Volume and Surface Area worksheet with problems involving unit fraction cubes and rectangular prisms.

Let's solve each problem step by step. The worksheet asks us to:

1. Determine how many unit fraction cubes fit into each solid.
2. Find the volume of each solid.

We are given a unit fraction cube in each case (e.g., $ \frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} $ or $ \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} $), and we need to use that to compute both the number of such cubes that fit into the larger solid and the volume.

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Problem 1a

Dimensions of the solid:
Length = $ 3\frac{2}{3} = \frac{11}{3} $
Width = $ 3 $
Height = $ 2\frac{1}{3} = \frac{7}{3} $

Unit fraction cube: $ \frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} $

#### Step 1: How many unit fraction cubes fit into the solid?

We divide each dimension by $ \frac{1}{3} $ to find how many $ \frac{1}{3} $-units fit along each side.

- Length: $ \frac{11}{3} \div \frac{1}{3} = 11 $
- Width: $ 3 \div \frac{1}{3} = 9 $
- Height: $ \frac{7}{3} \div \frac{1}{3} = 7 $

Total number of unit fraction cubes:
$$
11 \times 9 \times 7 = 693
$$

✔ Answer to 1a.1: 693 unit fraction cubes

#### Step 2: Volume of the solid

Volume = $ \text{length} \times \text{width} \times \text{height} $

$$
\frac{11}{3} \times 3 \times \frac{7}{3} = \frac{11 \times 3 \times 7}{3 \times 3} = \frac{231}{9} = 25\frac{6}{9} = 25\frac{2}{3}
$$

Alternatively, since each unit fraction cube has volume $ \left(\frac{1}{3}\right)^3 = \frac{1}{27} $, and there are 693 such cubes:

$$
693 \times \frac{1}{27} = \frac{693}{27} = 25\frac{18}{27} = 25\frac{2}{3}
$$

✔ Answer to 1a.2: $ 25\frac{2}{3} $ cubic units

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Problem 1b

Dimensions of the solid:
Length = $ 3\frac{1}{2} = \frac{7}{2} $
Width = $ \frac{1}{2} $
Height = $ 1\frac{1}{2} = \frac{3}{2} $

Unit fraction cube: $ \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} $

#### Step 1: How many unit fraction cubes fit into the solid?

Divide each dimension by $ \frac{1}{2} $:

- Length: $ \frac{7}{2} \div \frac{1}{2} = 7 $
- Width: $ \frac{1}{2} \div \frac{1}{2} = 1 $
- Height: $ \frac{3}{2} \div \frac{1}{2} = 3 $

Total number of unit fraction cubes:
$$
7 \times 1 \times 3 = 21
$$

✔ Answer to 1b.1: 21 unit fraction cubes

#### Step 2: Volume of the solid

$$
\frac{7}{2} \times \frac{1}{2} \times \frac{3}{2} = \frac{21}{8} = 2\frac{5}{8}
$$

Or using unit cubes: $ 21 \times \left(\frac{1}{2}\right)^3 = 21 \times \frac{1}{8} = \frac{21}{8} = 2\frac{5}{8} $

✔ Answer to 1b.2: $ 2\frac{5}{8} $ cubic units

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Problem 2a

Dimensions of the solid:
All sides = $ 2\frac{1}{2} = \frac{5}{2} $

So it's a cube with side $ \frac{5}{2} $

Unit fraction cube: $ \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} $

#### Step 1: How many unit fraction cubes fit into the solid?

Each side: $ \frac{5}{2} \div \frac{1}{2} = 5 $

So total cubes:
$$
5 \times 5 \times 5 = 125
$$

✔ Answer to 2a.1: 125 unit fraction cubes

#### Step 2: Volume of the solid

$$
\left(\frac{5}{2}\right)^3 = \frac{125}{8} = 15\frac{5}{8}
$$

Or: $ 125 \times \frac{1}{8} = \frac{125}{8} = 15\frac{5}{8} $

✔ Answer to 2a.2: $ 15\frac{5}{8} $ cubic units

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Problem 2b

Dimensions of the solid:
Length = $ 3\frac{2}{3} = \frac{11}{3} $
Width = $ 2\frac{1}{3} = \frac{7}{3} $
Height = $ 1\frac{1}{3} = \frac{4}{3} $

Unit fraction cube: $ \frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} $

#### Step 1: How many unit fraction cubes fit into the solid?

Divide each dimension by $ \frac{1}{3} $:

- Length: $ \frac{11}{3} \div \frac{1}{3} = 11 $
- Width: $ \frac{7}{3} \div \frac{1}{3} = 7 $
- Height: $ \frac{4}{3} \div \frac{1}{3} = 4 $

Total cubes:
$$
11 \times 7 \times 4 = 308
$$

✔ Answer to 2b.1: 308 unit fraction cubes

#### Step 2: Volume of the solid

$$
\frac{11}{3} \times \frac{7}{3} \times \frac{4}{3} = \frac{308}{27} = 11\frac{11}{27}
$$

Or: $ 308 \times \frac{1}{27} = \frac{308}{27} = 11\frac{11}{27} $

✔ Answer to 2b.2: $ 11\frac{11}{27} $ cubic units

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✔ Final Answers Summary:

#### 1a
1. 693 unit fraction cubes
2. Volume = $ 25\frac{2}{3} $ cubic units

#### 1b
1. 21 unit fraction cubes
2. Volume = $ 2\frac{5}{8} $ cubic units

#### 2a
1. 125 unit fraction cubes
2. Volume = $ 15\frac{5}{8} $ cubic units

#### 2b
1. 308 unit fraction cubes
2. Volume = $ 11\frac{11}{27} $ cubic units

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🔍 Explanation:

- A unit fraction cube is a small cube with side length equal to a fraction (like $ \frac{1}{2} $ or $ \frac{1}{3} $).
- To find how many fit into the solid, divide each dimension of the solid by the side of the unit cube and multiply.
- The volume can be found either by multiplying the dimensions directly or by multiplying the number of unit cubes by the volume of one unit cube.
- This method helps visualize volume as a count of smaller cubes, which is especially useful for fractional measurements.

Let me know if you'd like this formatted as a printable answer sheet!