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Mix It Up
Word Problems - Mixture

 
 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
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