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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 $$\quad 0.001 \text { cm} = \displaystyle \frac { 0.001 \times 10000000 \text { nm} } {1 } = 10000 \text { nm}$$