Volume Ratio Problem: 5 Spheres (10 cm³) Inside 125 cm³ Cube

Q: Why do I subtract the spheres' volume from the cube's volume?

Because the spheres occupy space inside the cube! The remaining volume is what's left empty after placing the spheres. Think of it like putting balls in a box - the empty space shrinks.

Q: What if the spheres don't physically fit in the cube?

This is a mathematical problem about volumes, not physical placement. We assume the spheres can fit perfectly inside the cube space, even if they overlap in reality.

Q: How do I know which volume goes in the numerator vs denominator?

Read carefully! The problem asks for spheres volume to remaining volume. First number mentioned = numerator, second = denominator. So it's $ \frac{\text{spheres}}{\text{remaining}} $.

Q: Can I simplify the fraction 50/75?

Yes! Both numbers are divisible by 25: $ \frac{50}{75} = \frac{50÷25}{75÷25} = \frac{2}{3} $. Always simplify ratios to their lowest terms.

Q: What if I calculated 75/50 instead?

That gives $ \frac{3}{2} $, which is the opposite ratio (remaining to spheres). Make sure you match the order in the question: spheres to remaining means spheres first!

Volume Ratios with Geometric Objects

Given a cube whose volume is equal to 125 cm3

We put into the cube 5 spheres, the volume of each sphere is 10 cm³.

What is the ratio between the total volume of the spheres and the volume remaining in the cube after inserting the spheres?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 What is the ratio of spheres volume to the remaining volume in the cube

00:03 Let's calculate the remaining volume in the cube

00:07 This volume equals the cube's volume minus 5 times the sphere's volume

00:12 Let's substitute appropriate values and solve for the remaining volume

00:27 This is the remaining volume in the cube

00:30 Let's find the volume ratio

00:36 Let's substitute appropriate values and solve for the volume ratio

00:45 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

Given a cube whose volume is equal to 125 cm3

We put into the cube 5 spheres, the volume of each sphere is 10 cm³.

What is the ratio between the total volume of the spheres and the volume remaining in the cube after inserting the spheres?

Step-by-step solution

To solve this problem, let's follow these steps:

Step 1: Calculate the total volume of the spheres.
Step 2: Determine the volume remaining in the cube after inserting the spheres.
Step 3: Calculate the ratio between the total volume of the spheres and the remaining volume of the cube.

Now, let's work through each step:

Step 1: The volume of each sphere is given as 10 cm $^3$ , and there are 5 spheres. Thus, the total volume of the spheres is:
$5 \times 10 = 50$ cm $^3$ .

Step 2: The volume of the cube is given as 125 cm $^3$ . After inserting the spheres, the remaining volume of the cube is:
$125 - 50 = 75$ cm $^3$ .

Step 3: Calculate the ratio of the total volume of the spheres to the remaining volume of the cube:
$\frac{50 \text{ cm}^3}{75 \text{ cm}^3} = \frac{2}{3}$ .

Therefore, the ratio between the total volume of the spheres and the volume remaining in the cube is $\frac{2}{3}$ .

Final Answer

$\frac{2}{3}$

Key Points to Remember

Essential concepts to master this topic

Rule: Ratio equals spheres volume divided by remaining cube volume
Technique: Total spheres volume: $5 \times 10 = 50$ cm³
Check: Verify: $\frac{50}{75} = \frac{2}{3}$ by cross-multiplying ✓

Common Mistakes

Avoid these frequent errors

Using total cube volume instead of remaining volume
Don't calculate $\frac{50}{125} = \frac{2}{5}$ ! This uses the original cube volume, not what's left empty. The spheres take up space, so you must subtract first: remaining volume = 125 - 50 = 75 cm³. Always use remaining volume for the denominator.

Practice Quiz

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A cube has a total of 14 edges.

FAQ

Everything you need to know about this question

Why do I subtract the spheres' volume from the cube's volume?

Because the spheres occupy space inside the cube! The remaining volume is what's left empty after placing the spheres. Think of it like putting balls in a box - the empty space shrinks.

What if the spheres don't physically fit in the cube?

This is a mathematical problem about volumes, not physical placement. We assume the spheres can fit perfectly inside the cube space, even if they overlap in reality.

How do I know which volume goes in the numerator vs denominator?

Read carefully! The problem asks for spheres volume to remaining volume. First number mentioned = numerator, second = denominator. So it's $\frac{\text{spheres}}{\text{remaining}}$ .

Can I simplify the fraction 50/75?

Yes! Both numbers are divisible by 25: $\frac{50}{75} = \frac{50÷25}{75÷25} = \frac{2}{3}$ . Always simplify ratios to their lowest terms.

What if I calculated 75/50 instead?

That gives $\frac{3}{2}$ , which is the opposite ratio (remaining to spheres). Make sure you match the order in the question: spheres to remaining means spheres first!

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