M45-18

Question

Anton, Beatrice, and Carl, each running at a constant rate, competed in a 30-kilometer race. The combined time Anton and Beatrice took to finish the race was exactly 3 hours longer than the time Carl took. Did Carl win the race?

(1) None of the three ran faster than 6 kilometers per hour.

(2) Anton finished before Beatrice.

Show Answer

Official Answer

A

(1) None of the three ran faster than 6 kilometers per hour.
 
(2) Anton finished before Beatrice.

Bunuel · Answer

Official Solution:

Anton, Beatrice, and Carl, each running at a constant rate, competed in a 30-kilometer race. The combined time Anton and Beatrice took to finish the race was exactly 3 hours longer than the time Carl took. Did Carl win the race? 
 
Let A, B, and C be the times, in hours, that Anton, Beatrice, and Carl took to finish the race, respectively. So, we are given that:
 
A + B = C + 3
 
and are asked to find whether C is the least of the three.
 
(1) None of the three ran faster than 6 kilometers per hour.
 
This implies that the minimum time anyone could take to complete the race is 30/6 = 5 hours. Thus, A + B ≥ 10.
 
Let's see if Carl could have won running at this maximum rate. If he ran at the fastest rate, he would finish in 5 hours, making A + B equal to C + 3 = 5 + 3 = 8 hours. However, this is impossible since we know that A + B ≥ 10. So, if Carl cannot win even running at the fastest rate, he could not have won at all. Sufficient.
 
(2) Anton finished before Beatrice.
 
This implies that Beatrice did not win. However, either Anton or Carl could have won. For example, if A = 1, B = 10, and C = 8, Anton won; but if A = 1.5, B = 2.5, and C = 1, then Carl won. Not sufficient.

Answer: A

hbhura · Answer

I like the solution - it’s helpful.

PocusFocus · Answer

I like the solution - it’s helpful.

Goldenfuture · Answer

To determine if Carl won the race, we need to know if his time (tc) was the shortest among the three.

Let the times taken by Anton, Beatrice, and Carl be ta, tb, and tc. The problem states: ta + tb = tc + 3

Carl wins if tc is less than ta AND tc is less than tb.

Analysis of the Equation
If we rewrite the equation to find tc: tc = ta + tb - 3

For Carl to win (tc < ta and tc < tb):

If tc < ta, then (ta + tb - 3) < ta. This simplifies to: tb < 3.

If tc < tb, then (ta + tb - 3) < tb. This simplifies to: ta < 3.

Therefore, Carl only wins if both Anton and Beatrice finished the race in less than 3 hours.

Evaluating Statement (1)
None of the three ran faster than 6 kilometers per hour. The race is 30 km. If the maximum speed is 6 km/h, then the minimum time any runner could take is: 30 km / 6 km/h = 5 hours

This means ta is 5 or more, and tb is 5 or more. Since ta and tb are both at least 5, it is impossible for them to be less than 3.

If ta = 5 and tb = 5, then tc = 5 + 5 - 3 = 7. In this case, Carl takes 7 hours, while the others took 5. Carl loses. As long as ta and tb are 5 or greater, tc will always be the largest number.

Statement (1) is SUFFICIENT because we can definitively say "No, Carl did not win."

siddharth_ · Answer

I don’t quite agree with the solution. --if Carl cannot win even running at the fastest rate, he could not have won at all.

IS this statement a 'must be true'? Carl's winning depends not only on his running rate but also on the other's rate. In this solution we are only saying that when C cannot attain it's min value. But can there be a scenario where C is not at it's min but can still be less than A or B (A and B both have much higher values).

Bunuel · Answer

Use this:

Your doubt is that Carl may still win even if his time is not the minimum possible time, because Anton and Beatrice could both have much higher times.

But that misses the given condition: A + B = C + 3.

If Carl won, then both Anton and Beatrice would have to take more time than Carl, so A > C and B > C. Hence, A + B > 2C.

But since A + B = C + 3, we would need C + 3 > 2C, or C < 3.

Statement (1) gives C ≥ 5, so Carl cannot win.

M45-18

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