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

Calculate the volume of the rectangular prism below using the data provided.

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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