sana.noor wrote:A and B ran a race of 480 m. In the first heat, A gives B a head start of 48 m and beats him by 1/10th of a minute. In the second heat, A gives B a head start of 144 m and is beaten by 1/30th of a minute. What is B's speed in m/s?
(A) 12
(B) 14
(C) 16
(D) 18
(E) 20
OA is A
An alternate approach is to plug in the answers, which represent B's rate.
Since B's rate is almost certainly a factor of 480 -- the value of the distance -- the correct answer is probably A, C, or E.
In the first heat, A beats B by (1/10)*60 = 6 seconds.
In the second heat, B beats A by (1/30)*60 = 2 seconds.
Answer choice C: B = 16 meters per second.
First heat:
Time for B = d/r = 480/16 = 30 seconds.
Headstart given to A = 48/16 = 3 seconds.
Since A beats B by 6 seconds, total time for A = 30-(3+6) = 21 seconds.
21 is not a factor of 480.
Eliminate C.
Answer choice A: B = 12 meters per second.
First heat:
Time for B = d/r = 480/12 = 40 seconds.
Headstart given to A = 48/12 = 4 seconds.
Since A beats B by 6 seconds, total time for A = 40-(4+6) = 30 seconds.
Rate for A = d/t = 480/30 = 16 meters per second.
Second heat:
Headstart given to B = 144/12 = 12 seconds.
From the first heat: Time for B - Time for A = 40-30 = 10 seconds.
Since B is given a 12-second headstart, and the time for A is only 10 seconds less than the time for B, B beats A by 12-10 = 2 seconds.
Success!
The correct answer is
A.
Algebracially:
Let A = A's rate and B = B's rate.
A gives B a head start of 48 meters:
After the head start, the distance traveled by B = 480-48 = 432 meters.
A travels the entire distance.
Time for A to travel 480 meters = 480/A.
Time for B to travel 432 meters = 432/B.
A wins by 1/10 of a minute = 6 seconds.
In other words, the difference between B's time and A's time is 6 seconds:
432/B - 480/A = 6.
A gives B a head start of 144 meters:
After the head start, the distance traveled by B = 480-144 = 336 meters.
A travels the entire distance.
Time for A to travel 480 meters = 480/A.
Time for B to travel 336 meters = 336/B.
B wins by 1/30 of a minute = 2 seconds.
In other words, the difference between A's time and B's time is 2 seconds:
480/A - 336/B = 2.
Adding the two equations, we get:
(432/B - 480/A) + (480/A - 336/B) = 6+2
96/B = 8
B = 12.
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