Problem 12 - Prealgebra

Question

On a set of architectural drawings for a new school, the scale is \(\frac{1}{2}\) inch \(=\) 9 feet. Find the actual length of each room. $$\begin{array}{|l|c|}\hline \quad \quad \quad \text { Room } & \text { Drawing Distance } \\ \hline \text { Exercise Room } & 3 \frac{3}{4} \text { inches } \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

The actual length of the Exercise Room is 67.5 feet.

Step 1: Understand the Scale

The problem states that \( \frac{1}{2} \) inch on the drawing equals 9 feet in reality. This means for every \( \frac{1}{2} \) inch on the drawing, the actual length is 9 feet.

Step 2: Convert Mixed Number to Improper Fraction

The drawing distance given for the Exercise Room is \( 3 \frac{3}{4} \) inches. First, convert this mixed number to an improper fraction.\[3 \frac{3}{4} = \frac{15}{4} \text{ inches}\]

Step 3: Calculate Actual Length

Use the scale to find the actual length by setting up a proportion: \( \frac{1}{2} \) inch corresponds to 9 feet. Therefore, \( \frac{15}{4} \) inches will correspond to \( x \) feet:\[\frac{1}{2} : 9 = \frac{15}{4} : x\] Solve for \( x \) by cross-multiplying and simplifying: \[\left( \frac{1}{2} \right) x = 9 \times \frac{15}{4} \\frac{x}{2} = \frac{135}{4} \x = \frac{135}{4} \times 2\x = \frac{270}{4} = 67.5\]

Step 4: Interpret the Result

The calculated value means the actual length of the Exercise Room is 67.5 feet according to the scale.

Key Concepts

Understanding Proportion

A proportion is a statement that two ratios are equal. It acts as a mathematical bridge that helps us understand relationships between different quantities. In our exercise, we use the given scale to create a proportion. Each part of the scale represents a real world distance compared to a distance on a map or drawing. This is essential in fields such as architecture and engineering, where you might want to know the real-life size based on a model or plan. To set up a proportion, follow these steps:

• Identify the relationship or scale. For example, in the exercise: 1/2 inch equals 9 feet.
• Convert the drawing measurement to something comparable. This might require changing a mixed number to an improper fraction, as is necessary in this problem.
• Finally, solve for the unknown by cross-multiplying and simplifying the expression, which helps keep the relationship balanced.

Using proportions lets us extrapolate or determine any missing value when one side of a ratio and a relation are known.

Converting Mixed Numbers to Improper Fractions

When you're dealing with mixed numbers in complex problems, it's often easier to convert them into improper fractions. A mixed number is composed of a whole number and a fraction, making calculations cumbersome.Here's how to convert a mixed number into an improper fraction:

• Multiply the whole number by the denominator of the fraction part.
• Add this product to the numerator of the fraction part.
• The total becomes the new numerator, while the denominator remains the same.

For instance, to convert the mixed number \(3 \frac{3}{4}\) into an improper fraction:

• Multiply the whole number 3 by the denominator 4, which gives 12.
• Add the numerator 3 to that product, reaching a sum of 15.
• This sum, 15, becomes the new numerator, over the original denominator, 4, to form \(\frac{15}{4}\).

This conversion makes subsequent calculations, like setting up a proportion, much more straightforward.

Cross-Multiplying in Proportions

Cross-multiplying is a useful technique in solving proportions. By cross-multiplying, you eliminate the fractions, which makes it easier to solve for the unknown variable.To cross-multiply, follow these straightforward steps:

• Set up your proportion, with two fractions set equal to each other.
• Multiply the numerator of the first fraction by the denominator of the second fraction.
• Then, multiply the denominator of the first fraction by the numerator of the second fraction.

Take, for example, the proportion from the exercise:\[ \frac{1}{2} : 9 = \frac{15}{4} : x\]Cross-multiply to solve for \(x\):

• Multiply \(\frac{1}{2} \times x\), which is \(\frac{x}{2}\).
• Then multiply \(9 \times \frac{15}{4}\) to get \(\frac{135}{4}\).
• Finally, isolate \(x\) by multiplying \(\frac{135}{4}\) by 2, resulting in \(x = \frac{270}{4} = 67.5\).

This process cuts through the complexity, allowing you to find the solution with clarity.