WP pg99, Problem 5

[ZacharyS701]

I don't understand the logic behind this problem:
On a particular exam, the boys in a history class averages 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?

The Solution:
The boy's in the class scored 4 points higher on average than the entire class. Similarly, the girls scored 2 points lower on average than the class. You can draw a teeter-totter to answer the question. Set up the starting info:

Boys (Less) +4% \ Class Average\ Girls (More) -2%

There are more girls because a 2-points difference is smaller than a 4-point difference. What's the actual ratio?
The "length" of the line is 4+2=6. The girl's side "pulls" the average away from boys by 4 points, and so girls are responsible for 4/6 of the overall length.

But wait! That's a fraction, not a ratio! 4/6 shows the part-to-whole relationship: 4 out of 6 points in the score spread are attributed to the girls. The boys are responsible for the other 2 out of 6 points in the spread. So the ratio of boys to girls is 2 to 4, or 1:2

My Solution:
80w+86m=82(m+w)
80w+86m=82m+82w
86m=2w
4m=2w
2m:1w
*I know that this is IMPOSSIBLE given the lean of the problem.

What I don't understand
Why didn't my calculation work? I understand the math as to why it didn't work, but where is the logic flagging? I also don't understand their solution. Why did they form the relationship between the boys and the girls to figure the 1:2 ratio?

[Sage Pearce-Higgins]

Both solutions are valid here. However, you're making a mistake with interpreting your alegbra.

My Solution:
80w+86m=82(m+w)
80w+86m=82m+82w
4m=2w
2m=1w

That's fine: you've set up the right equation and simplified it. (By the way, a table can help out here, with "men" and "women" as rows, and average, number, etc. as columns).
The trouble you're having is that you can't turn an equation into a ratio. Take some time to consider what the equation "2m = 1w" means. Answer: it means 'the number of men multiplied by 2 equals the number of women'. This agrees with your other solution path.

Takeaway: ratios and equations are different animals.