runners dropping out of a marathon race

[Nekkamath]

During the Great Marathon, 50% of the runners dropped out before reaching he first water station. By the second water station, 1/3 of the remaining runners dropped out. At the third water station, 25% of the remaining runners dropped out. By the fourth water station, 90% of the remaining runners dropped out. If the remaining six runners finished the marathon, how many runners started the race?

I started with the 6 runners. 60 minus 90% equals 54. 60 minus 54 is 6. So at the 4th station, the number of runners is 60. 80 minus 25% = 20 dropped out leaving 60. so there were 80 runners at the 3rd station. 120 minus 1/3 runners = 40 runners dropped out leaving 80. so at the 2nd water station there were 120. And if half the runners dropped out by the 1st water station, the beginning number is 120 times 2 = 240.

240 runners started the race.

 

[stapel]

During the Great Marathon, 50% of the runners dropped out before reaching he first water station. By the second water station, 1/3 of the remaining runners dropped out. At the third water station, 25% of the remaining runners dropped out. By the fourth water station, 90% of the remaining runners dropped out. If the remaining six runners finished the marathon, how many runners started the race?

i) After the first water station, only 1/2 of the runners remain.

ii) After the second water station, 2/3 of the above 1/2 remain; that is, (2/3)(1/2) = 1/3 of the original number remain.

iii) After the third water station, 3/4 of the above 1/3 remain; that is, (3/4)(1/3) = 1/4 of the original number remain.

iv) After the fourth water station, 1/10 of the above 1/4 remain; that is, (1/10)(1/4) = 1/40 of the original number remain.

v) Since 1/40 of the total is 6, then the total must be 6(40) = 240.

The answer to any "solving" problem may be checked by plugging it back into the original exercise:

i) (1/2)(240) = 120 dropped out, so 120 remain.

ii) (1/3)(120) = 40 dropped out, so 80 remain.

iii) (1/4)(80) = 20 dropped out, so 60 remain.

iv) (9/10)(60) = 54 dropped out, so 6 remain.

The solution checks!

Eliz.