#### USING CONVERSION FACTORS

(English-to-English, Metric-to-Metric, English-to-Metric or Metric-to-English.)

CONVERSION FACTORS (CF).

CF’s are derived from two quantities that are equal or that have been defined as equivalent and are expressed as a fraction. One quantity is placed in the numerator; one quantity is placed in the denominator, and it is optional as to which quantity is placed in either. The quantity in the numerator is equivalent to the quantity in the denominator; however each has a specific scalar value and units.  The respective values are not interchangeable. The CF is equivalent to “1”; thus, a quantity may be multiplied by (or divided by) a CF without changing its true value.

USING CONVERSION FACTORS:

For our purpose, a CF is used to convert the units of a known measurement in a specific dimension to different units in that same dimension.  This is done by multiplying the known measurement by the CF. The CF must contain a quantity with the units that are being converted as well as a quantity with the units that are desired. The choice of which quantity is placed in the numerator or denominator is made such that the undesired units will cancel and the desired units are maintained.

MANIPULATION OF CONVERSION FACTOR:

The dimension of the CF must be the same dimension as the known measurement.  It is often required to legally manipulate the “memorized” CF in order to obtain the required dimension.  The CF can be squared, cubed, square rooted, etc as a whole without changing its equality OR its equivalency of “1”. The selected operation may be applied simultaneously to both the numerator and denominator or it may be applied to the quotient (after the denominator has been divided into the numerator).

KNOWN MEASUREMENT:

The Known Measurement is the original measurement that is to be converted to another unit of measurement.  This measurement has a specific value; it CANNOT be squared, cubed, square rooted, etc.   It can, however, be multiplied (or divided) by a CF in order to be converted to various legal units without changing its true value.

DIMENSIONS:

The term dimension is a category of measurement expressed in specific units. The four basic dimension categories are: distance, volume, mass, and time. Other dimensions are some combination of these.  Such as, acceleration is a combination of a distance measurement divided by a time measurement squared (a = d/t2); force is a combination of a mass measurement multiplied by acceleration (f = ma); velocity is a combination of a distance measurement divided by a time measurement (v = d/t).     2nd dimension and & 3rd dimension refer to units that are squared and units that are cubed respectively.

Each dimension is expressed in a specific type of unit.  You must memorize the specific type of units that are legal for each of the four basic dimensions for the English and Metric systems. You must memorize the list of 21 basic conversion factors that are provided for the English and Metric systems. A specific dimension measurement can only be converted to that same dimension. Such as: mass units can only be converted into other mass units;  distance units can only be converted into other distance units; volume units can only be converted into other volume units, velocity units must be converted to velocity units, etc.

1. Determine the known measurement given and its units and the final units that are sought. This will lead to determining the first and last conversion factors.
2. It is convenient if the known measurement is expressed as a numerator with the denominator of “1”.
3. The CF will come from your list of conversions.  CF’s are multiplied in an            orientation that will cancel undesired units and maintain the desired unit.  If the            units that are to be converted are expressed in the numerator, the CF should have            the same units in its denominator in order to cancel the undesired unit.  If the units            that are to be converted are expressed in the denominator, the CF should have            the same units in its numerator in order to cancel the undesired unit.   Some measurements have units expressed in the numerator as well as in the            denominator. Pay close attention as to which unit is being converted.
4. Choosing the first CF.  If the units that are to be converted are expressed in the numerator, the CF should have the same units in its denominator in order to cancel the undesired unit. The quantity for the numerator of the first CF should now be predictable based on your memorized list of CF’s.
5. Choosing the last CF if more than one CF is required:  The unit of measurement that is sought dictates the unit of the numerator of the final CF.  The value of the denominator of the last CF should be predictable based on your memorized list of CF’s.
6. Determine the conversion category.    It should be: English-to-English, Metric-to-Metric, or English-to-Metric or Metric-to-English.
7. If an English-Metric or Metric-to-English conversion is required, you can direct your choice of conversion factors to the specific conversion memorized for that dimension. You should have memorized the three Metric-to-English conversions for the mass, volume, and distance dimensions. Remember:  the distance conversion factors can be manipulated to accommodate an area measurement (by squaring) or a volume measurement (by cubing).
8. For any conversion that involves converting a pre-fixed Metric unit, it is helpful if you first convert the pre-fixed unit to the basic unit.

EXAMPLE 1:     Please convert 30 in. to cm.

1. It should be understood that 30 inches is the known measurement.  It is to be converted into the final unit of centimeters.  This is a distance dimension.
2. It is convenient to express the known measurement with the understood

denominator of “1”.

.

30 in

1

1. The unit of inches in the numerator dictates the unit for the denominator of the first conversion factor.   This in turn limits the choices for the numerator of this conversion factor. This is an English-to-Metric conversion for a distance measurement.  We already know that inches belongs in the denominator.  It therefore follows that the “1 in = 2.54 cm” is one of the three CF’s memorized for this category.  It will be used for our first and final conversion factor.

We will multiply our known measurement by the conversion factor:  2.54 cm

1 in.

We select the orientation of the conversion factor so that the original units of “in.” will cancel.

30 in        X          2.54 cm  =      76.2 cm    =   76.2 cm

1                        1 in.                    1

Mathematical solution for these kinds of factors:  multiply all of the numerator values and multiply all of the denominator values, then divide the denominator  product into the numerator product.    It may be convenient, to legally cancel any values in the denominator with legal values in the numerator, but not required.  It is required, however, that all units that will cancel must be cancelled to achieve the desired units.

EXAMPLE 2:     Please convert 30 in2 into cm2.

1. It should be understood that 30 in2 is the known measurement.  It is to be converted into the unit of cm2.  This is an area dimension.
2. It is convenient to express the known measurement with the understood

denominator of “1”.

.

30 in2

1

1. The unit of in2 in the numerator dictates the unit for the denominator of the first conversion factor.   This in turn limits the choices for the numerator of this conversion factor. This is an English-to-Metric conversion for a distance measurement.  We already know that in2 belongs in the denominator.  It therefore follows that the “1 in = 2.54 cm”  is one of the three CF’s memorized for this category.  The units in2  is a 2nd dimensional unit.  It dictates that the conversion factor be manipulated to accommodate the units of the known measurement.  It therefore must be squared because the known unit is squared.   It will be used for our first and only conversion factor.

(1 in = 2.54 cm)2   =   1 in2 = 6.45 cm2

We will multiply our known measurement by the conversion factor:  6.45 cm2

1 in2.

We select the orientation of the conversion factor so that the original units of “in2.” will cancel.

30 in2     X          6.45 cm2  =      193.5 cm2    =  193.5 cm2

1                        1 in2                  1

Solution: multiply all of the numerator values and multiply all of the denominator values, then divide the denominator  product into the numerator product. “ in2   will cancel leaving the desired units.

EXAMPLE 3:     Please convert 30 in3 into cm3.

1. It should be understood that 30 in3 is the known measurement.  It is to be converted into the unit of cm3.    This is a volume dimension.
2. It is convenient to express the known measurement with the understood

denominator of “1”.

.

30 in3

1

1. The unit of in3 in the numerator dictates the unit for the denominator of the first conversion factor.   This in turn limits the choices for the numerator of this conversion factor. This is an English-to-Metric conversion for a distance measurement.  We already know that in3 belongs in the denominator.  It therefore follows that the “1 in = 2.54 cm”  is one of the three CF’s memorized for this category.  The units in3  is a 3rd dimensional unit.  It dictates that the conversion factor be manipulated to accommodate the units of the known measurement.  It therefore must be cubed because the known unit is cubed.   It will be used for our first and only conversion factor.

(1 in = 2.54 cm)3   =   1 in3 = 16.39 cm3

We will multiply our known measurement by the conversion factor:  16.39 cm3

1 in3.

We select the orientation of the conversion factor so that the original units of “in3.” will cancel.

30 in3     X          16.39 cm3  =      491.7 cm3    =  491.7 cm3

1                        1 in3                  1

Solution: multiply all of the numerator values and multiply all of the denominator values, then divide the denominator  product into the numerator product. “ in3   will cancel leaving the desired units.

EXAMPLE 4:     Please convert 30 in3 into HL.

1. It should be understood that 30 in3 is the known measurement.  It is to be converted into the unit of Hectoliters.  This is a volume dimension.
2. It is convenient to express the known measurement with the understood

denominator of “1”.

.

30 in3

1

1. The unit of in3 in the numerator dictates the unit for the denominator of the first conversion factor.   This in turn limits the choices for the numerator of this conversion factor. This is an English-to-Metric conversion for a distance measurement.  We already know that in3 belongs in the denominator.  It therefore follows that the “1 in = 2.54 cm”  is one of the three CF’s memorized for this category.  The units in3  is a 3rd dimensional unit.  It is a volume dimension.  It dictates that the conversion factor be manipulated to accommodate the units of a volume measurement.  It therefore must be cubed.   It will be used for our first conversion factor.   The final conversion factor will have  HL as the numerator.  Since it is a pre-fixed metric unit,  it is best to convert to the basic unit.  The denominator unit will be Liter.

(1 in = 2.54 cm)3   =   1 in3 = 16.39 cm3

We will multiply our known measurement by the conversion factor:  16.39 cm3

1 in3.

We select he orientation of the conversion factor so that the original units of “in3.” will cancel.

The final conversion factor will be:         1  HL

102 L

The known measurement, the first and last conversions factors are presented below:

First conversion          Last conversion

30 in3     X          16.39 cm3     ………..   1 HL    =

1                        1 in3                             102 L

The second conversion factor is easily determined because we recognize that the cm3    is a volume measurement.  It has the equivalent of mL.  The second conversion factor will be:

1 mL

1 cm3

First conversion       2nd conversion ……..last conversion

30 in3     X          16.39 cm3     X        1 mL ……?………....   1 HL    =

1                        1 in3                       1 cm3                            102 L

After the 2nd conversion factor is applied, it should be clear that a 3rd conversion is required to bridge the last conversion factor:

Since the mL is a pre-fixed metric, the third factor should be going from the pre-fixed value to the basic unit.  It should also be noted that the mL in the 2nd conversion needs to cancel thus  mL will be in the denominator to cancel the mL in the 2nd conversion factor.  The  L in the denominator of the last conversion needs to cancel thus L will be in the numerator to cancel the L in the denominator of the last conversion.

First conversion       2nd conversion     3rd conversion          last conversion

30 in3     X          16.39 cm3     X        1 mL             X     10-3 L                X               1 HL    =

1                        1 in3                       1 cm3                    1 mL                                102 L

30 x 16.39  x  1  x    10-3   x   1  HL    =   4.917  x  10-3  HL

1 x 1 x 1 x 1 x 102

Solution: multiply all of the numerator values and multiply all of the denominator values, then divide the denominator  product into the numerator product. “ in3, cm3, mL, & L  will cancel leaving the desired units, HL.

EXAMPLE 5:     Please convert 30 lbs/ in3 into g/cc.

1. It should be understood that 30 lbs/in3 is the known measurement.  It is to be converted into the units of  g/cc..  This is a density dimension.

It is convenient to express the known measurement as numerator & denominator.  .

30 lbs

in3

1. This measurement requires the conversion of units in both the numerator and the

denominator.  It requires a conversion for “30 lbs. to grams” for the numerator term as

well as a conversion for “1 in3 to cm3   or  cc”,  for the denominator term.  It is convenient

to convert each of these units with separate conversion factors.

3.         The unit of in3 in the denominator dictates that the unit for the numerator of the first

conversion factor will be “ in3 “ in order to cancel.  This limits and guides us toward the choices for the denominator. This is an English-to-Metric conversion, it therefore follows that the “1 in = 2.54 cm”  is one of the three CF’s memorized for this category.  The units in3  is a 3rd dimensional unit, volume.  This dictates that the conversion factor be manipulated to accommodate the units of the known measurement.  It therefore must be cubed to obtain the volume dimension.

(1 in = 2.54 cm)3    ?  1 in3 = 16.39 cm3

It will be used for our first conversion factor.

30 lbs           X

in3

We will multiply our known value by the conversion factor:       1 in3         .

16.39 cm3

We select the orientation of the conversion factor so that the original units of “in3.” will cancel.

First conversion

30 lbs              X     1 in3         .

1 in3                         16.39 cm3

The 30 lbs in the numerator dictates the unit for the denominator for the first conversion factor for converting “lbs”  to grams; this will be our second CF.. This is an English-to-Metric conversion, it therefore follows that the “1 lb = 453.6 g”  is one of the three CF’s memorized for this category and is the only mass dimension memorized in this category.

“1 lb = 453.6 g”  will be used for the conversion factor for  “30 lb”

The known measurement, the first and last conversions factors are presented below:

First conversion           Second & final conversion

30 lbs              X     1 in3         .   .        X                 453.6 g   .           =

1 in3                         16.39 cm3                                           1 lb

30 x  1  x  453.6   g    =   13608 g       =   820.26 g/cm3

1 x 16.39 cm3                      16.39 cm3

Solution: multiply all of the numerator values and multiply all of the denominator values, then divide the denominator  product into the numerator product. “ in3,  & lb will cancel leaving the desired units, g/cm3.