Mixture Problem: Marbles Displace Water

What This Problem Teaches

• Understanding volume displacement—when objects occupy space, less room remains for liquid
• Finding unit rates (volume per marble) from given information
• Applying proportional reasoning to scale from one scenario to another
• Working with conservation principles—total volume stays constant while components change
• Checking answers using inverse relationships (more marbles → less water)

Solution: Method 1 — Volume Displacement Analysis

The key insight is that the tank's total capacity never changes—what changes is how that space gets divided between marbles and water. When William adds more marbles, they displace more water.

Step 1 — Find the volume displaced by 85 marbles

We know the tank holds 26 liters total. With 85 marbles, William needs 20.9 liters of water to fill it completely. This means the marbles must be taking up the remaining space:

Step 2 — Calculate the volume of one marble

Since all marbles have the same volume, we can find the volume per marble by dividing:

Step 3 — Find the volume displaced by 200 marbles

Now we can calculate how much space 200 marbles will occupy:

Step 4 — Calculate the water needed

The remaining space in the tank will need to be filled with water:

Solution: Method 2 — Proportional Relationship Setup

We can also solve this by setting up a proportion based on the relationship between marble count and water volume.

Step 1 — Establish the pattern

Let's define variables. If m = number of marbles and w = liters of water, then:

Step 2 — Use the known values to find the marble volume

From the given information: 85 marbles require 20.9 liters of water. Let v = volume per marble:

Step 3 — Apply to the new scenario

With 200 marbles and volume per marble = 0.06 liters:

Verification

Does This Make Sense?

Let's check our answer against reasonable expectations:

Watch Out For These Mistakes

The Underlying Pattern

This problem demonstrates the general displacement formula:

For any displacement problem where identical objects take up space in a container:

1. Find the volume per object using one known scenario
2. Calculate total displacement for the new object count
3. Subtract from container capacity to find remaining space

This pattern appears in many contexts: pebbles in a graduated cylinder, books in a box with packing material, or even people in an elevator (where each person reduces cargo space).

Real Applications

Laboratory measurements: Chemists use displacement to measure irregular solid volumes. Drop the object in a graduated cylinder of water—the volume increase equals the object's volume.

Shipping and logistics: When loading cargo containers, you need to account for the space displaced by packaging materials, pallets, and securing equipment. The "useful volume" is always less than the container's total volume.

Aquarium management: Fish tank owners must consider how decorations, gravel, and equipment reduce the actual water volume. This affects filtration capacity and fish bioload calculations.