Converting Between English Units

The ratio rule for unit conversion: put the unit you want on top so the old one cancels, and why the number grows going to smaller units and shrinks going to larger ones.

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Once you know that a foot holds 12 inches, converting between units is mechanical. The trick is to write the conversion as a fraction equal to 1, arranged so the unit you want to lose cancels out.

Concept The conversion factors
Length: 1 foot = 12 inches, 1 yard = 3 feet, 1 mile = 5280 feet.
Mass: 1 pound = 16 ounces, 1 ton = 2000 pounds.

Every conversion below rests on one of these five facts.

Theorem The ratio rule

Build a fraction with the unit you want on top and the unit you want to remove underneath:

\text{conversion ratio} = \frac{\text{unit you want}}{\text{unit you are removing}}

Because the top and bottom describe the same physical amount, this fraction equals 1 — so multiplying by it changes the units without changing the quantity.

Concept Which way the number moves
big unit → small unit: number grows small unit → big unit: number shrinks 2.5 ft = 30 in
Smaller units need more of them to cover the same distance, so the number rises. Larger units need fewer, so the number falls. Use this as a check on every answer.
Example 2.5 feet into inches
We want inches, so inches goes on top:
2.5 \text{ ft} \times \frac{12 \text{ in}}{1 \text{ ft}}
The feet cancel, leaving inches.
2.5 \times 12 = 30
⟹ 30 inches

The number grew from 2.5 to 30, exactly as expected when moving to a smaller unit.

Example Three quarters of a ton into pounds
\frac{3}{4} \text{ ton} \times \frac{2000 \text{ lb}}{1 \text{ ton}}
The tons cancel, leaving pounds.
\frac{3}{4} \times 2000 = 1500
⟹ 1500 pounds
Example 18 inches into feet

This time we are moving to a larger unit, so feet goes on top.

18 \text{ in} \times \frac{1 \text{ ft}}{12 \text{ in}}
The inches cancel, leaving feet.
18 \div 12 = 1.5
Check: 1.5 \times 12 = 18
⟹ 1.5 feet

Here the number fell from 18 to 1.5, which is right for a move to a larger unit.

Note Mistakes to avoid
Writing the ratio upside down so the units do not cancel.
Multiplying when the move to a larger unit calls for dividing.
Mixing a length factor into a mass conversion.
Leaving the answer without its unit.
Skipping the reverse check.
Summary
  1. Put the unit you want on top and the unit you are removing underneath.
  2. That ratio equals 1, so the quantity is unchanged.
  3. Cancel the repeated unit and multiply.
  4. Big to small: the number grows. Small to big: it shrinks.
  5. Check by converting back the other way.
231. Arithmetic Sequences and Series
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