overlapping set

[guerrero]

Among 200 people, 56% like strawberry jam, 44% like apple jam, and 40% like raspberry jam. If 30% of the people like both strawberry and apple jam, what is the largest possible number of people who like raspberry jam but do not like either strawberry or apple jam?
A)20
B)60
c)80
D)86
E)92

How to use 3 set overlapping formula to solve this ?

OAB

[Atekihcan]

We can use 3 overlapping set formula and solve this but there is a better and less time consuming method to solve this.

(56 + 44 - 30)% = 70% people like either strawberry or apple jam
So, at the most (100 - 70)% = 30% people can be in the category "like raspberry jam but do not like either strawberry or apple jam"

30% 0f 200 = 60

Answer : B

[Atekihcan]

Using formula of 3 overlapping set...

Total = S + A + R - SA - SR - AR + ASR + N
Where, S = number of people who like strawberry jam, SA = number of people who like both strawberry and apple jam, ASR = number of who like all of them, N = number of people who like none of them and so on.

Now, Total = 200, S = 2*56 = 112, A = 2*44 = 88, R = 2*40 = 80, and SA = 2*30 = 60

We need to find out the largest possible number of people who like raspberry jam but do not like either strawberry or apple jam, i.e. the maximum value of [R - (AR + SR) + ASR] = X (say)

Now, total = S + A + R - SA - SR - AR + ASR + N
So, total = [R - (AR + SR) + ASR] + S + A - SA + N
So, total = X + S + A - SA + N
So, X = total - [S + A - SA + N]
So, X = 200 - [112 + 88 - 60 + N]
So, X = 200 - [140 + N]
So, X = 60 - N

Now, X will be maximum when N is minimum.
Minimum possible value of N is 0.
So, maximum possible value of X is 60

Answer : B

[GMATGuruNY]

Among 200 people, 56% like strawberry jam, 44% like apple jam, and 40% like raspberry jam. If 30% of the people like both strawberry and apple jam, what is the largest possible number of people who like raspberry jam but do not like either strawberry or apple jam?
A)20
B)60
c)80
D)86
E)92

How to use 3 set overlapping formula to solve this ?

OAB

To MAXIMIZE the number who like only raspberry, all of the 200 people must like at least one of the 3 flavors.
Let S = strawberry, A = apple, and R = raspberry.

T = S + A + R - (SA + SR + AR) - 2(SAR)

The big idea with overlapping group problems is to SUBTRACT THE OVERLAPS.
When we add together everyone in S, everyone in A, and everyone in R:
Those who like exactly 2 flavors (SA+SR+AR) are counted twice, so they need to be subtracted from the total ONCE.
Those who like all 3 flavors (SAR) are counted 3 times, so they need to be subtracted from the total TWICE.
By subtracting the overlaps, we ensure that no one is overcounted.

We are told the following:
S = 56%
A = 44%
R = 40%.
SA = 30%.

Let T = 100%.
Plugging these values into the formula, we get:

100 = 56 + 44 + 40 - (30 + SR + AR) - 2(SAR)
100 = 110 - (SR + AR) - 2(SAR)
SR + AR + 2(SAR) = 10.

Since SR + AR + 2(SAR) represents all of the overlaps that include R, the percentage who like R and at least one other flavor = 10%.
Thus, the maximum percentage who could like ONLY R = (total percentage who like R) - (percentage who like R and at least one other flavor) = 40-10 = 30%.
Since there are 200 people, we get:
Maximum number who could like only R = .3(200) = 60.

The correct answer is B.

[makhija1]

Hello,

Could someone please explain the difference between these two overlapping statements and how they both yield the same result?

Total = S + A + R - (SA + SR + AR) - 2(SAR)
and
Total = S + A + R - SA - SR - AR + ASR + N

I am getting confused as to which to use when.

Thanks,
Adi