I21-03

Question

A class of boys and girls has more girls than boys, with a total of 50 students. On a particular day, 20 students were absent, leaving more boys than girls present.
 
Select for  Maximum  the maximum possible number of girls that could have been in the original class of 50 students, and select for  Minimum  the minimum possible number of girls who could have been absent on that day. These two values could correspond to separate scenarios and are not necessarily consistent with each other. Make only two selections, one in each column.

Bunuel · Answer

Official Solution:

Given that the total number of students is 50, we have: G + B = 50
 
To determine the  maximum possible number of girls in the original class , we need to ensure that after 20 students were absent, the number of boys present was greater than the number of girls present.
 
If there were 49 girls and 1 boy, even if all 20 absent students were girls, the remaining students would be 29 girls and 1 boy, meaning boys would not outnumber girls. Therefore, we need to find the highest possible G such that G - 20 < B, while still satisfying G + B = 50.
 
Substituting B = 50 - G into G - 20 < B, we get G - 20 < 50 - G, which simplifies to 2G < 70, or G < 35. Thus, the maximum possible G is 34.
 
For example, if G = 35, then B = 15. Even if all 20 absent students were girls, the remaining students would be 15 girls and 15 boys, meaning boys would not outnumber girls. Therefore, the number of girls must be less than 35, making 34 the maximum possible.
 
 To minimize the number of girls who could have been absent on the day  when the number of boys exceeded the number of girls, we should start with the smallest possible number of girls, which is 26. Since 20 students were absent, 30 students were present in the class that day. To achieve the smallest possible number of absent girls, we need to distribute the 30 students as evenly as possible while ensuring that the number of boys exceeds the number of girls. This approach maximizes the number of girls present, minimizing the difference between 26 (the initial number of girls) and the number of girls present. The closest distribution that meets this condition is 16 boys and 14 girls, meaning the minimum number of girls who could have been absent is 26 - 14 = 12.

Correct answer:

Maximum  "34"

Minimum  "12"

Bunuel · Answer

I have revised the question, solution, and formatting by adding more details to enhance clarity.

vaishnavi2498 · Answer

I did not quite understand the solution. Why are we taking the minimum number of girls in 2nd part? Is it because we want to keep the minimum number of girls absent then total girls should be considered less?

Bunuel · Answer

We take the  minimum number of girls in the second part  because we are trying to  minimize the number of absent girls . Since 20 students were absent, the number of girls absent depends on how many girls were in the class  to begin with .

To minimize the number of absent girls, we start with the  smallest possible total number of girls (26)  because a lower total means fewer can be absent while still ensuring that  more boys than girls were present . This setup leads to the smallest possible number of absent girls.

KDM91 · Answer

I did not quite understand the solution. I did not understand the second part of the solution (finding minimum possible number of girls absent)

Bunuel · Answer

To minimize the number of absent girls, we start with the  smallest possible total number of girls (26)  because a lower total means fewer can be absent while still ensuring that  more boys than girls were present . This setup leads to the smallest possible number of absent girls.

Luca46 · Answer

I like the solution - it’s helpful.

I21-03

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