Question

The owner of Snack Shack mixes cashews worth $5.75 a pound with peanuts worth $2.30 a pound to get a half-pound, mixed-nut bag worth $1.90, How much of each kind of nut is included in the mixed bag?

Answer 1

`5/23` pounds of cashews, `13/46` pounds of peanuts

Explanation:

I haven't been doing the undated ones lately, but I like nuts.

Let `x` be the amount of cashews in pounds, so `1/2 -x` is the amount of peanuts.

We have

`5.75 x + 2.30 (1/2 -x ) = 1.90`

`575 x + 115 - 230 x = 190`

`345 x = 75`

`x = 75/345 = 5/23` pounds of cashews

`1/2-x = 23/46-10/46=13/46` pounds of peanuts

Check:

`5.75 (5/23) + 2.30 (13/46 ) = 1.9 quad sqrt`

Answer 2

Cashew nuts `5/23 lb`
Peanuts `13/46lb`

Explanation:

Final blend `->1/2 color(white)()^("lb")" at "$1.90`

Let the weight in pound of cashew nuts be `C` at `($5.75)/(1^("lb"))`

Let the weight in pound of peanuts be `P` at `("$2.30")/(1^("lb"))`

We know that by weight: `P+C=1/2 color(white)()^("lb")`

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`color(blue)("The calculation")`

`color(brown)("I will show why this works afterwards - see the graph")`
`color(brown)("Basically it is using the slope of a straight line graph")`

The slope of part is the same as the slope of all

Slope `=("Change in cost")/("change in cashew content by weight")`

From the above: `1/2color(white)(.) lbcolor(white)("d") C = $2.875`
From the above `1/2color(white)(.) lbcolor(white)("d") P = $1.15`

So if all cashew nuts the cost is `$2.875` alternatively if all peanuts the cost is `$1.15`

The `1/2` lb ( 0.5 lb ) of blend cost will be in between at $1.90

`(2.875-1.15)/(0.5) = (1.90-1.15)/C`

Turn everything up the other way to get the `C` on the top
`color(brown)("Using fractions to give an exact value")`

`(1/2)/1.725=C/(3/4)`

`C=1/2xx3/4xx[1/1.724xx1000/1000]`

`C=cancel(3)^1/8xx1000/(cancel(1725)^575)=5/23 lb`

Thus `P=1/2-5/23 = 13/46lb`
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