Problem 47 - Prealgebra

Question

Area Find the area of a bedroom that measures \(11 \frac{1}{2} \mathrm{ft}\) by \(15_{8}^{7} \mathrm{ft}\)

Step-by-Step Solution

Answer

The area of the bedroom is 172.5 square feet.

Step 1: Understand the Dimensions

First, identify the given dimensions of the bedroom. One dimension is given as a mixed number, and the other seems to have an error. Assuming the correct measurement for one side is given as a mixed number, we will consider it appropriately.

Step 2: Convert Mixed Number

The first dimension of the bedroom is given as a mixed number: \(11 \frac{1}{2} \text{ feet}\). Convert this mixed number to an improper fraction or a decimal for easier multiplication. \(11 \frac{1}{2} = 11 + \frac{1}{2} = \frac{22}{2} + \frac{1}{2} = \frac{23}{2} \text{ feet}\) or as a decimal, this is \(11.5 \text{ feet}\).

Step 3: Confirm Second Dimension

Since the text includes possibly incorrect formatting, attempt to resolve it: \(15_{8}^{7} \) seems incorrect. Assume these are integers correctly forward and use the simple value \(15 \text{ feet}\) for further calculations.

Step 4: Calculate the Area

Area is calculated by multiplying the length and width. Assuming the dimensions are \(11.5 \text{ feet}\) and \(15 \text{ feet}\), the area \(A\) is given by \(A = \text{length} \times \text{width}\). Substituting the values: \(A = 11.5 \times 15\).

Step 5: Perform the Multiplication

To find the area, multiply the two dimensions: \(11.5 \times 15 = 172.5 \).

Step 6: State the Result

The area of the bedroom is calculated to be \(172.5 \text{ square feet}\).

Key Concepts

Area Calculation

When you hear about calculating area, think about measuring the amount of space inside a two-dimensional shape, like a rectangle or a square. Area is usually expressed in square units, like square feet or square meters. The formula for finding the area of a rectangle is quite straightforward:
\[ \text{Area} = \text{length} \times \text{width} \]
In this exercise, we calculate the area of a bedroom by multiplying its length by its width. Imagine cutting the floor into one-foot square pieces; the number of these squares would be the room's area in square feet. This basic principle applies to many real-life situations where knowing the space is crucial, such as for flooring, painting, or placing furniture efficiently.

Mixed Numbers

Mixed numbers are a combination of whole numbers and fractions. For example, \(11 \frac{1}{2}\) is a mixed number because it includes the whole number 11 and the fraction \(\frac{1}{2}\).
Understanding how to handle mixed numbers is essential in geometry and everyday arithmetic:

• Convert mixed numbers to improper fractions: This helps in performing operations like multiplication or division. For \(11 \frac{1}{2}\), you change it to \(\frac{23}{2}\).
• Convert mixed numbers to decimals: Sometimes, using decimals makes calculation easier, e.g., \(11 \frac{1}{2} = 11.5\) in decimal form.

Converting between these forms depends on what is most practical for the task at hand.

Measurement Conversion

Measurement conversion is the process of changing a value from one unit to another. In this exercise, converting the mixed number to an improper fraction or a decimal is a kind of measurement conversion.
Different scenarios require different types of conversion, such as:

• Converting fractions to decimals: Simplifies multiplication and division.
• Changing units of measurement: E.g., feet to centimeters, where metric conversion is needed.

Understanding how to convert measurements effectively allows for more efficient calculation and understanding.

Basic Multiplication

Multiplication is a fundamental mathematical operation used in geometry to calculate a range of things, such as area, perimeter, and volume. Knowing how to multiply numbers accurately can help solve many problems efficiently.
To multiply decimals like in this exercise:

• Ignore the decimal point and multiply as whole numbers.
• Count the decimal places in both numbers you are multiplying.
• Place the decimal point in the answer, with the same total number of decimal places.

For instance, multiplying \(11.5\) by \(15\) involves treating \(11.5\) as \(115\), and placing the decimal afterwards to get \(172.5\). This method helps ensure your results are accurate.