Before yesterday

He merchant adds a sale sign in his shop window to the showed pair of shoes in the morning: "Today by p% cheaper than yesterday. " After a while, however, he decided that the sign saying: "Today 62.5% cheaper than the day before yesterday". Determine the number p.

Correct result:

p =  25 %

Solution:

x⋅ (1−2p/100)⋅ (1−p/100)=x⋅ (1−62.5/100)  (1−2p/100)⋅ (1−p/100)=1−62.5/100  (100−2p)∗(100−p)=100∗(100−62.5)  (100−2p)⋅ (100−p)=100⋅ (100−62.5) 2p2−300p+6250=0  a=2;b=−300;c=6250 D=b2−4ac=3002−4⋅2⋅6250=40000 D>0  p1,2=−b±D2a=300±400004 p1,2=300±2004 p1,2=75±50 p1=125 p2=25   Factored form of the equation:  2(p−125)(p−25)=0  0<p<100  p=p2=25=25%x \cdot \ (1-2p/100) \cdot \ (1-p/100)=x \cdot \ (1 - 62.5/100) \ \\ \ \\ (1-2p/100) \cdot \ (1-p/100)=1 - 62.5/100 \ \\ \ \\ (100-2p)*(100-p)=100*(100 - 62.5) \ \\ \ \\ (100-2p) \cdot \ (100-p)=100 \cdot \ (100 - 62.5) \ \\ 2p^2 -300p +6250=0 \ \\ \ \\ a=2; b=-300; c=6250 \ \\ D=b^2 - 4ac=300^2 - 4\cdot 2 \cdot 6250=40000 \ \\ D>0 \ \\ \ \\ p_{1,2}=\dfrac{ -b \pm \sqrt{ D } }{ 2a }=\dfrac{ 300 \pm \sqrt{ 40000 } }{ 4 } \ \\ p_{1,2}=\dfrac{ 300 \pm 200 }{ 4 } \ \\ p_{1,2}=75 \pm 50 \ \\ p_{1}=125 \ \\ p_{2}=25 \ \\ \ \\ \text{ Factored form of the equation: } \ \\ 2 (p -125) (p -25)=0 \ \\ \ \\ 0<p<100 \ \\ \ \\ p=p_{2}=25=25 \%

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