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01 Jul 2025, 09:01
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We want to know if Carl’s time is the shortest of the three.
Let their race times be A, B, and C (in hours), with corresponding constant speeds over 30 km. We know:
A+B=C+3
Statement (1): None ran faster than 6 km/h
If speed ≤ 6 km/h, then time ≥ 30/6 = 5 hours for each runner. So:
A,B,C≥5
Given their sum relation:
A+B=C+3
(A+B)≥10
C+3≥10
C≥7
So:
Could Carl be the smallest?
Statement (2): Anton finished before Beatrice
We know:
A<B
and
A+B=C+3
C=A+B−3
Without further constraints, can Carl’s time be smallest?
Example 1: Let A=1, B=2 then C=0. Carl wins.
Example 2: Let A=4, B=5 → C=6. Carl loses again.
So (2) is insufficient.
Hence, A) Statement (1) alone is sufficient.
Let their race times be A, B, and C (in hours), with corresponding constant speeds over 30 km. We know:
A+B=C+3
Statement (1): None ran faster than 6 km/h
If speed ≤ 6 km/h, then time ≥ 30/6 = 5 hours for each runner. So:
A,B,C≥5
Given their sum relation:
A+B=C+3
(A+B)≥10
C+3≥10
C≥7
So:
- C≥7C
- A+B=C+3 => A+B≥10
Could Carl be the smallest?
- If C=7C = 7C=7, then A+B=10. That allows one of A,B to be < 5?
No, they both must be ≥ 5 from the same statement, so that forces A=B=5. Then times are: A=5, B=5, C=7. Carl is slower → he doesn’t win. - If C>7, then A+B>10. But with each ≥5, that still splits into at least {5, >5}. So both A and B ≥ 5. And C is even larger. Thus Carl's time is always greater than or equal to both A and B and Carl is never the winner.
Statement (2): Anton finished before Beatrice
We know:
A<B
and
A+B=C+3
C=A+B−3
Without further constraints, can Carl’s time be smallest?
Example 1: Let A=1, B=2 then C=0. Carl wins.
Example 2: Let A=4, B=5 → C=6. Carl loses again.
So (2) is insufficient.
Hence, A) Statement (1) alone is sufficient.
01 Jul 2025, 09:01
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Statement (1) says no one ran faster than 6 km/h, so no one finished in less than 5 hours. That means Anton and Beatrice each took at least 5 hours, so together they took at least 10 hours. That makes Carl’s time at least 7 hours, meaning he was slower than both—so he definitely didn’t win. Statement (1) is sufficient. Statement (2) only says Anton finished before Beatrice, which tells us nothing about Carl’s time, so it’s not sufficient.
01 Jul 2025, 09:08
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We are asked:
Did Carl win the 30-kilometer race?
Meaning: Was Carl’s time less than both Anton’s and Beatrice’s?
We are told:
Anton and Beatrice together took 3 more combined hours than Carl did:
TA + TB = TC +3
We are to determine if TC < TA and TC < TB
Let’s evaluate the statements.
Statement (1):
None of the three ran faster than 6 km/h.
So:
Their minimum possible time for 30 km is 30/6 =5 hours
(since max speed = 6 km/h → minimum time = 5 hours)
So each person took at least 5 hours to finish.
Let’s test values. Suppose Carl ran at 6 km/h:
TC =5 hours
Then
TA + TB =5+3=8 hours
Average = 4 hours per person → But that contradicts the minimum of 5 hours per person.
So Anton and Beatrice must have gone slower than 6 km/h, meaning longer time.
Try:
Carl:
TC =6 hrs
-> TA + TB =9 hrs
-> So one of A or B could’ve run in 4, one in 5. But 4 is not allowed (since 30 ÷ 4 = 7.5 km/h > 6)
-> So you’re forced to pick realistic values satisfying:
TA ≥5,
TB ≥5, and TA + TB = TC +3
You can find multiple such values, but you cannot determine definitively whether Carl’s time is less than both A and B.
Example:
Carl: 6 hours
Anton: 5 hours
Beatrice: 4 hours -> But Not allowed (>6 km/h)
So that combination invalid
Try:
Carl: 6
Anton: 5.5
Beatrice: 3.5 → 3.5 hrs = 8.57 km/h
Eventually you'll find that while certain combinations are invalid, you still can’t guarantee that Carl was faster than both.
Thus, Statement (1) is not sufficient.
Statement (2):
Anton finished before Beatrice
So, TA < TB
Combined with original info:
TA + TB = TC +3
We now know:
TA < TB
So , TB > (TC+ 3)/2
But we still don’t know if
TC < TA and TC < TB
Counterexample:
Suppose Carl took 6 hours
-> A + B = 9 hours
Let A = 4, B = 5 → A < B , but Carl = 6 -> He is not faster than A -> So Carl didn’t win.
Now suppose:
Carl = 5
-> A + B = 8
Let A = 3.9, B = 4.1 -> A < B , but 3.9 hr = ~7.7 km/h but violates rate limit
So again, we don’t get definitive info from this clue.
Thus, Statement (2) is not sufficient.
Combine (1) and (2):
We now know:
Speeds ≤ 6 km/h -> minimum time per runner = 5 hours
TA + TB = TC +3
TA <TB
Let’s test if this gives a definite conclusion.
Try:
Carl = 5 hours -> then A + B = 8 hours
Try A = 3.9, B = 4.1 -> B > A
But A = 3.9 hours → speed = 30 ÷ 3.9 ≈ 7.7 km/h ->But violates (1)
Try A = 5.1, B = 2.9 -> B faster -> invalid
Try A = 4.5, B = 3.5 -> both under 5 -> invalid
Eventually you find that only valid combinations under speed constraints do not guarantee that Carl was faster than both A and B. It’s possible for Carl to be slower than A and faster than B — still satisfying all conditions.
Therefore:
Even together, statements (1) and (2) are NOT sufficient.
Final Answer:
(E) Statements (1) and (2) TOGETHER are not sufficient.
Did Carl win the 30-kilometer race?
Meaning: Was Carl’s time less than both Anton’s and Beatrice’s?
We are told:
Anton and Beatrice together took 3 more combined hours than Carl did:
TA + TB = TC +3
We are to determine if TC < TA and TC < TB
Let’s evaluate the statements.
Statement (1):
None of the three ran faster than 6 km/h.
So:
Their minimum possible time for 30 km is 30/6 =5 hours
(since max speed = 6 km/h → minimum time = 5 hours)
So each person took at least 5 hours to finish.
Let’s test values. Suppose Carl ran at 6 km/h:
TC =5 hours
Then
TA + TB =5+3=8 hours
Average = 4 hours per person → But that contradicts the minimum of 5 hours per person.
So Anton and Beatrice must have gone slower than 6 km/h, meaning longer time.
Try:
Carl:
TC =6 hrs
-> TA + TB =9 hrs
-> So one of A or B could’ve run in 4, one in 5. But 4 is not allowed (since 30 ÷ 4 = 7.5 km/h > 6)
-> So you’re forced to pick realistic values satisfying:
TA ≥5,
TB ≥5, and TA + TB = TC +3
You can find multiple such values, but you cannot determine definitively whether Carl’s time is less than both A and B.
Example:
Carl: 6 hours
Anton: 5 hours
Beatrice: 4 hours -> But Not allowed (>6 km/h)
So that combination invalid
Try:
Carl: 6
Anton: 5.5
Beatrice: 3.5 → 3.5 hrs = 8.57 km/h
Eventually you'll find that while certain combinations are invalid, you still can’t guarantee that Carl was faster than both.
Thus, Statement (1) is not sufficient.
Statement (2):
Anton finished before Beatrice
So, TA < TB
Combined with original info:
TA + TB = TC +3
We now know:
TA < TB
So , TB > (TC+ 3)/2
But we still don’t know if
TC < TA and TC < TB
Counterexample:
Suppose Carl took 6 hours
-> A + B = 9 hours
Let A = 4, B = 5 → A < B , but Carl = 6 -> He is not faster than A -> So Carl didn’t win.
Now suppose:
Carl = 5
-> A + B = 8
Let A = 3.9, B = 4.1 -> A < B , but 3.9 hr = ~7.7 km/h but violates rate limit
So again, we don’t get definitive info from this clue.
Thus, Statement (2) is not sufficient.
Combine (1) and (2):
We now know:
Speeds ≤ 6 km/h -> minimum time per runner = 5 hours
TA + TB = TC +3
TA <TB
Let’s test if this gives a definite conclusion.
Try:
Carl = 5 hours -> then A + B = 8 hours
Try A = 3.9, B = 4.1 -> B > A
But A = 3.9 hours → speed = 30 ÷ 3.9 ≈ 7.7 km/h ->But violates (1)
Try A = 5.1, B = 2.9 -> B faster -> invalid
Try A = 4.5, B = 3.5 -> both under 5 -> invalid
Eventually you find that only valid combinations under speed constraints do not guarantee that Carl was faster than both A and B. It’s possible for Carl to be slower than A and faster than B — still satisfying all conditions.
Therefore:
Even together, statements (1) and (2) are NOT sufficient.
Final Answer:
(E) Statements (1) and (2) TOGETHER are not sufficient.
