Problem Solving

In a box of 40 cookies, 24 of the cookies were round and 20 of them were made of chocolate. If 12 cookies were neither round nor made of chocolate, how many round chocolate cookies were in the box?
(A) 4
(B) 8
(C) 16
(D) 20
(E) 28

Is the OA D?

nakul_anand wrote:Is the OA D?

Let's first know how

Well,

The question stem states that out of 40 cookies, 24 were round and 20 of THEM were made out of chocolate.

So there must be 20 round, chocolate cookies right? I am assuming that THEM refers to the 'round' cookies as opposed to the 'total' (40) cookies.

this is a simple venn diagram problem. Where 40 - 12 = 24 + 20- Round&Chocolate cookies. This gives 16 as the answer!... just corrected myself here

nakul_anand wrote:Well,

The question stem states that out of 40 cookies, 24 were round and 20 of THEM were made out of chocolate.

So there must be 20 round, chocolate cookies right? I am assuming that THEM refers to the 'round' cookies as opposed to the 'total' (40) cookies.

I differ with [spoiler](D)[/spoiler]

Could you please explain why do you differ with D?

i get 16.

Givens: 12 not chocolate and not round
20 chocolate
24 round



set up a 2x2 box

chocolate not chocolate total
round 16 8 24
not round 4 12 16
total 20 20 40

hence round and chocolate = 16

alternatively using a venn diagram

total cookies = 40
12 not chocolate or round so 28 remaining cookies are chocolate, round, or both
create venn for chocolate and round
chocolate = 20 - x
round = 24-x
x= round and chocolate
20-x+x+24-x = 28
x=16

nakul_anand wrote:Could you please explain why do you differ with D?

My friend, Mr. Theo, explains it as under, isn't that too long?

Box contains 40 cookies.
24 of them are round.
20 of them are chocolate.
12 are neither round nor chocolate.

If 24 are round, then 16 must be not round.
If 20 are chocolate, then 20 must be not chocolate.

The count is therefore:

24 round and 16 not round
20 chocolate and 20 not chocolate

Since we have 12 not round and not chocolate, we'll move them to their own category to get:

24 round and 4 not round
20 chocolate and 8 not chocolate
12 not round and not chocolate

The 4 remaining that are not round must be chocolate. Otherwise they would have been counted in the not round and not chocolate.

We'll move them to their own category to get a revised count of:

24 round
16 chocolate and 8 not chocolate
12 not round and not chocolate
4 not round and chocolate

The 8 that are not chocolate must be round. Otherwise they would have been counted in the not round and not chocolate also.

We'll move them to their own category to get a revised count of:

16 round
16 chocolate
12 not round and not chocolate
4 not round and chocolate
8 round and not chocolate = 8

The 16 that are round and the 16 that are chocolate must be the same cookies.

We'll move them to their own category to get a revised count of:

12 not round and not chocolate = 12
4 not round and chocolate = 4
8 round and not chocolate = 8
16 round and chocolate = 16

The total number of cookies is equal to 12 + 4 + 8 + 16 = 40 as it should be.

Your answer is that the number of cookies that are round and chocolate is 16.

That would be selection [spoiler]C[/spoiler].

I will try to reduce the hard work, if my explanation is still needed.

this is how i approached it :
total cookies + 40
Round cookies = only round + round & chocolate = 28
choc cookies = only choc + round & chocolate = 20

total cookies = 40
neither round nor choc = 12
Therefore, only round + round & choc + only choc = 40 - 12 = 28

The 4 choices are values of round and choc. using them one by one....
only round / round & choc / only choc....only choice c satisfies all above conditions
eg>24/4/16 .. sum is not equal to 28

A-->Number of cookies that is round

B--> Number of choclate cookies


Number of cookies that is either round or choclate (ie)

P(AuB)= 40- 12 =28

We know that,

P(AuB)=P(A)+P(B)-P(AnB)


28=24+20-x

x=16

24/40 are round
20/40 are chocolate
12/40 are not round and not chocolate

24+12 = round cookies + not round cookies (that are not chocolate) = 36
40-36 = the number of cookies left that have to be chocolate and not round = 4

20-4 = the number of chocolate cookies less the number of chocolate not round cookies = 16, which is the number of round cookies.

no need for complicated algebra.
out of 40, 12 is neither roundn or choco. so
20+24-x = 28
x= 16

total = 40
round=24
chocolate=20
neither=12
total = round+chocolate-Both+neither
40=24+20-both+12
both = 16

analyst218 wrote:no need for complicated algebra.
out of 40, 12 is neither roundn or choco. so
20+24-x = 28
x= 16

so cute looking work when on test