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But if B is beaten by 19 seconds, that means when A reaches the finishing line, B is still 19 seconds behind. So 19 seconds should be added if we want to equate both sides, shouldn't we?
The way I understand it is, if X beats Y by 19 seconds, that means that X is faster and Y is slower. Hence X reaches finishing line in less than (19 seconds) than Y. That means that Y will still travel 19 MORE seconds after X has already reached the finishing line. So if you are comparing time, you need to take out 19 seconds from the slower guy to equal them up.
If you look at the statement "In a 1000 m race, A gives B a start of 40 m and beats him by 19 seconds". That means that A has beaten B in the race. That means that both A and B NEED TO COMPLETE the race if one of them has to be beaten.
If I am not wrong, your interpretation of the statement is, at the time when A reaches finishing line, B is "19 seconds behind A" and race is over. But if you interpret it like that, that means that B still has not completed the race. If B has not completed the race yet, he cannot be beaten. He can only be beaten when he actually completes the race (atleast in our situation )
I really thought about this, and this is the conclusion I came to. If A beats B in time, than we need to take time out of B to equal their TIMES. Alternatively, if A beats B is distance, than we need to add distance to B to equal their DISTANCES.
That's a really interesting way to look at it. I got a little confused when the question started going into how much time B is beaten. I just kept thinking whether to add or not. I guess this method is a great way to remember !