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# A surprise inspection of 200 companies revealed that 80 of

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A surprise inspection of 200 companies revealed that 80 of [#permalink]

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19 Jul 2006, 14:54
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A surprise inspection of 200 companies revealed that 80 of them had evaded paying taxes, 60 had bribed public officials and 90 had filed fraudulent Social Security claims. How many of the 200 companies inspected were not guilty of any of these offenses?

(1) 90 companies were guilty of at least two of these offenses.
(2) 5 companies were guilty of all three offenses.
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19 Jul 2006, 15:05
I am tempted to go with C ...I will explain if that is correct
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20 Jul 2006, 01:52
First look tells me to go with C.
But Kevin's pride would be hurt if he gave such an easy question.

Ok let me try

I think A is sufficient.

A says there are 90 people with at least 2 of the offenses.
Hence it means that these 90 also include the number of people who have commited all three offenses.

Therefore total number who have not commited any of the offense
200 = 90+80+60 - 90

That is 60.
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20 Jul 2006, 01:59
Drawing a venn diagram, we get:

#(all three offences) = n
#(tax + social security) = a
#(social security + bribe) = c
#(bribe + tax) = b
#(tax only) = 80-n-a-b
#(social security only) = 90-n-a-c
#(bribe only) = 60-n-b-c
#(neither) = x

St1:
Total number = 200 = 230-2n-a-b-c + x
200 = 140-n+x
x-n = 60

Insufficient as we have no information about the # of companies that committed all 3 offences.

St2:
Total number = 200 = 230-2(5)-a-b-c+x
200 = 220-a-b-c+x
a+b+c-20 = x

Insufficient as we do not have information about # of comapnies that committed 2 offences.

Using St1 and St2:
x-n = 60
x-5 = 60
x = 65

Sufficient.

Ans C
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21 Jul 2006, 08:57
"at least 2" ...so A is sufficient.
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21 Jul 2006, 10:31
C

A+B+C-AB-BC-CA+ABC + NONE = Total

St1: 80+60+90-(AB+BC+CA+ABC)+2ABC + x = 200
230- 90 + 2ABC + x = 200
we don't know ABC so can't find x.: INSUFF

St2: 80+60+90-(AB+BC+CA)+ 5 + x = 200 Clearly INSUFF

Combined:
230- 90 + 2ABC + x = 200 becomes
140+10+x = 200
x = 50: SUFF
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21 Jul 2006, 15:05
St1: 80+60+90-(AB+BC+CA+ABC)+2ABC + x = 200

Does AB mean A intersect B or A intersect B intersect not C

I think from what you wrote earlier, it means the former.
If this is true, we are NOT given AB+BC+CA+ABC.
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21 Jul 2006, 15:39
kevincan wrote:
St1: 80+60+90-(AB+BC+CA+ABC)+2ABC + x = 200

Does AB mean A intersect B or A intersect B intersect not C

I think from what you wrote earlier, it means the former.
If this is true, we are NOT given AB+BC+CA+ABC.

I meant former. Ok my bad.
Lets do it again

A+B+C-AB-BC-CA+ABC + NONE = Total
i.e A+B+C-AB-BC-CA+2ABC-2ABC+ ABC + NONE = Total
i.e A+B+C-(AB+BC+CA-2ABC)-2ABC+ ABC + NONE = Total
i.e A+B+C-(AB+BC+CA-2ABC)-ABC+ NONE = Total

St1: We are given (AB+BC+CA-2ABC) = 90
so 80+60+90 - 90 + ABC + x = 200
ABC + x = 60: INSUFF

St2: we are given ABC = 5: Clearly INSUFF

Together:
5 + x = 60
x = 55

Is this correct????
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21 Jul 2006, 16:22
Correct! It's too bad we can't draw circles to illustrate, huh?
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21 Jul 2006, 20:14
kevincan wrote:
Correct! It's too bad we can't draw circles to illustrate, huh?

I did with circles but made a small mistake.
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21 Jul 2006, 20:14
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