#### USING CONVERSION FACTORS

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

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**CONVERSION FACTORS (CF).**

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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: **

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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.

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**MANIPULATION OF CONVERSION FACTOR:**

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**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/t^{2}); 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). 2^{nd} dimension and & 3^{rd} dimension refer to units that are squared and units that are cubed respectively.

**HELPFUL HINTS:**

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.

**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.**It is convenient if the known measurement is expressed as a numerator with the denominator of “1”.****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**Some measurements have units expressed in the numerator as well as in the denominator. Pay close attention as to which unit is being converted.****Choosing the first CF.**If the**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.**Determine the conversion category.**It should be: English-to-English, Metric-to-Metric, or English-to-Metric or Metric-to-English.**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).****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.

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**EXAMPLE 1: Please convert 30 in. to cm.**

- 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.
- It is convenient to express the known measurement with the understood

denominator of “1”.

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30 in

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 in ^{2} into cm^{2}.**

- It should be understood that 30 in
^{2}is the known measurement. It is to be converted into the unit of cm^{2}. This is an area dimension. - It is convenient to express the known measurement with the understood

denominator of “1”.

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30 in^{2}

1

- The unit of in
^{2}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 in^{2}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 in^{2}is a 2^{nd}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 in^{2} = 6.45 cm^{2}

We will multiply our known measurement by the conversion factor: 6.45 cm^{2}

1 in^{2}.

We select the orientation of the conversion factor so that the original units of “in^{2}.” will cancel.

30 in^{2} X 6.45 cm^{2} = 193.5 cm^{2} = 193.5 cm^{2}

1 1 in^{2} 1

**Solution:** multiply all of the numerator values and multiply all of the denominator values, then divide the denominator product into the numerator product. “ in^{2} will cancel leaving the desired units.

**EXAMPLE 3: Please convert 30 in ^{3} into cm^{3}.**

- It should be understood that 30 in
^{3}is the known measurement. It is to be converted into the unit of cm^{3}. This is a volume dimension. - It is convenient to express the known measurement with the understood

denominator of “1”.

.

30 in^{3}

1

- The unit of in
^{3}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 in^{3}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 in^{3}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 in^{3} = 16.39 cm^{3}

We will multiply our known measurement by the conversion factor: 16.39 cm^{3}

1 in^{3}.

We select the orientation of the conversion factor so that the original units of “in^{3}.” will cancel.

30 in^{3} X 16.39 cm^{3} = 491.7 cm^{3} = 491.7 cm^{3}

1 1 in^{3} 1

**Solution:** multiply all of the numerator values and multiply all of the denominator values, then divide the denominator product into the numerator product. “ in^{3} will cancel leaving the desired units.

**EXAMPLE 4: Please convert 30 in ^{3} into HL.**

- It should be understood that 30 in
^{3}is the known measurement. It is to be converted into the unit of Hectoliters. This is a volume dimension. - It is convenient to express the known measurement with the understood

denominator of “1”.

.

30 in^{3}

1

- The unit of in
^{3}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 in^{3}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 in^{3}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 in^{3} = 16.39 cm^{3}

We will multiply our known measurement by the conversion factor: 16.39 cm^{3}

1 in^{3}.

We select he orientation of the conversion factor so that the original units of “in^{3}.” will cancel.

The final conversion factor will be: 1 HL

10^{2} L

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

First conversion Last conversion

30 in^{3} X 16.39 cm^{3} ……….. 1 HL =

1 1 in^{3} 10^{2} L

The second conversion factor is easily determined because we recognize that the cm^{3} is a volume measurement. It has the equivalent of mL. The second conversion factor will be:

1 mL

1 cm^{3}

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First conversion 2^{nd} conversion ……..last conversion

30 in^{3} X 16.39 cm^{3} X 1 mL ……?……….... 1 HL =

1 1 in^{3} 1 cm^{3} 10^{2} L

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After the 2^{nd} conversion factor is applied, it should be clear that a 3^{rd} 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 2^{nd} conversion needs to cancel thus mL will be in the denominator to cancel the mL in the 2^{nd} 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 2^{nd} conversion 3^{rd} conversion last conversion

30 in^{3} X 16.39 cm^{3} X 1 mL X 10^{-3} L X 1 HL =

1 1 in^{3} 1 cm^{3} 1 mL 10^{2} 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 10^{2}

**Solution:** multiply all of the numerator values and multiply all of the denominator values, then divide the denominator product into the numerator product. “ in^{3}, cm^{3}, mL, & L will cancel leaving the desired units, HL.

**EXAMPLE 5: Please convert 30 lbs/ in ^{3} into g/cc.**

- It should be understood that 30 lbs/in
^{3}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

in^{3}

- 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 in^{3} to cm^{3 }or cc”, for the denominator term. It is convenient

to convert each of these units with separate conversion factors.

3. The unit of in^{3} in the denominator dictates that the unit for the numerator of the first

conversion factor will be “ in^{3} “ 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 in^{3} 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 in^{3} = 16.39 cm^{3}

It will be used for our first conversion factor.

30 lbs X

in^{3}

We will multiply our known value by the conversion factor: 1 in^{3 }.

16.39 cm^{3}

We select the orientation of the conversion factor so that the original units of “in^{3}.” will cancel.

First conversion

30 lbs X 1 in^{3 }.

1 in^{3} 16.39 cm^{3}

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 in^{3 }. . X 453.6 g . =

1 in^{3} 16.39 cm^{3 }1 lb

30 x 1 x 453.6 g = 13608 g = 820.26 g/cm^{3}

1 x 16.39 cm^{3 }16.39 cm^{3}

**Solution:** multiply all of the numerator values and multiply all of the denominator values, then divide the denominator product into the numerator product. “ in^{3}, & lb will cancel leaving the desired units, g/cm^{3}.