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Concept

Conversion Factor

A conversion factor is a fraction where the numerator and denominator represent the same quantity with different units.

Example Conversion Factor \frac{6 0 minutes}{1 hour}

Recall that

1

hour and

60

minutes represent the same quantity. Multiplying a quantity by a conversion factor changes the quantity to an equivalent quantity in different units. Examine how to convert

2

hours to minutes using the above conversion factor.

Given Quantity	Conversion	Result
$2$ $hours$	$2 hours \cdot \frac{6 0 minutes}{1 hour}$	$120$ $minutes$

Although the final result is in minutes, both quantities represent the same amount of time. Note that the opposite conversion, from minutes to hours, has a conversion factor of $\frac{1 hour}{6 0 minutes} .$ If the task was to convert $120$ minutes to hours, $120$ minutes would be multiplied by this conversion factor.

Given Quantity	Conversion	Result
$120$ $hours$	$120 minutes \cdot \frac{1 hour}{6 0 minutes}$	$2$ $hours$

As shown in the examples above, the process of including units of measurement as factors is called dimensional analysis. Dimensional analysis can also be used when deciding which conversion factor will produce the desired units. In the table, some common conversion factors are used to convert the given measures.

Given Quantity	Conversion	Result
$3$ $pounds$	$3 pounds \cdot \frac{1 6 ounces}{1 pound}$	$48$ $ounces$
$160$ $ounces$	$160 ounces \cdot \frac{1 pound}{1 6 ounces}$	$10$ $pounds$
$1$ $mile$	$1 mile \cdot \frac{1 7 6 0 yards}{1 mile}$	$1760$ $yards$

Some common conversions involve distance, mass, area, volume, time, and temperature.

Why

The Reason That the Quantities Are Equivalent

The numerator and denominator of the conversion factor represent the same quantity. That means their quotient equals $1 .$ Then, by the Identity Property of Multiplication, the amount of the given quantity does not change when multiplied by the conversion factor.

When converting from one unit to another, the desired unit needs to be in the numerator of the conversion factor while the given unit needs to be in the denominator. That way when the quantity is multiplied by the conversion factor, the given unit will cancel out and the desired unit will remain.

Keep in mind that, despite the given quantity and the new quantity have different values, they represent the same amount.

Exercises

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