Volume with Fractional Edge Lengths

Finding the volume of a rectangular prism when the sides are fractions or mixed numbers works exactly the same way as when the sides are whole numbers. You can solve these problems using the standard volume formula or by visualizing how many fractional unit cubes fit inside.

Using the Volume Formula

The formula for the volume of a rectangular prism is:

$V = l \times w \times h$

Where $l$ is length, $w$ is width, and $h$ is height.

Example: Find the volume of a rectangular prism with dimensions $2\frac{1}{2} \times 3 \times 1\frac{1}{4}$.

Step 1: Convert all numbers, including mixed numbers, to improper fractions.

• $2\frac{1}{2} = \frac{5}{2}$
• $3 = \frac{3}{1}$
• $1\frac{1}{4} = \frac{5}{4}$

Step 2: Multiply the fractions. Multiply the numerators together, and the denominators together:

$V = \frac{5}{2} \times \frac{3}{1} \times \frac{5}{4} = \frac{5 \times 3 \times 5}{2 \times 1 \times 4} = \frac{75}{8}$

Step 3: Convert back to a mixed number (if needed).

$\frac{75}{8} = 9\frac{3}{8}$

The volume is $9\frac{3}{8}$ cubic units.

Packing with Fractional Unit Cubes

Sometimes, you are asked to find volume by figuring out how many smaller, fractional cubes can pack into a larger box.

Example: How many $\frac{1}{2}$-inch cubes fit inside a rectangular box measuring $3$ inches by $2$ inches by $1$ inch?

Step 1: Find out how many cubes fit along each side. To do this, divide each side length of the box by the side length of the small cube ($\frac{1}{2}$ inch).

• Length: $3 \div \frac{1}{2} = 3 \times 2 = 6$ cubes
• Width: $2 \div \frac{1}{2} = 2 \times 2 = 4$ cubes
• Height: $1 \div \frac{1}{2} = 1 \times 2 = 2$ cubes

Step 2: Multiply the number of cubes. Now, multiply the number of cubes that fit along the length, width, and height:

$6 \times 4 \times 2 = 48$

Exactly $48$ of the $\frac{1}{2}$-inch cubes will fit perfectly inside the box.