Question

Nuts Galore sells a trail mix containing dried banana chips, roasted almonds, and chocolate chips that sells for $6.52 per pound. The dried banana chips costs $4 per pound, the roasted almonds cost $8 per pound, and the chocolate chips cost $6 per pound. The amount of roasted almonds used is three times the amount of chocolate chips used. If they plan to make 100 pounds of this trail mix, how many pounds of each ingredient should be used

Answer 1

`x = 28` ----- Banana

`y = 54` ----- Almonds

`z = 18` ---- Chocolate

Explanation:

Let

`x \to` banana

`y \to` almonds

`z \to` chocolate

So, we have:

`x + y + z = 100` --- total used

`4x + 8y + 6z = 652` -- i.e. 6.52 * 100 --- total sales from 100 pounds

`y = 3z`

Required

The number of pounds of each (i.e. x, y and z)

Substitute
`y = 3z` in
`x + y + z = 100`

`x + 3z + z = 100`

`x + 4z = 100`

Make x the subject

`x = 100 - 4z`

Substitute
`y = 3z` in
`4x + 8y + 6z = 652`

`4x + 8 * 3z + 6z = 652`

`4x + 24z + 6z = 652`

`4x + 30z = 652`

Substitute
`x = 100 - 4z`

`4(100 - 4z) + 30z = 652`

`400 - 16z + 30z = 652`

Collect like terms

`- 16z + 30z = 652-400`

`14z = 252`

Divide by 14

`z = 18`

Substitute
`z = 18` in
`x = 100 - 4z`

`x = 100 - 4 * 18`

`x = 100 - 72`

`x = 28`

To calculate y, we have:

`y = 3z`

`y = 3 * 18`

`y = 54`