Pencil Costs, Dividing Property

Date: 6/25/96 at 4:50:14
From: Susan Goh 
Subject: Pencil Costs and Dividing Property

Dear Dr. Math,

Drawing pencils cost 8 cents each and colored pencils cost 11 cents 
each. Two dozen assorted pencils cost $2.16. How many colored pencils 
are there?

Another one, but this is more of a puzzle:

A man died, leaving behind 23 cows to his three children. His will 
states that the eldest child should have 1/2 of the fortune, the 
second child 1/3 and the youngest 1/8 of the total fortune. How can 
the three children divide the animals among themselves so that it is 
not necessary to kill any animal?

A wise man came to the scene. He brought along his only cow and put it 
with the other 23 cows to give a total of 24 cows. He gave 1/2 of the 
24 cows to the eldest child, i.e. 12 cows; 1/3 of the 24 cows to the 
second child, and 1/8 of the 24 cows to the youngest. He went away 
with his only cow. Can you discover the clue to this solution?

SOS.   Jean

Date: 6/25/96 at 12:3:7
From: Doctor Lisa
Subject: Re: Pencil Costs and Dividing Property

In this problem, let x = the number of colored pencils and 24-x = the 
number of drawing pencils.  This is possible because the colored 
pencils and the drawing pencils have to add up to 24.  If you knew you 
had 10 colored pencils, then to get the number of drawing pencils, 
you'd take 24-10 = 14.  So the equation set up would be (using cents):

8(24-x) + 11(x) = 216.

The remainder of the work goes as follows:

192 - 8x + 11x = 216 (distributing the 8 in the first parentheses)
192 + 3x = 216       (combining like terms)
3x = 24              (subtracting 192 from both sides)
x = 8                (dividing both sides by 3)

So you would have 8 colored pencils (and 16 drawing pencils).

In the second problem, the part with the wise man coming to the scene 
makes the problem clearer.  If you use 24, it is very easy to come up 
with 1/2, 1/3, and 1/8 of that number since 2, 3, and 8 all go into 24 
evenly. This means that the eldest would get 12 cows, the second would 
get 8 cows, and the youngest would get 3 cows.  If you add up 12, 8, 
and 3, you get 23 (and thus the wise man can take his cow with him).

Hope this helps!

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

Date: 6/25/96 at 12:27:40
From: Doctor Anthony
Subject: Re: Pencil Costs and Dividing Property

Yes, if you add 1/2 , 1/3 and 1/8 you get 23/24.  So the old man is 
dividing his estate in the RATIO 1/2 : 1/3 : 1/8, which is 12 : 8 : 3  
and by good fortune that adds up to 23.  1/24th part of his estate was 
not allocated to anyone, so if he had had 24 cows, there would have 
been one cow left over.

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