Exposed Industrial Ceilings

Exposed ceilings are really hip these days, creating an industrial warehouse feel in office spaces and restaurants.  They don’t cover up the wiring and beams, so you can see what’s going on.

Source: http://imgarcade.com/1/industrial-office-ceiling/

How is this related to math?

Well, try to simplify the fraction 275/11.  If you’re not sure where to start, use the Exposed Ceiling Technique.  Break everything down so you can see the metaphorical beams:

275/11 = (25*11)/11

Now it’s easy to notice the 11’s and cancel them out.

This is also useful when solving problems.

Often you’ll be plugging in values and you’ll want to multiply them out into one big number.  Resist this temptation.  Don’t rush to plaster things over.  Show your exposed ceilings some love and they will love you back.

Suppose you’re plugging values into an equation and you end up with,

(3*11*12*x)/(34) = (121)/(17)

Using the Exposed Ceiling Technique, i.e. without multiplying or calculating anything, we have:

x = (121*34)/(3*11*12*17)

…instead of rushing to multiply, expose it further…

x = (11*12*17*2)/(3*11*12*17)

Now we can eliminate 11, 12, and 17:

x = 2/3.

 

Exposed ceilings enable you to see patterns that would otherwise be obscured.

Source: http://tinyurl.com/qxd4aqn

Suppose you are exploring the concept of compound interest for the first time.  You are trying to derive a formula.  You come across a problem:

$2000 is invested in a savings account with 1% interest, compounded annually.  How much is in the account after 5 years?

After some thought, you decide that you need to repeat a process each year: write down the amount you had at the beginning of the year, and take 101% percent of it to find out how much you have at the end of the year.  Taking 101% of a number is the same as multiplying it by 1.01.

Let’s see how this works.

First, let’s do this without exposed ceilings:

starting amount: $2000

after 1 year: $2000*(1.01) = $2020 <– this is $2000 plus an extra 1% of $2000

after 2 years: $2020*(1.01) = $2040.20

after 3 years: $2040.20*(1.01) = $2060.60

after 4 years: $2060.60*(1.01) = $2081.21

after 5 years: $2081.21*(1.01) = $2102.02

final amount: $2102.02

You were able to solve the problem, but no obvious formula popped out of this work.

Now let’s do this with exposed ceilings:

This time, don’t multiply anything out until the end.  This is also a convenient approach if you don’t have a calculator.

starting amount: $2000

after 1 year: $2000*(1.01) <– this is $2000 plus an extra 1% of $2000

after 2 years: $2000*(1.01)*(1.01)

after 3 years: $2000*(1.01)*(1.01)*(1.01) = $2000*(1.01)3

after 4 years: $2000*(1.01)*(1.01)*(1.01)*(1.01) = $2000*(1.01)4

after 5 years: $2000*(1.01)*(1.01)*(1.01)*(1.01)*(1.01) = $2000*(1.01)5

final amount: ok, now you can multiply it out. $2000*(1.01)= $2102.02


You got the same answer, but this time you also exposed an underlying pattern: to find the amount after the nth year, just take the original amount, and multiply it by 1.01n.  In other words,

final amount after n years = (starting amount)*(1.01)n

Not only did your exposed ceilings save you from boring extra calculations, they gave you a glimpse at the underlying structure, which enabled you to write a general formula.

How would you adjust this formula for an annual interest rate of 3%?  How about 0.95%?  How about p%?