What’s Goin’ On In Room 133

Posts Tagged ‘ratio’

Check out page 291, sport fans!  Tackle questions 33 and 39.  And take a look at “Keeping Things in Perspective” below. 

Essayons!

Ever hear that phrase?  Keeping things in perspective is like saying don’t get carried away.  Don’t take things too far.  Sort of like ‘keeping it real’.  Well, mathematically speaking, keeping things in perspective – or in proportion – is very much what we are dealing with now. 

If we look at a football field from end zone to end zone, we can see that it has a set distance – 100 yards, not counting the end zone-dance floor areas.  Well, suppose we were strapped in a helicopter high over the field and we took a perfectly flat (not tilted) photograph of the field looking straight down.  Our photo would show a perfect image of the football field with one major difference – it would be a LOT smaller than the field itself.  But you would be able to make out the features - the goal posts, the bleachers, the parking lots – because the photo would have everything in perspective.  Large areas on the ground (like the playing field) would take up more room on the photo, and smaller areas on the ground (like the press box or a seating section) would take up less room. 

Well, guess what?  That characteristic of perspective (proportion) can be expressed mathematically.  (You knew I’d get there, didn’t you?)  That characteristic is based on the concept of scale, just like the scale of a map – or an overhead photo!  The scale is the link between how big something is on the ground and how small it is on the map or photo.  Sometimes it is written in words such as “One inch equals thirty feet (ten yards),”  or it can be written using the equal sign  (1 inch = 30 feet) or even more often, the colon “:” sign (1 inch : 30 feet). 

Quite often, the mapmaker or the photo interpreter will set up the scale so that there are no words or units of measure in the description.  How?  Well, knowing our conversion values, we can see that the scale above translates into 1 inch = (30 ft x 12) inches which gives us 1 inch = 360 inches.  This means that 1 inch of measurement on the photo (say from one 10-yd line to the next) represents a total distance of 360 inches on the field (10 yards, right?).

And this is where proportion kicks in.  Because if 1 inch on the photo measures 360 inches across the same distance on the ground, then we also know that 1 foot of measure on the photo will represent 360 feet of distance on the ground.  And 1 centimeter of photo distance will represent 360 centimeters of ground distance.  And so on and so on.  So the scale can be written without any units of measure, such as “1 : 360″.  Written in this form, it is also called a ratio.  It can also be written as a fraction, so it could look like “1/360″.  Here it would be called a representative fraction. 

And guess what?  If it’s a fraction, then we use it the same way we’ve been using fractions all week!  Now that’s keeping it real!