MHB Ratio of Boys & Girls in Class: 24 Boys & 16 Girls?

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The discussion revolves around determining the ratio and number of boys and girls in a class of 40 students, where there are 24 boys and 16 girls. The mean marks for boys and girls are 7.1 and 8.1, respectively, leading to an overall mean of 7.5. Through solving simultaneous equations, it is confirmed that the calculations support the conclusion of 24 boys and 16 girls. Alternative methods, including weighted averages and proportional reasoning, also validate this result. The consensus is that the correct distribution is indeed 24 boys and 16 girls.
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In a class, the mean of boys' mark is 7.1 while the mean of girls' mark is 8.1. Their overall mean is 7.5. If there are 40 students in the class, determine:
a. The ratio of boys and girls
b. The number of boys and girls

I got this question from someone I tutored. I got the number of boys as 24 and the number of girls as 16. However, his teacher got vice-versa. Which one was right?
 
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Let's let $B$ be the number of boys, and so the number of girls is $G=40-B$. Let $S_i$ be the sum of the respective marks. From what we are given, we know:

$$\frac{S_B}{B}=\frac{71}{10}$$

$$\frac{S_G}{G}=\frac{81}{10}$$

$$\frac{S_B+S_G}{40}=\frac{15}{2}$$

Thus, we have the system:

$$B+G=40$$

$$71B=10S_B$$

$$81G=10S_G$$

$$S_B+S_G=300$$

Now, the 2nd and 3rd equations, when added and simplified, give us

$$S_B+S_G=\frac{71B+81G}{10}$$

And so this implies, when considering the 4th equation:

$$71B+81G=3000$$

Then using the 1st equation, and substitution, we obtain:

$$71B+81(40-B)=3000$$

$$71B+3240-81B=3000$$

$$B=24\implies G=16$$

I agree with you. :D
 
Equivalently, let a be the number of boys and b the number of girls. Then the weighted average of grades is \frac{7.1a+ 8.1b}{a+ b}= 7.5. Of course, a+ b= 40 so that is \frac{7.1a+ 8.1b}{40}= 7.5 or 7.1a+ 8.1b= 7.5(40)= 300. We need to solve the simultaneous equations 7.1a+ 8.1b= 300 and a+ b= 40. Multiplying the second equation by 7.1, 7.1a+ 7.1b= 7.1(40)= 284. Subtracting that from the first equation eliminates a leaving b= 16. Then a= 40- 16= 24.

Yet another way: from 7.1 to 8.1 is a difference of 1.0. From 7.1 to 7.5 is a difference of 0.4. 1.0- 0.4= 0.6. The 40 students must be divided into boys and girls the same way 1.0 is divided into 0.6 and 0.4. 0.6(40)= 24, 0.4(40)= 16.
 
Thank you guys.:D
 
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