Cutting An Equal Number of Slices of Cake

Date: 07/13/98 at 17:13:39
From: Katherine Gonzalez
Subject: Question

I have had difficulties solving and understanding the following math 
problem and that's why I writing you, to ask for your help.
This is the problem:
                      
Before checking with the caterer, a cook cuts a cake into 35 equal 
pieces and an identical cake into 42 equal pieces. The caterer, 
however, insists that the cakes be cut exactly alike into the same
number of pieces. 

a) How many pieces will each cake now have to have? 
b) Which does this problem have to do with LCM or GCD?   

I think the answer is 210, but I can not explain or understand the
process of solving the problem.

Date: 07/13/98 at 18:31:32
From: Doctor Wallace
Subject: Re: Question

Dear Katherine,

You are correct. The answer is 210. Let's see why.  

One favorite problem-solving strategy of mine is to invent a similar 
problem, but with much simpler numbers. Then when you see how to solve 
the simpler problem, you apply the very same method to the bigger one.

So let's pretend that the cook cut one cake into 2 pieces and the 
other one into 3 pieces.  

How many pieces must the cakes be cut into, to make them identical?

Well, if I cut each of the 2-piece cake pieces in half, then I have 4 
pieces. But I can't make the 3-piece cake into 4 pieces, since 3 won't 
divide evenly into 4. So instead, let's cut the 2-piece cake into 6 
pieces (by cutting each of the two pieces into 3 equal pieces). Then I 
can cut the 3-piece cake into 6 pieces by cutting each into halves.  
So the answer is 6 pieces.

What do we notice about this method? Since we are always cutting cakes 
equally, we have to use division to solve the problem, but there can 
be no remainder. What we are looking for is a number that both of the 
given cake pieces divide into evenly. Now, your problem doesn't say it 
has to be the lowest number possible, but it would be nice if the cook 
didn't have to do any extra cutting. So what you are looking for is 
the least common multiple.

A good method for finding this is to start with the bigger number 
(since we know the LCM has to be at least this big) and list its 
multiples. Do this by making a multiplication chart. Here are the 
multiples of 3:

   3, 6, 9, 12, 18, 21, 24, 27, 30, 33, 36, 39....

This is just 3x1, 3x2, 3x3, 3x4 ... and so on.

Then, go down the list, and divide the other number you have from 
the problem (2 in our cake example) into each of the multiples on 
the list. The first one you come to that it goes evenly into is the 
LCM. In this case, 2 won't go into 3, but it will go into 6, so 6 is 
the LCM.

6 is a multiple of both 2 (it is 2x3) and 3 (it is 3x2).

If you do this with 35 and 42, you will find out that 210 is the LCM, 
as you wrote.

By the way, this method will also always find the least common 
denominator for two fractions.

I hope this helps you.

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