2014年AMC12竞赛A第11题 2014年AMC10竞赛A第15题
$$2014-AMC10A-15$$  David drives from his home to the airport to catch a flight. He drives $$35$$ miles in the first hour, but realizes that he will be $$1$$ hour late if he continues at this speed. He increases his speed by $$15$$ miles per hour for the rest of the way to the airport and arrives $$30$$ minutes early. How many miles is the airport from his home?  大卫从他家开车去机场赶飞机。 他在第一个小时内行驶了 35 英里，但他意识到如果他继续保持这个速度，他将迟到 1小时。 在前往机场的剩余路程中，他将速度提高了每小时 15英里，并提前了 30分钟。 机场离他家多少英里？
课内体系->知识点->等式与不等式->等式->方程组的解集 美国AMC10/12->Knowledge Point->Algebra->Application->Equation Word Problems
Note that he drives at $$50$$ miles per hour after the first hour and continues doing so until he arrives. Let $$d$$ be the distance still needed to travel after the frst $$1$$ hour.~ We have that $$\dfrac{d}{50}+1.5= \dfrac{d}{35}$$, ~where the $$1.5$$ comes from $$1$$ hour late decreased to $$0.5$$ hours early. Simplifying gives $$7d+525=10 d$$, or $$d=175$$. Now, we must add an extra $$35$$ miles traveled in the frst hour, giving a total or $$(\rm C) 210$$ miles.  Instead of spending time thinking about how one can set up an equation to solve the problem, one can simply start checking the answer choices. Quickly checking, we know that neither choice $$(\rm A)$$ or choice $$(\rm B)$$ work, but $$(\rm C)$$ does. We can verify as follows. After l hour at $$35 \rm mph$$, David has $$175$$ miles left. This then takes him $$3.5$$ hours at $$50 \rm mph$$. But $$210/35=6$$ hours. Since $$1+3.5=4.5$$ hours is $$1.5$$ hours less than $$6$$ our anwer is $$(\rm C)210$$.  Let us call the total number of miles after David changes speed $$x$$. We realize that if David is driving $$50$$ miles an hour, and he arrives $$30$$ minutes early, $$x≡25 (\rm mod 50)$$ The only solution that fits the descripion is $$210$$. 