How much of each type of nut is in the package?

Question

A company mixes peanuts, cashews, and almonds to obtain 17 oz package worth 4.63 dollars. If peanuts, cashews, and almonds cost 17, 42, and 50 cents per oz respectively, and the amount of cashews is equal to the amount of almonds, how much of each is in the package?

 

_____oz of peanuts

 

_____oz of cashews

 

_____oz of almonds

Answer 1

Hi Morgan, set this up as a system of linear equations. Here is the information provided:

 

Let p = peanuts, which cost 17 cents per ounce

Let c = cashews, which cost 42 cents per ounce

Let a = almonds, which cost 50 cents per ounce

 

The number of cashews equals the number of almonds.

 

The package weight is 17 ounces, or p + c + a = 17

 

The package costs $4.63, or .17p + .42c + .50a = 4.63

 

Our two equations are:

 

     p +      c +     a =    17

.17p + .42c + .50a = 4.63

 

This gives two equations in three unknowns, which we cannot solve. We have to use the fact the number of cashews equals the number of almonds to eliminate one of the unknowns.

 

The revised equations become:

 

     p +     c +      c =    17
.17p + .42c + .50c = 4.63

 

This gives us two equations in two unknowns, which has like terms to combine.

 

     p +    2c = 17
.17p + .92c =   4.63

 

I do not like to work with fractions or decimals, therefore, I will multiply the second equation by 100. This gives:

 

    p +   2c  =   17
17p + 92c  = 463

 

Multiply the first equation by -17 and add it to the second equation:

 

-17p  - 34c   = -289

 17p  + 92c  =  463

----------------------, subtract the equations gives:

           58c   =  174

 

Divide both sides by 58, gives c = 3. Since there are the same number of almonds as cashews, there are 3 almonds in the mix.

 

To find the number of peanuts, solve the original first equation substituting for c and a:

 

p + 3 + 3 = 17

 

Combine like terms:

 

p + 6 = 17

 

Subtract 6 from both sides gives:

 

p = 11

 

The mixture will have 11 ounces of peanuts, 3 ounces of cashews, and 3 ounces of almonds.

 

You can check your work by replacing the number of ounces for each nut in the second equation:

 

.17 * 11 + .42 * 3 + .50 * 3 = 4.63

 

1.87 + 1.26 + 1.50 = 4.63

 

4.63 = 4.63, the numbers check!

 

 

 

Answer 2

P = # ounces of peanuts

C = # ounces of cashews

A = # ounces of almonds

 

What we are given:

 

1. amount of cashews equals amount of almonds.  Then 

     C=A

 

2. Total amount in package is 17 oz. Then

     P + C + A = 17

   But, since C=A

   Then: P + A + A = 17, or P + 2 * A = 17

Then: P = 17 - 2*A

 

3. Cost per ounce of each is:

    17 cents for peanuts

    42 cents for cashews

    50 cents for almonds

 

4. Cost of 17 ounce package is 4.63 dollars = 463 cents

 

Then 17 * P + 42 * C + 50 * A = 463

since P = 17 - 2*A and A = C

Then 17 *(17 - 2*A)  + 42 * A + 50 * A = 463

17*17 - 34*A + A*(42+50) = 463

289  + A *(92 - 34) = 463

58*A = 463 - 289

58*A = 174

A = 174 / 58 = 3

So the amount of Almonds is 3 ounces

Since the amount of cashews equals the amount of almonds, the amount of cashews is 3 ounces

 

amount of peanuts = 17 - 2*amount of almonds = 17 - 2*3 = 11 ounces

 

So, to summarize

 

__11___oz of peanuts

__3___oz of cashews

__3___oz of almonds

 

Answer 3

Let's assign variables to the unknowns:

"P" for the number of oz. of peanuts

"C" for the number of oz. of cashews

"A" for the number of oz. of almonds

 

Now use the words to write equations...

"the amount of cashews is equal to the amount of almonds" can be written as:

(Equation 1) C = A

 

knowing the package is a "17 oz package" means:

(Equation 2) P + C + A = 17

 

knowing the package is "worth 4.63 dollars" "If peanuts, cashews, and almonds cost 17, 42, and 50 cents per oz respectively", then we write:

(Equation 3) .17P + .42C + .50A = 4.63

 

Now solve the set of 3 equations with 3 unknown variables.

1.)  Use Equation 1 to substitute in Equation 2:     P + A + A = 17

                                                                        P + 2A = 17

                                                                        P = 17 - 2A

2.) Now that you can replace 'C' with 'A' from Equation 1, and 'P' with '17-2A' from equation 2, then making these substitutions in Equation 3 will solve for A:

 -Making substitutions:                                     .17(17-2A) + .42A +.5A = 4.63

 -Distributing the .17:                                     2.89 - .34A + .42A + .5A = 4.63

 -Combining all the like terms with 'A's:           2.89 + .58A = 4.63

 -Subtracting 2.89 from both sides:                 .58A = 1.74

 -Dividing by .58 on both sides:                           A = 3

 

3.) Now if A=3, Equation 1 says C=3.

4.) Plug in A=3 and C=3 into Equation 2 to find:    P+3+3 = 17

                                                                             P+6 = 17

                                                                                 P = 11

 

5.) Check to make sure your answer is consistent by plugging in A=3, C=3 and P=11 into Equation 3:

                                                   .17(11) + .42(3) + .50(3) = 4.63

                                                    1.87    + 1.26    +  1.50  = 4.63

                                                                                4.63  = 4.63 ... which means we got the right answer!

Answer 4

Simultaneous system of three equations:

17p + 42c + 50a = 463  (prices in pennies)

p +c +a =17

a = c

 

Using equations (2) and (3), we can substitute a for c and get p+ 2a = 17, thus

p = 17 - 2a

 

using the a = c substitution, equation 1 is now 17p + 42a + 50a = 463, so

17p + 92a = 463

 

Using 17 - 2a to substitute for p, we have

17x(17 - 2a) +92a = 463

289 - 34a + 92a =463

58a = 174

a=3

 

since a = c then c = 3

 

p + (3) + (3) = 17, which resolves to 

p = 11

 

Double check the work:

17(11) + 42(3) + 50(3) = 463

187 + 126 + 150 + 463

 

Thus, 11 oz of peanuts, 3 oz of cashews, and 3 oz of almonds.