Metric Units of Length Conversion

Solution to Example 3
Using the factors Table 1 above going from \( \text{cm} \) to \( \text{dam} \), we write \[ 1 \text { dam} = 10 \times 10 \times 10 \text{ cm} = 1000 \text{ cm} \] which gives the factors of conversion \[ \displaystyle \frac{1 \text { dam}}{1000 \text{ cm}} = 1 \quad (I) \] and \[ \displaystyle \frac{1000 \text{ cm}} {1 \text { dam}} = 1 \quad (II) \]
We are given \( \text { dam} \) we therefore use the factor of conversion (II) since it has \( \text { dam} \) in the denominator which may be cancelled.
Convert \( 170 \; \text{dam} \) as follows
\[ 170 \; \text{dam} = 170 \; \text{dam} \times \displaystyle \frac{1000 \text{ cm}} {1 \text { dam}} \]
Cancel \( \text{dam} \)
\[ 170 \; \text{dam} = 170 \cancel{\text{dam}} \times 1000 \displaystyle \frac{\text{cm}}{\cancel{\text{dam}}} \]
Evaluate
\[ 170 \; \text{dam} = 170 \times 1000 \text{ cm} = 170000 \text{ cm}\]

Solution to Example 4
Using the factors from \( \text{mm} \) to \( \text{hm} \) in Table 1 above, we write \[ 1 \text { hm} = 10 \times 10 \times 10 \times 10 \times 10 \text{ mm} = 100000 \text{ mm} \] which gives the factors of conversion \[ \displaystyle \frac{1 \text { hm}}{100000 \text{ mm}} = 1 \quad (I) \] and \[ \displaystyle \frac {100000 \text{ mm}} {1 \text{ hm}} = 1 \quad (II) \]
We are given \( \text{ mm} \) which needs to be canceled, we therefore use the factor of conversion (I), because it has \( \text{ mm} \) in the denominator.
Convert \( 12500 \; \text{mm} \) as follows
\[ 12500 \; \text{mm} = 12500 \text{ mm} \times \displaystyle \frac{1 \text { hm}}{100000 \text{ mm}} \]
Cancel \( \text{mm} \) on the right
\[ 12500 \; \text{mm} = 12500 \cancel{\text{mm}} \times \displaystyle \frac{1 \text { hm}}{100000 \cancel{\text{mm}}} \]
Simplify and evaluate
\[ 12500 \; \text{mm} = \displaystyle \frac{12500}{100000} \text{hm} \]
Divide
\[ 12500 \; \text{mm} = 0.125 \text{ hm}\]

Solution to Example 5
Using the factors Table 1 above from \( \text{nm} \) to \( \text{cm} \), we write \[ 1 \text { cm} = 1000 \times 1000 \times 10 \text{ nm} = 10000000 \text{ nm} \] which gives the factors of conversion \[ \displaystyle \frac{1 \text { cm}}{10000000 \text{ nm}} = 1 \quad (I) \] and \[ \displaystyle \frac {10000000 \text{ nm}} {1 \text { cm}} = 1 \quad (II) \]
We are given \( \text{cm} \) which needs to be canceled and we therefore use the factor of conversion (II), because it has \( \text { cm} \) in the denominator.
Convert \( 0.0023 \; \text{cm} \) as follows
\[ 0.0023 \; \text{cm} = 0.0023 \; \text{cm} \times \times 10000000 \displaystyle \frac{\text{nm}}{\text{cm}} \]
Cancel \( \text{cm} \) on the right
\[ 0.0023 \; \text{ cm} = 0.0023 \cancel{\text{cm}} \times 10000000 \displaystyle \frac{\text{nm}}{\cancel{\text{cm}}}\]
Simplify and evaluate
\[ 0.0023 \; \text{ cm} = 0.0023 \times 10000000 \text{ nm} = 23000 \text{ nm}\]

Solution to Example 6
Using the factors Table 1 above from \( \; \mu\text{m} \) to and \( \text{ hm} \), we write \[ 1 \text { hm} = 1000 \times 10 \times 10 \times 10 \times 10 \times 10 \; \mu\text{m} = 100000000 \; \mu\text{m} \] which gives the factors of conversion \[ \displaystyle \frac{1 \text { hm}}{100000000 \mu\text{m}} = 1 \quad (I) \] and \[ \displaystyle \frac{100000000 \mu\text{m}} {1 \text { hm}} = 1 \quad (II) \]
We are given \( \mu\text{m} \) which needs to be canceled and we therefore use the factor of conversion (I), because it has \( \mu\text{m} \) in the denominator.
Convert \( 890000 \; \mu\text{m} \) as follows \[ 890000 \; \mu m = 890000 \; \mu m \times \displaystyle \frac{1}{100000000} \displaystyle \frac{\text{hm}}{\mu m} \]
Cancel \( \mu\text{m} \)
\[ 890000 \; \mu m = 890000 \; \cancel{\mu\text{m}} \times \displaystyle \frac{1}{100000000} \displaystyle \frac{\text{hm}}{\cancel{\mu\text{m}}} \]
Simplify and evaluate
\[ 890000 \; \mu m = \displaystyle \frac{890000 \times 1}{100000000} \text{ hm} = 0.0089 \text{ hm}\]

Solutions to the Above Exercises

Part A

1. 
   
   Convert \( \quad 0.0102 \text { Mm} \) to \( \text{hm} \).
   
   From Table 1, we have \[ 1 \text { Mm} = 10 \times 1000 \text{ hm} = 10000 \text{ hm} \] Hence \[ 1 \text { Mm} = 10000 \text{ hm} \] Factors of conversion \[ \displaystyle \frac{1 \text { Mm}}{10000 \text{ hm}} = 1 \quad (I) \] and \[ \displaystyle \frac {10000 \text{ hm}} {1 \text { Mm}} = 1 \quad (II) \] Use the factor of conversion (II) because it has \( \text { Mm} \) in the denominator we are converting \( \text { Mm} \). Hence \[ \quad 0.0102 \text { Mm} = \quad 0.0102 \text { Mm} \times \displaystyle \frac {10000 \text{ hm}} {1 \text { Mm}} \] Cancel \( \text { Mm} \) on the right \[ \quad 0.0102 \text { Mm} = \quad 0.0102 \cancel{\text { Mm}} \times \displaystyle \frac {10000 \text{ hm}} {1 \cancel{\text { Mm}}} \] Simplify and evaluate \[ \quad 0.0102 \text { Mm} = \quad \displaystyle \frac {0.0102 \times 10000 \text{ hm}} {1 } = 102 \text{ hm} \]
   
   
2. 
   
   Convert \( \quad 0.3 \text { nm} \) to \( \text{mm} \).
   
   From Table 1, we have \[ 1 \text { mm} = 1000 \times 1000 \text { nm} = 1000000 \text{ nm} \] Hence \[ 1 \text { mm} = 1000000 \text{ nm} \] Factors of conversion \[ \displaystyle \frac{1 \text { mm}}{1000000 \text{ nm}} = 1 \quad (I) \] and \[ \displaystyle \frac {1000000 \text{ nm}} {1 \text { mm}} = 1 \quad (II) \] Use the factor of conversion (I) because it has \( \text { nm} \) in the denominator and we are converting \( \text { nm} \). Hence \[ \quad 0.3 \text { nm} = \quad 0.3 \text { nm} \times \displaystyle \frac{1 \text { mm}}{1000000 \text{ nm}} \] Cancel \( \text { nm} \) on the right \[ \quad 0.3 \text { nm} = \quad 0.3 \cancel{\text { nm}} \times \displaystyle \frac{1 \text { mm}}{1000000 \cancel{\text{ nm}}} \] Simplify and evaluate \[ \quad 0.3 \text { nm} = \displaystyle \frac{ \quad 0.3 \times 1 \text { mm}}{1000000} = 0.0000003 \text{ mm} \]
   
   
3. 
   
   Convert \( \quad 1200 \text { m} \) to \( \text{Mm} \).
   
   From Table 1, we have \[ 1 \text { Mm} = 10 \times 10 \times 10 \times 1000 \text { m} = 1000000 \text{ m} \] Hence \[ 1 \text { Mm} = 1000000 \text{ m} \] Factors of conversion \[ \displaystyle \frac{1 \text { Mm}}{1000000 \text{ m}} = 1 \quad (I) \] and \[ \displaystyle \frac {1000000 \text{ m}} {1 \text { Mm}} = 1 \quad (II) \] We are given \( \text { m} \) to convert to \( \text { Mm} \) hence we the factor of conversion (I) because it has \( \text { m} \) in the denominator. \[ \quad 1200 \text { m} = \quad 1200 \text { m} \times \displaystyle \frac{1 \text { Mm}}{1000000 \text{ m}} \] Cancel \( \text { m} \) on the right \[ \quad 1200 \text { m} = \quad 1200 \cancel{\text { m}} \times \displaystyle \frac{1 \text { Mm}}{1000000 \cancel{\text { m}}} \] Simplify and evaluate \[ \quad 1200 \text { m} = \displaystyle \frac{ \quad 1200 \times 1 \text { Mm}}{1000000 } = 0.0012 \text { Mm} \]
   
   
4. 
   
   Convert \( \quad 13200 \text { dm} \) to \( \text{km} \).
   
   From Table 1, we have \[ 1 \text { km} = 10 \times 10 \times 10 \times 10 \text{ dm} = 10000 \text{ dm} \] Hence \[ 1 \text { km} = 10000 \text{ dm} \] Factors of conversion \[ \displaystyle \frac{1 \text { km}}{10000 \text{ dm}} = 1 \quad (I) \] and \[ \displaystyle \frac {10000 \text{ dm}}{1 \text{ km}} = 1 \quad (II) \] Given \( \text { dm} \) to convert to \( \text { km} \), we therefore use the factor of conversion (I) because it has \( \text { dm} \) in the denominator. \[ \quad 13200 \text { dm} = 13200 \text { dm} \times \displaystyle \frac{1 \text { km}}{10000 \text{ dm}} \] Cancel \( \text { dm} \) on the right \[ \quad 13200 \text { dm} = 13200 \cancel{\text { dm}} \times \displaystyle \frac{1 \text { km}}{10000 \cancel{\text{ dm}}} \] Simplify and evaluate \[ \quad 13200 \text { dm} = \displaystyle \frac{13200 \times 1 \text { km}}{10000 } = 1.32 \text { km}\]
   
   
5. 
   
   Convert \( \quad 0.002 \text { Gm} \) to \( \text{km} \).
   
   From Table 1, we have \[ 1 \text { Gm} = 1000 \times 1000 \text{ km} = 1000000 \text{ km} \] Hence \[ 1 \text { Gm} = 1000000 \text{ km} \] Factors of conversion \[ \displaystyle \frac{1 \text { Gm}}{1000000 \text{ km}} = 1 \quad (I) \] and \[ \displaystyle \frac {1000000 \text{ km}} {1 \text { Gm}} = 1 \quad (II) \] Given \( \text { Gm} \) to convert to \( \text { km} \), we therefore use the factor of conversion (II) because it has \( \text { Gm} \) in the denominator. \[ \quad 0.002 \text { Gm} = 0.002 \text { Gm} \times \displaystyle \frac {1000000 \text{ km}} {1 \text { Gm}} \] Cancel \( \text { Gm} \) on the right \[ \quad 0.002 \text { Gm} = 0.002 \cancel{\text { Gm} } \times \displaystyle \frac {1000000 \text{ km}} {1 \cancel{\text { Gm}}} \] Simplify and evaluate \[ \quad 0.002 \text { Gm} = \displaystyle \frac {0.002 \times 1000000 \text{ km}} {1 } = 2000 \text{ km} \]
   
   
6. 
   
   Convert \( \quad 13200 { \; \mu} \text m \) to \( \text{dm} \).
   
   From Table 1, we have \[ 1 \text { dm} = 1000 \times 10 \times 10 {\; \mu} \text m = 100000 {\; \mu} \text m \] Hence \[ 1 \text { dm} = 100000 { \; \mu} \text m \] Factors of conversion \[ \displaystyle \frac{1 \text { dm}}{100000 {\; \mu} \text m } = 1 \quad (I) \] and \[ \displaystyle \frac{100000 { \; \mu} \text m } {1 \text { dm}} = 1 \quad (II) \] Given \( \; \mu \) to convert to \( \text{dm} \), we therefore use the factor of conversion (I) because it has \( \; \mu \) in the denominator. \[ \quad 13200 { \; \mu} \text m = 13200 { \; \mu} \text m \times \displaystyle \frac{1 \text { dm}}{100000 {\; \mu} \text m }\] Cancel \( \; \mu \) on the right \[ \quad 13200 { \; \mu} \text m = 13200 \cancel{{ \; \mu} \text m } \times \displaystyle \frac{1 \text { dm}}{100000 \cancel{{ \; \mu} \text m } }\] Simplify and evaluate \[ \quad 13200 { \; \mu} \text m = \displaystyle \frac{13200 \times 1 \text { dm}}{100000 } = 0.132 \text{ dm}\]
   
   
7. 
   
   Convert \( \quad 0.003 \text { dam} \) to \( \text{mm} \).
   
   From Table 1, we have \[ 1 \text { dam} = 10 \times 10 \times 10 \times 10 \text { mm} = 10000 \text { mm} \] Hence \[ 1 \text { dam} = 10000 \text { mm} \] Factors of conversion \[ \displaystyle \frac{1 \text { dam}}{10000 \text { mm} } = 1 \quad (I) \] and \[ \displaystyle \frac{10000 \text { mm} }{1 \text { dam}} = 1 \quad (II) \] Given \( \text{dam} \) to convert to \( \text { mm} \), we use the factor of conversion (II) because it has \( \text { dam} \) in the denominator. \[ \quad 0.003 \text { dam} = 0.003 \text { dam} \times \displaystyle \frac{10000 \text { mm} }{1 \text { dam}} \] Cancel \( \text { dam}\) on the right \[ \quad 0.003 \text { dam} = 0.003 \cancel{\text { dam}} \times \displaystyle \frac{10000 \text { mm} }{1 \cancel{\text { dam}} } \] Simplify and evaluate \[ \quad 0.003 \text { dam} = \displaystyle \frac{0.003 \times 10000 \text { mm} }{1 } = 300 \text{mm} \]
   
   
8. 
   
   Convert \( \quad 0.001 \text { cm} \) to \( \text{nm} \).
   
   From Table 1, we have \[ 1 \text { cm} = 1000 \times 1000 \times 10 \text { nm} = 10000000 \text { nm} \] Hence \[ 1 \text { cm} = 10000000 \text { nm} \] Factors of conversion \[ \displaystyle \frac{1 \text { cm}}{10000000 \text { nm} } = 1 \quad (I) \] and \[ \displaystyle \frac {10000000 \text { nm} } {1 \text { cm}} = 1 \quad (II) \] We use the factor of conversion (II) because it has \( \text { cm} \) in the denominator and we are given \( \text { cm} \). \[ \quad 0.001 \text { cm} = 0.001 \text { cm} \times \displaystyle \frac {10000000 \text { nm} } {1 \text { cm}} \] Cancel \( \text { cm}\) on the right \[ \quad 0.001 \text { cm} = 0.001 \cancel{\text { cm} } \times \displaystyle \frac {10000000 \text { nm} } {1 \cancel{\text { cm}}} \] Simplify and evaluate \[ \quad 0.001 \text { cm} = \displaystyle \frac { 0.001 \times 10000000 \text { nm} } {1 } = 10000 \text { nm}\]