Measurement and Geometry Unit

3.4 Problem Solving with Units

Examples

• Consider paying $31.35 to fill a 15-gallon tank with gasoline. What is the price per gallon? How many gallons do you get for a dollar?

\(\begin{equation} \frac{\$ 31.35}{15 \text{ gal}}=\$ 2.09 \text{ per gallon} \end{equation} \)

\(\begin{equation} \frac{15 \text{ gal}}{\$ 31.35}=0.48 \text{ gallons per dollar} \end{equation} \)

• A 16 oz box of Great Grains Crunchy Pecan cereal is $3.46. What is the unit price? How many ounces per dollar?

\(\begin{equation} \frac{\$ 3.46}{16 \text{ oz}}=\$ 0.22 \text{ per ounce} \end{equation} \)

\(\begin{equation} \frac{16 \text{ oz}}{\$ 3.46}=4.62 \text { oz per dollar } \end{equation}\)

• If you left a lamp with a single 60-watt lightbulb turned on all day and night for a year, how much would the electricity cost if you pay 12 cents per kilowatt-hour? How much would you save for the year if you turned the light off at night for 8 hours while you sleep?

\(\begin{equation} \frac{60 \text{ w}}{1 \text{ bulb}} \times \frac{1 \text{ kw}}{1000 \text{ w}} \times \frac{24 \text{ hrs}}{1 \text{ day}} \times \frac{365 \text{ days}}{1 \text{ yr}} \times \frac{\$ 0.12}{1 \text{ kwh}} = \$ 63.07 \text{ per year} \end{equation} \)

\(\begin{equation} \frac{60 \text{ w}}{1 \text{ bulb}} \times \frac{1 \text{ kw}}{1000 \text{ w}} \times \frac{16 \text{ hrs}}{1 \text{ day}} \times \frac{365 \text{ days}}{1 \text{ yr}} \times \frac{\$ 0.12}{1 \text{ kwh}} = \$ 42.05 \text{ per year} \end{equation} \)

Savings: $63.07-$42.05=$21.02 per year

Group Activity

Use unit analysis and your problem-solving skills to answer the following questions.

  1. You plan to go to Myrtle Beach during spring break with two of your friends. It’s 412 miles to the beach from Knoxville. You will be driving a Honda CR-V which averages 28 miles per gallon on the highway. Gas prices are currently $2.15 per gallon. How much should you budget for gasoline on the trip if you split the cost equally with your friends?

    2. \( \begin{equation} \frac{824 \text{ mi}}{1 \text{ trip}} \times \frac{1 \text{ trip}}{3 \text{ friends}} \times \frac{1 \text{ gal}}{28 \text{ mi}} \times \frac{\$ 2.15}{1 \text{ gal}}= \$ 20.11 \text{ per friend} \end{equation} \)

  2. Your bathtub measures 5 feet by 2 feet by 1.25 feet. The flow rate of water through your bathtub spout is 4 gallons per minute. How long will it take to fill your bathtub ¾ of the way to the top?

    3. \(V=lwh\)

    \(V=(5)(2)(1.25)=12.5 \text{ft}^{3}\)

    \( \begin{equation} \frac{12.5 \text{ ft}^{3}}{1 \text{ tub}} \times \frac{7.48 \text{ gal}}{1 \text{ ft}^{3}} \times\frac{1 \text{ min}}{4 \text{ gal}} \times \frac{3}{4}= 17.5 \text{ min} \end{equation} \)

  3. You have an old clothes dryer that uses 4000 watts of power. You are thinking about replacing it with a new, more energy-efficient model. A new dryer uses only 2000 watts of power but costs $650 to purchase. Your electric company charges 11 cents per kilowatt-hour of electricity. If you run your dryer 1 hour each day, how long will it take you to recoup the purchase cost of the new dryer.

    4. \(\begin{equation} \frac{4000 \text{ w}}{1 \text{ hr}} \times \frac{1 \text{ kw}}{1000 \text{ w}} \times \frac{\$ 0.11}{1 \text{ kwh}}=\$ 0.44 \text{ per day} \end{equation} \)

    \(\begin{equation} \frac{2000 \text{ w}}{1 \text{ hr}} \times \frac{1 \text{ kw}}{1000 \text{ w}} \times \frac{\$ 0.11}{1 \text{ kwh}}=\$ 0.22 \text{ per day} \end{equation} \)

    difference of $0.22 per day

    \(\begin{equation} \frac{\$ 650}{\$ 0.22}=2955 \end{equation} \)

    It will take 2955 days to recover the cost of the new dryer.

  4. You want to spray weed killer on the weeds by the curb near your house. You have a bottle of weed killer concentrate that must be mixed with water before spraying. The directions say to mix 6 fluid ounces of weed killer concentrate per gallon of water. You want to mix the weed killer in a 1-quart spray bottle. How many tablespoons of concentrate should you mix with a quart of water? (1 oz = 2 tbsp)

    5. \(\begin{equation} \frac{6 \text{ oz}}{1 \text{ gal}} \times \frac{1 \text{ gal}}{4 \text{ qts}} \times \frac{2 \text{ tbsp}}{1 \text{ oz}}=3 \text{ tbsp/qt} \end{equation} \)

  5. Which is the better buy: 27 double rolls of Scott bath tissue with 1000 sheets per roll (2829 square feet total) for $16.97 OR 24 double rolls of Walmart brand bath tissue with 1000 sheets per roll (2528 square feet total) for $12.52.

    6. \(\begin{equation} \frac{\$ 16.97}{2829 \text{ ft}^{2}}=\$ 0.006 \text{/ft} ^{2} \end{equation} \)

    \(\begin{equation} \frac{\$ 12.52}{2528 \text{ ft}^{2}}=\$ 0.005 \text{ /ft}^{2} \end{equation} \)

    The Walmart brand bath tissue is the better buy.

  6. Your young nephew has a cold and you need to give him some liquid Benadryl for his symptoms. He weighs 45 pounds, so the dosage recommendation is 7.5 milliliters every 6 hours. How many teaspoons of Benadryl should you give him for one dose? If the 100 ml bottle of Benadryl is only half full, how many days will the bottle last if you give your nephew three doses a day?(1 mL = 0.203 tsp)

    7. \(\begin{equation} \frac{7.5 \text{ ml}}{1 \text { dose }} \times \frac{0.203 \text { tsp }}{1 \text{ ml}}=1.5 \text { tsp/dose } \end{equation} \)

    \(\begin{equation} \frac{50\;mL}{1\;dose} \times \frac{1\; dose}{7.5\;mL} \times \frac{1\;day}{3\;doses}=\frac{50\;days}{22.5}= 2.2\;days \end{equation} \)

  7. The population of the U.S. is 325,552,686 people. The land area of the U.S. is 3.797 million square miles. What is the population density of the U.S.? China has 3.705 million square miles and a population of 1,385,923,186 people. How many times greater is the population density of China compared to the U.S.?

    8. \(\begin{equation} \frac{325,552,686 \text { people }}{3.797 \text { million mi} ^{2}}=86 \text { people/mi}^{2} \end{equation} \)

    \(\frac{1,385,923,186 \text { people }}{3.705 \text { million mi}^{2}}=374 \text { people/mi }^{2} \)

    \(\begin{equation} \frac{374}{86}=4.35 \end{equation} \)

    China’s population density is 4.35 times that of the U.S.

More Dimensional Analysis

  1. Jamir is training for a race and is running laps around a field. If the distance around the field is 300 yards, how many complete laps would he need to run to run at least 2 miles?

    \(\begin{equation} \frac{2 \text{ miles}}{} \times \frac{1760 \text { yards }}{1 \text{ mile}}\times \frac{1 \text { lap }}{300 \text { yards }}=11.73 \text{ laps} \end{equation} \)

    12 laps

  2. A standard elevator in a midrise building can hold a maximum weight of about 1.5 tons. Assuming an average adult weighs 160 pounds, what is the maximum number of adults who can safely ride in the elevator?

    \(\begin{equation} \frac{1.5 \text{ tons}}{} \times \frac{2000 \text { lbs }}{1 \text{ ton}}\times \frac{1 \text { person}}{160 \text { lbs }}=18.75 \text{ people} \end{equation} \)

  3. A local hospital recently conducted a blood drive where they collected a total of 80 pints of blood from donors. The hospital was hoping to collect a total of 8 gallons. Did they meet their goal? How much more or less than their goal did the hospital collect?

    \(\begin{equation} \frac{80 \text{ pints}}{} \times \frac{1 \text { quart }}{2 \text{ pints}}\times \frac{1 \text { gal}}{4 \text { quarts }}=10 \text{ gal} \end{equation} \)

    They collected 2 more gallons than their goal.

  4. A car’s gas tank holds 12 gallons and is ¼ full. The car gets 20 miles/gallon. You see a sign saying “Next gas 82 miles”. Can you make it to the gas station before running out of gas?

    \(12 \times \frac{1}{4}= 3 \text { gallons }\)

    \(\begin{equation} \frac{3 \text{ gal}}{} \times \frac{20 \text { miles }}{1 \text{ gal }} =60 \text{ miles} \end{equation} \)

    You cannot make it to the gas station before running out of gas.

  5. The re-entry speed of the Apollo 10 space capsule was 11.0 km/s. How many hours would it have taken for the capsule to fall through the 25,000 miles of stratosphere?

    \(\begin{equation} \frac{25,000 \text{ miles}}{} \times \frac{1.61 \text { km }}{1 \text{ mile}}\times \frac{1 \text { sec}}{11 \text { km }}\times \frac{1 \text { min}}{60 \text { sec }}\times \frac{1 \text { hour}}{60 \text { min }}=1.02 \text{ hours} \end{equation} \)

  6. You find 13,406,190 pennies. If each penny weights 4 grams, how much did the money weigh in pounds?

    \(\begin{equation} \frac{13,406,190 \text{ miles}}{} \times \frac{4 \text { grams }}{1 \text{ penny}}\times \frac{1 \text { kg}}{1000 \text { grams }}\times \frac{2.2 \text { lbs}}{1 \text { kg }}=117,974.47 \text{ lbs} \end{equation} \)