Ratio and Proportion: Beaches and Hawks

Date: 10/26/98 at 18:39:53
From: Rebecca Tom
Subject: Word problems

On a tour guide map of Madagascar, the scale states that 3 inches 
represent 125 miles. Two beaches are 5.2 inches apart on the map.  
What is the approximate distance in miles between the two beaches? 
Round your answer to the nearest tenth.

A ornithologist is studying hawks in the Adirondack Mountains. She 
catches 24 hawks over a period of one month, tags them, and frees them 
back into the wild. The next month, she catches 20 hawks and finds 
that 12 are already tagged. Estimate the number of hawks in this part 
of the mountains.

In this problem I set it up like this, but I don't know if it is right 
or not. I got 40 hawks as the answer.

Date: 10/27/98 at 13:38:06
From: Doctor Peterson
Subject: Re: Word problems

Hi, Rebecca. Let's see what we can do for you. These are both ratio 
problems. I'll suggest a way to picture them, which might help.

For the first problem, a map is proportional to the real world it 
represents. Let's represent this by making two bars, one representing 
a distance in the real world, and the other the same distance on the 
map:

    Real:   125 mi
    +--------------------+
    |                    |
    +--------------------+

    Map:     3 in
    +--------------------+
    |                    |
    +--------------------+

Now we have a distance on the map, and want to know what it is in the 
real world:

            scale                        beaches

    Real:   125 mi                         ? mi
    +--------------------+    +-----------------------------+
    |                    |    |                             |
    +--------------------+    +-----------------------------+

    Map:     3 in                         5.2 in
    +--------------------+    +-----------------------------+
    |                    |    |                             |
    +--------------------+    +-----------------------------+

Because the map is proportional, the ratio of corresponding lengths is 
always the same, so we can write:

           scale   beaches

    Real   125 mi    ? mi
    ---- = ------ = ------
    Map     3 in    5.2 in

Now you just have to solve this for the unknown length. Just multiply 
both ratios by 5.2 in, and you'll get:

           125 mi            125 * 5.2
    ? mi = ------ * 5.2 in = --------- mi = 216.66 mi
            3 in                 3

For the second problem, you got the right answer, but I can't tell how 
you did it. I'll just explain the whole thing anyway.

This one is a little less obvious, but it's still a ratio. First we're 
told that 24 out of all the hawks in the area have been tagged:

    Tagged:  24
     ________/\__________
    /                    \
    +--------------------+--------------+
    |                    |              |
    +--------------------+--------------+
    \_________________  ________________/
                      \/
    Total birds:      ?

Then we're told that 12 out of 20 caught the next month are found to 
be already tagged:

    Tagged:  12
     ________/\__________
    /                    \
    +--------------------+--------------+
    |                    |              |
    +--------------------+--------------+
    \_________________  ________________/
                      \/
    Total birds:      20

If we assume that the birds were caught randomly, so the two ratios 
are about the same, we can write a proportion:

             tagged total

    First      24      ?
    ------  =  --  =  --
    Second     12     20

Now you can solve this ratio for the unknown.

You could also think of this another way, which is more natural to me. 
The second picture shows 12/20 of the total number of birds tagged, 
and the first picture shows 24/? tagged. We can set these two ratios 
equal:

             first  second

    Tagged     24     12
    ------  =  --  =  --
    Total       ?     20

You'll get the same answer either way.

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