Answer:
10 pounds
Step-by-step explanation:
You want to know the number of pounds of peanuts in a mix of $5/lb peanuts and $6/lb almonds that gives 14 pounds of nuts for $74.
Setup
Let p represent the number of pounds of peanuts in the mix. The (14-p) is weight of almonds, and the total cost is ...
5p +6(14 -p) = 74
Solution
Simplifying the equation gives ...
-p +84 = 74
10 = p . . . . . . . . add p-74
You should buy 10 pounds of peanuts to make the 14 lb bag cost $74.
__
Additional comment
You notice that simplifying the equation gives p a negative coefficient. If you choose the variable to represent the more expensive contributor, then the coefficient of that comes out positive. (We like positive coefficients, as they tend to reduce errors.)
Here, the problem is requesting the amount of the lower cost contributor, so we used the variable to represent that.
Often, you are asked to solve mixture problems using a system of equations. Those would use a variable for each of the quantities, and the equations would be ...
- p + a = 14 . . . . . . . total weight
- 5p +6a = 74 . . . . . total cost
You notice that substituting for 'a' gives the equation we used above.
- a = 14 -p . . . . . . . . . . . . . write an expression for 'a'
- 5p +6(14 -p) = 74 . . . . . use the expression for 'a'
Jumping directly to this step saves a bit of written work, though the mental work is the same.
