From a list of integers, all the multiples of 5 are sorted into Set A

Question

From a list of integers, all the multiples of 5 are sorted into Set A and all the even integers are sorted into Set B. If 60% of the integers in Set A and 50% of the integers in Set B are not divisible by 10, which of the following statements must be true?

I. The number of integers in Set A is less than the number of integers in Set B
II. The number of integers in Set B that are divisible by 10 is greater than the corresponding number in Set A
III. The number of odd multiples of 5 in Set A is greater than the number of integers in Set B that are not divisible by 10.

A. I only
B. II only
C. III only
D. I, II and III
E. None of the above

yashikaaggarwal · Answer

Let Set A = { 5,10,15,20,25,30,35,40,45,55} out of which 60% = 6 no. are not divisible by 10
(Since Set A and B are subpart of a list of integers set b will consist of multiple of 5 which are even)
Set B = { 4,8,10,20,24,28,30,40}out of which 50% are not divisible by 10}

Statement 1: The number of integers in Set A is less than the number of integers in Set B
The 40% of set A = 50% of set B
That shows The no. Of integers in set A is greater than set B not the other way round (false)

Statement 2: The number of integers in Set B that are divisible by 10 is greater than the corresponding number in Set A.
The 40% of set A divisible by 10 = 50% of set B divisible by 10
(Both are equal) (false)

Statement 3: The number of odd multiples of 5 in Set A is greater than the number of integers in Set B that are not divisible by 10.
If 40% of set A = 50% of set B
Than 60% of set A (the number of odd multiple of 5 is greater than the even no. 50% of set B not divisible by 10) (true)

Only statement 3 is true.
IMO C

Posted from my mobile device

TestPrepUnlimited · Answer

Note we know nothing about this list of integers, so we can use any numbers we'd like.

The key here is that the number of multiples of 10's in the original list dominates how many elements we have in each set. Let x = {# of multiples of 10's in the original list}.

Then A would have  \frac{x }{ 0.4} = 2.5x  elements. B would have  2x  elements since half are divisible by 10.

Now looking at the statements:
I. A has 2.5x elements, B has 2x elements,  2.5x < 2x . So this is false. 
II.  x > 2.5x . This is false. 
III. The number of odd multiples of 5 in A is 2.5x - x = 1.5. And this statement says  1.5x > x . True.

So only statement 3 is true. Ans: C

ScottTargetTestPrep · Answer

Solution:

We can let A = {5, 10, 15, 20, 25} (notice that 60% of them are not divisible by 10). Similarly, we can let B = {2, 4, 10, 20} (notice that 50% of them are not divisible by 10).

From the two sets we have designed, we see that statements I and II are not true. Statement III is true. However, one could argue that it might be true for the two sets we have designed and might be true for two sets that others might have designed. Therefore, we are going to prove that any two sets A and B one could design, it’s always true that the number of odd multiples of 5 in set A is greater than the number of integers in set B that are not divisible by 10. Here is the proof:

Let x = the number of multiples of 10 in the list of the integers. Notice that the multiples of 10 are the common elements in both sets A and B since they are both multiples of 5 and even integers. Then x must be 40% of the elements in set A and 50% of the elements in set B. In other words, set A has x/(0.4) = x/(2/5) = 5x/2 elements and set B has x/(0.5) = x/(1/2) = 2x elements. The number of odd multiples of 5 in set A is the number of elements in set A minus the number of even multiples of 5 (i.e., multiples of 10). Therefore, the number of odd multiples of 5 in set A is 5x/2 - x = 3x/2. Similarly, the number of integers in set B that are not divisible by 10 is the number of elements in set B minus the number of integers that are divisible by 10 (i.e., multiples of 10). Therefore, the number of integers in set B that are not divisible by 10 is 2x - x = x. Since 3x/2 > x for any positive value of x, we see that statement III must be true.

Answer: C

chacinluis · Answer

A={all multiples of 5}
B={all even integers}
A,B={all multiples of 10}
(A,B-complement)={all multiples of 5 not divisible by 10}
(A-complement,B)={all even numbers not divisible by 10}

.60A is not divisible by 10. Therefore .60A is the portion of A not divisible by 10.
.60A=(A,B-complement)
Therefore 
(A,B)=.40A

.50B is not divisible by 10. Therefore .50B is the portion of B not divisible by 10.
.50B=(A-complement,B)
Therefore
(A,B)=.50B

Therefore
.40A=.50B
A=(5/4)B
A>B

Choice I: The number of integers in Set A is less than the number of integers in Set B
 FALSE . We just showed A>B

II. The number of integers in Set B that are divisible by 10 is greater than the corresponding number in Set A
 FALSE .
The number of integers in B divisible by 10, and the number of integers in A divisible by 10 are exactly the numbers both A and B share. So it is the same for both A and B.

III. The number of odd multiples of 5 in Set A is greater than the number of integers in Set B that are not divisible by 10.
 TRUE 
Number of odd multiples of 5 in Set A=(A,B-complement)=.60A
Number of integers in Set B that are not divisible by 10= (A-complement, B)=.50B

A>B
.60A>.60B
.60A>.60B>.50B
.60A>.50B

Final Answer: C

saro · Answer

HI i considered the below set for case III. 
Set A = 5,15,25,10,20
Set B = 10,20,30,2,4,6
In this case number of integers not divisible by 10 is equal in both sets and hence III is not must be true case. Am i going wrong somewhere?

Fdambro294 · Answer

took me a while to realize that this was really an Overlapping Sets Question cleverly disguised.

I just set up terms according to the Constraints given and Drew a Venn Diagram in which

Set A = all multiples of 5

Set B = all Even Integers

Union of Set A and B Together = Even Integers that are Multiples of 5

Set A - 60% are NOT Divisible by 10. Means that the Multiples of 5 have to end in a Units Digit of 5.

Set A Contains: 
5 ---- 15 ---- 25 ----35 -----45 ----55 
6 Elements that are ONLY Part of Set A

10-----20-----30-----40
4 Elements that are part of BOTH Set A and Set B

Set B MUST Contain the 4 Elements that are part of BOTH Set A and Set B
Since 50% of Set B's Elements are NOT Divisible by 2 and only Even Integers, Set B can only have 4 Even Integers that are ONLY PART of Set B

Set B Contains:
2-----4-----8-----12
4 Elements that are ONLY Part of Set B

10-----20-----30-----40
4 Elements that are part of BOTH Set A and Set B

This scenario satisfies the conditions laid out in the problem. 60% of the Elements in Set A are NOT Divisible by 10. 50% of the Elements in Set B are NOT Divisible by 10.

Which of the following MUST be True?

I. this scenario shows that the Number of Integers in Set A (10) is greater than the Number of Integers in Set B (8)

I must not always be true

II. The Number of Integers in each set that are Divisible by 10 is the Intersection of the 2 Sets = 4 Elements. This is the Same No. of Elements in Each Set.

II. must not always be true.

III. The No. of Odd Multiple of 5 in Set A (which means those elements ONLY Part of Set A) is greater than the number of integers in Set that are NOT Divisible by 10 (which means those Even Integers ONLY Part of Set B)

Set A has 6 Elements ONLY Part of Set A > Set B has 4 Elements ONLY Part of Set B

III. must be true

Answer -C-

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From a list of integers, all the multiples of 5 are sorted into Set A

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From a list of integers, all the multiples of 5 are sorted into Set A

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