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15 Jan 2021, 09:00
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Minimum possible number
of students who take both: 35 Maximum possible number
of students who take both: 70
of students who take both: 35 Maximum possible number
of students who take both: 70
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
45%
(medium)
Question Stats:
70% (02:21) correct
30%
(02:39)
wrong
based on 1062
sessions
History
Date
Time
Result
Not Attempted Yet
The 160 students at a certain high school must choose at least one science course for the first semester. 110 choose to take chemistry and 70 choose to take physics. At least 15 students choose a different subject entirely—neither chemistry nor physics.
Using the information given, identify the minimum possible number of students who choose to take both chemistry and physics, and the maximum number of students who could take both. Make only two selections, one in each column.
Using the information given, identify the minimum possible number of students who choose to take both chemistry and physics, and the maximum number of students who could take both. Make only two selections, one in each column.
|
Minimum possible number of students who take both |
Maximum possible number of students who take both |
Number |
| 20 | ||
| 35 | ||
| 40 | ||
| 70 | ||
| 85 | ||
| 90 |
ShowHide Answer
Official Answer
Minimum possible number
of students who take both: 35 Maximum possible number
of students who take both: 70
of students who take both: 35 Maximum possible number
of students who take both: 70
16 Jan 2021, 09:30
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Official Explanation
This is an Overlapping Sets question. Initially, it might seem that there are three possible sets (physics chemistry, and some other unnamed subject). Because the problem is presented as choosing physics, chemistry, both, or neither, however, you can use a Double-Set Matrix to solve. You could also try to think through this type of problem logically—if you feel very comfortable with this type of math.
The at least language indicates that you will also have to use Maximize and Minimize principles in order to solve.
Logic
There are 160 students total, but at least 15 choose some other subject entirely, so at most 160 – 15 =145 students choose some combination of chemistry and physics.
The problem indicates that 110 choose chemistry and 70 choose physics, for an apparent total of 180 students. But since 145 is the actual maximum, at least 180 – 145 = 35 students must take both classes. This is the answer for the first column.
The problem states that a total of 70 students choose physics, so it can’t be the case that more than 70 take both chemistry and physics; eliminate answers 85 and 90. Since more students (110) take chemistry, it’s possible that all 70 students who take physics also take chemistry, so the maximum possible number of students who take both is 70. This is the answer for the second column.
Double-Set Matrix
Start by setting up the matrix with the information you are given as well as any additional information you can infer.
The center box (neither chemistry nor physics) must have at least 15, but is there a top limit? Since 50 students do not take chemistry, there cannot be more than 50 students who take neither subject. The “neither” box, therefore, has a range of 15 to 50.
Using each end of that range, find the possible range of students who took both chemistry and physics. For instance, in the middle row (NOT C), take the difference between each end of the range in the middle column and the figure in the Total column. The difference between 15 and 50 is 35, and the difference between 50 and 50 is 0, so the range for the “P but NOT C” box is 0 to 35.
If “P but NOT C is 0 to 35, then “C and P” must have a range of 35 to 70, in order for the Total P box to equal 70.
Therefore, the minimum possible number of students who take both C and P is 35, while the maximum possible number of students who take both is 70.
Note: You can also solve by calculating the range for the “C but NOT P” box (that range is 40 to 75) and then finding the range for the “C and P” box (which would still be 35 to 70).
Column 1: The correct answer is (B).
Column 2: The correct answer is (D).
This is an Overlapping Sets question. Initially, it might seem that there are three possible sets (physics chemistry, and some other unnamed subject). Because the problem is presented as choosing physics, chemistry, both, or neither, however, you can use a Double-Set Matrix to solve. You could also try to think through this type of problem logically—if you feel very comfortable with this type of math.
The at least language indicates that you will also have to use Maximize and Minimize principles in order to solve.
Logic
There are 160 students total, but at least 15 choose some other subject entirely, so at most 160 – 15 =145 students choose some combination of chemistry and physics.
The problem indicates that 110 choose chemistry and 70 choose physics, for an apparent total of 180 students. But since 145 is the actual maximum, at least 180 – 145 = 35 students must take both classes. This is the answer for the first column.
The problem states that a total of 70 students choose physics, so it can’t be the case that more than 70 take both chemistry and physics; eliminate answers 85 and 90. Since more students (110) take chemistry, it’s possible that all 70 students who take physics also take chemistry, so the maximum possible number of students who take both is 70. This is the answer for the second column.
Double-Set Matrix
Start by setting up the matrix with the information you are given as well as any additional information you can infer.
| P | Not P | Total | |
| C | 110 | ||
| Not C | ≥ 15 | 50 | |
| Total | 70 | 90 | 160 |
The center box (neither chemistry nor physics) must have at least 15, but is there a top limit? Since 50 students do not take chemistry, there cannot be more than 50 students who take neither subject. The “neither” box, therefore, has a range of 15 to 50.
| P | Not P | Total | |
| C | 110 | ||
| Not C | 15-50 | 50 | |
| Total | 70 | 90 | 160 |
Using each end of that range, find the possible range of students who took both chemistry and physics. For instance, in the middle row (NOT C), take the difference between each end of the range in the middle column and the figure in the Total column. The difference between 15 and 50 is 35, and the difference between 50 and 50 is 0, so the range for the “P but NOT C” box is 0 to 35.
| P | Not P | Total | |
| C | 35-70 | 110 | |
| Not C | 0-35 | 15-50 | 50 |
| Total | 70 | 90 | 160 |
If “P but NOT C is 0 to 35, then “C and P” must have a range of 35 to 70, in order for the Total P box to equal 70.
Therefore, the minimum possible number of students who take both C and P is 35, while the maximum possible number of students who take both is 70.
Note: You can also solve by calculating the range for the “C but NOT P” box (that range is 40 to 75) and then finding the range for the “C and P” box (which would still be 35 to 70).
Column 1: The correct answer is (B).
Column 2: The correct answer is (D).
General Discussion
16 Jan 2021, 00:26
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The provided information:
160 students take at least one science course .
110 chose to take Chemistry
70 chose to take Physics.
at least 15 took neither.
We can input these details in the table as below and try to vary [NON-CHEMISTRY & NON-PHYSICS] value from 15 to minimum of {maximum [NON-CHEMISTRY, NON-PHYSICS]} i.e. to ensure that sum of the column or row does accommodate the value.
We can't assign [NON-CHEMISTRY & NON-PHYSICS] more than 50, the minimum must be 15 as provided in the information.
Hence, we get the results of the field [CHEMISTRY & PHYSICS] as :
Minimum possible number of students who take both : 35
Maximum possible number of students who take both : 70
160 students take at least one science course .
110 chose to take Chemistry
70 chose to take Physics.
at least 15 took neither.
We can input these details in the table as below and try to vary [NON-CHEMISTRY & NON-PHYSICS] value from 15 to minimum of {maximum [NON-CHEMISTRY, NON-PHYSICS]} i.e. to ensure that sum of the column or row does accommodate the value.
We can't assign [NON-CHEMISTRY & NON-PHYSICS] more than 50, the minimum must be 15 as provided in the information.
Hence, we get the results of the field [CHEMISTRY & PHYSICS] as :
Minimum possible number of students who take both : 35
Maximum possible number of students who take both : 70
Attachments
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