story problem math please explain in detail

Question

she needs to prepare 130 pounds of coffee beans selling for $4.46 a pound. shes going to blend together a high quality bean costing 5.50 a pound, and a cheaper bean at 3.25 per pound. to the nearest pound find how much high-quality coffee bean and how much cheap coffee bean she should blend

Answer 1

This is a problem where you have two unknown variables:

  Q = the pounds of quality beans

  C = the pounds of cheap beans

 

One equation that is easily formed is  Q + C = 130 because after the blending, there will be 130 pounds of beans. You can't solve this ONE equation because it has TWO variables.

 

However, if you create a second equation, then you'll have TWO equations with TWO variables. This makes a system of equations. The number of variables you need to solve for requires that same number of equations. n unknown variables require n equations to solve.

 

Luckily, another equation can be formed. Let the total cost of all of the quality AND cheap beans equal the cost of the quality beans added to the cost of the cheap beans. (Omitting the dollar signs for the price in the equation for convenience), cost would be found by multiplying the number of beans at a certain price by that price:

 

130(4.46) = 5.50Q + 3.25C       now multiply

 

579.80 = 5.50Q + 3.25C and use the other equation Q+C=130 this way:

 

Q+C = 130  so  Q = 130-C   now substitute Q into the longer equation:

 

579.80 = 5.50(130-C) + 3.25C    now multiply and then simplify:

 

579.80 = 715 - 5.50C + 3.25C

579.80 = 715 - 2.25C

2.25C  =  715 - 579.80

2.25C  = 135.20   now there is ONE equation to solve with ONE unknown

 

C = 60.08888 beans

Q + C = 130  and  Q = 130 - C

Q = 69.11111 beans

 

It's highly likely that this math problem was created, and then transcribed incorrectly with one of the three bean prices off by one cent. I posted this result even though neither Q or C can be a fraction of a bean. These results are within 5/100 of 1% of:

 

60 cheap beans

70 quality beans. 

 

The process above does not contain any errors in the calculation. The process above does explain in great detail the proper way to solve a word problem with two unknowns which form two equations.