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

Answer:

Amount of pecans = 2 pounds

Amount of pistachios = 1 pound

Amount of dried bananas = 3 pounds

Step-by-step 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