How to Rearrange Density Formula

Master algebraic rearrangement to solve for density, mass, or volume

📅 Last updated: October 20, 2025 ⏱️ Reading time: ~8 minutes 📚 Difficulty: Beginner to Intermediate

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The Three Forms of the Density Formula

The beauty of the density formula is that it can be rearranged to solve for any variable.

Standard Form: D = M ÷ V (Density = Mass ÷ Volume)
Rearranged for Mass: M = D × V (Mass = Density × Volume)
Rearranged for Volume: V = M ÷ D (Volume = Mass ÷ Density)

Key Rule: When you move a variable across the equals sign, you flip the operation. Division becomes multiplication, and multiplication becomes division.

Solving for Mass (M = D × V)

When to Use This Form

Use this formula when you know density and volume, and need to find mass.

Step-by-Step Process

  1. Start with: D = M ÷ V
  2. Multiply both sides by V: D × V = M
  3. Rearrange: M = D × V

Example 1: Find Mass of Aluminum

Given: Density = 2.7 g/cm³, Volume = 50 cm³
Formula: M = D × V
Calculation: M = 2.7 g/cm³ × 50 cm³ = 135 grams

Example 2: Find Mass of Water

Given: Density = 1.0 g/mL, Volume = 250 mL
Calculation: M = 1.0 × 250 = 250 grams

Solving for Volume (V = M ÷ D)

When to Use This Form

Use this formula when you know mass and density, and need to find volume.

Step-by-Step Process

  1. Start with: D = M ÷ V
  2. Multiply both sides by V: D × V = M
  3. Divide both sides by D: V = M ÷ D

Example 1: Find Volume of Gold

Given: Mass = 193 grams, Density = 19.3 g/cm³
Formula: V = M ÷ D
Calculation: V = 193 g ÷ 19.3 g/cm³ = 10 cm³

Example 2: Find Volume of Mercury

Given: Mass = 68 grams, Density = 13.6 g/mL
Calculation: V = 68 g ÷ 13.6 g/mL = 5 mL

Solving for Density (D = M ÷ V)

When to Use This Form

This is the original formula. Use it when you know mass and volume, and need to find density.

Step-by-Step Process

This is already in the correct form: D = M ÷ V

  1. Simply divide mass by volume
  2. Include the proper units

Example 1: Find Density of Iron

Given: Mass = 78.7 grams, Volume = 10 cm³
Formula: D = M ÷ V
Calculation: D = 78.7 g ÷ 10 cm³ = 7.87 g/cm³

The Triangle Method (Visual Aid)

A quick way to remember all three forms is the density triangle:

To find any variable, cover it with your finger.
The remaining arrangement shows what to do:
M over D×V, D over M÷V, or V over M÷D

Common Mistakes When Rearranging

Mistake 1: Flipping the Wrong Operation

Wrong: D = M × V (this is incorrect!)

Correct: D = M ÷ V

Why: Division doesn't "flip" to multiplication without proper algebraic steps

Mistake 2: Forgetting Units

Wrong: M = 2.7 × 50 = 135

Correct: M = 2.7 g/cm³ × 50 cm³ = 135 grams

Why: Units are essential and show your work is correct

Mistake 3: Incorrect Unit Cancellation

Remember: When multiplying M = D × V, units multiply: (g/cm³) × (cm³) = g ✓

When dividing: V = M ÷ D gives: g ÷ (g/cm³) = cm³ ✓

Practice Problems

Problem 1: Find Mass

Mercury has a density of 13.6 g/mL. If you have 0.5 liters of mercury, what is its mass?

Answer: M = 13.6 g/mL × 500 mL = 6,800 grams

Problem 2: Find Volume

A copper block weighs 447.5 grams. Copper has a density of 8.96 g/cm³. What is the volume?

Answer: V = 447.5 g ÷ 8.96 g/cm³ = 50 cm³

Problem 3: Find Density

An object has a mass of 100 grams and a volume of 25 mL. What is its density?

Answer: D = 100 g ÷ 25 mL = 4 g/mL

Key Takeaways

  • The density formula can be rearranged three ways
  • D = M ÷ V (find density)
  • M = D × V (find mass)
  • V = M ÷ D (find volume)
  • Always include units in your answer
  • Check that units cancel properly
  • Verify your answer using the calculator