Before yesterday

The merchant adds a sale sign in his shop window to the pair of shoes shown 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 answer:

p =  25 %

Step-by-step explanation:

$$\begin{gathered} x\cdot (1-2p/100)\cdot (1-p/100)=x\cdot (1-62.5/100) \\ (1-2p/100)\cdot (1-p/100)=1-62.5/100 \\ (100-2p)\cdot (100-p)=100\cdot (100-62.5) \\ (100-2p)\cdot (100-p)=100\cdot (100-62.5) \\ 2p^2-300p+6250=0 \\ 2...\text{ prime number} \\ 300=2^2\cdot 3\cdot 5^2 \\ 6250=2\cdot 5^5 \\ \text{GCD}(2,300,6250)=2 \\ {p}^2-150p+3125=0 \\ a=1;b=-150;c=3125 \\ D=b^2-4ac=150^2-4\cdot 1\cdot 3125=10000 \\ D>0 \\ {p}_{1,2}=\frac{-b\pm \sqrt{D}}{2a}=\frac{150\pm \sqrt{10000}}{2} \\ {p}_{1,2}=\frac{150\pm 100}{2} \\ {p}_{1,2}=75\pm 50 \\ {p}_1=125 \\ {p}_2=25 \\ 0<p<100 \\ p=p_2=25=25\% \end{gathered}$$

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