lean_workbook_plus_80145

Let the escalater move at $A$ stairs in a time unit, person $A$ move $2B$ stairs in that same time unit and person $Z$ move $B$ stairs in the time unit. \n\nSo in the case of person $A$ , the escalator moves $n-27$ stairs in the same time that person $A$ moves $27$ stairs. Therefore, we can set up the equation that: \n\n $\frac{n-27}{A}=\frac{27}{2B}$ \n\nSimilarly, for person $Z$ , the escalator moves $n-18$ as person $Z$ moves $18$ stairs. Thus: \n\n $\frac{n-18}{A}=\frac{18}{B}$ \n\nDividing the second by the first gives the following. \n\n $\frac{n-18}{n-27}=\frac{18}{B} \cdot \frac{2B}{27}= \frac{4}{3}$ \n\nThis gives that, \n\n $3(n-18)=4(n-27)$ \n $n=4\cdot27 - 3\cdot18=54$ \n\nTherefore the answer is $n=\boxed{54}$ .