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- Fuel contamination levels at each of 20 petrol pumps P1, P2, …, P20 were recorded as either high, medium, or low. 1. Contamination levels at three
Fuel contamination levels at each of 20 petrol pumps P1, P2, …, P20 were recorded as either high, medium, or low.
1. Contamination levels at three pumps among P1 – P5 were recorded as high.
2. P6 was the only pump among P1 – P10 where the contamination level was recorded as low.
3. P7 and P8 were the only two consecutively numbered pumps where the same levels of contamination were recorded.
4. High contamination levels were not recorded at any of the pumps P16 – P20.
5. The number of pumps where high contamination levels were recorded was twice the number of pumps where low contamination levels were recorded.
Question:
Which of the following MUST be true?
The contamination level at P10 was recorded as high.
The contamination level at P13 was recorded as low.
The contamination level at P20 was recorded as medium.
The contamination level at P12 was recorded as high.
Answers (1)
Based on the given conditions 1 and 2, we can derive the following results:
| P1 | P2 | P3 | P4 | P5 | P6 |
| H | M | H | M | H | L |
Additionally, from condition 4, we encounter two possible cases:
| P16 | P17 | P18 | P19 | P20 |
| L | M | L | M | L |
| M | L | M | L | M |
According to condition 5, we have two scenarios:
If the total number of low (L) pipes is 3:
- Number of high (H) pipes: 6
- Number of medium (M) pipes: 11
If the number of low (L) pipes is 4:
- Number of high (H) pipes: 8
- Number of medium (M) pipes: 8
For P7 and P8, they can either be HH or MM, resulting in two distinct cases for P1 to P10.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| H | M | H | M | H | L | M | M | H | M |
| H | M | H | M | H | L | H | H | M | H |
Combining conditions 1 and 2, we find the following potential cases for P1 to P20:
Case 1: H M H M H L H H M H M H M H M L M L M L
- Number of (L) pipes: 4
- Number of (H) pipes: 8
- Number of (M) pipes: 8
Case 2: H M H M H L H H M H L M H M H M L M L M
- Number of (L) pipes: 4
- Number of (H) pipes: 8
- Number of (M) pipes: 8
Case 3: H M H M H L H H M H M L H M H M L M L M
- Number of (L) pipes: 4
- Number of (H) pipes: 8
- Number of (M) pipes: 8
First, we create a table and fill in the information provided with absolute certainty.
| Pumps | Contamination Level |
| P1 | |
| P2 | |
| P3 | |
| P4 | |
| P5 | |
| P6 | Low |
| P7 | |
| P8 | |
| P9 | |
| P10 | |
| P11 | |
| P12 | |
| P13 | |
| P14 | |
| P15 | |
| P16 | |
| P17 | |
| P18 | |
| P19 | |
| P20 |
According to statement 3, pumps P7 and P8 are the only consecutive pumps with the same contamination level recorded.
Statement 1 indicates that three pumps among P1 - P5 have high contamination levels. Therefore, pumps 1, 3, and 5 must have a high level of contamination. Additionally, P6 is the only pump among P1 - P10 with a recorded low contamination level. Hence, pumps 2 and 4 have a medium level of contamination.
| Pumps | Contamination Level |
| P1 | High |
| P2 | Medium |
| P3 | High |
| P4 | Medium |
| P5 | HIgh |
| P6 | Low |
| P7 | |
| P8 | |
| P9 | |
| P10 | |
| P11 | |
| P12 | |
| P13 | |
| P14 | |
| P15 | |
| P16 | |
| P17 | |
| P18 | |
| P19 | |
| P20 |
It is stated that high contamination levels were not recorded at any of the pumps P16 - P20. Thus, we can conclude that high contamination is limited to the first 15 pumps. The maximum number of pumps with high contamination can be '8' (considering consecutive pumps have different contamination levels, except for one case). Additionally, the number of pumps with high contamination is twice the number of pumps with low contamination. Therefore, the number of pumps with high contamination is an even number less than or equal to '8'.
Suppose there are '6' pumps with high contamination. In that case, there would be only '3' pumps with low contamination, leaving 11 pumps with a medium contamination level. However, the number of pumps of a single type cannot exceed 10 (due to consecutive pumps having different contamination levels, except for one case).
Hence, we can conclude that the number of pumps with high contamination is 8, the number of pumps with low contamination is 4, and the number of pumps with medium contamination is 12.
Since P7 and P8 are the only consecutive pumps with the same contamination level, if they both recorded a medium contamination level, there can be a maximum of 7 pumps (P1, P3, P5, P9, P11, P13, P15) with high contamination. Therefore, we can determine that pumps P7 and P8 recorded a high contamination level, allowing us to uniquely determine the contamination levels up to P10.
| Pumps | Contamination Level |
| P1 | High |
| P2 | Medium |
| P3 | High |
| P4 | Medium |
| P5 | High |
| P6 | Low |
| P7 | High |
| P8 | High |
| P9 | Medium |
| P10 | High |
| P11 | |
| P12 | |
| P13 | |
| P14 | |
| P15 | |
| P16 | |
| P17 | |
| P18 | |
| P19 | |
| P20 |
Considering that high contamination levels were not recorded in pumps P16 - P20, we can deduce that these 5 pumps recorded low and medium contamination levels. Two cases are possible:
Case 1: 3 pumps recorded low contamination and 2 pumps recorded medium contamination in pumps P16 - P20. In this scenario, P16, P18, and P20 would have low contamination levels.
| Pumps | Contamination Level |
| P1 | High |
| P2 | Medium |
| P3 | High |
| P4 | Medium |
| P5 | High |
| P6 | Low |
| P7 | High |
| P8 | High |
| P9 | Medium |
| P10 | High |
| P11 | Medium |
| P12 | High |
| P13 | Medium |
| P14 | High |
| P15 | Medium |
| P16 | Low |
| P17 | Medium |
| P18 | Low |
| P19 | Medium |
| P20 | Low |
Case 2: 2 pumps recorded low contamination and 3 pumps recorded medium contamination in pumps P16 - P20. In this situation, P16, P18, and P20 would have medium contamination levels.
Checking each option:
Option 1: The contamination level at P10 recorded as high. This is true for all cases.
Option 2: The contamination level at P13 recorded as low. This is not necessarily true, as in Case 2(a), P13 has a high contamination level.
Option 3: The contamination level at P20 recorded as medium. This is not necessarily true, as in Case 1, P20 has a low contamination level.
Option 4: The contamination level at P12 recorded as high. This is not necessarily true, as in Case 2(a), P12 has a medium contamination level.
Hence, option 1 is the correct answer.
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