Question

Darnel's Health Foods makes a mixed nut blend that contains 19% cashews. Today, Joe, oneof the workers, needs to make 74 pounds of this mix. If he will be using one nut mix thatcontains 21% cashews and another that contains 16% cashews, how much of each should hecombine?Write your answers as whole numbers or as decimals rounded to the nearest tenth.

Answer 1

Let A be the amount of mix that contains 21% cashews and B the amount of mix that contains 16% cashews that will be used to create a mix with 19% cashews.

The total amount of mix will be A+B.

Notice that the total amount of cashews on that mix will be equal to:

`(19)/(100)(A+B)`

On the other hand, the total amount of cashews on the 21% mix, is:

`(21)/(100)A`

While the total amount of cashews on the 16% mix, is:

`(16)/(100)B`

Then:

`(21)/(100)A+(16)/(100)B=(19)/(100)(A+B)`

On the other hand, since the total amount of mix will be 74 pounds, then:

`A+B=74`

These two equations form a 2x2 system of equations. Solve the sistem to find A and B.

Isolate A from the first equation and substitute the expression into the second equation:

`\begin{gathered} (21)/(100)A+(16)/(100)B=(19)/(100)(A+B) \\ \Rightarrow21A+16B=19A+19B \\ \Rightarrow21A-19A=19B-16B \\ \Rightarrow2A=3B \\ \Rightarrow A=(3)/(2)B \end{gathered}`
`\begin{gathered} A+B=74 \\ \Rightarrow(3)/(2)B+B=74_{} \\ \Rightarrow(5)/(2)B=74 \\ \Rightarrow B=(2)/(5)\cdot74 \\ \Rightarrow B=29.6 \end{gathered}`

Substitute the value of B into the expression of A to find A:

`\begin{gathered} A=(3)/(2)B \\ =(3)/(2)(29.6) \\ =44.4 \end{gathered}`

Therefore, 44.4 pounds of 21% cashews mix and 29.6 pounds of 16% cashews mix should be used to produce 74 pounds of 19% cashews mix.