Sweater and glasses
There are 40 students in the classroom.
- 50% of students wear both glasses and sweaters.
- Among the students who don't wear glasses, 75% of them wear sweaters.
If two students wearing both glasses and sweaters leave the classroom, then which of the following is a possiblity for the percentage of students who wear neither glasses nor sweaters?
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
2 solutions
Moderator note:
Good explanation. Why can't we find the exact number of people who do not wear glasses or sweaters?
Shouldn't the question be framed better? It needs to ask 'what could be the percentage' instead of 'what is'. According to your solution, 5 different solutions are possible. And, we are assuming an extreme case here. If z were to be 5, x would be 0, which is not the most desirable situation, taking into account the structure of the question.
Log in to reply
I absolutely agree. I was stumped at first because I did not see how we could reasonably deduce how many students do not wear glasses. Only by going through all the possibilities, can it be seen that only one answer is possibly correct, but not necessarily true.
Log in to reply
Thanks. I've updated the options to reflect actual possibilities.
Terrible problem. When the 20 students leave the room, the count starts from zero !! So 75% of 20 is 15. So 5 wear nothing. 5 / 20. is 20 %. !!
Log in to reply
What do you mean by "20 students leave the room"? Only two students leave the room.
Also, the 75% statistic only applies to the group before the students left, and not afterwards.
In the classroom there are 40 - 2 1 *40 = 20 students who wear
a = just glasses
b = just sweaters
c = neither
⟹ a + b + c = 20
⇔ c = 20 - a - b ( ∗ )
Number of students who don't wear glasses is 20 - a, so we can write
2 0 − a b = 4 3 ⇔ 4b = 60 - 3a ⇔ b = 15 - 4 3 a.
Substituting this to ( ∗ ) gives us
c = 2 0 − a − ( 1 5 − 4 3 a ) = 5 − 4 1 a ( ∗ ∗ )
Now we want to calculate the percentage of students who wear neither glasses nor sweaters. Two students left the room so this comes out as
3 8 c = ( ∗ ∗ ) 3 8 5 − 4 ⋅ 3 8 1 a ≈ 1 3 . 2 % − 1 5 2 1 a ≤ 1 3 . 2 %
So 13.2% is the percentage of students who wear neither glasses nor sweaters if nobody is wearing just glasses.
The answer, 13.2 percent, is not correct because the students who do wear glasses but do not wear sweater have not been taken into account.
There are 20 students who wear glasses and sweaters, that leaves 20 for the rest who wear glasses but no sweaters, no glasses and sweaters, and no glasses and no sweaters. Let's label those groups as x, y and z, respectively.
From the problem, we know that y=0.75(y+z), which gives us y=3z. Then we have x+y+z=20, which we can rewrite as x+4z=20. Now, since these are students, the values of x and z can only be integers, so if we try out a couple of numbers, x can be 0,4,8,12,16, while z can be 5,4,3,2,1, respectively. The percentage we are looking for is z/38 (40-2 students leaving). After trying out each of the 5 numbers, only 5/38 = 13.158% ~ 13.2% is among the solutions.