



There is
a type of problem that may fool you 


into
thinking that it is a percentage problem. 


It goes
something like this ... 





A barrel
contains 5 gallons of a solution that is 10% sugar. 


Another
barrel contains a solution that is 25% sugar. 


How much
of the 25% sugar solution 


must be
added to the 5 gallons of 10% sugar solution 


to make a
solution that is 15% sugar? 





The setup
of this problem looks like this ... 











We have 6
different things in this problem: 





Gallons of
the first solution = 
5 

Sugar in the
first solution = 
.10 
(.10
= 10%) 



Gallons of
the second solution = 
? 

Sugar in the
second solution = 
.25 
(.25
= 25%) 



Gallons of
the final mixture = 
? 

Sugar in the
final mixture = 
.15 
(.15
= 15%) 






So we
have a little problem. 


There are
two things in this problem that we don't know. 


The
gallons of the second solution AND 


the
gallons of the final mixture. 





So far,
we only know how to solve a problem 


that has
one unknown thing. 


We need a
new trick to deal with a problem 


that has
two unknowns. 





Good
news! 


We get
one! 





It goes like
this :






Gallons of the
first solution = 
5 
Gallons of the
second solution = 
X 






AND NOW
THE TRICKY PART ... 





Gallons of the
final mixture = 
X + 5 






Now the
only unknown is X. 


That
means we can solve this problem! 











Now we
move the X's to one side and the numbers to the other, 


and solve
for X ... 











We need
to add 2.5 gallons of the 25% sugar mixture 


to the 5
gallons of 10% sugar mixture 


to get a
mixture that is 15% sugar. 





These
problems can look way different, 


and still
use the same setup ... 





Example: 





A person
buys 2 products (A and B) and spends a total of $250. 


There are
mail in rebate coupons with each product. 





A rebate
is money that a company sends you for buying the product. 





The
rebate on product A is 10% of the purchase price. 


The
rebate on product B is 5% of the purchase price. 





The total
rebate received is $15. 





How much
did each product cost? 





The setup
for this problem is: 





(Rebate received
on A) + (Rebate received on B) = (Total Rebate) 





The
rebate on the product is equal to the cost of the product 


times the
rebate percentage. 





That
means we can write the setup as: 











Cost of
product A = 
? 

Rebate % on
product A = 
.10 
(.10
= 10%) 



Cost of
product B = 
? 

Rebate % on
product B = 
.05 
(.05
= 5%) 



Total Rebate
= 
$15 







Now we
use the same kind of trick on this problem: 





Cost of both
products together = 
$250 
Cost of
product A = 
X 
Cost of
product B = 
250
 X 












So
product A costs $50 


That
means product B costs $250  $50 = $200. 





copyright 2005 Bruce Kirkpatrick 
