In a shop, there are 500 sweet products. 210 are biscuit-based, 160

Question

In a shop, there are 500 sweet products. 210 are biscuit-based, 160 are milk-based and 160 are coconut-based. 80 have both milk and coconut but no biscuit, 70 have milk and biscuit but no coconut, 60 have biscuit and coconut but no milk. How many items have all three ingredients if 200 have none of these ingredients?

(A) 10
(B) 30
(C) 50
(D) 130
(E) Cannot be determined

Show Answer

Official Answer

A

(A) 10
(B) 30
(C) 50
(D) 130
(E) Cannot be determined

Abayomicontrol · Answer

Let the three be x
210-(60-x+x+70+x) =80+x
160-(70-x+x+80-x) =10+x
160-(60-x+x+80-x) =20+x
80+x+10+x+20+x+60-x+80-x+70-x+200+x=500
520+4x-3x=500
520+x=500
X=500-520
X=-20
Option E

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depil248 · Answer

Total = 500 
Biscuit based = 210
Milk based = 160 
Coconut based = 160 
Milk and coconut = 80 
Milk and biscuit = 70 
Biscuit and coconut = 60
All three = x 
None = 200

Now, the formula is Total – None = A + B + C – ( AB + BC + CA) – 2(ABC) 
500 – 200 = 210 + 160 + 160 – (80 + 70 + 60) – 2x 
300 = 530 – 210 – 2x 
2x = 320 – 300 
x = 20/2 = 10

The correct answer is A.

VirFortis · Answer

Total – None = A + B + C – ( AB + BC + CA) + 2(ABC) = 210+160+160-(70+60+80) +2x
500-200 = 530 - 210+2x
2x = 500 - 530 -200 +210 = -30+10 = 20
Ans -> 10.

lexjp1019 ,
For ref : Go through @bunuel's post ->  https://gmatclub.com/forum/formulae-for-3-overlapping-sets-69014.html

PolarisTestPrep · Answer

Hey guys

For questions like this I use a triple Venn diagram that I've included below

The total T is 500

Let:

A = 210 for the total number of biscuit based sweets
B = 160 for milk based
C = 150 for coconut based
n = 200 for the number that don't have any of those ingredients

In a triple venn diagram like this, the total T must be equal to all of the small component letters:

T = a + b + c + d + e + f + g + n

This makes sense because it is just adding up the sweets that only have biscuits, only have milk, only have coconut, have biscuits and coconut but no milk, have all the ingredients, have none of the ingredients, and so on

If you were to add up just A + B + C you would get 530, which is larger than the total

The reason for this is you would be counting the sweets that have two ingredients twice and the ones that have all three ingredients three times

d, e, f would be counted twice and g would be counted three times

In fact A + B + C = a + b + c + 2d + 2e + 2f + 3g

These formulas come in handy in some triple Venn diagram questions and hopefully help conceptualize the meaning behind the diagram better

If something is unclear please ask, these questions are tricky at first but easy once you get the hang of it

So essentially we are given many of the elements of the diagram and the equation and if we plug them in we can derive the ones that we need to answer the question

We are told that:

e = 80 (milk and coconut, no biscuit)
d = 70 (milk and biscuit, no coconut)
f = 60 (biscuit and coconut, no milk)

We are asked how many have all 3, we are asked for the value of g

T = a + b + c + d + e + f + g + n = 500

Let's plug in what we know

500 = a + b + c + 70 + 80 + 60 + g + 200

a + b + 210 + g = 300

a + b + c + g = 90

g = 90 - a - b - c

(D) is eliminated because it is too high

How do we find out the value of a, b, and c?

A = a + d + f + g = a + 70 + 60 + g = a + 130 + g = 210

210 = a + 130 + g

a + g = 80

B = b + d + e + g = b + 70 + 80 + g = b + 150 + g = 160

160 = b + 150 + g

b + g = 10

C = c + f + e + g = c + 60 + 80 + g = c + 140 + g = 160

160 = c + 140 + g

c + g = 20

You can now put a, b, and c in terms of g and plug it into the original equation

a = 80 - g

b = 10 - g

c = 20 - g

Recall from above:

a + b + c + g = 90

(80 - g) + (10 - g) + (20 - g) + g = 90

110 - 3g + g = 90

110 - 2g = 90

-2g = -20

g = 10

The answer is (A)

That's a lot of letters. No more alphabet soup for me

bumpbot · Answer

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In a shop, there are 500 sweet products. 210 are biscuit-based, 160

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