Proportional Relationships

Expressed as a rate, [latex]\displaystyle\frac{300\text{ miles}}{15\text{ gallons}}[/latex]. We can divide to find a unit rate:[latex]\displaystyle\frac{20\text{ miles}}{1\text{ gallon}}[/latex], which we could also write as [latex]\displaystyle{20}\frac{\text{miles}}{\text{gallon}}[/latex], or just 20 miles per gallon. This solution is worked out in the video below.

This proportion is asking us to find a fraction with denominator 6 that is equivalent to the fraction[latex]\displaystyle\frac{5}{3}[/latex]. We can solve this by multiplying both sides of the equation by 6, giving [latex]\displaystyle{x}=\frac{5}{3}\cdot6=10[/latex].

We can set up a proportion by setting equal two [latex]\displaystyle\frac{\text{map inches}}{\text{real miles}}[/latex] rates, and introducing a variable, x, to represent the unknown quantity—the mile distance between the cities.

[latex]\displaystyle\frac{\frac{1}{2}\text{map inch}}{3\text{ miles}}=\frac{2\frac{1}{4}\text{map inches}}{x\text{ miles}}[/latex]	Multiply both sides by x and rewriting the mixed number
[latex]\displaystyle\frac{\frac{1}{2}}{3}\cdot{x}=\frac{9}{4}[/latex]	Multiply both sides by 3
[latex]\displaystyle\frac{1}{2}x=\frac{27}{4}[/latex]	Multiply both sides by 2 (or divide by ½)
[latex]\displaystyle{x}=\frac{27}{2}=13\frac{1}{2}\text{ miles}[/latex]

Many proportion problems can also be solved using dimensional analysis, the process of multiplying a quantity by rates to change the units. This solution is worked out in the video below.

We could certainly answer this question using a proportion: [latex]\displaystyle\frac{300\text{ miles}}{15\text{ gallons}}=\frac{x\text{ miles}}{40\text{ gallons}}[/latex].

However, we earlier 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]\displaystyle40\text{ gallons}\cdot\frac{20\text{ miles}}{\text{gallon}}=\frac{40\text{ gallons}}{1}\cdot\frac{20\text{ miles}}{\text{gallons}}=800\text{ miles}[/latex]

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:

[latex]\displaystyle{50}\text{ miles}\cdot\frac{1\text{ gallon}}{20\text{ miles}}=\frac{50\text{ miles}}{1}\cdot\frac{1\text{ gallon}}{20\text{ miles}}=\frac{50\text{ gallons}}{20}=2.5\text{ gallons}[/latex]

If we knew 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 might start by converting the 20 seconds into hours.  To do this, we recall that there are 60 seconds in a minute and 60 minutes in an hour:

[latex]\displaystyle{20}\text{ seconds}\cdot\frac{1\text{ minute}}{60\text{ seconds}}\cdot\frac{1\text{ hour}}{60\text{ minutes}}=\frac{1}{180}\text{ hour}[/latex]

Now we can multiply by the 15 miles/hr

[latex]\displaystyle\frac{1}{180}\text{ hour}\cdot\frac{15\text{ miles}}{1\text{ hour}}=\frac{1}{12}\text{ mile}[/latex]

Finally, we will use the fact that 5280 ft = 1 mile to complete the final conversion from miles to feet:

[latex]\displaystyle\frac{1}{12}\text{ mile}\cdot\frac{5280\text{ feet}}{1\text{ mile}}=440\text{ feet}[/latex]

We could have also done this entire calculation in one long set of products:

[latex]\displaystyle20\text{ seconds}\cdot\frac{1\text{ minute}}{60\text{ seconds}}\cdot\frac{1\text{ hour}}{60\text{ minutes}}\cdot\frac{15\text{ miles}}{1\text{ hour}}\cdot\frac{5280\text{ feet}}{1\text{ mile}}=440 \text{ feet}[/latex]

In this case, while the width the room has doubled, the area has quadrupled. Since the number of tiles needed corresponds with the area of the floor, not the width, 400 tiles will be needed. We could find this using a proportion based on the areas of the rooms:

[latex]\displaystyle\frac{100\text{ tiles}}{100\text{ft}^2}=\frac{n\text{ tiles}}{400\text{ft}^2}[/latex]. This solution is worked out in the video below.

While it is tempting to say that they will gain 1000 new customers, it is likely that additional advertising will be less effective than the initial advertising. For example, if the company is a hot tub store, there are likely only a certain number of people interested in buying a hot tub, so there might not even be 1000 people in the town who would be potential customers. This solution is worked out in the video below.

Here we have a very large number, about $683,700,000,000 written out. Of course, imagining a billion dollars is very difficult, so it can help to compare it to other quantities.

1. If that amount of money was used to pay the salaries of the 1.4 million Walmart employees in the U.S., each would earn over $488,000.
2. There are about 300 million people in the U.S. The military budget is about $2,200 per person.
3. If you were to put $683.7 billion in $100 bills, and count out 1 per second, it would take 216 years to finish counting it.

You might begin by noticing that there is not enough information given to answer this question. We will need to find some data. Pause for a moment to think about what kind of data you might want. Then click “Show Solution” to reveal the required data and solution.

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 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: [latex]\displaystyle\frac{4,693,000,000,000\text{KWH}}{1,344,130,000\text{ people}}[/latex] ≈ 3491.5 KWH per person

Japan: [latex]\displaystyle\frac{859,700,000,000\text{KWH}}{127,817,277\text{ people}}[/latex] ≈ 6726 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.