a coffee retailer has two grades of decaffeinated coffee beans, one wholesales at $4.00 a pound and the other for 6.50 a pound. she wishes to blend the beans to form a 150 pound mixture that sells wholesale for 4.75 a pound. how many pounds of each grade of bean should she use in the mixture?

Step-by-step explanation:

To solve mixture problems like this the best way is to make a table. Here's the basic format of the table we will use, calling the first coffee C1 and the second coffee C2:

                 #lbs        x         $/lb         =       Total

C1

C2

Mix

This is the table we will fill in. First, we were given the cost per pound of each type of coffee, so we put that in first:

                #lbs        x         $/lb         =        Total

C1                                        4

C2                                     6.50

Mix

Next, we are told that she wants to make a mix of these coffees so she has a total number of pounds as 150 and that she wants it to cost $4.75 per pound. That goes in next:

                #lbs        x        $/lb        =        Total

C1                                         4

C2                                     6.50

Mix           150         *          4.75

That asterisk is the multiplication sign.

Here's where we are now responsible for filling out the rest of the chart. If she needs to mix these 2 coffees to get a mix that is `150 pounds, she can have x pounds of C1 which means that she has to have 150 - x of C2:

                #lbs       x        $/lb        =         Total

C1                x         *           4

C2         150 - x      *        6.50

Mix           150        *         4.75  

Now all we have left to do is what the table tells us to do, which is to multiply the first 2 columns and put that product under the Total column:

                #lbs        x        $/lb        =        Total

C1                x           *          4           =          4x

C2         150 - x        *       6.50        =     975 - 6.50x

Mix            150         *        4.75        =     712.50

Since we have to add the 2 coffees together to get the pounds of mix in the first column, we also have to add the cost of those coffees together to get the total cost of the mix. The equation looks like this:

4x + 975 - 6.50x = 712.50 and combining like terms:

-2.5x = -262.5 so

x = 105

That means that there is 105 pounds of coffee 1 and 150 - 105 pounds of coffee 2.

C1 = 105 pounds, C2 = 45 pounds