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 BII. The number of integers in Set B that are divisible by 10 is greater than the corresponding number in Set AIII. 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 onlyb)II onlyc)III onlyd)I, II and IIIe)None of the aboveCorrect answer is option 'C'. Can you explain this answer?

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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

Correct answer is option 'C'. Can you explain this answer?

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

Given:

Within a list of integers:
- Set A contains multiples of 5
  - Let the total number of integers in Set A be a
  - 60% of these a multiples of 5 are not divisible by 10 (= 2*5)
    - That is, 0.6a multiples of 5 are not divisible by 2 (hence, are odd)
    - And, the remaining 0.4a multiples of 5 are even
Set B contains even integers
- Let the total number of integers in Set B be b
- 50% of these b even integers are not divisible by 10 (= 2*5)
  - That is, 0.5b even integers are not multiples of 5
  - And, the remaining 0.5b even integers are multiples of 5
Representing the given and inferred information in a Venn Diagram:

To find: Which of the 3 statements must be true?

Approach:

Upon reading the 3 statements, we see that to evaluate them, we first need to know the actual value of a and b, or the value of a in terms of b.
Once we have a relation between a and b, we’ll evaluate the 3 statements one by one to see which is true without exceptions (that is, true for all values of a and b)

Working Out:

Finding a relation between a and b
- From the Venn Diagram, we notice that 0.4a = 0.5b
- That is, a=5b/4
Evaluating Statement I
- The number of integers in Set A is less than the number of integers in Set B
- Statement I says that a < b
- However, since a=5b/4, a>b
- So, clearly, Statement I is false
Evaluating Statement II
- The number of integers in Set B that are divisible by 10 is greater than the corresponding number in Set A
- (Number of integers in Set B that are divisible by 10) = (Overlap zone between Sets A and B) = (Number of integers in Set A that are divisible by 10)
- So, Statement II is false
Evaluating Statement 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
- Statement III says that 0.6a > 0.5b
- Now, we’ve determined that a=5b/4
- This means, 0.6a=0.6 ∗ 5b/4 = 3b/4 = 0.75b
  - Since b is a positive number, 0.75b will be greater than 0.5b
  - Therefore, 0.6a is indeed greater than 0.5b
  - Hence, Statement III is true for all values of a and b

Determining the correct answer choice
- Thus, we’ve determined that only Statement III is a must be true statement

Looking at the answer choices, we see that the correct answer is Option C

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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 BII. The number of integers in Set B that are divisible by 10 is greater than the corresponding number in Set AIII. 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 onlyb)II onlyc)III onlyd)I, II and IIIe)None of the aboveCorrect answer is option 'C'. Can you explain this answer?

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