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:

$$\begin{gathered} x\cdot (1-2p\mathrm{/}100)\cdot (1-p\mathrm{/}100)=x\cdot (1-62.5\mathrm{/}100) \\ (1-2p\mathrm{/}100)\cdot (1-p\mathrm{/}100)=1-62.5\mathrm{/}100 \\ (100-2p)\ast (100-p)=100\ast (100-62.5) \\ (100-2p)\cdot (100-p)=100\cdot (100-62.5) \\ 2p^2-300p+6250=0 \\ a=2;b=-300;c=6250 \\ D=b^2-4ac=300^2-4\cdot 2\cdot 6250=40000 \\ D>0 \\ {p}_{1,2}={\displaystyle \frac{-b\pm \sqrt{D}}{2a}}={\displaystyle \frac{300\pm \sqrt{40000}}{4}} \\ {p}_{1,2}={\displaystyle \frac{300\pm 200}{4}} \\ {p}_{1,2}=75\pm 50 \\ {p}_1=125 \\ {p}_2=25 \\ \text{ Factored form of the equation: } \\ 2(p-125)(p-25)=0 \\ 0<p<100 \\ p=p_2=25=25\mathrm{\%} \end{gathered}$$

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