Question

1. The percentage of boys in a math circle, rounded to an integer, is equal to 51%. The percentage
of girls in this math circle, rounded to an integer, is equal to 49%. What is the minimal possible
number of participants in the circle? Please properly answer ​

Answer 1

Given : The percentage of boys in a math circle, rounded to an integer, is equal to 51%. The percentage  of girls in this math circle, rounded to an integer, is equal to 49%

To Find : What is the minimal possible  number of participants in the circle?

Solution:

minimal possible  number of participants in the circle

if B = G + 1    

Total = B + G

= G + 1  + G

= 2G + 1

Girls Percentage  =  100G/(2G + 1)

rounded to an integer, is equal to 49%  

48.5 ≤ 100G/(2G + 1)  <  49.5

=> 97G  + 48.5  ≤  100G <  99G  + 49.5

=>  48.5  ≤  3G  => 16.16 ≤ G < 49.5

G = 17

G + 1 = 18

17 + 18 = 35

minimal possible  number of participants in the circle  = 35  

or we can solve by  Boys percentage

Boys percentage = 100 (G + 1)/ (2G + 1)

rounded to an integer, is equal to 51%  

50.5 ≤ 100(G+1)/(2G + 1)  <  51.5

=> 101G  + 50.5  ≤  100G + 100 <  103G  + 51.5

G ≤ 49.5

3G > 48.5  => G >  16.16

Girls = 17

Boys = 18

Total = 35

Girls %  = ( 17/35) * 100 = 48.57   = 49 %

Boys %  = ( 18/35) * 100 = 51.43   = 51 %

minimal possible  number of participants in the circle = 35

Another Ways   Difference between 48.5  and  51.5  is less than 3%  greater than 1 %

3 % = 1

Hence > 33

34 is even   so difference of one is not possible

so 35   Total  

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The percentage of boys in a math circle,

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