A food store makes a 11–pound mixture of peanuts, almonds, and raisins. The cost of peanuts is $1.50 per pound, almonds cost $3.00 per pound, and raisins cost $1.50 per pound. The mixture calls for twice as many peanuts as almonds. The total cost of the mixture is $21.00. How much of each ingredient did the store use?
3 lbs peanuts, 6 lbs almonds, 2 lbs raisins
8 lbs peanuts, 1 lb almonds, 2 lbs raisins
6 lbs peanuts, 3 lbs almonds, 2 lbs raisins
8 lbs peanuts, 2 lbs almonds, 1 lbs raisins

Question

A food store makes a 11–pound mixture of peanuts, almonds, and raisins. The cost of peanuts is $1.50 per pound, almonds cost $3.00 per pound, and raisins cost $1.50 per pound. The mixture calls for twice as many peanuts as almonds. The total cost of the mixture is $21.00. How much of each ingredient did the store use? 
    3 lbs peanuts, 6 lbs almonds, 2 lbs raisins
    8 lbs peanuts, 1 lb almonds, 2 lbs raisins
    6 lbs peanuts, 3 lbs almonds, 2 lbs raisins
    8 lbs peanuts, 2 lbs almonds, 1 lbs raisins

jim_thompson5910 · Answer

Start off by defining variables

jim_thompson5910 · Answer

let x = amount of peanuts (in pounds), y = amount of almonds (in pounds) and z = amount of raisins (in pounds)

anonymous · Answer

okey (:

jim_thompson5910 · Answer

If

         "A food store makes a 11–pound mixture of peanuts, almonds, and raisins"

then what does this mean?

anonymous · Answer

That they add up to 11 pounds haha

jim_thompson5910 · Answer

good, so x y and z add to 11, which means

x+y+z = 11

jim_thompson5910 · Answer

that's one equation

anonymous · Answer

so x+y+z=11 :o

jim_thompson5910 · Answer

yes

anonymous · Answer

But then we have to factor in how much they cost..

jim_thompson5910 · Answer

We also know that

             "The cost of peanuts is $1.50 per pound, almonds cost $3.00 per pound, and raisins cost $1.50 per pound"

So x lbs of peanuts cost 1.5x dollars, y lbs of almonds cost 3y dollars, and z lbs of raisins cost 1.5z dollars

They total 1.5x + 3y + 1.5z dollars

jim_thompson5910 · Answer

this total cost is given to be $21

so 

1.5x + 3y + 1.5z = 21

anonymous · Answer

so like, 1.50x+3.00y+1.50=21..

anonymous · Answer

Haha Okey you already said that x.x my bad

anonymous · Answer

1.50z* I meant btw haha. I get that indeed.

jim_thompson5910 · Answer

close, you mean to say 1.50x+3.00y+1.50z=21

but you got it

anonymous · Answer

And don't forget twice the peanuts!

jim_thompson5910 · Answer

that's your 2nd equation

jim_thompson5910 · Answer

twice the peanuts as what?

anonymous · Answer

it says, the recipe calls for twice as many peanuts as almonds O.o

jim_thompson5910 · Answer

good, so the equation for that is?

anonymous · Answer

ermm..  I dunno lol.. x(times2y)+y+z=11 O.O I have no idea.

jim_thompson5910 · Answer

not quite

anonymous · Answer

knew it was wrong x.x

jim_thompson5910 · Answer

Amount of peanuts = 2 times (amount of almonds)

jim_thompson5910 · Answer

y = 2 times x

or just

y = 2x

anonymous · Answer

oh pshh I should have thought of that x.x

jim_thompson5910 · Answer

so you have these 3 equations

x+y+z = 11
1.5x + 3y + 1.5z = 21
y = 2x

jim_thompson5910 · Answer

you can use them to solve for x, y and z

jim_thompson5910 · Answer

I'll get you started

anonymous · Answer

Lol I can't because I'm stupid, But smart people definitely can haha

jim_thompson5910 · Answer

x+y+z = 11

x+2x+z = 11

3x+z = 11

---------------------

1.5x + 3y + 1.5z = 21

1.5x + 3(2x) + 1.5z = 21

1.5x + 6x + 1.5z = 21

7.5x + 1.5z = 21

jim_thompson5910 · Answer

so after plugging in y = 2x, you now have the new system

3x+z = 11
7.5x + 1.5z = 21

anonymous · Answer

O.o That all made absolutely no sense to me but it seems to for you and that's good enough for me

jim_thompson5910 · Answer

where are you lost?

jim_thompson5910 · Answer

I took 

x+y+z = 11

plugged in y = 2x (ie replaced every 'y' with '2x') to get

x+2x+z = 11

then I combined like terms to get 

3x+z = 11

jim_thompson5910 · Answer

I did the same with 1.5x + 3y + 1.5z = 21 to get 7.5x + 1.5z = 21

anonymous · Answer

oh okay ! I see now, sorry it's really late and I'm super tired :/

jim_thompson5910 · Answer

so you have 

3x+z = 11 
7.5x + 1.5z = 21

now

jim_thompson5910 · Answer

solve the first equation for z to get this

3x+z = 11

3x+z - 3x = 11 - 3x

z = 11 - 3x

anonymous · Answer

Alrighty

jim_thompson5910 · Answer

Since z = 11 - 3x, you can plug it into the other equation and solve for x


7.5x + 1.5z = 21

7.5x + 1.5(11 - 3x) = 21

Solve for x

jim_thompson5910 · Answer

oh wait, i made a variable mistake...i switched x and y

one sec

jim_thompson5910 · Answer

Let me fix real quick

anonymous · Answer

I was thinking that haha x) ..

anonymous · Answer

uhh ? your comment disappeared .. :o

jim_thompson5910 · Answer

it should be x = 2y, not y = 2x 

so 

x+y+z = 11 

2y+y+z = 11 

3y + z = 11 

z = 11 - 3y 

--------------------------- 

1.5x + 3y + 1.5z = 21 

1.5(2y) + 3y + 1.5z = 21 

3y + 3y + 1.5z = 21 

6y + 1.5z = 21 

=========================== 

6y + 1.5z = 21 

6y + 1.5(11 - 3y) = 21 

Solve that last equation for y

jim_thompson5910 · Answer

sry

jim_thompson5910 · Answer

Solve 6y + 1.5(11 - 3y) = 21  for y

anonymous · Answer

haha it's okay

anonymous · Answer

Aw man I have to go to bed soon I'm dead tired x.x

jim_thompson5910 · Answer

What do you get when you solve that equation for y

anonymous · Answer

ergh I dunno x.x I'm brain dead

jim_thompson5910 · Answer

Then give it a break, get some sleep and you'll tackle it in the morning

anonymous · Answer

Oh no I keep losing connection >.<