Pretzels cost $3 per pound, dried fruit $4 per pound, and nuts $8 per pound. HOw many pounds of each should?

Question

be used to produce 140 pounds of trail mix costing $6 per pound in which there are twice as many pretzels (by weight) as dried fuit?

I have no clue what steps to take to get this solved or what equations to use. Thanks for any help!

Answer 1

Ok. So, let x=pounds of pretzels, y=pounds of dried fruit and z=pounds of nuts

Simply based on weight, we want the sum to be 140 pounds:

x + y + z = 140

Now, for cost, we want the following equation to hold, based on the price of each item, including the trailmix itself:

($3/lb)x+ ($4/lb)y + ($8/lb)z = 140($6/lb)

Also, since we know there are twice as many pretzels(by weight) as dried fruit:

x = 2y

We can take x = 2y and sub it into the second equation. Thus:

(2y) + y + z =140

We subtract 3y from each side and get:

z = 140 - 3y

Now, since we have z and x in terms of y, we substitute everything into the first equation:

3(2y) + 4(y) + 8(140 - 3y) = 140(6)
6y + 4y +8(140) - 24y = 140(6)
-14y= -2(140)

Divide by -14:

y = 20

Now plug that back into the other equations for z and x in terms of y:

x=2(y) = 2(20) = 40

z=140 - (3y) = 140 - 60 = 80

TA DA!

Answer 2

Let p = the pounds of pretzels needed for the mixture.
Let f = the pounds of fruit needed for the mixture.
Let n = the pounds of nuts needed for the mixture.

Since you want to end up with 140 pounds of the mixture,
p + f + n = 140. (Equation i)

Since 140 pounds of the mixture will cost $6 per pound, the total cost of the 140 pounds of mixture is $6 X 140 = $840. Thus
3p + 4f + 8n = 840. (Equation ii)

In the mixture, the weight of the pretzels is twice the weight of the fruit, that is, p = 2f. (Equation iii)

Substitute (iii) into (i) and get p + f + n = 2f + f + n = 3f + n = 140,
3f + n = 140 (Equation iv)

Substitute (iii) into (ii) and get 3p + 4f + 8n = 3(2f) + 4f + 8n =
6f + 4f + 8n = 10f + 8n = 840.
10f + 8n = 840 (Equation v)

Multiply both sides of (iv) by 8: 8(3f + n) = 8(140), or
24f + 8n = 1120 (Equation vi)

Subtract (v) from (vi): (24f + 8n) - (10f + 8n) = 1120 - 840, or
(24 - 10)f + (8 - 8)n = 280, or 14f + 0n = 14f = 280.
14f + 280 (Equation vii)

Divide (vii) by 14 : 14f / 14 = 280 / 14, or f = 20

Substitute this value for f into (iii) : p = 2f = 2(20) = 40

Substitute these values for f and p into (i) : f + p + n = 140, or
20 + 40 + n = 140. Then 60 + n = 140, or n = 140 - 60 = 80

Use 40 pounds of pretzels, 20 pounds of fruit, and 80 pounds of nuts.

To check, notice that p + f + n = 40 + 20 + 80 = 140 = total weight

3p + 4f + 8n = 3(4) + 4(20) + 8(80) = 120 + 80 + 640 = 840 = total cost, and 840 / 140 = 6 = cost per pound

And, finally, p = 40 = 2(20) = 2f

Answer 3

p = # of pounds of pretzels
f = # of pounds of fruit
n = # of pounds of nuts

The equations are:

"The total of weight of the mixture (pounds of pretzels plus pounds of fruit plus pounds of nuts) is 140 pounds."

p + f + n = 140

"The total cost of the mix ($3 per pound times the pounds of pretzels plus $4 per pound times the pounds of fruit plus $8 per pound times the pounds of nuts) is $6 per pound times 140 pounds, or $840"

3p + 4f + 8n = 140(6) = 840

"The pounds of pretzels are twice as large as the pounds of dried fruit"

p = 2f

So your three equations are:

p + f + n = 140
3p + 4f + 8n = 840
p = 2f

You can use the 3rd equation to simplify the first 2 by plugging in 2f for p:

2f + f + n = 140
3f + n = 140

3(2f) + 4f + 8n = 840
6f + 4f + 8n = 840
10f + 8n = 840 (and then, to make the numbers smaller, divide both sides by 2)
5f + 4n = 420

Now you only have 2 equations:

3f + n = 140
5f + 4n = 420

I think it's easiest to mutliply the top one by -4 and add:

5f + 4n = 420
-12f - 4n = -560
--------------------
-7f = -140
f = 20

Then

3(20) + n = 140
60 + n = 140
n = 80

And finally:

p = 2f = 2(20) = 40

So you'd have 40 lbs of pretzels, 20 lbs of fruit, and 80 pounds of nuts.

Double check:

40 + 20 + 80 = 140 pounds, check!

$3(40) + $4(20) + $8(80) = $120 + $80 + $640 = $840/140lbs = $6/lb, check!

40 lbs pretzels = 2(20 lbs fruit), check!

Answer 4

turn it into simultaneous equations: c = kind of cashew nuts w = kind of walnuts fee equation: (a million) 4c + 2.8w = 3.6*24 = 86.4 Weight equation: (2) c + w = 24 sparkling up like generic simultaneous equations: (2) c + w = 24 (2) 4c + 4w = ninety six (a million) 4c + 2.8w = 86.4 (2) - (a million) = (3) (3) a million.2w = 9.6 (3) w = 8 replace into (2): c + w = 24 c + 8 = 24 c = sixteen c = sixteen, w = 8