Eight figure skaters -four women: Fiona, Gloria, Heidi, and Jill; and four men: Ravi, Shigeru, Toby, and Vernon-will participate in a one-day skating exhibition consisting of four consecutively performed sets - set 1 through set 4. Each set will be performed in exactly one pair of skaters, one man and one woman. Each skater will perform by exactly one of the sets, subject to the following constraints:
Ravi skates in an earlier set than Vernon does.
Fiona skates in either set 1 or set 4.
Jill does not skate with Toby.
Shigeru skates with either Fiona or Gloria.
Which of the following could be the pairs of skaters who skate in each set, from set 1 through set 4?
Show Hint
- Set 1: Fiona, Ravi | Set 2: Jill, Toby | Set 3: Gloria, Shigeru | Set 4: Heidi, Vernon
- Set 1: Gloria, Shigeru | Set 2: Heidi, Ravi | Set 3: Fiona, Toby | Set 4: Jill, Vernon
- Set 1: Heidi, Shigeru | Set 2: Gloria, Ravi | Set 3: Jill, Vernon | Set 4: Fiona, Toby
- Set 1: Heidi, Toby | Set 2: Gloria, Shigeru | Set 3: Jill, Ravi | Set 4: Fiona, Vernon
- Set 1: Jill, Vernon | Set 2: Heidi, Ravi | Set 3: Gloria, Shigeru | Set 4: Fiona, Toby
The Correct Option is C
Solution and Explanation
This is a "could be true" or "acceptable arrangement" question. We must check each option against the rules until we find one that is valid.
Step 2: Detailed Explanation:
[(A)] (Fiona, Ravi), (Jill, Toby), (Gloria, Shigeru), (Heidi, Vernon).
Violates Rule 3 (Jill cannot skate with Toby).
[(B)] (Gloria, Shigeru), (Heidi, Ravi), (Fiona, Toby), (Jill, Vernon).
Violates Rule 2 (Fiona must be in set 1 or 4, but she is in set 3).
[(C)] (Heidi, Shigeru), (Gloria, Ravi), (Jill, Vernon), (Fiona, Toby).
F in 4 ✓, R before V ✓, J with V ✓, but Rule 4 is violated (Shigeru must be with Fiona or Gloria, not Heidi).
[(D)] (Heidi, Toby), (Gloria, Shigeru), (Jill, Ravi), (Fiona, Vernon).
- Rule 1: Ravi(3) before Vernon(4).
- Rule 2: Fiona in set 4.
- Rule 3: Jill is with Ravi, not Toby.
- Rule 4: Shigeru with Gloria.
All conditions satisfied.
\fbox{% \parbox{\dimexpr\linewidth-2\fboxsep-2\fboxrule}{ Final Answer: Option (D) is the only valid arrangement. }} % Correct Answer Correct Answer: (D) Set 1: Heidi, Toby | Set 2: Gloria, Shigeru | Set 3: Jill, Ravi | Set 4: Fiona, Vernon % Solution Solution:
Step 1: Checking the Rules against Option (D)
The proposed schedule is: - Set 1: Heidi and Toby - Set 2: Gloria and Shigeru - Set 3: Jill and Ravi - Set 4: Fiona and Vernon
Let's check each rule: 1. Ravi skates in an earlier set than Vernon does (R \textless V): Ravi is in set 3, Vernon is in set 4. \(3 \textless 4\), so this rule is satisfied. 2. Fiona skates in either set 1 or set 4: Fiona is in set 4. This rule is satisfied. 3. Jill does not skate with Toby: Jill skates with Ravi in set 3. This rule is satisfied. 4. Shigeru skates with either Fiona or Gloria: Shigeru skates with Gloria in set 2. This rule is satisfied.
Step 2: Conclusion
Since the arrangement in option (D) satisfies all four constraints, it is a possible schedule. Other options can be eliminated as they violate one or more rules (e.g., (A) violates Rule 3, (B) violates Rule 2, (C) violates Rule 4).
If Gloria skates with Toby in set 1, which of the following must be true?
Show Hint
- Vernon skates in set 2.
- Shigeru skates in set 4.
- Ravi skates in set 3.
- Jill skates in set 4.
- Heidi skates in set 3.
The Correct Option is B
Solution and Explanation
Step 1: Apply the Condition
We are given that Gloria and Toby are paired in Set 1.
- Set 1: (G, T)
Step 2: Make Deductions
- From the condition: Set 1 is $(G, T)$.
- By Rule 4, $S$ must skate with $F$.
- By Rule 2, $F$ must be in Set 1 or 4. Set 1 is already taken, so $(S, F)$ must be in Set 4.
- Remaining men: $R, V$. By Rule 1 ($R < V$), Ravi must be in Set 2 and Vernon in Set 3.
- Remaining women: $H, J$. They must pair with $R$ and $V$, giving two possible assignments: \[ \text{Set 2: } (H, R), \ (J, V) \quad \text{or} \quad \text{Set 2: } (J, R), \ (H, V) \]
- Rule 3 (Jill $\neq$ Toby) is already satisfied, since Toby is in Set 1 with Gloria.
Checking the options:
- (A) Vernon in Set 2: False. Vernon must be in Set 3.
- (B) Shigeru in Set 4: True. This always holds.
- (C) Ravi in Set 3: False. Ravi must be in Set 2.
- (D) Jill in Set 4: False. Jill must be in Set 2 or 3.
- (E) Heidi in Set 3: Possible, but not guaranteed (depends on assignment).
Final Answer:
\[ \boxed{\text{(B) Shigeru skates in Set 4. This is the only statement that must be true.}} \]
If Heidi skates in set 1 and Toby skates in set 2, which of the following must be true?
Show Hint
- Fiona skates with Ravi.
- Gloria skates with Ravi.
- Gloria skates with Shigeru.
- Gloria skates with Vernon.
- Jill skates with Vernon.
The Correct Option is
Solution and Explanation
We are given two pieces of information: - Set 1 woman is Heidi (H). - Set 2 man is Toby (T). This gives us a partial schedule: Set 1: (H, ?), Set 2: (?, T), Set 3: (?, ?), Set 4: (?, ?).
Step 2: Make Deductions
1. Rule 3 (J \(\neq\) T): Jill does not skate with Toby. Since Toby is the man in set 2, Jill cannot be the woman in set 2. 2. Rule 2 (F in 1 or 4): Fiona must be the woman in set 1 or 4. Heidi is the woman in set 1. Therefore, Fiona must be the woman in set 4. - Set 4 woman: F.
3. Placement of Women: We have placed H (set 1) and F (set 4). We know J is not in set 2. The only remaining woman is Gloria (G). The only remaining women's slots are in set 2 and set 3. Since J is not in set 2, G must be in set 2, and J must be in set 3. - Set 1: (H, ?) - Set 2: (G, T) - Set 3: (J, ?) - Set 4: (F, ?)
4. Rule 4 (S with F or G): Shigeru (S) skates with Fiona or Gloria. We have Gloria paired with Toby in set 2. So, Shigeru must be paired with Fiona in set 4. - Set 4 pair: (F, S)
5. Placement of Men: We have placed T (set 2) and S (set 4). The remaining men are Ravi (R) and Vernon (V). They must be the partners for H (set 1) and J (set 3). 6. Rule 1 (R \textless V): Ravi must skate before Vernon. Therefore, R must be in set 1, and V must be in set 3. - Set 1 pair: (H, R) - Set 3 pair: (J, V)
Step 3: Final Schedule and Evaluation
The complete and only possible schedule is: - Set 1: Heidi and Ravi (H, R) - Set 2: Gloria and Toby (G, T) - Set 3: Jill and Vernon (J, V) - Set 4: Fiona and Shigeru (F, S)
Now we check the options to see which one matches our deduced pairings. - (A) Fiona skates with Ravi. False. Fiona skates with Shigeru. - (B) Gloria skates with Ravi. False. Gloria skates with Toby. - (C) Gloria skates with Shigeru. False. Gloria skates with Toby. - (D) Gloria skates with Vernon. False. Gloria skates with Toby. - (E) Jill skates with Vernon. True. This is the pair in set 3.
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