From Section 2A:
8: "Square feet" is indeed a unit of area, so the statement isn't absurd on its face. If a house is 20 feet by 40 feet and has two floors, the area of each floor would be (20 feet)×(40 feet) = 800 square feet, and the total floor area in the house would be twice that, or 1600. So 1500 square feet is a reasonable floor-area for a house.
10: This is absurd: an acres is a unit of area, while the amount of water needed to fill the pool is measured in volume units (such as cubic feet).
12: If we divide distance (measured in miles) by speed (measured in miles per hour), we get an answer whose units are miles/(miles/hr) = hrs, which is a unit of time, so the statement isn't absurd on its face. If the airplane flies in a straight line, then the statement is correct, since the definition of speed is distance travelled divided by time elapsed. (Note that distance needs to be measured as the crow flies, and not as the mole digs!)
16: (a) 7/10. (b) 14/10 or 7/5 (either is acceptable).
19: dollars per pound or $/lb.
24: dollars per square foot or $/(ft2).
25: miles per gallon or mi/gal (sometimes written mpg).
32: 142 ounces times (1 lb / 16 ounces) = (142/16) ounces = 8.875 ounces.
36: (6 gallons)×(4 quarts / gallon)×(4 cups / quart)× (8 ounces / cup) = (6 × 4 × 4 × 8) ounces = 768 ounces.
44: (100 yards)×(60 yards) = 6000 square yards; (300 feet)×(180 feet) = 54000 square feet. (Check: there are 9 square feet in a square yard, and 9 × 6000 = 54000.)
54: ($100 US)×($1.336 CAN / $1 US) = $133.6 CAN. (Note: One good way to check your work in a conversion problem is to convert your answer in the other direction, to see if you get back to where you started. In this case, it would go like this: ($133.6 CAN)×($0.7483 US / $1 CAN) = $99.97 US, which is $100 US with a rounding error. It shouldn't surprise us that there's rounding-error involved, because the real exchange rates are not exact reciprocals of each other; they multiply to give something slightly less than 1. In real life, you'd be very lucky if you found exchange rates that let you change $100 US into Canadian money and then back into U.S. money and you only lost 3 cents in the process!)
65: Method 1: The area to be covered is 1530 square feet =
(1520 ft2)×(yard / 3 ft)×(yard / 3 ft)
= (1520/9) yard2 = 170 square yards.
So the cost is (170 square yard)×($18 / square yard) =
$(170 × 18) = $3060.
Method 2: The cost of the carpeting is
($18 / square yard)×(1 square yard / 9 square feet) = $2 / square foot.
So the cost is (1530 square feet)×($2 / square foot) = $3060.
77: (a) At 55 mph, the driving time is (2000 miles) / (55 miles / hr) = (2000/55) hr = 36.36 hours. At 70 mph, the driving time is (2000 miles) / (32 miles / gal) = (2000/32) hr = 62.50 hours. (b) At 55 mph, the cost is ($1.65 / gallon) (1 gallon / 38 miles) (2000 miles) = $86.84. At 70 mph, the cost is ($1.65 / gallon) (1 gallon / 32 miles) (2000 miles) = $103.12 (or $103.13).
(There's an alternate approach to problem 77 that uses ratios: 55 miles / 1 hr = 2000 miles / x hours, where x is the number of hours the trip would take at 55 mph; 55 × x = 1 × 2000, so x hours = 2000/55 hours = 36.36 hours. Computing the duration of the trip if you drive 70 mph is similar. For part (b), do the same using miles per gallon instead of miles per hour. For instance, 38 miles / 1 gallon = 2000 miles / x gallons, where x is the number of gallon the trip would use at 55 mph; 38 × x = 1 × 2000, so x gallons = 2000/38 gallons = 52.63 gallons; this gives gas expenses of (52.63 gallons) × ($1.65 / gallon) = $86.84. Computing the cost of the trip if you drive 70 mph is similar.)
From Section 2B:
8: (300 kg)×(2.205 lb / 1 kg) = 661.5 lb; such a heavy person is unlikely to be a professional bicyclist.
10: This does not make sense: liters are a unit of volume, not distance. (Meters per second would make sense as a measure of speed. But no human runner can run 35 meters per second, since 35 meters is over 100 feet.)
12: Since 1000 meters is a kilometer, 10,000 meters is 10 kilometers, which is (10 km)×(0.6214 mi / 1 km) = 6.214 miles. A good marathoner could maintain a speed of about 7 mph for an hour and hence run 6.214 miles in just under an hour.
22: 104 / 10-3 = 10(4)-(-3) = 107 or 10,000,000.
38: (30 nautical miles / hr)×(6076.1 feet / nautical mile)×(1 mile / 5280 feet) = 34.52 miles / hr. (Note: Don't confuse knots and nautical miles! Nautical miles are units of distance, whereas knots are units of speed. One knot is equal to one nautical mile PER HOUR.)
52: (150 lbs)×(1 kg / 2.205 lbs) = 68 kg.
53: (20 gallons)×(3.785 liters / gallon) = 76 liters.
58: (25 miles / hr)×(1 km / 0.6214 miles) = 40 km / hr.
69: (280 million people) / (3.5 million square miles) = (280/3.5) people / square mile = 80 people / square mile.
74: (a) (70 grams) / (6000 milliliters) (100 / 100) = 1.2 grams per 100 milliliters. This is way more than the legal limit of 0.08 grams per 100 milliliters; if all this alcohol were absorbed immediatley, the man would be in a coma. (b) After 4 hours, his body has metabolized 4 × 15 = 60 grams of the original 70, leaving only 10 grams. This would give a BAC of (10 grams) / 6000 milliliters) (100 / 100) = .17 grams per 100 milliliters. This is still twice the legal limit, so he should ask someone else to drive him home. (Another way to do part (b) is to note that the man's blood-alcohol is now 1/7 of what it was originally, so his BAC should be 1/7 of 1.2.)
83: ($.08 / kilowatt-hour)×(.350 kilowatts)×(24 hours / day) ×(365 days) = $245. (Note that there's no need to bring joules into the solution; that just introduces extra steps and increases the chances of making a computational error. A good general rule is to think about the units of the data you're given, and think about the units of the answer you're supposed to produce, and find a way to convert the former into the latter via as direct a route as possible.)
91: (1 household / 10,000 kilowatt-hours per year)×(200 kilowatts / turbine)×(24 hours / day)×(365 days / year) = 175 households / turbine. (Note the units here and how they cancel: 1 household / 10,000 kilowatt-hours per year can be written as 1 household / 10,000 (kw-hr / yr) which can be written as 1 household yr / 10,000 kilowatt-hours, so the unit "year" is in the numerator, and cancels off the year in the denominator of 365 days / year.)
A very common error in a problem like the last one is multiplying when you should divide and vice versa. One good way to catch errors of this kind is to ask yourself (after you've done the calculation) whether the answer would go up or down if the original data were altered. For instance: Suppose the energy usage of each house INCREASED from 10,000 kilowatt-hours per year to 20,000 kilowatt-hours per year. Since each house is using more energy, the number of households we can serve with a single turbine should DECREASE. Does our calculation predict this? Yes, because the number 10,000 occurred in the denominator (and when you make the denominator of a fraction bigger, the value of the fraction goes down). On the other hand, suppose the energy usage of each house stays at 10,000 kilowatt-hours per year, but the output of a turbine DOUBLES. Since each turbine creates twice as much energy, the number of households we can serve with a single turbine should DOUBLE as well. Does out calculation predict this? Yes, because the number 200 occurred in the numerator (and when you double the numerator of a fraction, the value of the fraction doubles). You can catch a lot of common mistakes this way!
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