Four teams – Aces, Bolts, Comets and Dashers – took part in a 4 × 400 m relay race. Each team had four runners, one for each leg. The table below shows the time (in seconds) taken by each team's runner in each leg; some entries are missing and are shown as '?'.
| Team | Leg 1 | Leg 2 | Leg 3 | Leg 4 |
|---|---|---|---|---|
| Aces | 52 | ? | 50 | 47 |
| Bolts | 50 | 53 | ? | 49 |
| Comets | ? | 46 | 52 | ? |
| Dashers | 51 | ? | 51 | ? |
The standings (first to last) at the end of each leg, based on the total time taken till then, were:
| After | 1st | 2nd | 3rd | 4th |
|---|---|---|---|---|
| Leg 1 | Bolts | Dashers | Aces | Comets |
| Leg 2 | Comets | Aces | Dashers | Bolts |
| Leg 3 | Bolts | Aces | Comets | Dashers |
| Leg 4 | Aces | Bolts | Comets | Dashers |
Additional information:
- Every leg time was a whole number of seconds from 45 to 55 (both inclusive).
- No two teams were ever level at the end of any leg.
- The winning margin (between 1st and 2nd at the finish) was exactly 1 second, and the last team finished exactly 5 seconds after the winner.
- For Comets, the difference between their slowest and fastest leg times was 8 seconds.
- Dashers' first three runners recorded identical times.
- Comets' final-leg runner was faster than Bolts' final-leg runner.
| Team | Leg 1 | Leg 2 | Leg 3 | Leg 4 |
|---|---|---|---|---|
| Aces | 52 | ? | 50 | 47 |
| Bolts | 50 | 53 | ? | 49 |
| Comets | ? | 46 | 52 | ? |
| Dashers | 51 | ? | 51 | ? |
The standings (first to last) at the end of each leg, based on the total time taken till then, were:
| After | 1st | 2nd | 3rd | 4th |
|---|---|---|---|---|
| Leg 1 | Bolts | Dashers | Aces | Comets |
| Leg 2 | Comets | Aces | Dashers | Bolts |
| Leg 3 | Bolts | Aces | Comets | Dashers |
| Leg 4 | Aces | Bolts | Comets | Dashers |
Additional information:
- Every leg time was a whole number of seconds from 45 to 55 (both inclusive).
- No two teams were ever level at the end of any leg.
- The winning margin (between 1st and 2nd at the finish) was exactly 1 second, and the last team finished exactly 5 seconds after the winner.
- For Comets, the difference between their slowest and fastest leg times was 8 seconds.
- Dashers' first three runners recorded identical times.
- Comets' final-leg runner was faster than Bolts' final-leg runner.
Question 1
Numerical answer
What was the time (in seconds) of Aces' runner in Leg 2?
Show answer and solution
Answer: 49
Completed table (cumulative times in brackets):
| Team | Leg 1 | Leg 2 | Leg 3 | Leg 4 | Total |
|---|---|---|---|---|---|
| Aces | 52 (52) | 49 (101) | 50 (151) | 47 (198) | 198 |
| Bolts | 50 (50) | 53 (103) | 47 (150) | 49 (199) | 199 |
| Comets | 54 (54) | 46 (100) | 52 (152) | 48 (200) | 200 |
| Dashers | 51 (51) | 51 (102) | 51 (153) | 50 (203) | 203 |
Key steps: Dashers' leg 2 = 51 (identical times). Comets last after leg 1 → leg 1 > 52; first after leg 2 → leg 1 + 46 < 101 → leg 1 ≤ 54; the 8-second spread with a 46 forces leg 1 = 54. After leg 2 order C(100) < A < D(102) < B(103) gives Aces' leg 2 = 49 (A = 101). After leg 3 Bolts lead Aces (151): 103 + x < 151 → x ≤ 47. Finish: Aces win by 1 s, and the runner-up is Bolts → Bolts total = 199 = 103 + x + 49 → x = 47 (150 after leg 3). Comets are 3rd: 199 < 152 + y and y < 49 (faster than Bolts' 49) → y = 48 (200). Dashers are last, 5 s behind the winner: 203 → leg 4 = 203 − 153 = 50.
Aces must be 2nd after Leg 2, between Comets (100) and Dashers (102): 52 + x = 101 → x = 49.
| Team | Leg 1 | Leg 2 | Leg 3 | Leg 4 | Total |
|---|---|---|---|---|---|
| Aces | 52 (52) | 49 (101) | 50 (151) | 47 (198) | 198 |
| Bolts | 50 (50) | 53 (103) | 47 (150) | 49 (199) | 199 |
| Comets | 54 (54) | 46 (100) | 52 (152) | 48 (200) | 200 |
| Dashers | 51 (51) | 51 (102) | 51 (153) | 50 (203) | 203 |
Key steps: Dashers' leg 2 = 51 (identical times). Comets last after leg 1 → leg 1 > 52; first after leg 2 → leg 1 + 46 < 101 → leg 1 ≤ 54; the 8-second spread with a 46 forces leg 1 = 54. After leg 2 order C(100) < A < D(102) < B(103) gives Aces' leg 2 = 49 (A = 101). After leg 3 Bolts lead Aces (151): 103 + x < 151 → x ≤ 47. Finish: Aces win by 1 s, and the runner-up is Bolts → Bolts total = 199 = 103 + x + 49 → x = 47 (150 after leg 3). Comets are 3rd: 199 < 152 + y and y < 49 (faster than Bolts' 49) → y = 48 (200). Dashers are last, 5 s behind the winner: 203 → leg 4 = 203 − 153 = 50.
Aces must be 2nd after Leg 2, between Comets (100) and Dashers (102): 52 + x = 101 → x = 49.
Question 2
Single correct option
What was the total time taken by Dashers?
- A. 201 seconds
- B. 203 seconds
- C. 204 seconds
- D. 202 seconds
Show answer and solution
Answer: (B) 203 seconds
Completed table (cumulative times in brackets):
| Team | Leg 1 | Leg 2 | Leg 3 | Leg 4 | Total |
|---|---|---|---|---|---|
| Aces | 52 (52) | 49 (101) | 50 (151) | 47 (198) | 198 |
| Bolts | 50 (50) | 53 (103) | 47 (150) | 49 (199) | 199 |
| Comets | 54 (54) | 46 (100) | 52 (152) | 48 (200) | 200 |
| Dashers | 51 (51) | 51 (102) | 51 (153) | 50 (203) | 203 |
Key steps: Dashers' leg 2 = 51 (identical times). Comets last after leg 1 → leg 1 > 52; first after leg 2 → leg 1 + 46 < 101 → leg 1 ≤ 54; the 8-second spread with a 46 forces leg 1 = 54. After leg 2 order C(100) < A < D(102) < B(103) gives Aces' leg 2 = 49 (A = 101). After leg 3 Bolts lead Aces (151): 103 + x < 151 → x ≤ 47. Finish: Aces win by 1 s, and the runner-up is Bolts → Bolts total = 199 = 103 + x + 49 → x = 47 (150 after leg 3). Comets are 3rd: 199 < 152 + y and y < 49 (faster than Bolts' 49) → y = 48 (200). Dashers are last, 5 s behind the winner: 203 → leg 4 = 203 − 153 = 50.
Winner Aces = 198 s; the last team (Dashers) finished 5 s later → 203 s.
| Team | Leg 1 | Leg 2 | Leg 3 | Leg 4 | Total |
|---|---|---|---|---|---|
| Aces | 52 (52) | 49 (101) | 50 (151) | 47 (198) | 198 |
| Bolts | 50 (50) | 53 (103) | 47 (150) | 49 (199) | 199 |
| Comets | 54 (54) | 46 (100) | 52 (152) | 48 (200) | 200 |
| Dashers | 51 (51) | 51 (102) | 51 (153) | 50 (203) | 203 |
Key steps: Dashers' leg 2 = 51 (identical times). Comets last after leg 1 → leg 1 > 52; first after leg 2 → leg 1 + 46 < 101 → leg 1 ≤ 54; the 8-second spread with a 46 forces leg 1 = 54. After leg 2 order C(100) < A < D(102) < B(103) gives Aces' leg 2 = 49 (A = 101). After leg 3 Bolts lead Aces (151): 103 + x < 151 → x ≤ 47. Finish: Aces win by 1 s, and the runner-up is Bolts → Bolts total = 199 = 103 + x + 49 → x = 47 (150 after leg 3). Comets are 3rd: 199 < 152 + y and y < 49 (faster than Bolts' 49) → y = 48 (200). Dashers are last, 5 s behind the winner: 203 → leg 4 = 203 − 153 = 50.
Winner Aces = 198 s; the last team (Dashers) finished 5 s later → 203 s.
Question 3
Single correct option
Of the sixteen individual leg times, how many were 50 seconds or more?
- A. 8
- B. 9
- C. 10
- D. 11
Show answer and solution
Answer: (C) 10
Completed table (cumulative times in brackets):
| Team | Leg 1 | Leg 2 | Leg 3 | Leg 4 | Total |
|---|---|---|---|---|---|
| Aces | 52 (52) | 49 (101) | 50 (151) | 47 (198) | 198 |
| Bolts | 50 (50) | 53 (103) | 47 (150) | 49 (199) | 199 |
| Comets | 54 (54) | 46 (100) | 52 (152) | 48 (200) | 200 |
| Dashers | 51 (51) | 51 (102) | 51 (153) | 50 (203) | 203 |
Key steps: Dashers' leg 2 = 51 (identical times). Comets last after leg 1 → leg 1 > 52; first after leg 2 → leg 1 + 46 < 101 → leg 1 ≤ 54; the 8-second spread with a 46 forces leg 1 = 54. After leg 2 order C(100) < A < D(102) < B(103) gives Aces' leg 2 = 49 (A = 101). After leg 3 Bolts lead Aces (151): 103 + x < 151 → x ≤ 47. Finish: Aces win by 1 s, and the runner-up is Bolts → Bolts total = 199 = 103 + x + 49 → x = 47 (150 after leg 3). Comets are 3rd: 199 < 152 + y and y < 49 (faster than Bolts' 49) → y = 48 (200). Dashers are last, 5 s behind the winner: 203 → leg 4 = 203 − 153 = 50.
Count ≥ 50: Aces 52, 50 (2); Bolts 50, 53 (2); Comets 54, 52 (2); Dashers 51, 51, 51, 50 (4) → 10.
| Team | Leg 1 | Leg 2 | Leg 3 | Leg 4 | Total |
|---|---|---|---|---|---|
| Aces | 52 (52) | 49 (101) | 50 (151) | 47 (198) | 198 |
| Bolts | 50 (50) | 53 (103) | 47 (150) | 49 (199) | 199 |
| Comets | 54 (54) | 46 (100) | 52 (152) | 48 (200) | 200 |
| Dashers | 51 (51) | 51 (102) | 51 (153) | 50 (203) | 203 |
Key steps: Dashers' leg 2 = 51 (identical times). Comets last after leg 1 → leg 1 > 52; first after leg 2 → leg 1 + 46 < 101 → leg 1 ≤ 54; the 8-second spread with a 46 forces leg 1 = 54. After leg 2 order C(100) < A < D(102) < B(103) gives Aces' leg 2 = 49 (A = 101). After leg 3 Bolts lead Aces (151): 103 + x < 151 → x ≤ 47. Finish: Aces win by 1 s, and the runner-up is Bolts → Bolts total = 199 = 103 + x + 49 → x = 47 (150 after leg 3). Comets are 3rd: 199 < 152 + y and y < 49 (faster than Bolts' 49) → y = 48 (200). Dashers are last, 5 s behind the winner: 203 → leg 4 = 203 − 153 = 50.
Count ≥ 50: Aces 52, 50 (2); Bolts 50, 53 (2); Comets 54, 52 (2); Dashers 51, 51, 51, 50 (4) → 10.
Question 4
Single correct option
What was Bolts' time in Leg 3, and by how many seconds did Comets finish ahead of Dashers?
- A. 47 seconds; 3 seconds
- B. 46 seconds; 4 seconds
- C. 47 seconds; 2 seconds
- D. 48 seconds; 3 seconds
Show answer and solution
Answer: (A) 47 seconds; 3 seconds
Completed table (cumulative times in brackets):
| Team | Leg 1 | Leg 2 | Leg 3 | Leg 4 | Total |
|---|---|---|---|---|---|
| Aces | 52 (52) | 49 (101) | 50 (151) | 47 (198) | 198 |
| Bolts | 50 (50) | 53 (103) | 47 (150) | 49 (199) | 199 |
| Comets | 54 (54) | 46 (100) | 52 (152) | 48 (200) | 200 |
| Dashers | 51 (51) | 51 (102) | 51 (153) | 50 (203) | 203 |
Key steps: Dashers' leg 2 = 51 (identical times). Comets last after leg 1 → leg 1 > 52; first after leg 2 → leg 1 + 46 < 101 → leg 1 ≤ 54; the 8-second spread with a 46 forces leg 1 = 54. After leg 2 order C(100) < A < D(102) < B(103) gives Aces' leg 2 = 49 (A = 101). After leg 3 Bolts lead Aces (151): 103 + x < 151 → x ≤ 47. Finish: Aces win by 1 s, and the runner-up is Bolts → Bolts total = 199 = 103 + x + 49 → x = 47 (150 after leg 3). Comets are 3rd: 199 < 152 + y and y < 49 (faster than Bolts' 49) → y = 48 (200). Dashers are last, 5 s behind the winner: 203 → leg 4 = 203 − 153 = 50.
Bolts lead after Leg 3 with Aces at 151, so 103 + x ≤ 150 → x ≤ 47. Bolts then finish 2nd at 1 second behind Aces (198) → 199 = 103 + x + 49 → x = 47. Comets 200 vs Dashers 203 → 3 seconds.
| Team | Leg 1 | Leg 2 | Leg 3 | Leg 4 | Total |
|---|---|---|---|---|---|
| Aces | 52 (52) | 49 (101) | 50 (151) | 47 (198) | 198 |
| Bolts | 50 (50) | 53 (103) | 47 (150) | 49 (199) | 199 |
| Comets | 54 (54) | 46 (100) | 52 (152) | 48 (200) | 200 |
| Dashers | 51 (51) | 51 (102) | 51 (153) | 50 (203) | 203 |
Key steps: Dashers' leg 2 = 51 (identical times). Comets last after leg 1 → leg 1 > 52; first after leg 2 → leg 1 + 46 < 101 → leg 1 ≤ 54; the 8-second spread with a 46 forces leg 1 = 54. After leg 2 order C(100) < A < D(102) < B(103) gives Aces' leg 2 = 49 (A = 101). After leg 3 Bolts lead Aces (151): 103 + x < 151 → x ≤ 47. Finish: Aces win by 1 s, and the runner-up is Bolts → Bolts total = 199 = 103 + x + 49 → x = 47 (150 after leg 3). Comets are 3rd: 199 < 152 + y and y < 49 (faster than Bolts' 49) → y = 48 (200). Dashers are last, 5 s behind the winner: 203 → leg 4 = 203 − 153 = 50.
Bolts lead after Leg 3 with Aces at 151, so 103 + x ≤ 150 → x ≤ 47. Bolts then finish 2nd at 1 second behind Aces (198) → 199 = 103 + x + 49 → x = 47. Comets 200 vs Dashers 203 → 3 seconds.
Saved on this device. Sign in or create a free account to keep it on every device.