Unit Conversions
By the end of this lesson, you’ll be able to:
- Convert between units using conversion factors and dimensional analysis.
- Perform single-step and multi-step conversions accurately.
- Convert square and cubic units (cm² ↔︎ m², cm³ ↔︎ m³).
- Apply conversions to real-world contexts (rates, density, scale).
Key Ideas
A conversion factor is a ratio equal to 1:
\[ \frac{60\ \text{min}}{1\ \text{hour}} = 1 \]
Multiplying by this ratio changes units but not value.
General conversion structure:
\[ \text{quantity} \times \frac{\text{new units}}{\text{old units}} \]
Units must cancel like algebraic variables.
Multi-step conversions chain several conversion factors together.
Always write out the units. Unit-conversion traps rely on missing or mismatched units.
Common Problem Types
Single-Step Conversions
Convert 5 hours to minutes:
\[ 5\ \text{hr} \times \frac{60\ \text{min}}{1\ \text{hr}} = 300\ \text{min} \]
Multi-Step Conversions
Convert 90 km/hr to m/s:
\[ 90 \times \frac{1000\ \text{m}}{1\ \text{km}} \times \frac{1\ \text{hr}}{3600\ \text{s}} = 25\ \text{m/s} \]
Converting Units in Rates
Convert 12 ft/s to mph:
\[ 12 \times \frac{3600\ \text{s}}{1\ \text{hr}} \times \frac{1\ \text{ft}}{5280\ \text{ft}} = 8.18\ \text{mph} \]
Square Unit Conversions
Convert \(4\ \text{m}^2\) to \(\text{cm}^2\):
Since \(1\ \text{m} = 100\ \text{cm}\):
\[ 4\ \text{m}^2 \times (100\ \text{cm})^2 = 4 \times 10{,}000 = 40{,}000\ \text{cm}^2 \]
Cubic Unit Conversions
Convert \(0.003\ \text{m}^3\) to \(\text{cm}^3\):
\[ 0.003 \times (100\ \text{cm})^3 = 0.003 \times 1{,}000{,}000 = 3000\ \text{cm}^3 \]
Strategies
- Write conversion factors as fractions and cancel units step-by-step.
- Keep units attached throughout the work.
- Square or cube the conversion factor when converting area/volume.
- When stuck, convert to a unit rate (per 1).
- Check that your final units match the question.
Worked Examples
Example 1
Question: Convert 2500 grams to kilograms.
Solution:
\[ 2500\ g \times \frac{1\ kg}{1000\ g} = 2.5\ kg \]
Example 2
Question: A student ran 800 meters. How many kilometers is this?
Solution:
\[ 800 \times \frac{1\ km}{1000\ m} = 0.8\ km \]
Example 3
Question: Convert 2.4 m² to cm².
Solution:
\[ 2.4 \times (100\ \text{cm})^2 = 2.4 \times 10{,}000 = 24{,}000\ \text{cm}^2 \]
Example 4
Question: Convert 10 L to mL.
Solution:
\[ 10 \times 1000 = 10{,}000\ \text{mL} \]
- Not squaring or cubing conversion factors for area or volume.
- Reversing conversion factors (wrong numerator/denominator).
- Cancelling incorrect units or skipping unit cancellation.
- Mixing SI and English units incorrectly.
- Forgetting that fraction bars act as grouping symbols in expressions.
Practice Problems
- Convert 3 hours to seconds.
- Convert 250 cm to meters.
- Convert \(5\ \text{m}^2\) to \(\text{cm}^2\).
- Convert 72 km/hr to m/s.
- Convert 0.002 m³ to cm³.
1. Convert 3 hours to seconds.
\(1\text{ hr} = 60\text{ min}\), \(1\text{ min} = 60\text{ s}\).
\(3 \times 60 \times 60 = 10{,}800\) seconds.
Answer: \(10{,}800\ \text{s}\)
2. Convert 250 cm to meters.
\(250 \div 100 = 2.5\).
Answer: \(2.5\ \text{m}\)
3. Convert \(5\ \text{m}^2\) to \(\text{cm}^2\).
\(1\ \text{m}^2 = 10{,}000\ \text{cm}^2\).
\(5 \times 10{,}000 = 50{,}000\).
Answer: \(50{,}000\ \text{cm}^2\)
4. Convert 72 km/hr to m/s.
Shortcut: divide by 3.6 → \(72 \div 3.6 = 20\).
Answer: \(20\ \text{m/s}\)
5. Convert 0.002 m³ to cm³.
\(0.002 \times 1{,}000{,}000 = 2000\).
Answer: \(2000\ \text{cm}^3\)
Summary
- Multiply by conversion factors equal to 1 to change units without changing value.
- Units should cancel cleanly during each step.
- Multi-step conversions appear frequently in PSDA contexts.
- Area and volume conversions require squared or cubed factors.
- Keep units attached at every step — they guide the calculation.
- Multiply by fractions that equal 1 (like \(\frac{60\text{ min}}{1\text{ hr}}\)).
- Never convert area or volume using the linear factor — always square/cube it.
- For km/hr → m/s, divide by 3.6 as a clean shortcut.