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?

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#

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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