Volume and Surface Area worksheet featuring problems on calculating volume using unit fraction cubes.

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

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

We’ll use the formula for volume:
$$
\text{Volume} = \text{length} \times \text{width} \times \text{height}
$$

And since we're dealing with unit fraction cubes, we need to convert all dimensions into fractions and then calculate how many small cubes (of size $ \frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} $ or $ \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} $) can fit into the larger solid.

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1a.

Dimensions:
- 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?

We divide each dimension by $ \frac{1}{3} $:

- 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 $

So, total number of unit cubes:
$$
11 \times 9 \times 7 = 693
$$

✔ Answer 1: 693 unit fraction cubes fit.

#### Step 2: Volume of the solid

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

✔ Answer 2: Volume = $ 25\frac{2}{3} $ cubic units

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1b.

Dimensions:
- 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?

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 cubes:
$$
7 \times 1 \times 3 = 21
$$

✔ Answer 1: 21 unit fraction cubes fit.

#### Step 2: Volume of the solid

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

✔ Answer 2: Volume = $ 2\frac{5}{8} $ cubic units

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2a.

Dimensions:
- Length: $ 2\frac{1}{2} = \frac{5}{2} $
- Width: $ 2 $
- Height: $ 2\frac{1}{2} = \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?

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

- Length: $ \frac{5}{2} \div \frac{1}{2} = 5 $
- Width: $ 2 \div \frac{1}{2} = 4 $
- Height: $ \frac{5}{2} \div \frac{1}{2} = 5 $

Total cubes:
$$
5 \times 4 \times 5 = 100
$$

✔ Answer 1: 100 unit fraction cubes fit.

#### Step 2: Volume of the solid

$$
V = \frac{5}{2} \times 2 \times \frac{5}{2} = \frac{5 \times 2 \times 5}{2 \times 2} = \frac{50}{4} = 12.5 = 12\frac{1}{2}
$$

✔ Answer 2: Volume = $ 12\frac{1}{2} $ cubic units

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2b.

Dimensions:
- 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?

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 1: 308 unit fraction cubes fit.

#### Step 2: Volume of the solid

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

✔ Answer 2: Volume = $ 11\frac{11}{27} $ cubic units

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✔ Final Answers Summary:

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

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

#### 2a.
1. 100 unit fraction cubes fit.
2. Volume = $ 12\frac{1}{2} $ cubic units

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

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