Rates and Unit Conversion

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Worksheet Rates and Unit Conversion

1.

It’s given that $1 \text{ foot} = 12 \text{ inches}\text{.}$ Do the following unit conversions.

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$\displaystyle{ 5 \text{ feet} = }$ $\displaystyle{ \text{ inches} }$
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$\displaystyle{ 192 \text{ inches} = }$ $\displaystyle{ \text{ feet} }$
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Answer 1.

$60$

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Answer 2.

$16$

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Solution.

Cancelling Unit Method

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It’s given that $1 \text{ foot} = 12 \text{ inches}\text{.}$ This can be written as either $\frac{1 \text{ ft}}{12 \text{ in}}$ or $\frac{12 \text{ in}}{1 \text{ ft}}\text{.}$

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To convert $5$ feet to inches, we will use $\frac{12 \text{ in}}{1 \text{ ft}}\text{,}$ so the unit ft will cancel:

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$\displaystyle{ \begin{aligned}[t] \amp \phantom{{}=} 5 \text{ ft} \cdot \frac{12 \text{ in}}{1 \text{ ft}} \\ \amp = 5 \cdot \frac{12 \text{ in}}{1} \\ \amp = 5 \cdot 12 \text{ in} \\ \amp = 60 \text{ in} \end{aligned} }$

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So, $\displaystyle{ 5 \text{ ft} = 60 \text{ in} }$

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Next, to convert $192$ inches to feet, we will use $\frac{1 \text{ ft}}{12 \text{ in}}\text{,}$ so the unit in will cancel:

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$\displaystyle{ \begin{aligned}[t] \amp \phantom{{}=} 192 \text{ in} \cdot \frac{1 \text{ ft}}{12 \text{ in}} \\ \amp = 192 \cdot \frac{1 \text{ ft}}{12} \\ \amp = \frac{192}{1} \cdot \frac{1 \text{ ft}}{12} \\ \amp = \frac{192 \cdot 1}{1 \cdot 12} \text{ ft} \\ \amp = \frac{192}{12} \text{ ft} \\ \amp = 16 \text{ ft} \end{aligned} }$

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So, $\displaystyle{ 192 \text{ in} = 16 \text{ ft} }$

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Proportion Method

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It’s given that $1 \text{ feet} = 12 \text{ inches}\text{.}$ In the following proportions, we will write ft in numerators (top) and in in denominators (bottom).

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Assume $5 \text{ ft} = x \text{ in}\text{.}$ We have:

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$\displaystyle{\begin{aligned}[t] \frac{\text{ft}}{\text{in}} \amp = \frac{\text{ft}}{\text{in}} \\ \frac{1 \text{ ft}}{12 \text{ in}} \amp = \frac{5 \text{ ft}}{x \text{ in}} \\ \frac{1}{12} \amp = \frac{5}{x} \\ 1 \cdot x \amp = 12 \cdot 5 \\ x \amp = 60 \end{aligned} }$

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So, $\displaystyle{ 5 \text{ ft} = 60 \text{ in} }$

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Next, assume $x \text{ ft} = 192 \text{ in}\text{.}$ We have:

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$\displaystyle{\begin{aligned}[t] \frac{\text{ft}}{\text{in}} \amp = \frac{\text{ft}}{\text{in}} \\ \frac{1 \text{ ft}}{12 \text{ in}} \amp = \frac{x \text{ ft}}{192 \text{ in}} \\ \frac{1}{12} \amp = \frac{x}{192} \\ 12 \cdot x \amp = 1 \cdot 192 \\ 12x \amp = 192 \\ \frac{12x}{12} \amp = \frac{192}{12} \\ x \amp = 16 \end{aligned} }$

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So, $\displaystyle{ 192 \text{ in} = 16 \text{ ft} }$

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Shortcut

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It’s given that $1 \text{ feet} = 12 \text{ inches}\text{.}$ To convert between these two units, we either multiply or divide by $12\text{.}$

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For Question 1, to convert $5$ feet to inches, the number will become bigger. So we do:

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$\displaystyle{ 5 \text{ ft} = 5 \cdot 12 \text{ in} = 60 \text{ in} }$

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For Question 2, to convert $192$ inches to feet, the number will become smaller. So we do:

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$\displaystyle{ 192 \text{ in} = 192 \div 12 \text{ ft} = 16 \text{ ft} }$

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2.

It’s given that $1 \text{ mile} = 5280 \text{ feet}\text{.}$ Do the following unit conversions.

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$\displaystyle{ 0.059 \text{ miles} = }$ $\displaystyle{ \text{ feet} }$
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$\displaystyle{ 16896 \text{ feet} = }$ $\displaystyle{ \text{ miles} }$
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Answer 1.

$311.52$

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Answer 2.

$3.2$

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Solution.

Cancelling Unit Method

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It’s given that $1 \text{ mile} = 5280 \text{ feet}\text{.}$ This can be written as either $\frac{1 \text{ mi}}{5280 \text{ ft}}$ or $\frac{5280 \text{ ft}}{1 \text{ mi}}\text{.}$

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To convert $0.059$ miles to feet, we will use $\frac{5280 \text{ ft}}{1 \text{ mi}}\text{,}$ so the unit mi will cancel:

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$\displaystyle{ \begin{aligned}[t] \amp \phantom{{}=} 0.059 \text{ mi} \cdot \frac{5280 \text{ ft}}{1 \text{ mi}} \\ \amp = 0.059 \cdot \frac{5280 \text{ ft}}{1} \\ \amp = 0.059 \cdot 5280 \text{ ft} \\ \amp = 311.52 \text{ ft} \end{aligned} }$

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So, $\displaystyle{ 0.059 \text{ mi} = 311.52 \text{ ft} }$

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Next, to convert $16896$ feet to miles, we will use $\frac{1 \text{ mi}}{5280 \text{ ft}}\text{,}$ so the unit in will cancel:

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$\displaystyle{ \begin{aligned}[t] \amp \phantom{{}=} 16896 \text{ ft} \cdot \frac{1 \text{ mi}}{5280 \text{ ft}} \\ \amp = 16896 \cdot \frac{1 \text{ mi}}{5280} \\ \amp = \frac{16896}{1} \cdot \frac{1 \text{ mi}}{5280} \\ \amp = \frac{16896 \cdot 1}{1 \cdot 5280} \text{ mi} \\ \amp = \frac{16896}{5280} \text{ mi} \\ \amp = 3.2 \text{ mi} \end{aligned} }$

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So, $\displaystyle{ 16896 \text{ ft} = 3.2 \text{ mi} }$

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Proportion Method

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It’s given that $1 \text{ miles} = 5280 \text{ feet}\text{.}$ In the following proportions, we will write mi in numerators (top) and in in denominators (bottom).

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Assume $0.059 \text{ mi} = x \text{ ft}\text{.}$ We have:

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$\displaystyle{\begin{aligned}[t] \frac{\text{mi}}{\text{ft}} \amp = \frac{\text{mi}}{\text{ft}} \\ \frac{1 \text{ mi}}{5280 \text{ ft}} \amp = \frac{0.059 \text{ mi}}{x \text{ ft}} \\ \frac{1}{5280} \amp = \frac{0.059}{x} \\ 1 \cdot x \amp = 5280 \cdot 0.059 \\ x \amp = 311.52 \end{aligned} }$

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So, $\displaystyle{ 0.059 \text{ mi} = 311.52 \text{ ft} }$

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Next, assume $x \text{ mi} = 16896 \text{ ft}\text{.}$ We have:

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$\displaystyle{\begin{aligned}[t] \frac{\text{mi}}{\text{ft}} \amp = \frac{\text{mi}}{\text{ft}} \\ \frac{1 \text{ mi}}{5280 \text{ ft}} \amp = \frac{x \text{ mi}}{16896 \text{ ft}} \\ \frac{1}{5280} \amp = \frac{x}{16896} \\ 5280 \cdot x \amp = 1 \cdot 16896 \\ 5280x \amp = 16896 \\ \frac{5280x}{5280} \amp = \frac{16896}{5280} \\ x \amp = 3.2 \end{aligned} }$

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So, $\displaystyle{ 16896 \text{ ft} = 3.2 \text{ mi} }$

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Shortcut

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It’s given that $1 \text{ miles} = 5280 \text{ feet}\text{.}$ To convert between these two units, we either multiply or divide by $5280\text{.}$

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For Part a, to convert $0.059$ miles to feet, the number will become bigger. So we do:

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$\displaystyle{ 0.059 \text{ mi} = 0.059 \cdot 5280 \text{ ft} = 311.52 \text{ ft} }$

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For Part b, to convert $16896$ feet to miles, the number will become smaller. So we do:

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$\displaystyle{ 16896 \text{ ft} = 16896 \div 5280 \text{ mi} = 3.2 \text{ mi} }$

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3.

It’s given that $1 \text{ ton} = 2000 \text{ lb}\text{.}$ Do the following unit conversions.

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$\displaystyle{ 7 \text{ tons} = }$ $\displaystyle{ \text{ lb} }$
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$\displaystyle{ 18000000 \text{ lb} = }$ $\displaystyle{ \text{ tons} }$
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Answer 1.

$14000$

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Answer 2.

$9000$

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Solution.

Cancelling Unit Method

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It’s given that $1 \text{ ton} = 2000 \text{ lb}\text{.}$ This can be written as either $\frac{1 \text{ ton}}{2000 \text{ lb}}$ or $\frac{2000 \text{ lb}}{1 \text{ t}}\text{.}$

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To convert $7$ tons to lb, we will use $\frac{2000 \text{ lb}}{1 \text{ t}}\text{,}$ so the unit t will cancel:

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$\displaystyle{ \begin{aligned}[t] \amp \phantom{{}=} 7 \text{ t} \cdot \frac{2000 \text{ lb}}{1 \text{ t}} \\ \amp = 7 \cdot \frac{2000 \text{ lb}}{1} \\ \amp = 7 \cdot 2000 \text{ lb} \\ \amp = 14000 \text{ lb} \end{aligned} }$

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So, $\displaystyle{ 7 \text{ t} = 14000 \text{ lb} }$

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Next, to convert $18000000$ lb to tons, we will use $\frac{1 \text{ t}}{2000 \text{ lb}}\text{,}$ so the unit lb will cancel:

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$\displaystyle{ \begin{aligned}[t] \amp \phantom{{}=} 18000000 \text{ lb} \cdot \frac{1 \text{ t}}{2000 \text{ lb}} \\ \amp = 18000000 \cdot \frac{1 \text{ t}}{2000} \\ \amp = \frac{18000000}{1} \cdot \frac{1 \text{ t}}{2000} \\ \amp = \frac{18000000 \cdot 1}{1 \cdot 2000} \text{ t} \\ \amp = \frac{18000000}{2000} \text{ t} \\ \amp = 9000 \text{ t} \end{aligned} }$

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So, $\displaystyle{ 18000000 \text{ lb} = 9000 \text{ t} }$

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Proportion Method

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It’s given that $1 \text{ ton} = 2000 \text{ lb}\text{.}$ In the following proportions, we will write t in numerators (top) and lb in denominators (bottom).

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Assume $7 \text{ t} = x \text{ lb}\text{.}$ We have:

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$\displaystyle{\begin{aligned}[t] \frac{\text{t}}{\text{lb}} \amp = \frac{\text{t}}{\text{lb}} \\ \frac{1 \text{ t}}{2000 \text{ lb}} \amp = \frac{7 \text{ t}}{x \text{ lb}} \\ \frac{1}{2000} \amp = \frac{7}{x} \\ 1 \cdot x \amp = 2000 \cdot 7 \\ x \amp = 14000 \end{aligned} }$

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So, $\displaystyle{ 7 \text{ t} = 14000 \text{ lb} }$

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Next, assume $x \text{ t} = 18000000 \text{ lb}\text{.}$ We have:

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$\displaystyle{\begin{aligned}[t] \frac{\text{t}}{\text{lb}} \amp = \frac{\text{t}}{\text{lb}} \\ \frac{1 \text{ t}}{2000 \text{ lb}} \amp = \frac{x \text{ t}}{18000000 \text{ lb}} \\ \frac{1}{2000} \amp = \frac{x}{18000000} \\ 2000 \cdot x \amp = 1 \cdot 18000000 \\ 2000x \amp = 18000000 \\ \frac{2000x}{2000} \amp = \frac{18000000}{2000} \\ x \amp = 9000 \end{aligned} }$

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So, $\displaystyle{ 18000000 \text{ lb} = 9000 \text{ t} }$

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Shortcut

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It’s given that $1 \text{ ton} = 2000 \text{ lb}\text{.}$ To convert between these two units, we either multiply or divide by $2000\text{.}$

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For Part a, to convert $7$ tons to lb, the number will become bigger. So we do:

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$\displaystyle{ 7 \text{ t} = 7 \cdot 2000 \text{ lb} = 14000 \text{ lb} }$

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For Part b, to convert $18000000$ lb to tons, the number will become smaller. So we do:

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$\displaystyle{ 18000000 \text{ lb} = 18000000 \div 2000 \text{ t} = 9000 \text{ t} }$

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4.

It’s given that $1 \text{ kilometer} = 1000 \text{ meters}\text{.}$ Do the following unit conversions.

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$\displaystyle{ 8 \text{ kilometers} = }$ $\displaystyle{ \text{ meters} }$
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$\displaystyle{ 16000 \text{ meters} = }$ $\displaystyle{ \text{ kilometers} }$
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Answer 1.

$8000$

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Answer 2.

$16$

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Solution.

Cancelling Unit Method

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It’s given that $1 \text{ kilometer} = 1000 \text{ meters}\text{.}$ This can be written as either $\frac{1 \text{ km}}{1000 \text{ m}}$ or $\frac{1000 \text{ m}}{1 \text{ km}}\text{.}$

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To convert $8$ kilometers to meters, we will use $\frac{1000 \text{ m}}{1 \text{ km}}\text{,}$ so the unit km will cancel:

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$\displaystyle{ \begin{aligned}[t] \amp \phantom{{}=} 8 \text{ km} \cdot \frac{1000 \text{ m}}{1 \text{ km}} \\ \amp = 8 \cdot \frac{1000 \text{ m}}{1} \\ \amp = 8 \cdot 1000 \text{ m} \\ \amp = 8000 \text{ m} \end{aligned} }$

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So, $\displaystyle{ 8 \text{ km} = 8000 \text{ m} }$

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Next, to convert $16000$ meters to kilometers, we will use $\frac{1 \text{ km}}{1000 \text{ m}}\text{,}$ so the unit m will cancel:

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$\displaystyle{ \begin{aligned}[t] \amp \phantom{{}=} 16000 \text{ m} \cdot \frac{1 \text{ km}}{1000 \text{ m}} \\ \amp = 16000 \cdot \frac{1 \text{ km}}{1000} \\ \amp = \frac{16000}{1} \cdot \frac{1 \text{ km}}{1000} \\ \amp = \frac{16000 \cdot 1}{1 \cdot 1000} \text{ km} \\ \amp = \frac{16000}{1000} \text{ km} \\ \amp = 16 \text{ km} \end{aligned} }$

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So, $\displaystyle{ 16000 \text{ m} = 16 \text{ km} }$

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Proportion Method

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It’s given that $1 \text{ kilometer} = 1000 \text{ meters}\text{.}$ In the following proportions, we will write km in numerators (top) and m in denominators (bottom).

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Assume $8 \text{ km} = x \text{ m}\text{.}$ We have:

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$\displaystyle{\begin{aligned}[t] \frac{\text{km}}{\text{m}} \amp = \frac{\text{km}}{\text{m}} \\ \frac{1 \text{ km}}{1000 \text{ m}} \amp = \frac{8 \text{ km}}{x \text{ m}} \\ \frac{1}{1000} \amp = \frac{8}{x} \\ 1 \cdot x \amp = 1000 \cdot 8 \\ x \amp = 8000 \end{aligned} }$

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So, $\displaystyle{ 8 \text{ km} = 8000 \text{ m} }$

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Next, assume $x \text{ km} = 16000 \text{ m}\text{.}$ We have:

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$\displaystyle{\begin{aligned}[t] \frac{\text{km}}{\text{m}} \amp = \frac{\text{km}}{\text{m}} \\ \frac{1 \text{ km}}{1000 \text{ m}} \amp = \frac{x \text{ km}}{16000 \text{ m}} \\ \frac{1}{1000} \amp = \frac{x}{16000} \\ 1000 \cdot x \amp = 1 \cdot 16000 \\ 1000x \amp = 16000 \\ \frac{1000x}{1000} \amp = \frac{16000}{1000} \\ x \amp = 16 \end{aligned} }$

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So, $\displaystyle{ 16000 \text{ m} = 16 \text{ km} }$

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Shortcut

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It’s given that $1 \text{ kilometer} = 1000 \text{ meters}\text{.}$ To convert between these two units, we either multiply or divide by $1000\text{.}$

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Recall that multiplying $1000$ causes the decimal point to move right $3$ times, like in $3 \cdot 1000 = 3000\text{.}$

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Dividing by $1000$ causes the decimal point to move left $3$ times, like in $3000 \div 1000 = 3\text{.}$

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For Part a, to convert $8$ kilometers to meters, the number will become bigger. So we do:

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$\displaystyle{ 8 \text{ km} = 8 \cdot 1000 \text{ m} = 8000 \text{ m} }$

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For Part b, to convert $16000$ meters to kilometers, the number will become smaller. So we do:

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$\displaystyle{ 16000 \text{ m} = 16000 \div 1000 \text{ km} = 16 \text{ km} }$

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5.

It’s given that $1 \text{ meter} = 100 \text{ centimeters}\text{.}$ Do the following unit conversions.

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$\displaystyle{ 890 \text{ meters} = }$ $\displaystyle{ \text{ centimeters} }$
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$\displaystyle{ 280000 \text{ centimeters} = }$ $\displaystyle{ \text{ meters} }$
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Answer 1.

$89000$

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Answer 2.

$2800$

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Solution.

Cancelling Unit Method

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It’s given that $1 \text{ meter} = 100 \text{ centimeters}\text{.}$ This can be written as either $\frac{1 \text{ m}}{100 \text{ cm}}$ or $\frac{100 \text{ cm}}{1 \text{ m}}\text{.}$

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To convert $890$ meters to centimeters, we will use $\frac{100 \text{ cm}}{1 \text{ m}}\text{,}$ so the unit m will cancel:

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$\displaystyle{ \begin{aligned}[t] \amp \phantom{{}=} 890 \text{ m} \cdot \frac{100 \text{ cm}}{1 \text{ m}} \\ \amp = 890 \cdot \frac{100 \text{ cm}}{1} \\ \amp = 890 \cdot 100 \text{ cm} \\ \amp = 89000 \text{ cm} \end{aligned} }$

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So, $\displaystyle{ 890 \text{ m} = 89000 \text{ cm} }$

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Next, to convert $280000$ centimeters to meters, we will use $\frac{1 \text{ m}}{100 \text{ cm}}\text{,}$ so the unit cm will cancel:

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$\displaystyle{ \begin{aligned}[t] \amp \phantom{{}=} 280000 \text{ cm} \cdot \frac{1 \text{ m}}{100 \text{ cm}} \\ \amp = 280000 \cdot \frac{1 \text{ m}}{100} \\ \amp = \frac{280000}{1} \cdot \frac{1 \text{ m}}{100} \\ \amp = \frac{280000 \cdot 1}{1 \cdot 100} \text{ m} \\ \amp = \frac{280000}{100} \text{ m} \\ \amp = 2800 \text{ m} \end{aligned} }$

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So, $\displaystyle{ 280000 \text{ cm} = 2800 \text{ m} }$

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Proportion Method

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It’s given that $1 \text{ meter} = 100 \text{ centimeters}\text{.}$ In the following proportions, we will write m in numerators (top) and cm in denominators (bottom).

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Assume $890 \text{ m} = x \text{ cm}\text{.}$ We have:

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$\displaystyle{\begin{aligned}[t] \frac{\text{m}}{\text{cm}} \amp = \frac{\text{m}}{\text{cm}} \\ \frac{1 \text{ m}}{100 \text{ cm}} \amp = \frac{890 \text{ m}}{x \text{ cm}} \\ \frac{1}{100} \amp = \frac{890}{x} \\ 1 \cdot x \amp = 100 \cdot 890 \\ x \amp = 89000 \end{aligned} }$

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So, $\displaystyle{ 890 \text{ m} = 89000 \text{ cm} }$

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Next, assume $x \text{ m} = 280000 \text{ cm}\text{.}$ We have:

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$\displaystyle{\begin{aligned}[t] \frac{\text{m}}{\text{cm}} \amp = \frac{\text{m}}{\text{cm}} \\ \frac{1 \text{ m}}{100 \text{ cm}} \amp = \frac{x \text{ m}}{280000 \text{ cm}} \\ \frac{1}{100} \amp = \frac{x}{280000} \\ 100 \cdot x \amp = 1 \cdot 280000 \\ 100x \amp = 280000 \\ \frac{100x}{100} \amp = \frac{280000}{100} \\ x \amp = 2800 \end{aligned} }$

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So, $\displaystyle{ 280000 \text{ cm} = 2800 \text{ m} }$

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Shortcut

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It’s given that $1 \text{ meter} = 100 \text{ centimeters}\text{.}$ To convert between these two units, we either multiply or divide by $100\text{.}$

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Recall that multiplying $100$ causes the decimal point to move right once, like in $3 \cdot 100 = 300\text{.}$

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Dividing by $100$ causes the decimal point to move left once, like in $3000 \div 100 = 30\text{.}$

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For Part a, to convert $890$ meters to centimeters, the number will become bigger. So we do:

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$\displaystyle{ 890 \text{ m} = 890 \cdot 100 \text{ cm} = 89000 \text{ cm} }$

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For Part b, to convert $280000$ centimeters to meters, the number will become smaller. So we do:

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$\displaystyle{ 280000 \text{ cm} = 280000 \div 100 \text{ m} = 2800 \text{ m} }$

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6.

It’s given that:

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$\displaystyle{\begin{aligned} 1\text{mile} \amp = 5280\text{ft} \\ 1\text{ft} \amp = 12\text{in} \\ 1\text{in} \amp = 2.54\text{cm} \\ 1\text{m} \amp = 100\text{cm} \\ 1\text{km} \amp = 1000\text{m} \end{aligned} }$

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Do the following unit conversion. Use decimals in your answer when needed.

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$\displaystyle{ 0.01 \frac{\text{mile}}{\text{second}} = }$ $\displaystyle{ \frac{\text{km}}{\text{hr}} }$

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Answer.

$57.9364$

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Solution.

To convert $0.01 \frac{\text{mile}}{\text{second}}$ to $\frac{\text{km}}{\text{hr}}\text{,}$ we will first convert the unit mile to km:

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$\displaystyle{ \begin{aligned}[t] \amp \phantom{{}=} 0.01 \frac{\text{mile}}{\text{second}} \cdot \frac{5280 \text{ ft}}{1 \text{ mile}} \cdot \frac{12 \text{ in}}{1 \text{ ft}} \cdot \frac{2.54 \text{ cm}}{1 \text{ in}} \cdot \frac{1 \text{ m}}{100 \text{ cm}} \cdot \frac{1 \text{ km}}{1000 \text{ m}} \\ \amp = \frac{0.01\cdot 5280\cdot 12 \cdot 2.54 \text{ km}}{100\cdot 1000 \text{ second}} \\ \amp = 0.01609344 \frac{\text{ km}}{\text{second}} \end{aligned} }$

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Next, convert the unit second to hr:

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$\displaystyle{ \begin{aligned}[t] \amp \phantom{{}=} 0.01609344 \frac{\text{ km}}{\text{second}} \cdot \frac{60 \text{ seconds}}{1 \text{ min}} \cdot \frac{60 \text{ min}}{1 \text{ hr}} \\ \amp = \frac{0.01609344\cdot 60\cdot 60 \text{ km}}{1 \text{ hr}} \\ \amp = 57.936384 \frac{\text{ km}}{\text{second}} \end{aligned} }$

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So $\displaystyle{ 0.01 \frac{\text{mile}}{\text{second}} = 57.936384 \frac{\text{km}}{\text{hr}} }$

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7.

It’s given that:

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Do the following unit conversion. Use decimals in your answer when needed.

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$\displaystyle{ 1103 \frac{\text{km}}{\text{hr}} = }$ $\displaystyle{ \frac{\text{miles}}{\text{second}} }$

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Answer.

$0.190381$

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Solution.

To convert $1103 \frac{\text{km}}{\text{hr}}$ to $\frac{\text{miles}}{\text{second}}\text{,}$ we will first convert the unit km to mi:

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$\displaystyle{ \begin{aligned}[t] \amp \phantom{{}=} 1103 \frac{\text{km}}{\text{hr}} \cdot \frac{1000 \text{ m}}{1 \text{ km}} \cdot \frac{100 \text{ cm}}{1 \text{ m}} \cdot \frac{1 \text{ in}}{2.54 \text{ cm}} \cdot \frac{1 \text{ ft}}{12 \text{ in}} \cdot \frac{1 \text{ mile}}{5280 \text{ ft}} \\ \amp = \frac{1103\cdot 1000\cdot 100 \text{ mile}}{2.54\cdot 12 \cdot 5280 \text{ hr}} \\ \amp = 685.372425037779 \frac{\text{ miles}}{\text{hr}} \end{aligned} }$

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Next, convert the unit hr to second:

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$\displaystyle{ \begin{aligned}[t] \amp \phantom{{}=} 685.372425037779 \frac{\text{ miles}}{\text{hr}} \cdot \frac{1 \text{ hr}}{60 \text{ min}} \cdot \frac{1 \text{ min}}{60 \text{ seconds}} \\ \amp = \frac{685.372425037779 \text{ miles}}{60 \cdot 60 \text{ second}} \\ \amp = 0.190381229177161 \frac{\text{ km}}{\text{second}} \end{aligned} }$

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So $\displaystyle{ 1103 \frac{\text{km}}{\text{hr}} = 0.190381229177161 \frac{\text{miles}}{\text{second}} }$

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8.

Ken purchased $2.6$ pounds of apples at the total cost of ${\$5.46}\text{.}$ If he purchases $8.8$ pounds of apples at this store, how much would it cost?

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It would cost to purchase $8.8$ pounds of apples.

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Answer.

$\$18.48$

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Solution.

Method 1:

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We will use rate to solve this problem. To find the cost of $8.8$ pounds of apples, we first need to find the cost of each pound of apples:

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$\displaystyle{ \frac{{\$5.46}}{2.6\text{ lb}} = \frac{{\$2.10}}{1 \text{ lb}}}$

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The unit price of apples is ${\$2.10}$ per pound. Next, we do:

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$\displaystyle{\begin{aligned}[t] \amp \phantom{{}=}8.8 \text{ lb} \cdot \frac{{\$2.10}}{1 \text{ lb}} \\ \amp = 8.8 \cdot \frac{{\$2.10}}{1} \\ \amp = 8.8 \cdot {\$2.10} \\ \amp = {\$18.48} \end{aligned} }$

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It would cost ${\$18.48}$ to purchase $8.8$ pounds of apples.

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Method 2

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We will use proportion to solve this problem. Assume it would cost $x$ dollars to purchase $8.8$ pounds of apples.

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In the proportion, the top (numerators) have the cost in dollars, and the bottom (denominators) have the weight in pounds. We have:

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$\displaystyle{\begin{aligned}[t] \frac{\text{cost of apples in dollars}}{\text{amount of apples in lb}} \amp =\frac{\text{cost of apples in dollars}}{\text{amount of apples in lb}} \\ \frac{{\$5.46}}{2.6 \text{ lb}} \amp = \frac{x \text{ dollars}}{8.8 \text{ lb}} \\ 2.6x \amp = 5.46 \cdot 8.8 \\ 2.6x \amp = 48.048 \\ \frac{2.6x}{2.6} \amp = \frac{48.048}{2.6} \\ x \amp = 18.48 \end{aligned} }$

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It would cost ${\$18.48}$ to purchase $8.8$ pounds of apples.

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9.

Set up a proportion to solve the application problem.

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Jesse’s car gets 18 miles per gallon of gas. If Las Vegas is 180 miles away, how many gallons of gas are needed to get there and then home? If gas is $2.52 per gallon, what is the total cost of the gas for the trip?

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Gas Needed: gallons

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Cost of Gas: dollars

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Answer 1.

$20$

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Answer 2.

$50.4$

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Solution.

Jesse get 18 miles per 1 gallon of gasoline. We want to know how many gallons of gas she will need to drive to Las Vegas and back. Las Vegas is 180 miles away, so the roundtrip will be $2 \cdot 180 = 360$ miles.

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Let x = the number of gallons of gas need to drive 360 miles. We can use the proportion:

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$\displaystyle{\frac{18}{1}=\frac{360}{x}}$

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Set the cross products equal:

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$18x = 360\cdot 1$

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$18x = 360$

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$x = 360 \div 18 = 20$

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If gas is $$2.52$ per gallon, she will spend:

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$2.52\cdot 20 = 50.4$

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Jesse will need 20 gallons of gas and will spend $50.4 on gasoline.

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10.

A team of geologists are observing a sink hole in a remote area. When they arrived at the scene, the hole was $5$ feet deep. Three days later, the hole sank $6$ inches further. How deep was the sink hole three days later? Answer this question first in inches, and then in feet.

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The sink hole was $\displaystyle{ \text{ inches} }$ deep three days later.
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The sink hole was $\displaystyle{ \text{ feet} }$ deep three days later. Use fractions in your answer. Don’t use decimal.
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Answer 1.

$66$

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Answer 2.

$5{\textstyle\frac{1}{2}}$

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Solution.

We will use the formula $1 \text{ foot} = 12 \text{ inches}\text{.}$

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Part a: We need to change $5$ feet into inches:

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$\displaystyle{ 5 \text{ ft} = 5 \cdot 12 \text{ in} = 60 \text{ in} }$

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Next, we add up the distances:

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$\displaystyle{ 60 \text{ in} + 6 \text{ in} = 66 \text{ in} }$

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The sink hole was $66$ $\displaystyle{ \text{ inches} }$ deep three days later.

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Part b: We need to change $6$ inches into feet:

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$\displaystyle{ 6 \text{ in} = \frac{6}{12} \text{ ft} = {{\frac{1}{2}}} \text{ ft} }$

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Next, we add up the distances:

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$\displaystyle{ {{\frac{1}{2}}} \text{ ft} + 5 \text{ ft} = {5{\textstyle\frac{1}{2}}} \text{ ft} }$

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The sink hole was ${5{\textstyle\frac{1}{2}}}$ $\displaystyle{ \text{ feet} }$ deep three days later.

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11.

Hill A is $4.14$ kilometers in height, while Hill B is $860$ meters in height. What is the difference in their height? Answer this question in both meters and kilometers.

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The difference in these two hills’ height is meters.
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The difference in these two hills’ height is kilometers. Use decimal in your answer.
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Answer 1.

$3280$

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Answer 2.

$3.28$

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Solution.

We will use the formula $1 \text{ kilometer} = 1000 \text{ meters}\text{.}$

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Part a: We need to change Hill A’s height from $4.14$ kilometers to meters:

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$\displaystyle{ 4.14 \text{ km} = 4.14 \cdot 1000 \text{ m} = 4140 \text{ m} }$

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Next, we find their difference in height by subtraction:

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$\displaystyle{ 4140 \text{ m} - 860 \text{ m} = 3280 \text{ m} }$

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The difference in these two hills’ height is $3280$ meters.

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Part b: We need to change Hill B’s height from $860$ meters to kilometers:

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$\displaystyle{ 860 \text{ m} = 860 \div 1000 \text{ km} = 0.86 \text{ km} }$

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Next, we find their difference in height by subtraction:

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$\displaystyle{ 4.14 \text{ km} - 0.86 \text{ km} = 3.28 \text{ km} }$

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The difference in these two hills’ height is $3.28$ kilometers.

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12.

A handyman worked on a job ${2{\textstyle\frac{5}{6}}}$ hours in the morning, and then worked on the same job ${25}$ minutes in the afternoon. Altogether, how long did the handyman work on the job? First use hours, and then use minutes to answer this question.

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Question 1: The handyman worked on the job for a total of hours. Use fraction in your answer.

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Question 2: The handyman worked on the job for a total of minutes.

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Answer 1.

$3{\textstyle\frac{1}{4}}$

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Answer 2.

$195$

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Solution.

Question 1:

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We need to answer Question 1 with hours, so we will first convert ${25}$ minutes to hours:

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$\displaystyle{ \begin{aligned}[t] \amp \phantom{{}=} {25} \text{ min} \cdot \frac{1 \text{ hr}}{60 \text{ min}} \\ \amp = {25} \cdot \frac{1}{60} \text{ hr} \\ \amp = \frac{{25}}{1} \cdot \frac{1}{60} \text{ hr} \\ \amp = \frac{{5}}{1} \cdot \frac{1}{{12}} \text{ hr} \\ \amp = {{\frac{5}{12}}} \text{ hr} \\ \end{aligned} }$

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Next, we add up those two fractions:

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$\displaystyle{\begin{aligned}[t] \amp \phantom{{}=} {2{\textstyle\frac{5}{6}}} + {{\frac{5}{12}}} \\ \amp = 2 + {{\frac{5}{6}}} + {{\frac{5}{12}}} \\ \amp = 2 + \frac{5 \cdot 2}{6 \cdot 2} + \frac{5}{12} \\ \amp = 2 + \frac{10}{12} + \frac{5}{12} \\ \amp = 2 + \frac{10+5}{12} \\ \amp = 2 + \frac{15}{12} \\ \amp = 2 + \frac{5}{4} \\ \amp = 2 + {1{\textstyle\frac{1}{4}}} \\ \amp = {3{\textstyle\frac{1}{4}}} \end{aligned} }$

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Solution 1: The handyman worked on the job for a total of ${3{\textstyle\frac{1}{4}}}$ hours.

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Question 2:

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We need to answer Question 2 with minutes, so we will first convert ${2{\textstyle\frac{5}{6}}}$ hours to minutes:

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$\displaystyle{ \begin{aligned}[t] \amp \phantom{{}=} {2{\textstyle\frac{5}{6}}} \text{ hr} \cdot \frac{60 \text{ min}}{1 \text{ hr}} \\ \amp = \frac{17}{6} \cdot \frac{60}{1} \text{ min} \\ \amp = \frac{17}{1} \cdot \frac{10}{1} \text{ min} \\ \amp = 17 \cdot 10 \text{ min} \\ \amp = {170} \text{ min} \end{aligned} }$

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Next, we add up the number of minutes from the morning and the afternoon:

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$\displaystyle{ {170} \text{ min} + {25} \text{ min} = {195} \text{ min} }$

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Solution 2: The handyman worked on the job for a total of ${195}$ minutes.

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