Quantities & units
The idea
Behind every applied math problem sits a quantity: a number attached to a unit, like 54 m³ or 60 km/h. You already know how to measure, convert units, and work with rates; this concept turns those skills into a system called dimensional analysis, where units are treated like algebraic factors that multiply and cancel. Choosing the right unit and a sensible level of precision is often the difference between an answer you can defend and a number that means nothing.
The core move is multiplying by conversion factors that equal 1, such as 1000 L per 1 m³, arranged so the unwanted units cancel. If the leftover units are wrong, the setup is wrong — no need to recheck the arithmetic. The common misconception is that converting is just shifting a decimal point; that shortcut only works between metric prefixes of the same base unit. Converting a compound unit like km/h to m/s needs two factors, one for each part, and the units themselves tell you whether to multiply or divide.
Worked example
A pump moves water at 2.4 L/s into an empty tank that holds 54 m³. How long will the pump take to fill the tank, in hours?
- Get both quantities into compatible units first. One cubic meter is 1000 liters, so the tank holds 54 × 1000 = 54,000 L.
- Convert the rate to liters per hour so the time comes out in hours: 2.4 L/s × 3600 s/h = 8640 L/h. The seconds cancel, leaving liters per hour, which confirms the setup.
- Time equals amount divided by rate: 54,000 L ÷ 8640 L/h = 6.25 h. The liters cancel and hours remain — exactly the unit the question asks for.
- Sanity-check the size: 8640 L/h is roughly 8.6 m³ per hour, and 6 × 8.6 ≈ 52 m³, so taking a little over 6 hours to reach 54 m³ is believable.
Answer. The pump fills the tank in 6.25 hours, which is 6 hours 15 minutes.
Check your understanding
- Why does multiplying by a conversion factor like 1000 L per 1 m³ leave the actual quantity unchanged even though the number changes?
- How can checking only the leftover units catch a setup error before you do any arithmetic?
- What goes wrong if you convert km/h to m/s by moving the decimal point, and why does that shortcut fail for compound units?
- How would you decide how many significant figures to keep when a measured rate like 2.4 L/s drives the whole calculation?
Build the foundations first
Quantities & units builds on these concepts. If any feel shaky, start there.