applicaions of second-degree inequalities

[anas]

If The Price of a large candy bar rose by 10 cents, a buyer would receive 2 fewer candy bars for $6.00 than she does at the current price. What is the current price?

 

[arthur ohlsten]

let the current price be P and number of candy bars she receives be N
eq1 N=600/P P>0

eq2 N+2=600/[p-10] but N=600/P
600/P+2=600/[P-10] multiply both sides of equal sign with P
600+2P =600P/[p-10] multiply both sides by p-10
600[P-10] + 2P[P-10]=600P
600P-6000+2P^2-20P=600P
2P^2-20P-6000=0
P^2-10P-3000=0
P=10 +/-[100+12000]^1/2 all over 2
P=10+/- [12100]^12 all over 2
P=10+/- 110 all over 2
P=60 or P=-50 impossible
P=60 cents answer
Arthur

 

[soroban]

Hello, anas!

If the price of a large candy bar rose by 10 cents,
a buyer would receive 2 fewer candy bars for $6.00 than she does at the current price.
What is the current price?

Let \(\displaystyle P\) = current unit price (in cents).
Let \(\displaystyle N\) = number of bars bought now.
Then: \(\displaystyle \,NP\,=\,600\;\;\Rightarrow\;\;N\:=\:\frac{600}{P}\;\) [1]

The new unit price is: \(\displaystyle P\,+\,10\)
and the new number of bars is: \(\displaystyle N\,-\,2\)
Then: \(\displaystyle \,(P\,+\,10)(N\,-\,2)\:=\:600\;\) [2]

Substitute [1] into [2]: \(\displaystyle \,\left(\frac{600}{P}\,-\,2\right)(P\,+\,10)\:=\:600\)

This simplifies to: \(\displaystyle \,P^2\,+\,10P\,-\,3000\:=\:0\)

. . which factors: \(\displaystyle \,(P\,-\,50)(p\,+\,60)\:=\:0\)

. . and has roots: \(\displaystyle \,P \:=\:50,\:\not{-}\not{6}\not{0}\)


Therefore, the current price is 50¢.