I used the d=rt to figure it out.. still I cannot find a simple way to solve this problem..
yolanda and yoko ran in a 100-yd dash. when yolanda crossed the finish line, yoko was 10-yd behidn her. The girls then repeated the race, with yolanda
10-yd behind the starting line.
A- if each girl ran at the same rate as before, who won the race?
B- by how many yards?
C- Assuming they ran at the same rate as before, how far behind the starting line should yolanda be in order for the two to finish in a tie?
Okay well I'll just say it took them t amount of time (in seconds) to finish. So Yolanda ran at 100/t yd/second and Yoko 90/t.
Now Yolanda again runs at 100/t for a distance of 110 yards and Yoko at 90/t for a distance of 100 yards.
Use r x t = d so we have d / r = t.
110 / (100/t) = (11/10)t for Yolanda's time and (10/9)t for Yoko's time. In other words it takes Yolanda (11/10)t seconds to run 110 yards while it takes Yoko (10/9)t seconds to run 100 yards.
11/10 = 1.1 while 10/9 = 1.1111...
So it takes Yoko (1/90)t more seconds to finish the race than it takes Yolanda. That means Yolanda wins since it takes her less time to run 110 yards than it takes Yoko to run 100 yards.
How many yards does she win by? Well if Yoko needs (1/90)t more seconds to complete her race and she is running at 90/t rate, use r x t to get your distance. You get (1/90)t * 90/t = 1 yard. She has 1 yard to go.
Yolanda won the second race by 1 yard.
How far behind the starting line should Yolanda be to have a tie?
Well it should take them the same time to finish. We're keeping Yoko at 100 yards and she's running at 90/t rate. So it takes her 100 / (90/t) = (10/9)t seconds to finish the race. So we want Yolanda to take (10/9)t seconds to finish the race. We know her rate = 100/t. So we use r x t to get the distance. You get (100/t) * (10/9)t = (1000/9) yards which is 111.111... yards, or 111 and 1/9 yards. Since a yard is 36 inches, we can say that she needs to start at exactly 111 yards and 4 inches.
====
That's keeping things non-specific. What if you chose an actual time that it took them to finish? Let's say it takes them 10 seconds to finish.
So you have 100 yards in 10 seconds for Yolanda = 10yd/s and 90 yards in 10 seconds for Yoko = 9yd/s.
Now Yolanda needs to run 110 yards and Yoko needs to run 100 yards at the same rates as before.
So Yolanda needs 110yd/(10yd/s) = 11 seconds and Yoko needs 100yd/(9yd/s) = 100/9 seconds = 11 and 1/9 seconds. So Yoko needs an extra 1/9 second to finish up after Yolanda wins. Well in 1/9 second she can go (9yd/s)[(1/9)s] = 1 yard. So Yolanda won by a yard.
How to make them tie? Well they need to finish in the same time. It takes Yoko 11 and 1/9 seconds (100/9 seconds) to finish so it should take Yolanda that same amount of time. So she needs to be at a distance of (10yd/s)[(100/9)s] = 1000/9 yards = 111 and 1/9 yards = 111 yards and 4 inches.
====
Which is the better method to use? I prefer the first because it's as non-specific as you can get, which means you KNOW your answer is correct. Plus it's a nice way to test your math skills, not only solving a problem with variables but setting it up.
The advantages of the second method are obvious. You have clear cut numbers. No variables. You can choose numbers which make the problem easier to solve (how much harder would it have been if I told you they finished in 13 seconds?). As such it probably takes less time to do. Of course when you choose a random time that it takes them to finish, it might leave you wondering in the end if your answer truly is right, or did your choice of numbers affect the answer you got?
I don't think there's really a shorter way to go about doing this. Any 'short cuts' you take might end up being just as long as this way.
==
Edit: A different way to do part 3 would be to say that since Yoko needs another (1/90)t seconds to finish up, then what if it took Yolanda that much more time to finish her race? How many extra yards could she run in (1/90)t seconds?
Her rate is 100/t and the time is (t/90) so she can run (10/9) yards in the time it would take Yoko to finish that race. If we just add that on to her distance of 110 yards, we get 111 and 1/9 yards = our answer.
That might save you a small amount of time.
