Comparisons

...and unit conversions.

In order to make a decision,

We often need to make a comparison between two quantities,

We'd like to know which quantity is smaller / larger, but it doesn't always have to be exact.

But the units of both quantities have to be the same.

traveladdicts.net
Does gas in Iceland cost more or less than in the U.S.??

Buying groceries


Which one is the better deal? What do you need to calculate to make the comparison?

show / hide

$$\frac{\$3.89}{13\text{ oz}}=0.299 \frac{\$}{\text{oz}}= \frac{\$0.299}{\text{1 oz}}$$ $$\frac{\$7.39}{26.5\text{ oz}}=0.279 \frac{\$}{\text{oz}}=\frac{\$0.279}{\text{1 oz}}$$


Changing the units

What if I want to change the units? Usually people talk about amounts less than 1 dollar in cents, and you know that: $$1\ \$=100\text{ cents}.$$ Watch this: Divide both sides of this equation by 1 \$, and you have... $$1 = \frac{\text{100 cents}}{\text{1 } \$}.$$

That ratio on the right is a "conversion factor". Since it's equal to 1, I can multiply that by any other number, and it won't change the underlying quantity. The larger container of Nutella cost \$0.299/oz, which I'll write like this... $$\frac{\$0.299}{\text{1 oz}}\cdot\frac{\text{100 cents}}{1 \$}=\frac{\text{29.9 cents}}{\text{1 oz}}.$$ I've cancelled out the dollar signs in the top and bottom, and I'm left with the the same quantity of money, but expressed in terms of cents/oz instead.

You can chain together several conversion factors. For example, how many seconds are there in 1/2 hour? Use 60 seconds = 1 minute and 60 minutes = 1 hour:

$$0.5\text{ hours}\cdot \frac{\text{60 minutes}}{\text{1 hour}}\cdot \frac{\text{60 seconds}}{\text{1 minute}}=0.5*60*60\text{ seconds} =\text{1800 seconds}.$$

The procedure is...

  1. Write down the number you have, with units on the left.
  2. Write out the units you'd like to have on the right of an equals sign.
  3. Multiply your quantity on the left by one (or more) conversion factor(s), which will cancel out the units you started with on the left, and leave you with the units on the right.

Try these:

  • Using 1 mile = 1.6 kilometers, find out how long a 5 km footrace is in miles.
  • Using 1 kg = 2.2 lbs, find your weight in kilograms (kg).
  • Find out what 90 kg is in pounds.
  • Using 1 inch = 2.54 cm, *and* 1 meter = 100 cm, figure out your height in meters. (You'll need to start by figuring out how many inches tall you are. 12 inches = 1 foot!)

Nowadays you can often use Google to do these kind of calculations. E.g. Google '5 km in miles'. But you should still know how to do this "by hand" as well, given some conversion factors.

Back to Iceland...


traveladdicts.net
Does gas in Iceland cost more or less than in the U.S.??

Online, I find that:

  • 128 Icelandic Króna (Kr) = \$1,
  • 1 gallon = 3.8 liters. So, starting with the price on the pump of 230.9 Kr/liter, and ending on the other side of the equation with \$ / gallon... $$\begineq\frac{\text{230.9 Kr}}{\text{1 L}}\cdot \frac{\text{3.8 L}}{\text{1 gal}}\cdot \frac{\text{\$1}}{\text{128 Kr}} &=\frac{230.9 \cdot 3.8}{128} \frac{\$}{\text{gal}}\\ &= \frac{\$6.86}{\text{gal}} \endeq$$