Calculating Percent Increases

Date: 10/29/96 at 23:22:23
From: Anonymous
Subject: Re: Calculating percent increases

Dr. Math, 

I am a college student, but my question is quite elementary.  I'm sure 
that you can help.  In order to explain how to find a percent 
increase, a gentleman gave me this as an example:

I want to find what a 5 percent increase of 1200 is:  

First:  .05 x 1200= 60

Second:  (.05 x 1200) + (1 x 1200) = 1260

This is a correct answer, but I am confused why this gentleman 
multiplied 1 x 1200 in the second parentheses.  Why didn't he just add 
the results from the first parentheses to 1200?  Instead, he used an 
extra step of multiplying 1 x 1200.  What does that number one 
represent?  Obviously, the one will not change the answer, but I would 
still like to know why he has that one in the equation. 

   Thanks!

Date: 10/30/96 at 13:41:46
From: Doctor Mike
Subject: Re: Calculating percent increases

Hi Jason,
  
Of course you are right that multiplying by one makes no difference in
the final result.  I can think of two possible reasons for doing this.  
You might refer to "the original amount" or "the whole original 
amount" depending on whether you wish to give emphasis by adding the 
extra word.  Multiplying 1200 by one emphasizes that in the end, you 
have the entire whole original amount plus a little bit more.  In 
other words, it's not a big deal.
  
Perhaps you have run across the Distributive Law of multiplication 
over addition, which has many forms, among which is 
A*X + B*X = (A+B)*X.  This is sometimes called the rule for 
multiplying out something, or the rule for factoring out something, 
depending on how you are using it. Here I want to use it to help with 
factoring out 1200 as follows: 
  
  (.05 * 1200) + (1 * 1200)  =  (.05 + 1) * 1200  =  1.05 * 1200
  
You probably would not notice this connection without the "one times" 
in there.  The benefit of this is that it shows a calculator shortcut 
for figuring out the total.  Instead of figuring out the increase as a 
first operation, and then adding in the principal as another 
operation, you can do the "1 + .05 = 1.05" part in your head, and then 
multiply the 1200 original amount by 1.05 as the one and only 
operation.  This is not going to save you enough hours to take off 
fishing, but it's not a bad thing to understand.
  
I hope this helps. 

-Doctor Mike,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 10/30/96 at 22:22:23
From: Anonymous
Subject: Re: Calculating percent increases

Dr. Math,

Thanks for getting back to me so quickly.  By the way, are you a math
professor?  Anyway, let me make sure I got this straight.  1.05 is 
the same as 105 percent, and essentially in doing a percent increase I 
am adding a certain percent to 100, correct?  I am still a bit 
confused, however, on why a 5 percent increase is shown as 1.05.  Is 
it because 1.05 converted from a decimal to a percent is 105 percent 
and that is essentially what I am looking for?  

Please write back.  I am still confused about what the one really 
represents.  Why wouldn't it be 100.05 "times" 1200?

Date: 11/06/96 at 13:41:46
From: Doctor Mike
Subject: Re: Calculating percent increases
  
Hello again Jason,
 
I'll try to get to all your questions. First, I am neither a prof nor 
a student, but a mathematician in industry. There are math doctors 
here with many different backgrounds and situations.  

Second, the task "take 105 percent of something" is the same as 
"multiply something by 1.05". Another way to think about what ONE 
represents is that 100 percent means "all of it", which is ONE 
complete quantity.  Another example that might help is that you could 
go into a grocery store and ask for "ONE dozen" eggs, which is in 
effect asking for 100 percent of a dozen eggs.  Does that help?  

Finally, I will ask you to think carefully about what 100.05 "times" 
1200 would mean.  That would give you something a little more than 
100 times 1200, and this is quite different from 100 percent of 1200.  
If a cooking recipe calls for 100 percent of a dozen eggs, and you put 
in 100 dozen eggs, then you will have one messy kitchen!   
  
I hope this straightens the percentage area out for you. 

Best regards,

-Doctor Mike,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/