# Separating three types of students into groups with the same number of students of each type

In the Mathematics department of a college, there are 60 first-year students, 84 second-year students and 108 third-year students. All of these students are to be divided into project groups such that each group has the same number of first-, second-, and third-year students. What is the smallest possible size of each group?

• You must start with finding the largest positive integer that divides $60$, $84$ and $108$. – drhab Jan 29 at 12:32

Hint:

The ratio of students in years $$Y_1:Y_2:Y_3$$ is $$60:84:108$$

Can you cancel down this ratio into its simplest form by dividing common factors out?

When you get a ratio of three coprime integers, the group size is their sum.

• Yep, gotcha, very simple, just the HCF of the 3 nos., 12 groups with 21 in each group – Avik Shakhari Jan 29 at 12:45
• 21 is the answer – Avik Shakhari Jan 29 at 12:45
• Actually, I'm preparing for a tough entrance exam in which only math is asked so I need your help to clear my doubts as I have got no other source for it, pls help me guys – Avik Shakhari Jan 29 at 12:46
• You got it! Nice work!! If you have other questions, you should ask them as new questions. Do try to include whatever work you already have done on the problems as well. – Rhys Hughes Jan 29 at 12:54
• f:R -> R be a continuous function such that for any 2 real nos. x and y, |f(x)-f(y)| ≤ 7|x-y|^201 Then A) f(101) = f(202) + 8 B) f(101) = f(201) + 1 C) f(101) = f(200) + 2 D) None – Avik Shakhari Jan 29 at 12:56

You could divide into project groups where each group had one first-year student, one second-year student and one third-year student. If you did this, you would have 60 groups, and each group would have the same number of first-years, the same number of second-years, and the same number of third-years, which is good. However, you would also have 24 second-year students and 48 third-year students without a group. This is bad, and we don't want that.

You could have one group of 60 first-years, one group of 84 second-years, and one group of 108 third-years. Then no student would be without a group, which is good. However, then the groups wouldn't have the same number of first-years, the same number of second-years, nor the same number of third-years. This is bad, and we don't want that.

You could divide into groups where each group has 30 first-years, 42 second-years and 54 third-years. Then you would have two groups with equal composition, and no students left over. This is perfect. However, those two groups turn out to be pretty large. Can you make the groups smaller, while still making sure that all students get a group, and all groups have the same composition of first, second and third-years?

This is what the question is about.

• It says, each group has same no. of 1st, 2nd and 3rd year students, that's the troublesome part here – Avik Shakhari Jan 29 at 12:36
• @AvikShakhari Yes, I added a paragraph about that. – Arthur Jan 29 at 12:36
• Ohhh, thanks a lot, I got u pal – Avik Shakhari Jan 29 at 12:38
• @AvikShakhari something tells me you've misunderstood the question. "Each group having the same number of Year 1, Year 2, Year 3" doesn't mean that the group are divided equally among the years, just that the years are divided equally among the groups. For example, if there were two groups there will be $54$ Y3 students in each, but there doesn't have to be $54$ students from Y1 and Y2, and there won't be. – Rhys Hughes Jan 29 at 12:51
• I got that, that's the point where I misunderstood – Avik Shakhari Jan 29 at 12:57