Multiply by 1, in disguise.
Every unit conversion uses the same elegant move: multiply by a fraction equal to 1, written so the unwanted unit cancels. Once you see it, you can convert anything.
A recipe calls for 16 ounces of flour, but your scale only reads grams. On a German Autobahn the speed limit is 130 km/h and your speedometer shows mph. Building a deck, the lumber is sold in feet but the plans are in meters.
Same problem every time: rewrite a number in different units. The trick behind every conversion you'll ever do is one of the most elegant moves in math: multiply by 1, cleverly disguised.
A conversion factor is a fraction that equals 1.
4 miles times "5280 ft / 1 mi" gives 21,120 ft. The miles unit cancels because it's on top in one factor and on the bottom in the other.
A conversion factor is a fraction of two equivalent quantities, written so that the fraction itself equals 1.
The vocabulary you'll see in word problems
- Conversion factor A fraction whose numerator and denominator are equivalent quantities in different units, e.g. 5280 ft / 1 mi or 2.54 cm / 1 in. Always equals 1.
- Dimensional analysis The formal name for this technique. Set up a chain of conversion factors so the original unit cancels and the target unit survives.
- Numerator / Denominator Top of the fraction / bottom of the fraction. Which slot a unit lives in determines whether it cancels or survives.
- Equivalent quantities Two measurements that name the same physical amount: 12 inches and 1 foot, 1000 grams and 1 kilogram, 4 quarts and 1 gallon.
- Unit cancellation When a unit appears on the top of one fraction and the bottom of another in the same product, it cancels. The point of dimensional analysis: set up the multiplication so every unit cancels except the one you want.
Convert 4 miles to feet.
"You ran 4 miles this morning. Your fitness app wants the distance in feet. How many feet did you run?"
Identify the conversion factor.
You need a fraction that relates miles and feet. The known equivalence: 1 mile = 5280 feet. That gives you a conversion factor.
Put the unwanted unit on the bottom.
You're starting with miles and you want to end with feet. So miles is the unwanted unit. Put it on the bottom of the fraction so it cancels with the miles in your starting quantity.
Multiply, and watch the units cancel.
Set up the multiplication. Notice the mi on top of "4 mi" and the mi on the bottom of the factor cancel each other out (top × bottom = 1). The ft on top is what survives.
Compute the number.
4 × 5280 = 21,120. The surviving unit is feet.
Sanity check.
4 miles is more than 1 mile, so the answer should be more than 5280 ft. It is (21,120 > 5280). And it should be a fairly big number, since feet are tiny compared to miles. 21,120 feels right.
Three problems. Three different conversions.
Convert 3 feet to inches. (1 ft = 12 in.)
A football field is 100 yards. How many feet is that? (1 yd = 3 ft.)
Convert 200 grams to ounces. (1 oz ≈ 28.35 g.) Round to two decimals.
Three fast questions before you move on.
Q1. Which of these conversion factors actually equals 1?
Q2. To convert 8 feet to inches, you should multiply by...
Q3. If you set up the conversion 100 cm × (1 in / 2.54 cm), what unit will the answer be in?
Three things to remember every conversion.
Unit conversions all work the same way. Three questions to ask yourself, every time.
In ALEKS, dimensional-analysis problems often give you the conversion factor in the question prompt. Your job is to set up the fraction in the right direction and run the multiplication. Read the prompt twice and circle the factor before you start.
In Excel, conversion factors live in their own cell that the rest of the sheet references with a dollar sign ($A$1). Change the factor in one place, the whole column updates. That pattern shows up all over Major Assignment 1.
Next: Lesson 05 chains conversion factors together (mph to m/s in three multiplications) and tackles compound units. Same trick, scaled up.
Continue to Lesson 05Different angle? Need another rep? These are optional — tap any that look helpful.