I don't understand a seemingly simple number of students in two classes ratio

[rasen58]

Problem:

of the students in the Math Club are in the Science Club, and

of the students in the Science Club are in the Math Club. There are 15 students in the Science Club. How many students are in the Math Club?

The solution to this seems to involve using the ratios and both facts.
But I don't understand why we even need to care about the second fact.

We are given from the first fact that 30% of the students in the math club are in the science club, and we know that there are 15 students in the science club.
So why can't we just do (0.3)M = 15 => M = 50? (where M = number of students in the math club).

The actual number of students in the math club is 40, but I don't understand why what I did is wrong. Would appreciate any help!

 

[skeeter]

So why can't we just do (0.3)M = 15 => M = 50? (where M = number of students in the math club).

because all 15 members of the science club are not in the math club.


80% of 15 science club members = 12 science club members are in the math club

30% of M, where M is the number of members in the math club, are in the science club = 12

30% of M = 12 means M = 40

 

[JeffM]

Assign variables to ALL the unknowns.

p = number in science club but not in math club
q = number in both science and math club
r = number in math club but not in science club
s = number in science club
t = number in math club

Five unknowns requires five equations.

What do we know:

[MATH]s = p + q.[/MATH]
[MATH]t = q + r.[/MATH]
[MATH]s = 15.[/MATH]
[MATH]0.8s = q.[/MATH]
[MATH]0.3t = q.[/MATH]
How do we use this information.

[MATH]0.8s = q \implies 0.8 * 15 = q \implies q = 12.[/MATH]
[MATH]s = p + q \implies 15 = 12 + p \implies p = 15 - 12 = 3.[/MATH]
[MATH]0.3t = q \implies t = \dfrac{12}{0.3} = 40.[/MATH]
[MATH]t = q + r \implies 40 = 12 + r \implies r = 40 - 12 = 28.[/MATH]
Now let's check all that.

[MATH]3 + 12 = 15.\ \checkmark[/MATH]
[MATH]12 + 28 = 40.\ \checkmark[/MATH]
[MATH]0.8 * 15 = 12.\ \checkmark[/MATH]
[MATH]0.3 * 40 = 12. \ \checkmark[/MATH]
It all makes sense.