Question about a word problem

An air rescue plane averages 300 miles per hour in still air. It carries enough fuel for 5 hours of flying time. If, upon takeoff it encounters a head wind of 30 mi/hr and the wind remains constant, how far can the plane fly and then return safely?

Let $$t =$$ flying time of the leg with the wind.

Let $$(5 - t) =$$ flying time of the leg against the wind

Let $$d =$$ distance with $$d = rate \times time$$:

against the wind $$d = (300 - 30)(5 - t)$$

with the wind $$d = (300 + 30) t$$

Thus: $$270(5 - t) = 330t$$ which yields the correct answer for $$t$$. However if you switch the time around and

Let $$t =$$ flying time of the leg against the wind.

Let $$(5 - t) =$$ flying time of the leg with the wind.

so that:$$270t = 330(5 - t)$$ You don't get the correct answer. Why? It doesn't seem that it should matter. Thanks

• Both methods give a distance of $\frac {1485}2$, there is no significant difference between them. In each case, you fly out for a total time of $\frac {11}4$.
– lulu
Oct 1, 2022 at 23:05
• Note: of course $t$ changes... but that's to be expected. In both cases, the flying time out is $\frac {11}4$. What variable you assign to that is up to you.
– lulu
Oct 1, 2022 at 23:06
• @JohnDouma: The question asks "how far can the plane fly and then return safely?" This does not remain the same with wind, so idk what you mean by saying it is a "trick question" Oct 2, 2022 at 7:30
• @lulu yes, that's what I was missing. The definition of $t$ changes so the results are different. Time with the wind is $2.25$ hours while against the wind it's $2.75$ hours. For some reason I was thinking the results had to be the same. Oct 3, 2022 at 21:13

Let $$f$$ = fraction of total flying time of the leg with the wind.
then $$(1-f)$$ = fraction of total time flying time of the leg against the wind
Then $$330(f) = 270(1-f) \Rightarrow f = \frac{9}{20}$$
Total distance $$5\times \left(\frac{9}{20}\cdot 330 +\frac{11}{20}\cdot 270\right) = 1485$$ miles
Distance out = distance in $$= 742.5$$ miles