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

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

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

Many proportion problems can also be solved using dimensional analysis, the process of multiplying a quantity by rates to change the units.

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

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:

$\displaystyle40\text{ gallons}\cdot\frac{20\text{ miles}}{\text{gallon}}=\frac{40\text{ gallons}}{1}\cdot\frac{20\text{ miles}}{\text{gallons}}=800\text{ miles}$

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:

$\displaystyle{50}\text{ miles}\cdot\frac{1\text{ gallon}}{20\text{ miles}}=\frac{50\text{ miles}}{1}\cdot\frac{1\text{ gallon}}{20\text{ miles}}=\frac{50\text{ gallons}}{20}=2.5\text{ gallons}$

Solution
Notice that the variable is in the denominator, so we will solve by finding the cross products and setting them equal.

Find the cross products and set them equal.		$4\cdot144=a\cdot9$
Simplify.		$576=9a$
Divide both sides by $9$.		${\Large\frac{576}{9}}={\Large\frac{9a}{9}}$
Simplify.		$64=a$
Check your answer.
	${\Large\frac{144}{a}}={\Large\frac{9}{4}}$
Substitute $a=\color{red}{64}$	${\Large\frac{144}{\color{red}{64}}}\stackrel{?}{=}{\Large\frac{9}{4}}$
Show common factors..	${\Large\frac{9\cdot16}{4\cdot16}}\stackrel{?}{=}{\Large\frac{9}{4}}$
Simplify.	${\Large\frac{9}{4}}={\Large\frac{9}{4}}\quad\checkmark$

Solution

Find the cross products and set them equal.
		$y\cdot52=91(-4)$
Simplify.		$52y=-364$
Divide both sides by $52$.		${\Large\frac{52y}{52}}={\Large\frac{-364}{52}}$
Simplify.		$y=-7$
Check:
	${\Large\frac{52}{91}}={\Large\frac{-4}{y}}$
Substitute $y=\color{red}{-7}$	${\Large\frac{52}{91}}\stackrel{?}{=}{\Large\frac{-4}{\color{red}{-7}}}$
Show common factors.	${\Large\frac{13\cdot4}{13\cdot4}}\stackrel{?}{=}{\Large\frac{-4}{\color{red}{-7}}}$
Simplify.	${\Large\frac{4}{7}}={\Large\frac{4}{7}}\quad\checkmark$

Solution

Identify what you are asked to find.	How many ml of acetaminophen the doctor will prescribe
Choose a variable to represent it.	Let $a=$ ml of acetaminophen.
Write a sentence that gives the information to find it.	If $5$ ml is prescribed for every $25$ pounds, how much will be prescribed for $80$ pounds?
Translate into a proportion.
Substitute given values—be careful of the units.	${\Large\frac{5}{25}}={\Large\frac{a}{80}}$
Multiply both sides by $80$.	$80\cdot{\Large\frac{5}{25}}=80\cdot{\Large\frac{a}{80}}$
Multiply and show common factors.	${\Large\frac{16\cdot5\cdot5}{5\cdot5}}={\Large\frac{80a}{80}}$
Simplify.	$16=a$
Check if the answer is reasonable.
Yes. Since $80$ is about $3$ times $25$, the medicine should be about $3$ times $5$.
Write a complete sentence.	The pediatrician would prescribe $16$ ml of acetaminophen to Zoe.

Solution

Identify what you are asked to find.	How many calories are in a whole bag of microwave popcorn?
Choose a variable to represent it.	Let $c=$ number of calories.
Write a sentence that gives the information to find it.	If there are $120$ calories per serving, how many calories are in a whole bag with $3.5$ servings?
Translate into a proportion.
Substitute given values.	${\Large\frac{120}{1}}={\Large\frac{c}{3.5}}$
Multiply both sides by $3.5$.	$(3.5)({\Large\frac{120}{1}})=(3.5)({\Large\frac{c}{3.5}})$
Multiply.	$420=c$
Check if the answer is reasonable.
Yes. Since $3.5$ is between $3$ and $4$, the total calories should be between $360 (3⋅120)$ and $480 (4⋅120)$.
Write a complete sentence.	The whole bag of microwave popcorn has $420$ calories.

Solution

Identify what you are asked to find.	How many Mexican pesos did Josiah get?
Choose a variable to represent it.	Let $p=$ number of pesos.
Write a sentence that gives the information to find it.	If $\text{\$1}$ U.S. is equal to $12.54$ Mexican pesos, then $\text{\$325}$ is how many pesos?
Translate into a proportion.
Substitute given values.	${\Large\frac{1}{12.54}}={\Large\frac{325}{p}}$
The variable is in the denominator, so find the cross products and set them equal.	$p\cdot{1}=12.54(325)$
Simplify.	$c=4,075.5$
Check if the answer is reasonable.
Yes, $\text{\$100}$ would be $\text{\$1,254}$ pesos. $\text{\$325}$ is a little more than $3$ times this amount.
Write a complete sentence.	Josiah has $4075.5$ pesos for his spring break trip.

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:

$\displaystyle\frac{100\text{ tiles}}{100\text{ft}^2}=\frac{n\text{ tiles}}{400\text{ft}^2}$

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 fixed 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.

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.

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

Japan: $\displaystyle\frac{859,700,000,000\text{KWH}}{127,817,277\text{ people}}$ ≈ 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.