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?

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Answer 1 · 2018-06-23T10:48:59

$\frac{5}{23}$ $5/23$ pounds of cashews, $\frac{13}{46}$ $13/46$ pounds of peanuts

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

Let $x$ $x$ be the amount of cashews in pounds, so $\frac{1}{2} - x$ $1/2 -x$ is the amount of peanuts.

We have

$5.75 x + 2.30 (\frac{1}{2} - x) = 1.90$ $5.75 x + 2.30 (1/2 -x ) = 1.90$

$575 x + 115 - 230 x = 190$ $575 x + 115 - 230 x = 190$

$345 x = 75$ $345 x = 75$

$x = \frac{75}{345} = \frac{5}{23}$ $x = 75/345 = 5/23$ pounds of cashews

$\frac{1}{2} - x = \frac{23}{46} - \frac{10}{46} = \frac{13}{46}$ $1/2-x = 23/46-10/46=13/46$ pounds of peanuts

Check:

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

Answer 2 · 2018-06-23T11:20:43

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

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: $P+C=1/2 color(white)()^("lb")$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$color(blue)("The calculation")$

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

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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Tony B