Racing

manik11 wrote:Donovan and Michael are racing around a circular 400-meter track. If Donovan runs each lap in 45 seconds and Michael runs each lap in 40 seconds, how many laps will Michael have to complete in order to pass Donovan, assuming they start at the same time?

A) 8

B) 9

C) 10

D) 11

E) 12

To catch up to Donovan, Michael must complete 1 more lap than Donovan in the same amount of time.
We can PLUG IN THE ANSWERS, which represent the number of laps that must be completed by Michael.
When the correct answer choice is plugged in, Michael will complete 1 more lap than Donovan.
Since the options for Michael's laps are INTEGERS, the number of laps run by Donovan must also be an integer.

Answer choice D: 11 laps
Time for Michael to complete 11 laps = 11*40 = 440 seconds.
Number of laps completed by Donovan in 440 seconds = 440/45 = non-integer.
Eliminate D.

Answer choice B: 9 laps
Time for Michael to complete 9 laps = 9*40 = 360 seconds.
Number of laps completed by Donovan in 360 seconds = 360/45 = 8.
Michael's laps - Donovan's laps = 9-8 = 1.
Success!

The correct answer is B.

manik11 wrote:Donovan and Michael are racing around a circular 400-meter track. If Donovan runs each lap in 45 seconds and Michael runs each lap in 40 seconds, how many laps will Michael have to complete in order to pass Donovan, assuming they start at the same time?

A) 8

B) 9

C) 10

D) 11

E) 12

OA : B
Source : Veritas Prep

Backsolving may be the way to go here, but if you're a fiend for algebra:

Michael's rate: 400 meters/40 seconds = 10 meters/second
Assuming he runs for t seconds, his distance will be 10t meters

Donovan's rate: 400 meters/45 seconds = 80 meters/9 seconds
Assuming he runs for t seconds, his distance will be 80t/9 meters

We want Michael to have completed one more lap than Donovan, meaning he'll have covered 400 meters more than Donovan will have covered. So

10t = 80t/9 + 400
10t - 80t/9 = 400
90t/9 - 80t/9 = 400
10t/9 = 400
t = 360 seconds

If Michael is completing a lap every 40 seconds, then in 360 seconds, he'll have completed 360/40 = 9 laps. Answer is B

manik11 wrote:Donovan and Michael are racing around a circular 400-meter track. If Donovan runs each lap in 45 seconds and Michael runs each lap in 40 seconds, how many laps will Michael have to complete in order to pass Donovan, assuming they start at the same time?

A) 8

B) 9

C) 10

D) 11

E) 12

An alternate approach is to rephrase the rates in terms of the SAME AMOUNT OF TIME.
The LCM of the two times -- 45 and 40 -- is 360.

Donovan's rate = (1 lap)/(45 seconds) = (8 laps)/(360 seconds).
Michael's rate = (1 lap)/(40 seconds) = (9 laps)/(360 seconds).

Every 360 seconds:
Donovan travels 8 laps.
Michael travels 9 laps and thus catches up to and passes Donovan.

Thus, the distance traveled by Michael when he first passes Donovan = 9 laps.

The correct answer is B.

I have another approach which seems to be simpler. If we consider that each 40 seconds for Michael equals 1 Lap and each 45 second for Donovan is 1 lap. then if we consider that the distance both have to run to meet each other will be be the point that Donovan and Michael have run a number of laps we can have this equation:

D1 = Donovan's distance
D2 = Michael's distance
T1 = Number of laps for Donovan (consider each 45 seconds as 1 lap)
T2 = Number of Laps for Michael (consider each 40 seconds as 1 lap)
s1 = Donovan's rate (45seconds)
S2 = Michael's rate (40 seconds)

D1 = D2 --> T1 * 45 = T2 * 40 --> T1/T2 = 45/40 --> we simplify the fraction and come up with:

T1 = 8 units
T2 = 9 units (Michael's)