+130515 This is one I pondered on for a little while.....I finally solved it with the same answer as Headingnorth....However, I saw it in a slghtly different manner....
The first assumption to be made is that both people step on consecutive steps.....they don't "skip" any steps.
Notice that the man steps on the first step at the bottom, and by the time he gets to the top.....he is on the top step of the escalator. Thus......there are 20 consecutive steps from the top of the escalator back to the first step that he stepped on. And between this step and the bottom of the escalator, we can let this number of steps = s. So...the total steps on the escalator = 20 + s
And the same situation develops for his wife. When she gets to the top of the escalator, there are 16 consecutive steps from the top of the escalator back to the first step she stepped on. But notce.......if "s" steps were fed by the escalator in 60 seconds from the bottom of the escalator to the man's first step [20 steps from the top], then (6/5)s steps must have been fed by the escalator to the woman's first step [16 steps from the top]. This is because it takes her 6/5 times as long [72 seconds / 60 seconds] to get from the bottom to the top as it does him.
So....from her point of view....there must be (6/5)s + 16 steps on the escalator......
So, since the number of steps are equal in each case.....we have....
s + 20 = (6/5)s + 16 subtract 16 and s from both sides
4 = (1/5)s multiply by 5 on both sides
20 = s
So there are, in the man's case .... s + 20 = 20 + 20 = 40 steps showing on the escalator.
Check that ... (6/5)(20) + 16 = 24 + 16 = 40 steps in the woman's case, too......!!!!!
+130515 5x^3 - 8x^2 - 5x + 2 = 0 ..this doesn't appear to factor....let's use the "Rational Zeroes" Theorem to find a root - if possible...
The possible roots are ± 1, ± 2 and ± 2/5
I see that 2 is a root......let's use synthetic division to find the next polynomial
2 5 -8 -5 2
10 4 -2
5 2 -1 0
And the next polynomial is
5x^2 + 2x -1 = 0
This will not factor...using the on-site solver [and the Quadratic Formula] ...the other two roots are :
$${\mathtt{5}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,-\,}}{\mathtt{1}} = {\mathtt{0}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\frac{\left({\sqrt{{\mathtt{6}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}}\right)}{{\mathtt{5}}}}\\
{\mathtt{x}} = {\frac{\left({\sqrt{{\mathtt{6}}}}{\mathtt{\,-\,}}{\mathtt{1}}\right)}{{\mathtt{5}}}}\\
\end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = -{\mathtt{0.689\: \!897\: \!948\: \!556\: \!635\: \!6}}\\
{\mathtt{x}} = {\mathtt{0.289\: \!897\: \!948\: \!556\: \!635\: \!6}}\\
\end{array} \right\}$$
Here's a graph......https://www.desmos.com/calculator/hpqotand0v
Mindblown.
