1.5: Problem Solving Using Rates and Dimensional Analysis .
- Page ID
- 139258
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A rate is the ratio of two quantities. A unit rate is a rate with a denominator of one.
Your car can drive 300 miles on a tank of 15 gallons.
Express this as a rate and as a unit rate, in miles per gallon.
Solution
Expressed as a rate, \(\dfrac{300 \text { miles }}{15 \text { gallons }}\).
We clan divide to find a unit rate: \(\dfrac{20 \text { miles }}{1 \text { gallon }}\), which we could also write as \(20 \dfrac{\text { miles }}{\text { gallon }}\), or just 20 miles per gallon.
Find the unit rates. If necessary, round your answers to the nearest hundredth. 6 pounds for $5.29
______________ dollars per pound ______________ pounds per dollar
- Answer
-
$0.88 per pound, 1.13 pounds per dollar
Compare the electricity consumption per capita in China to the rate in Japan.
Solution
To address this question, we will first need data. From the CIA \({ }^1\) website we can find the electricity consumption in 2011 for China was 4,693,000,000,000 KWH (kilowatt-hours), or 4.693 trillion KWH, while the consumption for Japan was \(859,700,000,000\), or 859.7 billion KWH. To find the rate per capita (per person), we will also need the population of the two countries. From the World \(\mathrm{Bank}^2\), we can find the population of China is \(1,344,130,000\), or 1.344 billion, and the population of Japan is \(127,817,277\), or 127.8 million.
Computing the consumption per capita for each country:
China: \(\dfrac{4,693,000,000,000 \mathrm{KWH}}{1,344,130,000 \text { people }} \approx 3491.5 \mathrm{KWH}\) per person
Japan: \(\dfrac{859,700,000,000 \mathrm{KWH}}{127,817,277 \text { people }} \approx 6726 \mathrm{KWH}\) per person
While China uses more than 5 times the electricity of Japan overall, because the population of Japan is so much smaller, it turns out Japan uses almost twice the electricity per person compared to China.
Many problems can also be solved by multiplying a quantity by rates to change the units. This is the foundation of the Factor-Label process that we have been using already in this chapter.
Your car can drive 300 miles on a tank of 15 gallons.
a. How far can it drive on 40 gallons?
b. How many gallons are needed to drive 50 miles?
Solution
We earlier found that 300 miles on 15 gallons gives a rate of 20 miles per gallon.
a. If we multiply the given 40 gallon quantity by this rate, the gallons unit "cancels" and we're left with a number of miles:
40 gallons \(\cdot \dfrac{20 \text { miles }}{\text { gallon }}=\dfrac{40 \text { gallons }}{1} \cdot \dfrac{20 \text { miles }}{\text { gallon }}=800\) miles
Notice that this could also have been achieved using the rate given in the problem:
40 gallons \(\cdot \dfrac{300 \text { miles }}{15 \text { gallons }}=\dfrac{40 \text { gallons }}{1} \cdot \dfrac{300 \text { miles }}{15 \text { gallons }}=\dfrac{40}{1} \cdot \dfrac{300 \text { miles }}{15}=\dfrac{1200}{15}\) miles \(=800\) miles
b. Notice if instead we were asked "how many gallons are needed to drive 50 miles?" we could answer this question by inverting the 20 mile per gallon rate so that the miles unit cancels and we're left with gallons:
50 miles \(\cdot \dfrac{1 \text { gallon }}{20 \text { miles }}=\dfrac{50 \text { miles }}{1} \cdot \dfrac{1 \text { gallon }}{20 \text { miles }}=\dfrac{50 \text { gallons }}{20}=2.5\) gallons
A bicycle is traveling at 15 miles per hour. How many feet will it cover in 20 seconds?
Solution
To answer this question, we need to convert 20 seconds into feet. If we know the speed of
the bicycle in feet per second, this question would be simpler. Since we don’t, we will need
to do additional unit conversions. We will need to know that 5280 ft = 1 mile. We might
start by converting the 20 seconds into hours:
20 seconds \(\cdot \dfrac{1 \text { minute }}{60 \text { seconds }} \cdot \dfrac{1 \text { hour }}{60 \text { minutes }}=\dfrac{1}{180}\) hour
Now we can multiply by the 15 miles/hr
\(\dfrac{1}{180}\) hour \(\cdot \dfrac{15 \text { miles }}{1 \text { hour }}=\dfrac{1}{12}\) mile
Now we can convert to feet
\(\dfrac{1}{12}\) mile \(\cdot \dfrac{5280 \text { feet }}{1 \text { mile }}=440\) feet
We could have also done this entire calculation in one long set of products:
\[
20 \text { seconds } \cdot \dfrac{1 \text { minute }}{60 \text { seconds }} \cdot \dfrac{1 \text { hour }}{60 \text { minutes }} \cdot \dfrac{15 \text { miles }}{1 \text { hour }} \cdot \dfrac{5280 \text { feet }}{1 \text { mile }}=440 \text { feet }
\]
You are walking through a hardware store and notice two sales on tubing.
- 3 yards of Tubing A costs $5.49.
- Tubing B sells for $1.88 for 2 feet.
Either tubing is acceptable for your project. Which tubing is less expensive?
Find
Solution
Find the unit price for each tubing. This will make it easier to compare.
| Tubing A: 3 yards = $5.49 | Tubing A is sold by the yard. |
| \(\dfrac{\$ 5.49 \div 3}{3 \text { yards } \div 3}=\dfrac{\$ 1.83}{1 \text { yard }}\) | Find the cost per yard of Tubing A by dividing the cost of 3 yards of the tubing by 3. |
| Tubing B: 2 feet = $1.88 | Tubing B is sold by the foot. |
| \(\dfrac{\$ 1.88 \div 2}{2 \text { feet } \div 2}=\dfrac{\$ 0.94}{1 \text { foot }}\) | Find the cost per foot by dividing $1.88 by 2 feet. |
To compare the prices, you need to have the same unit of measure. You can choose to use dollars per foot (like we have for Tubing B) or dollars per yard (like we have for Tubing A.
Either will work. For this example, we will go with dollars per yard.
Tubing A: \(\$ 1.83\) per yard
Tubing B: \(\dfrac{\$ 0.94}{1 \text { foot }} \cdot \dfrac{3 \text { feet }}{1 \text { yard }} \) Use the conversion factor \(\dfrac{3 \text { feet }}{1 \text { yard }}\)
\( \dfrac{\$ 0.94}{1 \text { fort }} \cdot \dfrac{3 \text { feet }}{1 \text { yard }}=\dfrac{\$ 2.82}{1 \text { yard }} \) Cancel and multiply.
$2.82 per yard
Compare prices for 1 yard of each tubing.
Tubing A: \(\$ 1.83\) per yard Tubing B: \(\$ 2.82\) per yard
Tubing A is less expensive than Tubing B.
The cost of gasoline in Arizona is about $2.05 per gallon. When you travel over the border into Mexico, gasoline costs 14.81 pesos per liter. Where is gasoline more expensive?
Note: This problem requires a currency conversion factor. Currency conversions are constantly changing, but at the time of print $1 = 18.68 pesos.
Solution
To answer this question, we need to convert from gallons to liters AND from U.S. dollars to Mexican pesos.
\[\begin{array}{c}\dfrac{\$ 2.05}{1 \text { gallon }} \cdot \dfrac{1 \text { gallon }}{3.8 \text { liters }} \cdot \dfrac{18.68 \text { pesos }}{\$ 1} \\ \dfrac{\$ 2.05}{1 \text { gallon }} \cdot \dfrac{1 \text { gallon }}{3.8 \text { liter } s} \cdot \dfrac{18.68 \text { pesos }}{\$ 1} \\ \dfrac{2.05}{1} \cdot \dfrac{1}{3.8 \text { liters }} \cdot \dfrac{18.68 \text { pesos }}{1}=\dfrac{2.05 \cdot 18.68}{3.8} \dfrac{\text { pesos }}{\text { liters }} \\ \dfrac{2.05 \cdot 18.68 \text { pesos }}{3.8 \text { liters }}=10.08 \text { pesos per liter }\end{array}\]
The price in Arizona of $2.05 per gallon is equivalent to 10.08 pesos per liter. Since the actual price in Mexico is 14.81 pesos per liter, gasoline is more expensive in Mexico.