Why is my unit conversion wrong?
Updated
The short answer
A common slip is using a conversion factor upside down, so the old unit never cancels and the number moves the wrong way. Write every factor as a fraction with its units, like 1000 m / 1 km, and set it up so the unit you're replacing cancels top and bottom.
What it looks like
A car is moving at 72.0 km/h. What is that speed in m/s?
| Wrong | Right |
|---|---|
| 72.0 km/h × 3.6 = 259.2 m/s | 72.0 km/h × (1000 m / 1 km) × (1 h / 3600 s) = 20.0 m/s |
On the left, the 3.6 has no units, so nothing on the page says whether to multiply or divide by it. On the right, every factor is a fraction equal to 1 with its units written in. The km in 72.0 km/h cancels the km on the bottom of the first factor, the h cancels the h on top of the second, and m/s is what's left. The numbers come from the SI prefixes, where kilo means 1000, and from 1 h = 3600 s.
In alai, an iPad app that reads your handwriting as you work, Ally, the tutor, would point at the 3.6 on the left and ask a question about it.
Why it happens
Conversion factors usually get memorized as bare numbers: 1000, 60, 3600, 3.6. A bare number has no direction, so you have to remember whether to multiply or divide, and that's an easy thing to get backward when you're rushing.
A rate makes it harder, because km/h has a unit on top and a unit on the bottom, and they behave in opposite ways. Making the top unit smaller makes the number bigger: kilometers to meters multiplies by 1000. Making the bottom unit smaller makes the number smaller: per hour to per second divides by 3600, because a car covers far less ground in one second than in one hour.
Writing units feels slow, so it's the first thing to go under time pressure. Without them, the one check that would catch a flipped factor is gone.
How to catch it yourself
Before you use the calculator, cross out each unit that appears once on top and once on the bottom, then ask:
What unit is left on your answer after everything cancels?
If it isn't the unit the question asked for, one of your factors is upside down.
Try these
- Convert 90.0 km/h to m/s.
- A runner moves at 6.00 m/s. What is that in km/h?
- Convert 625 cm² to m².
Answers
- 90.0 km/h × (1000 m / 1 km) × (1 h / 3600 s) = 25.0 m/s
- 6.00 m/s × (1 km / 1000 m) × (3600 s / 1 h) = 21.6 km/h
- 625 cm² × (1 m / 100 cm)² = 0.0625 m². The factor is squared because cm² has two centimeters to cancel.
How alai helps
alai is a notes app for iPad and the web. In Homework help, its tutor, Ally, follows each step you write, and when a factor is upside down, Ally points at that step and asks a question about it, so you make the fix yourself. Ally is also set up to treat a physics answer as unfinished until the number has its units.