Comparing Prices

Date: 09/29/2003 at 19:10:16
From: jeff
Subject: word problems

How do you calculate whether something is a better buy?  For example, 
a 6-ounce can of tuna that sells for $1.59, or a package of three
3-ounce cans for $2.19?

I'm not sure what the best way is to tackle the question. 

Date: 09/30/2003 at 13:30:17
From: Doctor Ian
Subject: Re: word problems

Hi Jeff, 

Whether something is a better buy depends on how it fits your needs. 
If you could get a can of tuna for $1.59, or a ton of tuna, all in one
big can, for $100, the ton would definitely be less expensive per
pound... but what could you do with it?  It would start spoiling as
soon as you opened it, so most of your $100 would be wasted. 

Having said that, note that when we divide two numbers with different
units, we get a ratio, e.g., 

  26 miles in 1/2 hour = 52 miles/hour

  38 grams in 19 cubic inches = 2 grams/cubic inch

Similarly, we can get a ratio like

  $1.59 in 6 ounces = (1.59/6) dollars/ounce

If we compute that for two different combinations, we can compare the
results to find out which is cheaper.  

Note that we can compute either dollars per ounce, or ounces per
dollar.  In the former case, we want the smaller ratio.  In the latter
case, we want the larger ratio.   

However, having said that, the _easiest_ way to compare two prices
like this is to convert them to a common quantity.  That is, if I buy
three 3-ounce cans for $2.19, that's 9 ounces for $2.19.  If I double
it, I get 18 ounces for $4.38. 

If I buy one 6-ounce can for $1.59, I can triple that to get 18 ounces
for 3 times $1.59.  Then the comparison is easy.  Which way do I pay
less for 18 ounces? 

Does this help? 

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