Complex Unit Conversion

Many magnitudes have a relation with more than one unit. For example, speed is measured in meters per second (m/s), density in grams per cubic centimeter (g/cm³), etc. To convert these complex units, we need to use a conversion factors for each individual unit.

Understanding Compound Units

When working with compound units like speed, remember that 100 km/h means:

$$ 100 \frac{ \text{km}}{ \text{h}} = \frac{100 \text{ km}}{1 \text{ h}} $$

Important: When converting time units in the denominator, the time unit must be placed on top in the conversion factor. For example, to convert hours to seconds: use 3600 s / 1 h, not 1 h / 3600 s. This ensures the unwanted unit cancels out properly.

Example of unit changes

Desired Conversion	Equivalence Relationship	Formula and Calculation
90 km/h to m/s	1 km = 1000 m, 1 h = 3600 s	$90 \frac{ \text{km}}{ \text{h}} \times \frac{1000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ h}}{3600 \text{ s}} = 25 \text{ m/s}$
3.5 g/cm³ to kg/m³	1 g = 0.001 kg, 1 cm³ = 10⁻⁶ m³	$3.5 \frac{ \text{g}}{ \text{cm}^3} \times \frac{0.001 \text{ kg}}{1 \text{ g}} \times \frac{1 \text{ cm}^3}{10^{-6} \text{ m}^3} = 3500 \text{ kg/m}^3$
25 L/min to mL/s	1 L = 1000 mL, 1 min = 60 s	$25 \frac{ \text{L}}{ \text{min}} \times \frac{1000 \text{ mL}}{1 \text{ L}} \times \frac{1 \text{ min}}{60 \text{ s}} = 417 \text{ mL/s}$
5 m/s² to km/h²	1 m = 0.001 km, 1 s = 1/3600 h	$5 \frac{ \text{m}}{ \text{s}^2} \times \frac{0.001 \text{ km}}{1 \text{ m}} \times \frac{(3600 \text{ s})^2}{1 \text{ h}^2} = 64800 \text{ km/h}^2$
200 mg/L to g/m³	1 mg = 0.001 g, 1 L = 0.001 m³	$200 \frac{ \text{mg}}{ \text{L}} \times \frac{0.001 \text{ g}}{1 \text{ mg}} \times \frac{1 \text{ L}}{0.001 \text{ m}^3} = 0.2 \text{ g/m}^3$
1.5 atm to Pa	1 atm = 101325 Pa	$1.5 \text{ atm} \times \frac{101325 \text{ Pa}}{1 \text{ atm}} = 151988 \text{ Pa}$
120 J/min to W	1 W = 1 J/s, 1 min = 60 s	$120 \frac{ \text{J}}{ \text{min}} \times \frac{1 \text{ min}}{60 \text{ s}} = 2 \text{ J/s} = 2 \text{ W}$
60 mph to km/h	1 mile = 1.609 km	$60 \frac{ \text{mile}}{ \text{h}} \times \frac{1.609 \text{ km}}{1 \text{ mile}} = 96.5 \text{ km/h}$
15 N/cm² to Pa	1 N/cm² = 10⁴ Pa	$15 \frac{ \text{N}}{ \text{cm}^2} \times \frac{10^4 \text{ Pa}}{1 \text{ N/cm}^2} = 1.5 \times 10^5 \text{ Pa}$

Solved Examples

Example 1: Convert 0.5 km³ to cm³

Method 1: Direct conversion

1 km = 1,000 m = 100,000 cm

So 1 km³ = (100,000 cm)³ = 10¹⁵ cm³

$$ 0.5 \text{ km}^3 \times \frac{10^{15} \text{ cm}^3}{1 \text{ km}^3} = 5 \times 10^{14} \text{ cm}^3 $$

Method 2: Step-by-step

$$ 0.5 \text{ km}^3 \times \frac{10^9 \text{ m}^3}{1 \text{ km}^3} \times \frac{10^6 \text{ cm}^3}{1 \text{ m}^3} = 0.5 \times 10^{15} \text{ cm}^3 = 5 \times 10^{14} \text{ cm}^3 $$

Result: 0.5 km³ = 5 × 10¹⁴ cm³ = 500,000,000,000,000 cm³

Example 2: A rectangular water tank is 25 m long and 12.5 m wide and 2 m high. Calculate its volume in liters and cubic feet.

Step 1: Calculate volume in m³

$$ \text{Volume} = 25 \text{ m} \times 12.5 \text{ m} \times 2 \text{ m} = 625 \text{ m}^3 $$

Step 2: Convert to liters

$$ 625 \text{ m}^3 \times \frac{1,000 \text{ L}}{1 \text{ m}^3} = 625,000 \text{ L} $$

Step 3: Convert to cubic feet

1 m³ ≈ 35.31 ft³

$$ 625 \text{ m}^3 \times \frac{35.31 \text{ ft}^3}{1 \text{ m}^3} ≈ 22,069 \text{ ft}^3 $$

Result: Tank volume = 625 m³ = 625,000 L ≈ 22,069 ft³

Example 3: A swimming pool is 15 m long and 800 cm wide and 150 cm deep. What is its volume in liters?

Method 1: Convert everything to m first

Length: 15 m

Width: 800 cm = 8 m

Depth: 150 cm = 1.5 m

$$ \text{Volume} = 15 \text{ m} \times 8 \text{ m} \times 1.5 \text{ m} = 180 \text{ m}^3 $$

Method 2: Convert to liters

$$ 180 \text{ m}^3 \times \frac{1,000 \text{ L}}{1 \text{ m}^3} = 180,000 \text{ L} $$

Result: Pool volume = 180,000 L

Example 4: A cylindrical tank has a diameter of 200 cm and height 150 cm. Find its volume in liters and gallons.

Step 1: Find radius and calculate volume

Radius = 200 cm ÷ 2 = 100 cm = 1 m, Height = 150 cm = 1.5 m

$$ V = \pi r^2 h = 3.14159 \times (1 \text{ m})^2 \times 1.5 \text{ m} = 4.71 \text{ m}^3 $$

Step 2: Convert to liters

$$ 4.71 \text{ m}^3 \times \frac{1,000 \text{ L}}{1 \text{ m}^3} = 4,710 \text{ L} $$

Step 3: Convert to gallons

$$ 4,710 \text{ L} \times \frac{1 \text{ gal}}{3.785 \text{ L}} ≈ 1,244 \text{ gal} $$

Result: Tank volume = 4.71 m³ = 4,710 L ≈ 1,244 gal

Example 5: A cube has sides of 20 cm. What is its volume in liters and milliliters?

Step 1: Calculate volume in cm³

$$ V = (20 \text{ cm})^3 = 8,000 \text{ cm}^3 $$

Step 2: Convert to liters

Since 1 L = 1,000 cm³:

$$ 8,000 \text{ cm}^3 \times \frac{1 \text{ L}}{1,000 \text{ cm}^3} = 8 \text{ L} $$

Step 3: Convert to milliliters

Since 1 cm³ = 1 mL:

$$ 8,000 \text{ cm}^3 = 8,000 \text{ mL} $$

Result: Cube volume = 8,000 cm³ = 8 L = 8,000 mL