Key takeaway: A conversion factor is a fraction equal to 1 built from two equal measurements, like 36 inches over 1 yard, that you multiply by to switch units without changing the size.
Grade 66.RP.A.3dRatios, Rates and Percents
Definition of Conversion Factor
A conversion factor records how two units are related. Since 1 yard and 36 inches are the same length, the fraction \(\frac{36\text{ in}}{1\text{ yd}}\) has the value 1.
Multiplying by 1 changes how an amount is written but never its size. Arrange the factor so the unit you want to get rid of sits in the denominator and cancels, leaving the unit you want.
Formula and notation
Write the equal measurements as a fraction and cancel the old unit:
\[\text{old amount} \times \frac{\text{new unit}}{\text{old unit}} = \text{new amount}\]
Each fact gives two versions that are reciprocals, such as \(\frac{36\text{ in}}{1\text{ yd}}\) and \(\frac{1\text{ yd}}{36\text{ in}}\). Pick the version that makes the unwanted unit cancel.
Key facts
- Top and bottom of a conversion factor name the same length or amount, so its value is exactly 1.
- Units cancel like numbers: the starting unit must end up in a denominator.
- A chain of factors converts in several steps, such as hours to minutes to seconds.
- Going from a larger unit to a smaller one gives a bigger number.
Conversion Factor example
A banner for the school fair is 4.5 yards long. How many inches long is it? (1 yard = 36 inches)
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Choose the version with yards in the denominator: \(\frac{36\text{ in}}{1\text{ yd}}\).
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Multiply: \(4.5\text{ yd} \times \frac{36\text{ in}}{1\text{ yd}}\). The yards cancel.
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Compute \(4.5 \times 36 = 162\).
The banner is 162 inches long.
Common mistake
Watch out
Ben converts 4.5 yards to inches by dividing: \(4.5 : 36 = 0.125\).
Why it is wrong
He used the conversion factor upside down, so the yards did not cancel. An inch is far shorter than a yard, so the count of inches must be larger than 4.5. Put the unit to be removed in the denominator and multiply.
Related terms and lessons
Browse the glossary: A–Z list · Grade 6 vocabulary · Ratios, Rates and Percents