If Danielle wants to make a profit of $0.52 per pound at a selling price of $5.61 per pound, the cost of the mix must be $5.61 -0.52 = $5.09 per pound.
The proportion of the higher-priced ingredient will be
.. ((cost of mix) -(cost of lower-priced ingredient))/(difference in ingredient prices)
.. = (5.09 -2.66)/(6.17 -2.66)
.. = 2.43/3.51 = 9/13
Danielle should use (9/13*13 lb = 9 lb of jellybeans and 4 lb of chocolates.
_____
It is time-saving to remember the proportion described above. If you don't want to do that, there are a couple of other ways to get there. In general, it is preferred to solve for the quantity of higher-contributing item (here, higher-priced ingredient), as the numbers stay positive that way.
Alternate method 1. Assume the entire output is made up of lower-contributing item. Here, we want the mix cost to be (5.09/lb)*(13 lb) = 66.17. If the whole 13 lb is made of chocolate, the cost will be (2.66/lb)*(13 lb) = 34.58. The difference between these numbers is 31.59, and it must be made up by the contribution of jellybeans.
Each pound of jellybeans that replaces a pound of chocolate adds (6.17 -2.66) = 3.51 to cost of the mix. We need to add 31.59/3.51 = 9 pounds of jellybeans to the mix. Since 9 pounds of chocolate have been replaced by jellybeans, 4 pounds remain. The mix is 9 lb jellybeans, 4 lb chocolate.
Alternate method 2. Assign variables and write equations. Let j and c represent pounds of jellybeans and chocolate, respectively.
.. j + c = 13
.. 6.17j +2.66c = 5.09*13
We choose to solve for "j", so we make a substitution for "c".
.. 6.17j +2.66(13 -j) = 5.09*13
.. J(6.17 -2.66) = 13*(5.09 -2.66) . . . . collect j terms, subtract 13*2.66
.. j = 13*(5.09 -2.66)/(6.17 -2.66) . . . . this proportion should look familiar
.. j = 9
.. c = 13 -9 = 4
