Ratios Rates Proportional Relationships and Units Pattern - Density Formula

Density Formula

Density is a rate that relates a quantity to the space it occupies: $\text{Density} = \dfrac{\text{Mass}}{\text{Volume}}$ (or $\dfrac{\text{Population}}{\text{Area}}$). This pattern asks you to compute density, or rearrange the formula to find mass or volume.

The Three Formulas

$D = \dfrac{M}{V}$, so $M = D \times V$, and $V = \dfrac{M}{D}$

For population density: $\text{PD} = \dfrac{\text{Population}}{\text{Area}}$

Worked Examples

Example 1. A metal has density $8.9$ g/cm$^3$. A block of this metal has volume $25$ cm$^3$. What is its mass?

$M = D \times V = 8.9 \times 25 = 222.5$ grams.

Example 2. A substance has mass $540$ grams and density $2.7$ g/cm$^3$. What is its volume?

$V = \dfrac{M}{D} = \dfrac{540}{2.7} = 200$ cm$^3$.
Gotcha: Don't multiply mass $\times$ density. When finding volume, you divide.

Example 3. A region has an area of $480$ square miles and a population of $120{,}000$. What is the population density?

$PD = \dfrac{120{,}000}{480} = 250$ people per square mile.

Example 4. An aluminum sphere has radius $r = 3$ cm and density $2.7$ g/cm$^3$. What is its mass? (Volume of sphere $= \dfrac{4}{3}\pi r^3$)

$V = \dfrac{4}{3}\pi(3)^3 = \dfrac{4}{3}\pi(27) = 36\pi \approx 113.1$ cm$^3$
$M = 2.7 \times 36\pi \approx 305.4$ grams
Gotcha: Harder density problems combine the density formula with a geometry formula. You need to find the volume first, then use $M = D \times V$.

Example 5. A farmer plants corn at a density of $32{,}000$ stalks per acre. If the field has $A$ acres, how many stalks are planted?

Stalks $= 32{,}000 \times A = 32{,}000A$

What to Do on Test Day

• $D = M/V$. Know all three rearrangements. Finding mass? Multiply. Finding volume? Divide mass by density.
• Population density works the same way: $\text{PD} = \text{Pop}/\text{Area}$.
• Harder problems add geometry. You'll need to compute volume first (sphere, cylinder, rectangular prism) before using the density formula.
• Units must match. If density is in g/cm$^3$, volume must be in cm$^3$ (not m$^3$).