Problem 11 - Elementary Algebra

Question

Calculate the area of a rectangle with dimensions \(41 / 2\) feet by 6 feet.

Step-by-Step Solution

Answer

The area of the rectangle is 123 square feet.

Step 1: Understanding the Problem

To find the area of a rectangle, we need to multiply its length by its width. The given dimensions are \(41/2\) feet and 6 feet for length and width respectively.

Step 2: Convert Mixed Number to Improper Fraction

The length is given as \(41/2\) which is a mixed number. To calculate, convert it to an improper fraction: \(41/2 = \frac{8 \times 5 + 1}{2} = \frac{41}{2}\).

Step 3: Multiply the Dimensions

Now, multiply the length and width to find the area: \[ \text{Area} = \frac{41}{2} \times 6 \] Simplify the multiplication: \[ \text{Area} = \frac{41 \times 6}{2} = \frac{246}{2} \].

Step 4: Simplify the Fraction

Now simplify the fraction: \(\frac{246}{2}\) simplifies to \(123\).

Step 5: State the Final Answer with Units

Thus, the area of the rectangle is \(123\) square feet.

Key Concepts

Multiplying Fractions

Whenever you want to determine the area of a rectangle using fractional dimensions, multiplying fractions is a must-learn skill. The formula for the area is straightforward: length multiplied by width. With fractional dimensions, you simply multiply the numerators together and the denominators together. This method makes it simple to work with fractions, whether they are proper or improper.

• Multiply the numerators: This gives the numerator of your product fraction.
• Multiply the denominators: This provides the denominator of your product fraction.

For instance, when you multiply \(\frac{41}{2}\) by 6, you treat 6 as \(\frac{6}{1}\). Therefore, you multiply the fractions like this: \[ \frac{41}{2} \times \frac{6}{1} = \frac{41 \times 6}{2 \times 1} \].Now, this process gives you the fraction \(\frac{246}{2}\), which you can further simplify. Understanding this concept ensures you can handle any fractional multiplication confidently.

Improper Fractions

Improper fractions are crucial when dealing with mixed numbers in calculations. A mixed number consists of a whole number and a fraction. In problems involving multiplication, especially for areas, it's often necessary to convert mixed numbers to improper fractions first. This makes the calculation process both easier and more systematic.
Here's how to convert a mixed number to an improper fraction:

• Multiply the denominator of the fraction by the whole number.
• Add the result to the numerator of the fraction.
• Write this sum over the original denominator. This is your improper fraction.

For example, the mixed number \(4\frac{1}{2}\) becomes \(\frac{9}{2}\) because \(4\times2+1=9\), making the fraction \(\frac{9}{2}\). When dealing with mixed numbers in any math problem, especially in finding areas, it's a good practice to handle improper fractions precisely.

Simplifying Fractions

The last important concept is simplifying fractions, which helps in achieving clean and understandable results. After multiplying fractions, you often need to simplify the result to its lowest form.
The process of simplification involves:

• Finding the greatest common divisor (GCD) of the numerator and the denominator.
• Dividing both the numerator and the denominator by this GCD.

For instance, you initially find \(\frac{246}{2}\) when calculating the area. Recognize that 2 is a divisor of 246, meaning you can simplify \(\frac{246}{2}\) to 123 by dividing both the numerator and the denominator by 2.
Understanding how to simplify fractions helps in making complex calculations much more straightforward and the answers easier to interpret.