News paper article regarding teenagers who smoke

[Probability]

A news paper published that a quarter of teenage girls and a six of teenage boys smoke regular, research had shown. The news paper said that these figures amounted to five twelves of the teenage population, which suggested that nearly half of the population of teenagers smoked?

I initially did the following math to see if the advertisement was correct;

[MATH]\frac{1}{4}+\frac{1}{6}=\frac{5}{12}[/MATH]
I kept reading it over as it came across and could not make my mind up if half the teenage population amounted to five twelves of the teenage population?

I started to ask myself some questions;

1 / How many girls smoke,
2 / How many boys smoke,
3 / What is the population size.

So I said to myself, [MATH]\frac{3}{4}[/MATH]of girls don't smoke and [MATH]\frac{1}{6}[/MATH]of boys smoke. That means 83% of the male population don't smoke, and 75% of girls don't smoke. Based on those figures the advertisement must have a mistake in it!

So then I asked myself, how do I find out what fraction of all teenagers smoke?

I came up with this mathematical model;

[MATH]\frac{3}{4}[/MATH]of girls do not smoke, hence

[MATH]\frac{3}{4}-\frac{1}{4}=\frac{1}{2}[/MATH] therefore [MATH]\frac{1}{2}\times\frac{1}{4}=\frac{1}{8}[/MATH]
then [MATH]\frac{1}{6}[/MATH]of boys smoke, hence

[MATH]\frac{1}{2}\times\frac{1}{6}=\frac{1}{12}[/MATH]
I then came to the conclusion that all teenagers that smoked must be;

[MATH]\frac{1}{8}+\frac{1}{12}=\frac{2}{24}+\frac{3}{24}=\frac{5}{24}[/MATH]
That amounts to about 21% of the teenage population.

What do you ladies/gents think?

 

[Dr.Peterson]

Assuming that half of teens are girls and half are boys, you have it exactly right. This is not a situation where you just add the two fractions as given. Notice that the answer is exactly half of the claimed answer, 5/12, because you are multiplying each fraction by 1/2.

So it's really about 1/4 (which would be 6/24), not about 1/2 (which would be 6/12).

By the way, the correct English names for these fractions are 5/12 = five twelfths, and 1/6 = a sixth, not "five twelves" and "a six".

 

[Steven G]

Suppose there were 120 teenagers---60 boys and 60 girls.

1/4 of the girls smoke = 15 smoke.
1/6 of the boys smoke = 10 smoke

So out of 120 teenagers, 15+10 =25. smoke. That is 25 out of 120 smoke which is the same as 5 out of 24 NOT 5/12!