difficult problem?

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dzelkas: Junior | Next Rank: 30 Posts; Posts: 13; Joined: Tue Jun 05, 2007 1:29 pm

difficult problem?

by dzelkas » Sun May 25, 2008 2:44 pm

I can't seem to figure out the below problem. Can someone please help
the correct answer should be 25%

Foodmart customers regularly buy at least one of the following products: milk, chicken, or apples. 60% of shoppers buy milk, 50% buy chicken, and 35% buy apples. If 10% of the customers buy all 3 products, what percentage of Foodmart customers purchase 2 of the above products?

a) 5%
b) 10%
c) 15%
d) 25%
e) 30%

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Source: Beat The GMAT — Problem Solving | Sun May 25, 2008 2:44 pm

netigen: Legendary Member; Posts: 631; Joined: Mon Feb 18, 2008 11:57 pm; Thanked: 29 times; Followed by:3 members

by netigen » Sun May 25, 2008 3:56 pm

Lets say,

Milk = M
Chicken = C
Apple = A

MAC = intersection of Milk, chicken and apple

similarly,
MA = intersection of Milk and apple
MC = intersection of Milk, chicken
AC = intersection of chicken and apple

then

100 = M + A + C - (MA+MC+AC) + MAC
(MA+MC+AC) = 60+50+35-100+10 = 45

We need to find all cases where only 2 items are purchased = (MA+MC+AC) - 2 x MAC = 45 - 2 x 10 = 25%

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luvaduva: Senior | Next Rank: 100 Posts; Posts: 84; Joined: Tue Feb 05, 2008 8:52 pm; Thanked: 8 times

by luvaduva » Sun May 25, 2008 5:05 pm

netigen wrote:Lets say,
We need to find all cases where only 2 items are purchased = (MA+MC+AC) - 2 x MAC = 45 - 2 x 10 = 25%

3. No of persons in exactly two of the sets: P(AnB) + P(AnC) + P(BnC) – 3P(AnBnC)

Should it be 15?

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netigen: Legendary Member; Posts: 631; Joined: Mon Feb 18, 2008 11:57 pm; Thanked: 29 times; Followed by:3 members

by netigen » Sun May 25, 2008 6:40 pm

I stand corrected:

No of persons in exactly two of the sets: P(AnB) + P(AnC) + P(BnC) – 3P(AnBnC)
No of persons in two or more of the sets: P(AnB) + P(AnC) + P(BnC) – 2P(AnBnC)

so answer should be 15%

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sukrant26: Senior | Next Rank: 100 Posts; Posts: 51; Joined: Sun Mar 23, 2008 6:54 pm; Location: San Francisco; Thanked: 2 times

by sukrant26 » Sun May 25, 2008 8:14 pm

60+50+35-100+10 = 55

Hence 55-3X10 = 25

Am I correct?

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Stuart@KaplanGMAT: GMAT Instructor; Posts: 3225; Joined: Tue Jan 08, 2008 2:40 pm; Location: Toronto; Thanked: 1710 times; Followed by:614 members; GMAT Score:800

Re: difficult problem?

by Stuart@KaplanGMAT » Sun May 25, 2008 10:13 pm

dzelkas wrote:I can't seem to figure out the below problem. Can someone please help
the correct answer should be 25%

Foodmart customers regularly buy at least one of the following products: milk, chicken, or apples. 60% of shoppers buy milk, 50% buy chicken, and 35% buy apples. If 10% of the customers buy all 3 products, what percentage of Foodmart customers purchase 2 of the above products?

a) 5%
b) 10%
c) 15%
d) 25%
e) 30%

Basically, if someone is in two groups they're counted twice, so we need to subtract them once; if someone is counted in three groups they're counted three times, so we need to subtract them twice.

So:

True # = total group a + total group b + total group c - (ab + ac + bc) - 2(abc)

100 = 60 + 50 + 35 - (doubles) - 2(triples)

100 = 145 - 2(10) - doubles
doubles = 145 - 20 - 100
doubles= 145 - 120 = 25

Note that if there were also some people in none of the 3 groups, the formula would have been:

True # = total group a + total group b + total group c - (ab + ac + bc) - 2(abc) + total in none of a/b/c

but in this question we know that every shopper buys at least one product.