Problem 46 - Algebra for College Students

Question

Set up an algebraic equation and solve each problem. A blueprint has a scale where 1 inch represents 5 feet. Find the dimensions of a rectangular room that measures \(3 \frac{1}{2}\) inches by \(5 \frac{3}{4}\) inches on the blueprint.

Step-by-Step Solution

Answer

The real dimensions are 17.5 feet by 28.75 feet.

Step 1: Understand the Scale

The blueprint is scaled such that 1 inch on the blueprint is equivalent to 5 feet in reality. This means that to find the actual dimensions of the room, we'll need to multiply each dimension given in inches by 5.

Step 2: Convert Dimensions to Improper Fractions

First, convert the mixed numbers on the blueprint to improper fractions. The width is given as \(3 \frac{1}{2}\). Convert it to an improper fraction:\[3 \frac{1}{2} = \frac{7}{2}\]The length is given as \(5 \frac{3}{4}\). Convert it to an improper fraction:\[5 \frac{3}{4} = \frac{23}{4}\]

Step 3: Calculate the Real Width

Multiply the width in inches by the scale factor to get the actual width in feet:\[\text{Width} = \frac{7}{2} \times 5 = \frac{35}{2} = 17.5 \text{ feet}\]

Step 4: Calculate the Real Length

Multiply the length in inches by the scale factor to find the actual length in feet:\[\text{Length} = \frac{23}{4} \times 5 = \frac{115}{4} = 28.75 \text{ feet}\]

Step 5: Present Final Dimensions

The actual dimensions of the room are 17.5 feet by 28.75 feet.

Key Concepts

Improper Fractions

Improper fractions are useful when dealing with mixed numbers, especially in mathematical operations like multiplication or division. A mixed number consists of an integer and a fraction, like \(3 \frac{1}{2}\). To convert a mixed number to an improper fraction:

• Multiply the whole number by the denominator of the fraction.
• Add the numerator to the result.
• Place the total over the original denominator.

For example, to convert \(3 \frac{1}{2}\) we calculate:

• Whole number 3 multiplied by denominator 2 gives 6.
• Adding the numerator 1 gives 7.
• Thus, \( \frac{7}{2} \) is the improper fraction.

This conversion makes it easier to multiply by the scale factor in this exercise, as improper fractions straightforwardly let us apply arithmetic operations.

Scale Conversion

Scale conversion is a method used to translate measurements from one context to another using a set scale. Often seen in blueprints or maps, a scale defines how many real-world units are represented by a single unit on a drawing or model.To perform scale conversion:

• Identify the scale (e.g., 1 inch equals 5 feet).
• Multiply every measurement on the blueprint by the scale factor to find the real-world measurement.

In our blueprint problem:

• Every 1 inch on the blueprint corresponds to 5 feet in reality.
• For example, to find the real width from a blueprint width of \(\frac{7}{2}\) inches, multiply by 5 to get \(17.5\) feet.

This technique ensures that the proportions of the design are accurately reflected in the actual world dimensions.

Length and Width Calculations

Calculating length and width from a scaled drawing requires understanding both the scale factor and the dimensions given. It involves converting measurements, often in the form of mixed numbers or improper fractions, into real-world units.Follow these steps:

• Convert mixed numbers to improper fractions for easier calculations.
• Multiply the improper fraction by the scale factor to find the real-world dimension.

For the room dimensions in our example:

• The width \(3 \frac{1}{2}\) converts to \(\frac{7}{2}\). Multiply by 5 to find the actual width: \(17.5\) feet.
• The length \(5 \frac{3}{4}\) converts to \(\frac{23}{4}\). Multiply by 5 for the real length: \(28.75\) feet.

These straightforward calculations ensure accuracy when translating a blueprint into its actual size, crucial for construction and design.