Solving a Fraction Problem Using Common Sense

Date: 03/21/2004 at 14:47:58
From: Corey
Subject: Algebra 1.5

When the gun sounded, only two-fifths of the racers began to run. If
460 racers began to run, how many racers were there in all?  Can you
show me how to figure out what a fraction is without knowing the total
number?  I just don't get it.

Date: 03/22/2004 at 11:28:52
From: Doctor Ian
Subject: Re: Algebra 1.5

Hi Corey,

Well, let's start by looking at a simpler example of the same kind of
problem.  Forget about math for a moment, and just let your common
sense guide you.  

   $5 is one-third of the money in my pocket.  How much
   money do I have in my pocket?

Like I said, don't think about equations, or fractions, or anything
"mathematical."  Just figure the answer out any way you can, and try
to explain how you went about finding it.  Then I'll show you how to
adapt what you already know to a more general kind of problem, okay?

Hope to hear from you soon,

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 03/23/2004 at 14:37:40
From: Corey
Subject: Algebra 1.5

You have $15 in your pocket.  I multiplied 5 times 3 and I got $15.

Date: 03/23/2004 at 16:15:30
From: Doctor Ian
Subject: Re: Algebra 1.5

Hi Corey, 

Thanks for getting back to me.  Your solution is interesting!  Now
let's try a slightly different version:

   $10 is two-thirds of the money in my pocket.  How much
   money do I have in my pocket now?

What do you get this time, and how? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 03/23/2004 at 20:16:38
From: Corey
Subject: Algebra 1.5

You still have $15 in your pocket. I got the answer by taking 5 from 
10 and then adding the 5 to the ten to make 3/3.

Date: 03/23/2004 at 21:05:36
From: Doctor Ian
Subject: Re: Algebra 1.5

Hi Corey,

Nicely done.  Now let's look at your original problem:

   460 racers is 2/5 of the all the racers in a race. 
   What's the total number of racers? 

It's basically the same problem, isn't it?  It's just that the numbers
are different.  Try the same strategy you just used, and let me know
what you come up with. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 03/24/2004 at 15:43:19
From: Corey
Subject: Algebra 1.5

I take 460 and divide it by 2 and get 330.  Then I muliply 330 by 5 
and I get 1,650.  The total number of racers is 1,650!  Right?

Date: 03/24/2004 at 22:32:32
From: Doctor Ian
Subject: Re: Algebra 1.5

Hi Corey,

Exactly right.  I guess you knew more than you thought!

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 03/25/2004 at 12:05:04
From: Corey
Subject: Algebra 1.5

WOW!  Thank you so much!  I guess I did know more than I thought!