Problem 3 - Prealgebra

Question

The formula for the area \(A \text { of a rectangle with length } I \text { and width } w \text { is } A=I \cdot w . \text { Find } A \text { if: [Examples } 1-4]\) \(I=\frac{3}{2}\) inch and \(w=\frac{3}{4}\) inch

Step-by-Step Solution

Answer

The area is \( \frac{9}{8} \) or 1.125 square inches.

Step 1: Understand the Formula

The formula for the area of a rectangle is given by \( A = I \times w \), where \( I \) is the length and \( w \) is the width of the rectangle.

Step 2: Substitute Values into the Formula

Substitute the given values of length and width into the formula: \( A = \frac{3}{2} \times \frac{3}{4} \).

Step 3: Multiply the Fractions

To multiply the fractions \(\frac{3}{2}\) and \(\frac{3}{4}\), multiply the numerators together and the denominators together: \( A = \frac{3 \times 3}{2 \times 4} = \frac{9}{8} \).

Step 4: Simplify the Answer

The fraction \( \frac{9}{8} \) is already in its simplest form, which is equivalent to 1.125 inches squared.

Key Concepts

Multiplication of Fractions

When multiplying fractions, the process is straightforward. You will multiply the numerators together to get the new numerator and multiply the denominators together to get the new denominator. This is a fundamental skill when working with areas in geometry, especially when dimensions involve fractions.

Consider the example from our exercise:

• One fraction is \( \frac{3}{2} \)
• Another fraction is \( \frac{3}{4} \)

To find the product, multiply the tops (numerators) and multiply the bottoms (denominators):

• Numerators: \( 3 \times 3 = 9 \)
• Denominators: \( 2 \times 4 = 8 \)

So, the product of these fractions is \( \frac{9}{8} \). This means the area of the rectangle is \( \frac{9}{8} \) or 1.125 square inches.

Geometry in Prealgebra

Geometry often involves shapes like rectangles, and understanding how to work with these shapes is crucial in prealgebra. In this problem, you are asked to find the area of a rectangle, which is a basic geometric concept. The area is a measure of how much surface the shape covers.

The formula for finding the area of a rectangle is:

• Area \( A = I \times w \)
• \( I \) is the length of the rectangle.
• \( w \) is the width of the rectangle.

To find the area, you simply multiply the length by the width. When working with fractions, this process helps reinforce skills in fraction multiplication. This is an integral part of geometry in prealgebra.
Learning to visualize and calculate dimensions builds a foundation for more complex geometry in later math courses.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form. This means the numerator and the denominator have no common factors other than one. In our example, the fraction \( \frac{9}{8} \) represents an improper fraction, where the numerator is larger than the denominator.

In some instances, you may need to simplify further by converting the improper fraction into a mixed number:

• Divide the numerator by the denominator: \( 9 \div 8 = 1\) remainder \(1\)
• This gives us the mixed number of \(1 \frac{1}{8}\).

Although \( \frac{9}{8} \) is not reducible in its fractional form, converting to a decimal (1.125) also shows another simplified representation, combining concepts of fraction, mixed numbers, and decimal equivalency.