Section 1.6: Problem Solving Using Rates & Dimensional Analysis

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: State your givens and your goal.

Given: Your car can drive 300 miles on a tank of 15 gallons.

Goal: Express this as a rate and as a unit rate, in miles per gallon.

1) As a rate, create a fraction with the two quantities with miles in the numerator and gallons in the denominator. Include units in your answer.

[latex]\frac{300~miles}{15~gallons}[/latex]

2) As a unit rate, divide the quantity in the numerator by the quantity in the denominator. Use the “units miles per gallon” (equivalent to miles per 1 gallon).

[latex]\frac{300~miles}{15~gallons}[/latex] = [latex]\frac{300}{15}[/latex] x [latex]\frac{miles}{gallons}[/latex] = 20 miles per gallons

Notice that, had we wanted to find the unit rate in gallons per mile, we would have had to invert the original rate:

[latex]\frac{15~gallons}{300~miles}[/latex] = [latex]\frac{15}{300}[/latex] x [latex]\frac{gallons}{miles}[/latex] = [latex]\frac{1}{20}[/latex] gallons per mile

Use the following information to compare the electricity consumption per capita in China to the rate in Japan.

From the CIA1 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 Bank2, we can find the population of China is 1,344,130,000, and the population of Japan is 127,817,277.

Solution: State your givens and your goal.

Given: China: 4.693 trillion KWH and 1,344,130,000 people. Japan: 859.7 billion KWH
and 127,817,277 people.

Goal: Computing the consumption per capita for each country and compare the results.

1) Find KWH per person for China and Japan by dividing the quantity KWH by the number of people.

China: [latex]\frac{4,693,000,000,000~KWH}{1,344,130,000~people}[/latex] = 3,491.49 KWH per person

Japan: [latex]\frac{859,700,000,000~KWH}{127,817,277~people}[/latex] = 6,726 KWH per person

Answer: 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.

Your car can drive 300 miles on a tank of 15 gallons. How far can it drive on 40 gallons? How many gallons are needed to drive 50 miles?

Solution: State your givens and your goal.

Given: Your car can drive 300 miles on a tank of 15 gallons.

Goals: a) How far can it drive on 40 gallons? b) How many gallons are needed to drive 50 miles?

a) We found that 300 miles on 15 gallons gives a rate of 20 miles per gallon. If we multiply the given 40 gallon quantity by this rate, the gallons unit “cancels” and we’re left with a number of miles.

[latex]\frac{40~\bcancel{gallons}}{1}[/latex] x [latex]\frac{20~miles}{1~\bcancel{gallon}}[/latex] = [latex]\frac{40~x~20}{1~x~1}[/latex] = 800 miles

b) We also found that 300 miles on 15 gallons gives a rate of [latex]\frac{1}{20}[/latex] gallons per mile. If we multiply the given 50 mile quantity by this rate, the miles unit “cancels” and we’re left with gallons.

[latex]\frac{50~\bcancel{miles}}{1}[/latex] x [latex]\frac{1~gallon}{20~\bcancel{miles}}[/latex] = [latex]\frac{50~x~1}{1~x~20}[/latex] gallons = 2.5 gallons

Answers: The car can drive 800 miles on 40 gallons of gas. The car needs 2.5 gallons of gas to drive 50 miles.

A bicycle is traveling at 15 miles per hour. How many feet will it cover in 20 seconds?

Solution: State your givens and your goal.

Given: 15 miles per hour.

Goal: How many feet will the bicycle cover in 20 seconds?

1) 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 feet = 1 mile. We might start by converting the 20 seconds into hours:

[latex]\frac{20~\bcancel{sec}}{1}[/latex] x [latex]\frac{1~\bcancel{min}}{60~\bcancel{sec}}[/latex] x [latex]\frac{1~hr}{60~\bcancel{min}}[/latex] hr = [latex]\frac{20}{3600}[/latex] hr = [latex]\frac{1}{180}[/latex] hr

2) Now we can multiply by the 15 miles/hr

[latex]\frac{1~\bcancel{hr}}{180}[/latex] x [latex]\frac{15~mi}{1~\bcancel{hr}}[/latex] = [latex]\frac{15}{180}[/latex] miles = [latex]\frac{1}{12}[/latex] miles

3) Now we can convert to feet.

[latex]\frac{1~\bcancel{mile}}{12}[/latex] x [latex]\frac{5280}{1~\bcancel{mile}}[/latex] = [latex]\frac{5280}{12}[/latex] ft = 440 ft

4) Note we could have also done this entire calculation in one long set of products:

[latex]\frac{20~\bcancel{sec}}{1}[/latex] x [latex]\frac{1~\bcancel{min}}{60~\bcancel{sec}}[/latex] x [latex]\frac{1~\bcancel{hr}}{60~\bcancel{min}}[/latex] x [latex]\frac{15~\bcancel{miles}}{1~\bcancel{hr}}[/latex] x [latex]\frac{5280~ft}{1~\bcancel{mile}}[/latex] = [latex]\frac{20~x~15~x~5280}{60~x~60}[/latex] ft = 440 ft

Answer: The bicycle covers 440 feet in 20 seconds.

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?

Solution: State your givens and your goal.

Given: 3 yards of Tubing A costs $5.49. Tubing B sells for $1.88 for 2 feet.

Goal: Which tubing is less expensive?

1) Find the unit price per yard for each tubing. This will make it easier to compare.

Tubing A: 3 yards for $5.49

Unit price per yard: [latex]\frac{$5.49}{3~yds}[/latex] = [latex]\frac{5.49}{3}[/latex] [latex]\frac{dollars}{yards}[/latex] = $1.83 per yard

Tubing B: 2 feet for $1.88

2) Compare prices for 1 yard of each tubing.

Tubing A: $1.83 per yard

Tubing B: $2.82 per yard

Answer: 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: State your givens and your goal.

Given: Gasoline in Arizona is about $2.05 per gallon. Gasoline in Mexico costs 14.81 pesos per liter. $1 = 18.68 pesos

Goal: Where is gasoline more expensive?

1) We will convert the Arizona price from dollars to pesos and gallons to liters to obtain pesos per liter.

[latex]\frac{$2.05}{1~gallon}[/latex] x [latex]\frac{18.68~pesos}{$1}[/latex] x [latex]\frac{1~gallon}{3.8~L}[/latex] = [latex]\frac{2.05~x~18.68}{3.8~L}[/latex] [latex]\frac{pesos}{L}[/latex] = 10.08 pesos per liter

2) Compare the pesos per liter values.

Arizona: 10.08 pesos per liter
Mexico: 14.81 pesos per liter

Answer: Gasoline is more expensive in Mexico.