SOLUTION: Hi! I have this word problem that I've been trying to figure out for awhile. My professor wants us to solve it by completing the square, which I am comfortable doing, I just am n

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Question 1128574: Hi!
I have this word problem that I've been trying to figure out for awhile. My professor wants us to solve it by completing the square, which I am comfortable doing, I just am not sure how to set up the equation. The problem says:
If the price of eggs rises 10 cents per dozen, one will be able to get 2 dozen fewer eggs with $6.00 than was possible at the lower price. What is the lower price?
Thank you!
Found 3 solutions by ankor@dixie-net.com, MathLover1, ikleyn:
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
If the price of eggs rises 10 cents per dozen, one will be able to get 2 dozen fewer eggs with $6.00 than was possible at the lower price.
What is the lower price?
:
let p = the lower price
original doz - high price doz = 2 doz
6%2Fp - 6%2F%28%28p%2B.10%29%29 = 2
multiply equation by p(p+.10), cancel the denominators
6(p+.10) - 6p = 2p(p+.10)
6p + .6 - 6p = 2p^2 + .2p
Combine on the right to form a quadratic equation
0 = 2p^2 + .2p - .6
simplify, divide by 2
p^2 + .1p - .3 = 0
Using completing the square to solve
p^2 + .1p + ____ = .3
(.1/2)^2 = .0025
p^2 + .1p + .0025 = .3 + .0025
(p + .05)^2 = .3025
p + .05 = +/-sqrt%28.3025%29
the positive solution is what we want here
p = -.05 + .55
p = .50 a dozen is the lower price
:
:
We can check this by finding the no. of dozen at each pric
6/.50 = 12
6/.60 = 10

Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!


Let L be the lower price per dozen of eggs, in dollars (which is under the question).
Then the new price is %28L%2B0.1%29+dollars per dozen. (Notice 0.1+dollars+=+10+cents)

At the lower price, the buyer could buy 6%2FL dozens for 6+dollars.

At the new price, it can buy only 6%2F%28L%2B0.1%29 dozens.

The condition says that this difference is 2 dozens:
6%2FL+-+6%2F%28L%2B0.1%29=+2

To solve this equation, multiply both sides by+L%2A%28L%2B0.1%29. You will get
6%2A%28L%2B0.1%29+-+6L+=+2%2AL%2A%28L%2B0.1%29

Simplify it step by step:
6L+%2B+0.6+-+6L+=+2L%5E2+%2B+0.2L,
0.6+=+2L%5E2+%2B+0.2L
2L%5E2+%2B+0.2L+-+0.6+=+0 .............. divide by 2 both sides
L%5E2+%2B+0.1L++=+0.3............complete square:
%28L%5E2+%2B+0.1L%2Bb%5E2%29-b%5E2++=+0.3
coefficient a=1, 2ab=0.1=>2%2A1b=0.1=>b=0.1%2F2->b=0.05

%28L%5E2+%2B+0.1L%2B0.05%5E2%29-0.05%5E2++=+0.3
%28L+%2B0.05%29%5E2-0.0025++=+0.3
%28L+%2B0.05%29%5E2+=+0.3%2B0.0025+
%28L+%2B0.05%29%5E2+=+0.3025+
L+%2B0.05+=sqrt%28+0.3025%29+
L++=+%28-0.05%2B-sqrt%28+0.3025%29%29+
L++=+%28-0.05%2B-0.55%29+

so, roots are:
L≈-0.6
L≈0.5 -> we need only positive root because price cannot be negative number

Answer. The lower price was $0.5 per dozen of eggs.

Check:
6%2F0.5+=+12
6%2F%280.5%2B0.1%29+=+6%2F0.6+=+10
+++12+-+10+=+2 ! Correct !


Answer by ikleyn(52879) About Me  (Show Source):
You can put this solution on YOUR website!
.

For the solution of this and many other similar problems see the lessons
    - Had they sold . . .
    - Challenging word problems solved using quadratic equations
in this site.

These lessons were specially written with the goal to teach students on how to setup such complicated problems.

-----------------

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