Question

A food store makes 6- pound mixture of pecans, pistachios , and dried bananas. The cost of pecans is $1.50 per pound, pistachios cost $1.00 per pound, and dried bananas cost$1.50 per pound. The mixture calls for twice as many pecans as pistachios. The total cost of the mixture is$8.50. How much of each ingredient did the store use?

Answer 1

Amount of pecans = 2 pounds

Amount of pistachios = 1 pound

Amount of dried bananas = 3 pounds

Explanation:

Let
`x,y` and
`z` denote amount of pecans, pistachios , and dried bananas in 6-pound mixture.

Amount of mixture = 6 pounds

So,

`x+y+z=6\,\,\,...(i)`

The cost of pecans is $1.50 per pound, pistachios cost $1.00 per pound, and dried bananas cost$1.50 per pound and the total cost of the mixture is$8.50.

`1.50x+y+1.50z=8.50\,\,\,...(ii)`

The mixture calls for twice as many pecans as pistachios.

`x=2y\,\,\,...(iii)`

Put
`x=2y` in (i)

`2y+y+z=6\\3y+z=6\,\,\,...(iv)`

Put
`x=2y` in (ii)

`1.50(2y)+y+1.50z=8.50\\3y+y+1.50z=8.50\\4y+1.50z=8.50\,\,\,(v)`

Multiply (iv) by 4 and (v) by 3

`12y+4z=24\,\,\,(vi)`

`12y+4.5z=25.5\,\,\,(vii)`

Subtract (vi) from (vii)

`0.5z=1.5\\z=3`

Put
`z=3` in (iv)

`3y+3=6\\3y=3\\y=1`

Put
`y=1` in (iii)

`x=2(1)=2`

Therefore,

Amount of pecans = 2 pounds

Amount of pistachios = 1 pound

Amount of dried bananas = 3 pounds