Two turkeys weight 20 lbs together. The little one sells for

[KingAce]

PROBLEM: Together, two turkeys weigh 20 pounds. The little one sells for ten cents a pound more than the big one. You can buy the little one for $3.56, while the big one would cost a total of $12.64. Find the weight AND the cost per pound of each turkey.

Before I can find the cost per pound, I need to find the weight of each turkey. After thinking of how to go about solving the problem (40 min.), I could think of nothing but guess-and-check and simply guessed a weight for each turkey, found the price per pound of the big turkey, and used this plus $.10 times the weight I guessed for the small turkey to try and hope this matched my prediction. This may sound a bit confusing, and it is, as well as a completely wrong way to go about solving the problem. Is there an easier route I can take or hint you can give me to help me find the weight of the turkeys? Thanks.

 

[soroban]

Re: TURKEY

Hello, KingAce!!

Together, two turkeys weigh 20 pounds.
The little one sells for ten cents a pound more than the big one.
You can buy the little one for $3.56, while the big one would cost a total of $12.64.
Find the weight AND the cost per pound of each turkey.

Let \(\displaystyle S\) = weight of the small turkey.
Then \(\displaystyle 20\,-\,S\) = weight of the large turkey.

Let \(\displaystyle P\) = unit price of the large turkey (in cents).
Then \(\displaystyle P\,+\,10\) = unit price of the small turkey (in cents).

The small turkey weighs \(\displaystyle S\) pounds and sells for \(\displaystyle P+10\) cents/lb.
. . Its cost is: \(\displaystyle \,S(P\,+\,10)\:=\:356\;\) [1]

The large turkey weighs \(\displaystyle 20\,-\,S\) pounds and sells for \(\displaystyle P\) cents/lb.
. . Its cost is: \(\displaystyle \,(20\,-\,S)P \:=\:1264\;\) [2]


Solve [1] for \(\displaystyle S:\;\;S\:=\:\frac{356}{P\,+\,10}\)

Solve [2] for \(\displaystyle S:\;\;S\:=\:\frac{20P\,-\,1264}{P}\)

Equate: \(\displaystyle \,\frac{356}{P\,+\,10} \:=\:\frac{20P\,-\,1264}{P}\)

. . which simplifies to the quadratic: \(\displaystyle \,P^2\,-\,71P\,-\,632\:=\:0\)

. . which factors: \(\displaystyle \,(P\,-\,79)(P\,+\,8)\:=\:0\)

. . and has roots: \(\displaystyle \,P\:=\:79,\,-8\)


Hence, the large turkey costs \(\displaystyle 79\not{c}\text{ per pound}\)
. . and weighs \(\displaystyle \frac{1264}{79}\:=\:16\text{ pounds.}\)

And the small turkey costs \(\displaystyle 89\not{c}\text{ per pound}\)
. . and weighs \(\displaystyle \frac{356}{89} \:=\:4\text{ pounds.}\)

 

[KingAce]

Re: TURKEY

Equate: \(\displaystyle \,\frac{356}{P\,+\,10} \:=\:\frac{20P\,-\,1264}{P}\)

. . which simplifies to the quadratic: \(\displaystyle \,P^2\,-\,71P\,-\,632\:=\:0\)

. . which factors: \(\displaystyle \,(P\,-\,79)(P\,+\,8)\:=\:0\)

. . and has roots: \(\displaystyle \,P\:=\:79,\,-8\)

HOW DO I FACTOR THE QUADRATIC AND FIND ITS ROOTS? THANKS.

 

[galactus]

Re: TURKEY

Equate: \(\displaystyle \,\frac{356}{P\,+\,10} \:=\:\frac{20P\,-\,1264}{P}\)

. . which simplifies to the quadratic: \(\displaystyle \,P^2\,-\,71P\,-\,632\:=\:0\)

. . which factors: \(\displaystyle \,(P\,-\,79)(P\,+\,8)\:=\:0\)

. . and has roots: \(\displaystyle \,P\:=\:79,\,-8\)

HOW DO I FACTOR THE QUADRATIC AND FIND ITS ROOTS? THANKS.

What 2 numbers when added equal 71 and when multiplied equal -632?.
79 and -8.

 

[KingAce]

Thanks! But what is the significance of the -8 then? It's not used in the final equations of the problem.

 

[jonboy]

Thanks! But what is the significance of the -8 then? It's not used in the final equations of the problem.

There is no significance to it ; Whenever you solve a quadratic you always get two roots, and sometimes you have to ignore a negative one since it is not applicable to this type of problem.