Nested Cuboids with 1/3 Ratio: Calculate Volume Relationship

Q: Why do I need to cube the ratio instead of just using it once?

Because volume is three-dimensional! When all three dimensions (length, width, height) are scaled by $ \frac{1}{3} $, you multiply $ \frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} = \left(\frac{1}{3}\right)^3 $.

Q: What if the ratios for length, width, and height were different?

Then you'd multiply the individual ratios together. For example, if dimensions were $ \frac{1}{2} $, $ \frac{1}{3} $, and $ \frac{1}{4} $, the volume ratio would be $ \frac{1}{2} \times \frac{1}{3} \times \frac{1}{4} = \frac{1}{24} $.

Q: Can I work backwards from the answer 27?

Yes! Since 27 small cuboids fit, the small volume is $ \frac{1}{27} $ of the large volume. Taking the cube root: $ \sqrt[3]{\frac{1}{27}} = \frac{1}{3} $, which matches our dimension ratio.

Q: What does the 1/3 ratio tell us about the small cuboid?

It tells us the small cuboid is a scaled-down version of the large one. Each edge of the small cuboid is exactly one-third the length of the corresponding edge of the large cuboid.

Volume Scaling with Fractional Dimensions

Shown below is a large cuboid with a small cuboid inside it.

$AB=\frac{1}{3}AC$

$AD=\frac{1}{3}AE$

$BF=\frac{1}{3}CK$

How many times does the small cuboid fit into the large cuboid?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 How many times does the small box fit in the large one

00:03 Let's use the formula for calculating box volume

00:07 width times height times length

00:11 This is the expression for the large box volume

00:25 Let's find the expression for the small box volume

00:46 Let's calculate the common denominator

00:58 Let's calculate the volume ratio

01:05 Let's substitute the volume expressions and solve

01:18 Multiply numerator by denominator and vice versa

01:21 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Shown below is a large cuboid with a small cuboid inside it.

$AB=\frac{1}{3}AC$

$AD=\frac{1}{3}AE$

$BF=\frac{1}{3}CK$

How many times does the small cuboid fit into the large cuboid?

Step-by-step solution

To solve this problem, we'll calculate the volumes of both cuboids:

Step 1: Determine the volume of the larger cuboid
Step 2: Determine the volume of the smaller cuboid
Step 3: Calculate how many smaller cuboids fit inside the larger cuboid

Now, let's work through each step:
Step 1: Let the dimensions of the larger cuboid be $(L, W, H)$ . The volume is $V_{large} = L \times W \times H$ .
Step 2: The dimensions of the smaller cuboid are $\left(\frac{L}{3}, \frac{W}{3}, \frac{H}{3}\right)$ . Thus, the volume is $V_{small} = \frac{L}{3} \times \frac{W}{3} \times \frac{H}{3} = \frac{L \times W \times H}{27}$ .
Step 3: To find the number of smaller cuboids that fit inside the larger cuboid, divide the larger volume by the smaller volume:
$\frac{V_{large}}{V_{small}} = \frac{L \times W \times H}{\frac{L \times W \times H}{27}} = 27$

Therefore, the number of times the smaller cuboid fits into the larger cuboid is 27.

Final Answer

Key Points to Remember

Essential concepts to master this topic

Volume Rule: Volume scales as the cube of linear dimensions
Technique: Each dimension is $\frac{1}{3}$ , so volume ratio is $\left(\frac{1}{3}\right)^3 = \frac{1}{27}$
Check: If small volume is $\frac{1}{27}$ of large, then 27 small cuboids fit ✓

Common Mistakes

Avoid these frequent errors

Adding dimension ratios instead of multiplying
Don't add the three $\frac{1}{3}$ ratios to get $\frac{3}{3} = 1$ = same volume! Volume requires multiplying all three dimensions together. Always multiply: $\frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} = \frac{1}{27}$ of the original volume.

Practice Quiz

Test your knowledge with interactive questions

A rectangular prism has a base measuring 5 units by 8 units.

The height of the prism is 12 units.

Calculate its volume.

FAQ

Everything you need to know about this question

Why do I need to cube the ratio instead of just using it once?

Because volume is three-dimensional! When all three dimensions (length, width, height) are scaled by $\frac{1}{3}$ , you multiply $\frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} = \left(\frac{1}{3}\right)^3$ .

What if the ratios for length, width, and height were different?

Then you'd multiply the individual ratios together. For example, if dimensions were $\frac{1}{2}$ , $\frac{1}{3}$ , and $\frac{1}{4}$ , the volume ratio would be $\frac{1}{2} \times \frac{1}{3} \times \frac{1}{4} = \frac{1}{24}$ .

How do I remember that volume scales with the cube?

Think about a simple example: if you double each dimension of a cube, you get 2 × 2 × 2 = 8 times the volume. The scaling factor gets used once for each dimension!

Can I work backwards from the answer 27?

Yes! Since 27 small cuboids fit, the small volume is $\frac{1}{27}$ of the large volume. Taking the cube root: $\sqrt[3]{\frac{1}{27}} = \frac{1}{3}$ , which matches our dimension ratio.

What does the 1/3 ratio tell us about the small cuboid?

It tells us the small cuboid is a scaled-down version of the large one. Each edge of the small cuboid is exactly one-third the length of the corresponding edge of the large cuboid.

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