Solutions

1. \(\frac{382\:\text{mi}}{1}\times\frac{1\:\text{gal}}{25\:\text{mi}}\times\frac{\$3.75}{1\:\text{gal}} = \$57.30\) (Written Solution)

   Step 1: Start with what you know

   We know that it is 382 miles to San Francisco from Los Angeles

   Step 2: Determine what you want to get in the end. (Figure out what the end units should be):

   We need to find out how much the trip will cost so the units we are looking for are US Dollars.

   Step 3: Determine what conversion factor(s) to use:

   25 mi = 1 gal

   $3.75 = 1 gal

   Step 4: Multiply by 1 in the form of the conversion factor that cancels out the unwanted units.

   We are starting with 382 miles and need to find the cost of our trip, so we will multiply 382 miles by the conversion factors that will cancel out miles and gallons and leave us with dollars:

   \(\frac{382\:\text{mi}}{1}\times\frac{1\:\text{gal}}{25\:\text{mi}}\times\frac{\$3.75}{1\:\text{gal}}\)

   Note that multiplying by each of the two conversion factors, \(\frac{1\:\text{gal}}{25\:\text{mi}}\) and \(\frac{\$3.75}{1\:\text{gal}}\), is the same as multiplying by 1 since the numerator is equal to the denominator in both cases. This is why we can do unit conversions.

   Then we cancel out miles and gallons:

   \(\frac{382\:\cancel{{\color{Red} \text{mi}}}}{1}\times\frac{1\:\cancel{{\color{Blue} \text{gal}}}}{25\:\cancel{{\color{Red} \text{mi}}}}\times\frac{\$3.75}{1\:\cancel{{\color{Blue} \text{gal}}}}\)

   Using the zig-zag method we make the calculations in a zig-zag pattern. Remember, any time we move to the denominator we divide and any time we move to the numerator, we multiply:

   

   382 ÷ 1 × 1 ÷ 25 × $3.75 ÷ 1 = $57.3

   So it will cost $57.30 to drive from Los Angeles to San Francisco.



2. \(\frac{23\:\text{mi}} {1}\times\frac{10\:\text{gal}}{120\:\text{mi}}\times\frac{\$3.55}{1\:\text{gal}} = \$6.80\)


3. \(\frac{3\:\text{oz}}{1\:\text{night}}\times\frac{1\:\text{gal}}{128\:\text{oz}}\times\frac{\$9}{1\:\text{gal}}= \$0.21\) per night, or \(\frac{\$0.21}{1\:\text{night}}\) (Written Solution)

   Step 1: Start with what you know

   We know that Pam uses 3 ounces of oil per night

   Step 2: Determine what you want to get in the end. (Figure out what the end units should be):

   We need to find out how much the trip will cost so the units we are looking for are US dollars.

   Step 3: Determine what conversion factor(s) to use:

   $9 = 1 gal

   3 oz = 1 night

   1 gal = 128 oz

   Step 4: Multiply by 1 in the form of the conversion factor that cancels out the unwanted units.

   We are starting with 3 ounces per night. To find the cost, we multiply \(\frac{3\:\text{oz}}{1\:\text{night}}\) by the conversion factors that will enable us to cancel out ounces and gallons and leave us with dollars:

   \(\frac{3\:\text{oz}}{1\:\text{night}}\times\frac{1\:\text{gal}}{128\:\text{oz}}\times\frac{\$9}{1\:\text{gal}}\)

   Note that each of the two conversion factors, \(\frac{1\:\text{gal}}{128\:\text{oz}}\) and \(\frac{\$9}{1\:\text{gal}}\), are theoretically equal to 1 because anytime you divide something by itself it equals 1. Since the numerator is equivalent to the denominator .

   Then we cancel out ounces and gallons:

   \(\frac{3\:\cancel{{\color{Red} \text{oz}}}}{1\:\text{night}}\times\frac{1\:\cancel{{\color{Blue} \text{gal}}}}{128\:\cancel{{\color{Red} \text{oz}}}}\times\frac{\$9}{1\:\cancel{{\color{Blue} \text{gal}}}}\)

   Using the zig-zag method we make the calculations in a zig-zag pattern. Remember, any time we move to the denominator we divide and any time we move to the numerator, we multiply:

   

   3 ÷ 1 night × 1 ÷ 128 × $9 ÷ 1 = \(\frac{$0.2109}{1\:\text{night}}\)

   So it will cost about $0.21 per night.

   
   


4. \(\frac{9\:\text{ml}}{100\:\text{m}}\times\frac{1\:\text{L}}{1000\:\text{ml}}\times\frac{1000\:\text{m}}{1\:\text{km}} = \frac{.09\:\text{L}}{1\:\text{km}} \) (Fuel efficiency is often written as liters per 100 kilometers. If we want to see it this way, we can multiply the numerator and denominator by 100.) \( \frac{.09\:\text{L}}{1\:\text {km}}\times\frac{100}{100} = \frac {9\:\text{L}}{100\:\text{km}}\)
   


5. \(\frac{45\: {\text{km}}}{1}\times\frac{5\: {\text{L}}}{100\:{\text{km}}}\times\frac{€1.50}{1\: {\text{L}}} = €3.38\)