Residuals and Ratios

[sarahmoore.2009]

Question: On a particular exam, the boys in a history class averaged 86 points and the girls in the class averaged 80 points. If the overall class average was 82 points, what was the ratio of boys to girls in the class?

Answer: 1:2
Imagine that every boy has exactly an 86 and every girl has exactly an 80. Then, if the class average is 82, every boy has a positive residual of +4, and every girl has a negative residual of -2. Since the residuals must sum to zero, by definition, there must be 2 girls for every boy.

Can someone please help me understand why the residual being over by 2 for every boy is indicative of a 1 to 2 ratio? I get that there must be more girls than boys since the residual is over by 2 I just don't understand how that amounts to knowing the exact proportion between boys and girls.

[mondegreen]

Question: On a particular exam, the boys in a history class averaged 86 points and the girls in the class averaged 80 points. If the overall class average was 82 points, what was the ratio of boys to girls in the class?

Answer: 1:2
Imagine that every boy has exactly an 86 and every girl has exactly an 80. Then, if the class average is 82, every boy has a positive residual of +4, and every girl has a negative residual of -2. Since the residuals must sum to zero, by definition, there must be 2 girls for every boy.

Can someone please help me understand why the residual being over by 2 for every boy is indicative of a 1 to 2 ratio? I get that there must be more girls than boys since the residual is over by 2 I just don't understand how that amounts to knowing the exact proportion between boys and girls.

Hmm. Imagine a single boy in the class and one girl. What is the residual for the boy : 86-82 = +4.
Now, the residual for the girl = 80-82 = -2. Now, the sum of the residuals must be zero(which is +4-2 = 2 now). Now, if you introduce one more boy, the sum of residuals is only going to increase by +4 for each additional boy. The only way you can make the summation of residual zero, is by introducing one more girl --> The sum of residual = +4-2-2 = 0. Thus, in the class, we have 1 boy, 2 girls,ergo the ratio : 1:2. Now , you can keep adding a group of 1 boy+2 girls, without changing the residual, and the ratio.

[sarahmoore.2009]

Thank you SO much for the incredibly clear reply. I actually understand the problem now!

[RonPurewal]

cool.