Volume of Rectangular Prisms with Fractional and Mixed-Number Dimensions

🔑 Key Concepts

• The volume of a rectangular prism is found using \(V=lwh\).
• Fractional and mixed-number dimensions are multiplied in the same way as other rational numbers.
• Rename each mixed number as an improper fraction before multiplying.
• Look for common factors that can be simplified before multiplying the numerators and denominators.
• The final answer is written in cubic units because three dimensions are multiplied.
• The order of the length, width, and height factors does not change the product.

✏️ Worked Examples

Example 1: One Whole-Number Dimension

Find the volume of a rectangular prism measuring \(2\frac{2}{3}\) inches by \(3\frac{1}{2}\) inches by \(6\) inches.

Begin with the volume formula:

\(V=lwh\)

\(\displaystyle V= 2\frac{2}{3} \times 3\frac{1}{2} \times 6 \)

Rename the mixed numbers as improper fractions:

\(\displaystyle 2\frac{2}{3}=\frac{8}{3} \qquad 3\frac{1}{2}=\frac{7}{2} \)

\(\displaystyle V= \frac{8}{3} \times \frac{7}{2} \times \frac{6}{1} \)

Simplify common factors before multiplying:

\(\displaystyle \frac{6}{3}=2\)

\(\displaystyle \frac{2}{2}=1\)

\(V=8\times7\)

\(V=56\)

Volume:
\(\boxed{56\text{ in}^3}\)

Example 2: Three Mixed-Number Dimensions

Find the volume of a rectangular prism measuring \(2\frac{2}{3}\) inches by \(5\frac{5}{8}\) inches by \(1\frac{3}{5}\) inches.

\(V=lwh\)

\(\displaystyle V= 2\frac{2}{3} \times 5\frac{5}{8} \times 1\frac{3}{5} \)

Rename all three mixed numbers:

\(\displaystyle 2\frac{2}{3}=\frac{8}{3} \)

\(\displaystyle 5\frac{5}{8}=\frac{45}{8} \)

\(\displaystyle 1\frac{3}{5}=\frac{8}{5} \)

\(\displaystyle V= \frac{8}{3} \times \frac{45}{8} \times \frac{8}{5} \)

Simplify before multiplying:

\(\displaystyle \frac{8}{8}=1, \qquad \frac{45}{5}=9, \qquad \frac{9}{3}=3 \)

\(V=1\times3\times8\)

\(V=24\)

Volume:
\(\boxed{24\text{ in}^3}\)

Review: Multiplying Mixed Numbers

1. Rename

Change each mixed number into an improper fraction.

\(\displaystyle a\frac{b}{c} = \frac{ac+b}{c} \)

2. Simplify

Divide common factors from numerators and denominators before multiplying.

\(\displaystyle \frac{\cancel{8}}{3} \times \frac{45}{\cancel{8}} \)

3. Multiply

Multiply the remaining numerators and denominators.

\(\displaystyle \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd} \)

Why Is the Answer Measured in Cubic Inches?

Volume multiplies three one-dimensional measurements:

\(\text{inches}\times\text{inches}\times\text{inches}\)

\(\text{in}\times\text{in}\times\text{in}=\text{in}^3\)

The exponent \(3\) indicates that the measurement describes three-dimensional space rather than a one-dimensional length or two-dimensional area.

🧠 Math Vocabulary

• Rectangular prism: A three-dimensional solid with six rectangular faces.
• Dimension: A measurable direction, such as length, width, or height.
• Volume: The amount of three-dimensional space contained inside a solid.
• Cubic unit: A unit used to measure volume, such as a cubic inch or \(\text{in}^3\).
• Fractional dimension: A length, width, or height represented by a fraction rather than a whole number.
• Mixed number: A number containing a whole-number part and a fractional part.
• Improper fraction: A fraction whose numerator is greater than or equal to its denominator.
• Rename: To write a number in an equivalent form, such as renaming a mixed number as an improper fraction.
• Common factor: A number that divides evenly into two or more values.
• Simplify: To rewrite an expression in an equivalent but less complicated form.
• Product: The result of multiplication.
• Rational number: A number that can be written as a ratio of two integers.

💡 Main Idea

The formula \(V=lwh\) applies even when the dimensions of a rectangular prism are fractions or mixed numbers. Rename mixed numbers as improper fractions, simplify common factors, multiply the remaining values, and label the final answer with cubic units.

📚 What You Should Already Know

Students should know how to identify the dimensions of a rectangular prism, rename mixed numbers as improper fractions, multiply fractions, simplify common factors, and distinguish between linear, square, and cubic units.

🚀 What Comes Next

Students can extend this reasoning by solving for missing prism dimensions, comparing prisms with equal volumes, working with decimal edge lengths, and applying \(V=Bh\) to other prisms and cylinders.

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