In the Stafford Public School students had an option to study none or one or more of three foreign languages viz French, Spanish an German. The total student strength in school was 2116 students out of which 1320 students studied French and 408 students studied both French and Spanish. The number of people who studied German was found to be 180 higher than the number of students who studied Spanish. It was also observed that 108 students studied all three subjects.
[a] What is the maximum possible number of students who did not study any of the three languages?
[b] What is the minimum possible number of students who did not study any of the three languages?
[c] If the number of students who used to speak only French was 1 more than the number of people who used to speak only German, then what could be the maximum number of people who used to speak only Spanish ?
Answers
Answer:
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Step-by-step explanation:
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a] 796 is the maximum possible number of students who did not study any of the three languages.
[b] 0 is the minimum possible number of students who did not study any of the three languages.
[c] The maximum number of people who used to speak only Spanish is 431.
Given:
The total student strength in school= 2116
students out of which 1320 students studied French
408 students studied both French and Spanish.
The number of people who studied German was found to be 180 higher than the number of students who studied Spanish.
It was also observed that 108 students studied all three subjects.
To find:
a] What is the maximum possible number of students who did not study any of the three languages?
[b] What is the minimum possible number of students who did not study any of the three languages?
[c] If the number of students who used to speak only French was 1 more than the number of people who used to speak only German, then what could be the maximum number of people who used to speak only Spanish ?
Solution:
You must first consider the fundamental details in the question in order to consider the potential of the maximum and/or least number of individuals who could be learning none of the three languages.
In order to keep the limitations the question imposed on the relative numbers in the figure, we can now try to fit the remaining numbers into this Venn diagram. To accomplish this, we must consider the goal with which we must complete the remaining numbers in the figure. Currently, while you fill up the remaining numbers, you must keep the following two restrictions in mind:
(a) The remainder of the French circle must equal 912 (1320 - 408);
(a) There must be 180 more in the German circle than in the Spanish circle.
When we attempt to input the appropriate quantity to reach the maximum number of students who did not study any of the three subjects:
By trying to envision how you would want to distribute the final 912 in that circle, you may start by filling the French circle. We would need to use the smallest number of people inside the three circles—while making sure that all the constraints are met—if we wanted to increase the percentage of students who studied none of the three.
Since we must squeeze 912 into the remaining spaces of the French circle, we must determine if we can also meet the second limitation while doing so.
This way of thinking would enable you to consider the following potential solutions:
In this instance, we have made sure that the German total is 180 points higher than the Spanish total as required and that the French circle has also reached the required 1320 points. As a result, there can be 2116-1320 = 796 students who don't study any of the three subjects (at maximum).
The goal would be to utilise the greatest number of students possible in order to adhere to the fundamental limits while minimising the number of students who have not studied any of the three disciplines. In the following circumstances, the answer in this situation can be lowered to zero: Hence, minimum number of students will be zero.
We start by using the 912 in the "only French" area as we consider the numbers in this situation. There are currently 796 students that need to be distributed. By starting with the single German as 300 + 180, we may first build the German circle 180 more than the Spanish circle. We now have 316 additional pupils, who can be divided evenly as 316/2 between the "only German" and "only Spanish" regions.
If we attempt to incorporate the remaining restrictions in this case, we would obtain:
As a result, there are 796 extra persons that must be distributed equally since we cannot upset the balance of German speaking people having exactly 180 more than Spanish speaking people.
When you consider this circumstance, you see that if we decrease the only French area and reallocate the decrease into the "only French" and German area, Spanish might really increase. It is possible to see a reduction of 10 from the "only French" area as follows:
In this instance, as you can see from the accompanying figure, there are now 5 more students who only study Spanish (which is half of 10).
Since there is still a difference between the "only German" and "only French" areas in the figure, we should try to minimise the "only French" area in order to reduce the gap.
When we do that, the following solution figure would become apparent:
Thus, 431 is the highest number that can be used for the only Spanish area.
Hence, the answer of a part is 796, b part is o and c part is 431.
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