Joseph's solution involved trial and
First I decided that the number of pupils in the class had to be a
mulitple of 10 so that I could work out the percentages with no
fractions of people.
I started with 20 pupils in the class. This meant that 14 passed
one question (70%) and 12 the other (60%). As 9 pupils passed both
questions this meant that 5 got only the first one right (14-9) and
3 the second one, making a total of 9 + 5 + 3 = 17 pupils. This is
not enough as I started my working with 20 pupils.
Next I tried 30 pupils in the class. This time 21 passed one
question (70% of 30) and 18 passed the other question (60%). Taking
out the 9 pupils who passed both gave:
21-9 = 12 passing one question,
18-9 = 9 passing the other
9 passing both.
This makes a total of 30 (12+9+9), which is right
There were 30 pupils in the class.
This is Zi Heng's solution:
|70% + 60%
|130% - 100%
||= 9 pupils
||= 9/30 * 100
||= 30 pupils.
30 pupils took the exam.
Andrei, School 205, Bucharest, Romania solved
this problem using a Venn diagram.
Let A be the set of solvers of the first problem, and B the set
of solvers of the second problem and the number in set A be written
$n(A)$ etc. Their intersection has 9 elements: $$n(A\cap B)=9$$
Their union contains all students. It is evident that: $$n(A\cup
B)=n(A)+n(B)-n(A\cap B)$$ If x is the number of students
participating in the exam, then A has 70 per cent of x elements, B
has 60 per cent of x elements, and relation (2) can be re-written
as $x=0.7x+0.6x-9$ or $x=30$. So, 30 pupils came to the exam, 21
solved the first problem and 18 the second one.
Prateek , James , Alan , Jenny and Robert also
sent in good solutions. Joseph's solution to the second part of the
As all the pupils solved at least one problem 44% solved both
and this is 11/25 in its lowest form.
72% - 44% = 28% = 7/25
This means that the number of pupils in the group must be a
multiple of 25.
So, if 25 pupils took the exam14 solved one problem (7+7) and
If 50 pupils took the exam 28 solved one problem and 22 both
and so on...