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grouping question gmat:using ratios

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by viny » Sun Aug 11, 2013 6:04 pm
Orange Computers is breaking up its conference attendees into groups. Each
group must have exactly one person from Division A, two people from Division B,
and three people from Division C. There are 20 people from Division A, 30 people
from Division B, and 40 people from Division C at the conference. What is the
smallest number of people who will not be able to be assigned to a group?

I thought of solving it with the help of ratios.Considering the ratio to be 1x:2x:3x and for every number for group there is 1 in ,2 in B and 3 in C
now these all should be integers ie my total number should be such that when i divide them in this ratio I get an integer as the number of groups this is possible if only my X is divisible by 6(lcm of 3 and 2)
now for that i can consider options as 6,12,18
Only 12 fits the criteria according to the max number of available people in each group as 6 would lead to less number of people being assigned I would choose that as my answer,and if I consider my x to be 18 then 2x would exceed the max number of people in my group B.
leaving me option 12!!
so 12:24:36 leaving a the number of 20+30+40 - (12+24+36) = 8+6+4 =18
but the answer is 12,something is wrong with either my understanding of ratios or I cannot do it using them i do not know.
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by GMATGuruNY » Sun Aug 11, 2013 7:29 pm
viny wrote:Orange Computers is breaking up its conference attendees into groups. Eachgroup must have exactly one person from Division A, two people from Division B, and three people from Division C. There are 20 people from Division A, 30 people from Division B, and 40 people from Division C at the conference. What is the smallest number of people who will not be able to be assigned to a group?
This is a MAX/MIN problem.
To MINIMIZE the number of people not assigned to a group, we must MAXIMIZE the number of people who ARE assigned to a group.

Each group must include 3 people from Division C.
Division C consists of 40 people.
Since 13*3 = 39, at most 13 groups can be formed that each include 3 people from Division C.
Thus, the MAXIMUM number of groups that can be formed = 13.

Division A:
Since each of the 13 groups will include 1 person from Division A, the maximum number of people from Division A assigned to a group = 1*13 = 13.
Since Division A consists of 20 people, the minimum number of people from Division A NOT assigned to a group = 20-13 = 7.

Division B:
Since each of the 13 groups will include 2 people from Division B, the maximum number of people from Division B assigned to a group = 2*13 = 26.
Since Division B consists of 30 people, the minimum number of people from Division B NOT assigned to a group = 30-26 = 4.

Division C:
Since each of the 13 groups will include 3 people from Division C, the maximum number of people from Division C assigned to a group = 3*13 = 39.
Since Division C consists of 40 people, the minimum number of people from Division C NOT assigned to a group = 40-39 = 1.

Thus, the least number of people not assigned to a group = 7+4+1 = 12.
Last edited by GMATGuruNY on Mon Aug 12, 2013 12:00 am, edited 1 time in total.
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by viny » Sun Aug 11, 2013 8:13 pm
yes but cant this be solved using ratios.Whats wrong is solving this way
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by [email protected] » Sun Aug 11, 2013 10:58 pm
Hi Viny,

Using a ratio can work on this question, BUT you have to account for IF the ratio actually fits all 3 groups of people (remember that you're bounded by the fact that Group A is 20 people, Group B is 30 people and Group C is 40 people).

In this question, you can either max out group A, group B or Group C

If you max out Group A, you'd have:
20 people from A
40 people from B
60 people from C
However, this is IMPOSSIBLE, since there aren't enough total people in Group B or Group C to make this math work

If you max out Group B, you'd have:
15 people from A
30 people from B
45 people from C
However, this TOO is IMPOSSIBLE, since there aren't enough total people in Group C to make this math work

If you max out Group C, you'd have:
13 people from A
26 people from B
39 people from C
This IS POSSIBLE; the number of people who AREN'T in a group is 7 (from A) + 4 (from B) + 1 (from C) = 12

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by GMATGuruNY » Mon Aug 12, 2013 12:16 am
viny wrote:yes but cant this be solved using ratios.Whats wrong is solving this way
Here's how we could use ratios.

Let x = the total number of groups.
In each group, A:B:C = 1:2:3.
Thus:
The TOTAL number of people from A assigned to a group = x.
The TOTAL number of people from B assigned to a group = 2x.
The TOTAL number of people from C assigned to a group = 3x.
Determine the MAXIMUM POSSIBLE value of x.

Division A:
Since there are a total of 20 people in A, x≤20.
Division B:
Since there are a total of 30 people in B, 2x≤30, implying that x≤15.
Division C:
Since there are a total of 40 people in C, 3x≤40, implying that x≤40/3.

The greatest integer value that satisfies all 3 inequalities is x=13.
Thus, the greatest number of groups that can be formed = 13.
From here, we can proceed as I did in my solution above.
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