CAT Quantitative Aptitude Questions | CAT Set theory Questions - Union and Intersection

The question is from CAT Set theory. It combines number theory with set theory. Our task is to find out the number of elements in all three sets. CAT exam is known to test on basics rather than high funda ideas. CAT also tests multiple ideas in the same question, and 2IIMs CAT question bank provides you with CAT questions that can help you gear for CAT Exam CAT 2023. Set Theory (especially constructing venn diagrams) is a frequently tested topic. Make sure you know the basics from this chapter.

Question 8: Set A comprises all three digit numbers that are multiples of 5, Set B comprises all three–digit even numbers that are multiples of 3 and Set C comprises all three–digit numbers that are multiples of 4. How many elements are present in A ∪ B ∪ C?

1. 420
2. 405
3. 555
4. 480

Video Explanation

Explanatory Answer

Method of solving this CAT Question from Set theory: This question is more of a Number Systems question than a Set Theory one!!

Set A = {100, 105, 110, ….995} ↦ {5 * 20, 5 * 21, ….. 5 * 199} ↦ 180 elements.
Set B = {102, 108, 114, ……996} ↦ {6 * 17, 6 * 18, 6 * 19, ….. 6 * 166} ↦ 150 elements.
Set C = {100, 104, 108, …..996} ↦ {4 * 25, 4 * 26, ….. 4 * 249} ↦ 225 elements.

A ∩ B = {120, 150, 180, …..990} ↦ All 3-digit multiples of 30 ↦ 30 elements.
B ∩ C = {108, 120, 132, …..996} ↦ All 3-digit multiples of 12 ↦ 75 elements.
C ∩ A = {120, 140, 160, …..980} ↦ All 3-digit multiples of 20 ↦ 45 elements.

A ∩ B ∩ C = {120, 180, …..960} ↦ All 3-digit multiples of 60 ↦ 15 elements.
A ∪ B ∪ C = A + B + C – A ∩ B – B ∩ C – C ∩ A + A ∩ B ∩ C
= 180 + 150 + 225 – 30 – 75 – 45 + 15 = 420

The question is "How many elements are present in A ∪ B ∪ C?"

Hence, the answer is "420".

Choice A is the correct answer.