving Steps

Following clear problem-solving steps makes any problem much easier to tackle. Here's a structured approach adapted from the exercise above:

• Understand the problem: Identify what is given and what is required. In this scenario, a scale of 1 inch equating to 2 feet was the starting point.
• Convert units where necessary: Translate the units to match, ensuring calculations are based on consistent measurements, such as converting feet into inches.
• Express in ratio form: Once conversion is complete, state the ratio in the simplest form possible. This often involves writing it in \(a: b\) form, like 1:24 in this context.
• Verify the result: Double-check the conversion and applied methods to ensure the ratio correctly reflects the information provided.

By following these steps, you not only reach the correct solution but also understand the reasoning behind each step. This builds confidence in tackling similar problems in the future.