How Many Freshmen Originally Tried Out?

Date: 01/15/2003 at 20:39:14
From: Rosie
Subject: Addition and Subtraction Equations

Some freshmen were trying out for the school track team.

After Round 1, 1/2 of those remaining were eliminated.
After Round 2, 1/3 of those remaining were eliminated.
After Round 3, 1/4 of those remaining were eliminated.
After Round 4, 1/5 of those remaining were eliminated.
After Round 5, 1/6 of those remaining were eliminated.

The 10 who remained became the track team. How many freshmen 
originally tried out?

I cannot figure out how to approach it.

Date: 01/16/2003 at 09:34:36
From: Doctor Ian
Subject: Re: Addition and Subtraction Equations

Hi Rosie,

You can approach it from the front, or from the rear.  

To approach from the front, you have to use algebra. Let N be the
number who tried out. After the first round, 1/2 are eliminated, so
the number remaining is N/2. 

  remaining = N/2

After the second round, 1/3 are eliminated, which means the number
remaining is 2/3 of N/2:

  remaining = (2/3)(N/2)

and so on, until you have a product of fractions, whose value has to
be equal to 10. Solve for N, and you know how many people tried out.

To approach from the rear, you start with the 10 people who made the
team, and reconstruct what must have happened in each round.  

In round 5, 1/6 of the team was eliminated, so 10 people must be 5/6
of the number who entered that round.  And 10 is 5/6 of 12, so 12
people must have entered round 5. 

In round 4, 1/5 of the team was eliminated, so 12 people must be 4/5
of the number who entered that round. And 12 is 4/5 of 15, so 15
people must have entered round 4.  

And so on. 

Is this enough to get started? 

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