Ratios and Rates

Objective: When done with this lesson, you will have demonstrated how to find ratios and unit rates.

Washington State GLE: 1.2.6, 4.2.2, 5.1.1

Approximate completion time: 2 hours

 

Have you ever seen the message "the film you are about to watch has been reformatted to fit your television screen"?
Have you ever wondered what that meant?
Have you ever noticed something looks funny or missing when you were watching a movie?
Have you ever noticed the opening credits or storyline in a movie were too wide to fit your TV? (Next time you watch an original Star Wars movie, look closely at the storyline.)

Current widescreen televisions don't have this issue. That's because the widescreen TV has the same "aspect ratio" as the movie screen in the theater. Do you know what would happen if the movie wasn't reformatted to fit your television screen? Or what would happen if a TV show were to be fit to a movie screen? Take a look at these pictures.

correct
imaging

TV ratio of 4:3

Theater ratio of 16:9
distorted
imaging

Theater ready image fit into a TV ratio of 4:3

TV ready image fit into a theater ratio of 16:9
  Technically, "the 16:9 format adopted for HDTV is actually narrower than commonly-used cinematic widescreen formats. Anamorphic widescreen (2.39:1) and American theatrical standard (1.85:1) have wider aspect ratios, while the European theatrical standard (1.66:1) is just slightly less. " (from wikipedia)

Do you notice that in the top two images, the theater ratio is a wider image? That is, more can be fit into the scene in the horizontal direction? Some people could argue that less is fit in vertically, yet I don't think they'd win the marketing wars with "shortscreen" TV's.

Do you notice that the bottom two images are distorted looking? The theater ready image fit into a TV ratio image looks like it was stretched vertically to fit, and the TV ready image fit into a theater ratio looks like it was stepped on to be shorter.

So to have the film you are about to watch reformatted to fit your television screen, the original images are cut off on the sides so the images are not distorted. Here's my recreation of the famous Star Wars "rollup" introduction:


"orignal" look
16:9 aspect ratio

cropped to fit
4:3 aspect ratio

 

So what exactly is a ratio? Simply put, a ratio is a comparison of two numbers. Ratios are technically fractions.

For example look at this LCD TV.
The aspect ration for this TV is the ratio of the screen width to the screen height. That would be .

In that fraction, the units "inches" divide out. Simplifying 32/18 gives 16/9.

So the aspect ratio for this TV is 16 to 9.
We also can write this ratio as 16:9.

If you divide 16 by 9, the result of 1.777... can be expressed as in the rounded ratio form of 1.778:1.

   

This LCD TV doesn't look widescreen at first glance.
The aspect ration for this TV is.

In that fraction, the units "inches" divide out. Simplifying 16/12 gives 4/3.

So the aspect ratio for this TV is 4 to 3.
We also can write this ratio as 4:3.

If you divide 4 by 3, the result of 1.333... can be expressed as in the rounded ratio form of 1.333:1.

 

Did you know there are some famous ratios? The golden ratio tops the list! That's the ratio frequently found in "beautiful things" like these:


The Nautilus

Flowers

Vatican Staircase

Galaxy M100

Pinecones

 

 

If the units of measure in a ratio are not the same, then the result is a rate.

For example look at these two boxes of Cheerios.
Suppose the price for the 15oz box is $2.55 while the 20oz box is $3.50.

Comparing the numbers in each box, you get the smaller box has a ratio of  . The dollars ($) and ounces (oz) units do not divide out. So this is actually a rate.

What rate is it? Dividing 2.55 by 15 gives 0.17.

That means the unit rate is  which is commonly said as "17 cents per ounce".

For the larger box, the unit rate is  = .

We'd say this unit rate as "17.5 cents per ounce".


Do you notice that these rates explain which box is the better deal? (Obviously the smaller box costs less per ounce which is better.)

If you happen to do your ratios "up-side-down", you'll still get a rate—just a different one. Take a look:

 and   .

So the smaller box gets you more ounces per dollar—still the better deal!

 

Rates are simply ratios with differing units. They could be almost anything.

 


For your assignment, download and complete this assignment.