Expert: Paul Klarreich - 5/11/2006

Question
I need your assistance on one more problem, Paul!  I've tried, and I'm stumped!

A coffee merchant has coffee beans that sell for $9 per pound and $12 per pound. The two types are to be mixed to create 100 lb of a mixture that will sell for $11.25 per pound. How much of each type of bean should be used in the mixture?

Thank you!

Answer
J.B.,
 
Subject:  Algebra Word Problem
Question:  I need your assistance on one more problem, Paul! I've tried, and I'm stumped!

A coffee merchant has coffee beans that sell for $9 per pound and $12 per pound.

The two types are to be mixed to create 100 lb of a mixture that will sell for $11.25 per pound. How much of each type of bean should be used in the mixture?

Thank you!
.............................
The old mixture problems.  Haven't seen these since I was a kid.  Anyway, there is a whole class of problems like these -- rate-time-distance problems, straight mixture problems (yours), percent solution problems, etc.

They all involve some multiplication principle, such as:

Rate * time = distance
Price-per-pound * weight = cost
Percent concentration * volume = actual amount.

And you set them up like this.  First you make a nice table.  (Not a chart, please -- that's for your history class.)

USE A FIXED-SIZE FONT TO VIEW THIS.


+----------+----------+----------+----------+
|          |          |          |          |
+----------+----------+----------+----------+
|          |          |          |          |
+----------+----------+----------+----------+
|          |          |          |          |
+----------+----------+----------+----------+

Then you put descriptive labels at the top.  DON'T BE LAZY -- MAKE GOOD ONES.

 Type of    Price per   Number     Actual
 Stuff       pound    of pounds     cost
+----------+----------+----------+----------+
|          |          |          |          |
+----------+----------+----------+----------+
|          |          |          |          |
+----------+----------+----------+----------+
|          |          |          |          |
+----------+----------+----------+----------+

Then you fill in some details

 Type of    Price per   Number     Actual
 Stuff       pound    of pounds     cost
+----------+----------+----------+----------+
| Cheap    |    9     |          |          |
+----------+----------+----------+----------+
| Good     |   12     |          |          |
+----------+----------+----------+----------+
| Mixture  |  11.25   |          |          |
+----------+----------+----------+----------+

Determine your variables, in this case:

x = number of pounds of $9 coffee
y = number of pounds of $12 coffee.

[Note that you don't do something lazy, like  "Let x = $9 coffee."  That won't tell you anything; you wind up doing the wrong thing. ]

Now put that into the table.

 Type of    Price per   Number     Actual
 Stuff       pound    of pounds     cost
+----------+----------+----------+----------+
| Cheap    |    9     |    x     |          |
+----------+----------+----------+----------+
| Good     |   12     |    y     |          |
+----------+----------+----------+----------+
| Mixture  |  11.25   |          |          |
+----------+----------+----------+----------+

Now use the information in the problem:

There are to be 100 pounds of coffee mixture

 Type of    Price per   Number     Actual
 Stuff       pound    of pounds     cost
+----------+----------+----------+----------+
| Cheap    |    9     |    x     |          |
+----------+----------+----------+----------+
| Good     |   12     |    y     |          |
+----------+----------+----------+----------+
| Mixture  |  11.25   |  100     |          |
+----------+----------+----------+----------+

Operate the multiplication process on each line:
Price per pound * number of pounds = actual cost


 Type of    Price per   Number     Actual
 Stuff       pound    of pounds     cost
+----------+----------+----------+----------+
| Cheap    |    9     |    x     |    9x    |
+----------+----------+----------+----------+
| Good     |   12     |    y     |   12y    |
+----------+----------+----------+----------+
| Mixture  |  11.25   |  100     |  1125    |
+----------+----------+----------+----------+

Now write equations from the table:

Total amount of coffee = total of pounds
Total cost of coffee = total of costs

x + y = 100
9x + 12y = 1125

Now solve these equations.

Comment:  Have you learned to solve systems of equations yet?  I mean, can you handle equations with two variables?  If not, then you must set things up this way:

Let x = number of pounds of $9 coffee.
THEN  100 - x = number of pounds of $12 coffee.

And the table will appear this way:

 Type of    Price per   Number     Actual
 Stuff       pound    of pounds     cost
+----------+----------+----------+----------+
| Cheap    |    9     |    x     |    9x    |
+----------+----------+----------+----------+
| Good     |   12     | 100-x    |12(100-x) |
+----------+----------+----------+----------+
| Mixture  |  11.25   |  100     |  1125    |
+----------+----------+----------+----------+

And there is only one equation:

9x + 12(100-x) = 1125