Sets, students

[francoisph]

In a class of 120 students numbered 1 to 120, all even numbered students opt for Physics, whose numbers are divisible by 5 opt for Chemistry and those whose numbers are divisible by 7 opt for Math. How many opt for none of the three subjects?

[Patrick_GMATFix]

Rephrase: "How many integers from 1 to 120 are not divisible by 2, 5 or 7?"

We start with 120 integers. Let's remove those that are divisible by any of the 3 factors above

Remove all multiples of 2 (60 evens). This leaves us with 60 odds. These are 1, 3, 5...119.

Of these, we should remove all odd multiples of 5. These are 5*1, 5*3, 5*5, 5*7...5*23 (115). There are 12 of them. Removing these leaves us with 120-60-12 integers left.

Of the 60 odds, we should also remove all odd multiples of 7. These are 7*1, 7*3, 7*5...7*17 (119). There are 9 of them. However, not all 9 need to be removed. We've already removed the multiples of 5 (7*5, 7*10, 7*15) so we cannot remove them twice. This means that we only remove 6 additional integers. Of the original 120 integers there are 120-60-12-6 left. These numbers are not multiples of 2, 5 or 7.

The answer is 42.

[amising6]

In a class of 120 students numbered 1 to 120, all even numbered students opt for Physics, whose numbers are divisible by 5 opt for Chemistry and those whose numbers are divisible by 7 opt for Math. How many opt for none of the three subjects?

basically you need to find number between 1 to 120 which are not divisible by 2,5,7

there is a straight formula for that
number=120(1-1/2)(1-1/5)(1-1/7)
120*1/2*4/5*6/7
=(12*4*6)/7=288/7=41 ignore remainder

[Rahul@gurome]

Let P be the students who opted for Physics.
Let C be the students who opted for Chemistry.
Let M be the students who opted for Math.

First let us find the number of students who opted for all 3 subjects.
Then P = 120/2 = 60
C = 120/5 = 24
M = 120/7 = 17 (we take the integer part only)
Now, the numbers ending in "0" (10, 20, 30, 40 and so on till 120) that are divisible by 5 are also divisible by 2(even numbers). So, students who opted for both P and C = 120/LCM(2, 5) = 120/10 = 12

Similarly, Students who opted for both P and M = 120/LCM(2, 7) = 120/14 = 8
Students who opted for both C and M = 120/LCM(5, 7) = 120/35 = 3

Now, Students who opted for all 3 subjects = 120/ LCM (3, 5, 7) = 120/70 = 1

Therefore, students who opt for none of the three subjects = 120 - {60 + 24 + 17 - (12 + 8 + 3) + 1} = 41

[Patrick_GMATFix]

there is a straight formula for that
number=120(1-1/2)(1-1/5)(1-1/7)
120*1/2*4/5*6/7
=(12*4*6)/7=288/7=41 ignore remainder

Thanks for sharing the formula!

[jube]

Total Physics = 60
Total Chem = 24
Total Maths = 17

P&C = 11
P&M = 7
C&M = 2
P&C&M = 1

P Only = 41
C Only = 10
M Only = 7

Therefore none of the 3 = 120 - (11+7+2+1+41+10+7) = 120 - 79 = 41

[pharmxanthan]

In a class of 120 students numbered 1 to 120, all even numbered students opt for Physics, whose numbers are divisible by 5 opt for Chemistry and those whose numbers are divisible by 7 opt for Math. How many opt for none of the three subjects?

basically you need to find number between 1 to 120 which are not divisible by 2,5,7

there is a straight formula for that
number=120(1-1/2)(1-1/5)(1-1/7)
120*1/2*4/5*6/7
=(12*4*6)/7=288/7=41 ignore remainder

Thanks amising6 for the formula. How does the formula work? Where did you find the formula? I am interested to know the logic behind the formula....

[amising6]

In a class of 120 students numbered 1 to 120, all even numbered students opt for Physics, whose numbers are divisible by 5 opt for Chemistry and those whose numbers are divisible by 7 opt for Math. How many opt for none of the three subjects?

basically you need to find number between 1 to 120 which are not divisible by 2,5,7

there is a straight formula for that
number=120(1-1/2)(1-1/5)(1-1/7)
120*1/2*4/5*6/7
=(12*4*6)/7=288/7=41 ignore remainder

Thanks amising6 for the formula. How does the formula work? Where did you find the formula? I am interested to know the logic behind the formula....

let start by the simple concept


c if you take number 1,2,3,4,5,6,7,8,9,10 and find number divisible by 5 .now every 5 th number in this will be divisible by 5
so in number 1 to 10 2 number i.e 5,10 will be divisible by 5 .
so how many number wont be divisible by 5 =10(1-1/5)=8 so 8 number not divisible by 5
now let us extend the concept first find out all teh number divisible by 2 1.e 10(1/2)=5 now if there will be 5 number which is not divisble by 2 and u get from 10(1-1/2)=5

[outreach]

formula is grt

In a class of 120 students numbered 1 to 120, all even numbered students opt for Physics, whose numbers are divisible by 5 opt for Chemistry and those whose numbers are divisible by 7 opt for Math. How many opt for none of the three subjects?

basically you need to find number between 1 to 120 which are not divisible by 2,5,7

there is a straight formula for that
number=120(1-1/2)(1-1/5)(1-1/7)
120*1/2*4/5*6/7
=(12*4*6)/7=288/7=41 ignore remainder