nov15.wordpress presents QUANT FOR CAT 2009

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VENN DIAGRAMS

Venn Diagrams with 2 groups B A a b c n Only A Only B Not A or B Both A & B

Problem • Of all CAT aspirants, 80% spend time for Quant whereas 60% spend time for verbal. If only those users will crack CAT who spend time for both, what percentage of CAT aspirants • will crack CAT? • will not crack CAT?

Solution QA = 80 => a+c = 80 -> (1) VA = 60 => b+c = 60 -> (2) Now, a+b+c+n = 100 As n = 0 , a+b+c = 100 -> (3) Add (1) & (2) a+b+2c = 140 Now, from (3) - (4) a+b+2c = 140 a+b+c = 100 C = 40, These 40% qualify CAT So 60% don’t qualify CAT. QA = 80 VA = 60 a b c n= 0 Only QA Only VA Not QA or VA = 0 Both QA & VA

Venn Diagrams with 3 groups Both A & B Only B B A a b d None of A,B,C g c e f Both B & C C n Only A Both A & C Only C All A, B & C

Problem – CAT 2003 • A survey on a sample of 25 new cars being sold at a local auto dealer was conducted to see which of the three popular options - air conditioning, radio and power windows were already installed. The survey found: • 15 had air conditioning • 2 had air conditioning and power windows but no radios • 12 had radio • 6 had air conditioning and radio but no power windows • 11 had power windows • 4 had radio and power windows • 3 had all three options. • What is the number of cars that had none of the options?1. 4 2. 3 3. 1 4. 2

Solution • Lets fill in the Venn Diagram with the info given : Both A & R Only R Radio AC 4 2 6 None of A,R,P 3 5 2 1 Both R & P n = ? PW Only A Both A & P Only P All A, R & P

Solution contd… • As we know total number of cars = 25 a+b+c+d+e+f+g+n = 25 a+b+c+d+e+f+g = 23 • Filling in the required values we get n = 2

Problem : (CAT 2003- Leaked) • New Age Consultants have three consultants Gyani, Medha and Buddhi. • The sum of the number of projects handled by Gyani and Buddhi individually is equal to the number of projects in which Medha is involved. • All three consultants are involved together in 6 projects. • Gyani works with Medha in 14 projects. • Buddhi has 2 projects with Medha but without Gyani, and 3 projects with Gyani but without Medha. • The total number of projects for New Age Consultants is one less than twice the number of projects in which more than one consultant is involved.

Problem contd… • What is the number of projects in which Gyani alone is involved?1. 02. 1.3. 4.4. cannot be determined • What is the number of projects in which Medha alone is involved?1. 02. 1.3. 4.4. cannot be determined

Solution: • Filling in the values given: Both G & M Only M Medha Gyani a c 8 None of G,M,B 6 b 3 2 Both M & B n = 0 Buddhi Only G Both G & B Only B All G, M & B

Solution: contd… • We are left with the last line : • The total number of projects for New Age Consultants is one less than twice the number of projects in which more than one consultant is involved. Total = a+b+c+d+e+f+g+n=2(d+e+f+g) - 1 Now, n = 0, d = 8, e = 6, g = 2 a+c = b + 16 Also, Total no. = 2 (8+6+3+2) – 1 = 37 • We cannot determine a and c individually. • So, answer to Q1 is Option 4. – Cannot be determined

Solution: contd… But, from Total = a+b+c+d+e+f+g = 37, we get a+b+c = 37 – (d+e+f+g) = 37-(8+6+2+3) = 19 So, a+b+c = 19 and a+c = b – 16, • Using both the above equations, we get b = 1 • Answer to Q2 is Option 2 ie., 1,

Maxima and Minima Both A & B Only B B = 1 A = 1 0 0 1 None of A,B,C 0 0 0 0 Both B & C C = 0 n Only A Both A & C Only C All A, B & C

Maxima and Minima contd… Both A & B Only B B = 1 A = 1 0 0 0 None of A,B,C 1 0 0 0 Both B & C C = 1 n Only A Both A & C Only C All A, B & C

Maxima and Minima contd… • Adding 1 element to intersection of 2 sets gives surplus of 1 element. • Adding 1 element to intersection of 3 sets gives surplus of 2 elements.

Problem: • According to a survey, at least 70% of people like apples, at least 75% like bananas and at least 80% like cherries. What is the minimum percentage of people who like all three?

Solution • percentage of people who like apples + percentage of people who like bananas + percentage of people who like cherries = 70% + 75% + 80% = 225% • A surplus of 125%.

Solution contd… • Now this surplus can be accommodated by adding elements • to either intersection of only two sets or • to intersection of only three sets. • For minima of intersection of 3 sets => • Put 100% in intersection of 2 sets (this is max. for this area) • Left over 25% has to be accommodated in intersection of 3 sets. • So Minima of intersection of 3 sets = minimum %age of people who like all 3 fruits = 25%

Problem • In a college, where every student follows at least one of the three activities- drama, sports, or arts: • 65% follow drama, • 86% follow sports, and • 57% follow arts. • What can be the maximum and minimum percentage of students who follow 1· all three activities2· exactly two activities

Solution • Calculate surplus : • Drama + Sports + Arts = 65+86+57 = 208% • Surplus = 208 – 100 = 108% • Now, surplus can be accommodated in 2 ways : • Maximum in 2 only and minimum in all 3 • Maximum in All 3 and minimum in 2 only

Solution contd… • Maximum in 2 only and minimum in all 3: • 100% can be accommodated in 2 only • 8% will have to be accommodated in all 3. • Maximum percentage who follow 2 events = 100 • Some Answers : • Minimum percentage who follow all 3 = 8 • Max. percentage who follow exactly 2 = 100 – 8 = 92%

Solution contd… • Maximum in All 3 and minimum in 2 only: • As 1 in All 3 area causes increase of surplus as 2. • If we put 54% in all 3 -> surplus = 54 x 2 = 108. • Some Answers : • Max. who follow all 3 = 54% • Min. no. who follow exactly 2 = 0%

Exercises • Do Check out the exercise given along with the lesson for more practice problems of this type. http://nov15.wordpress.com

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nov15.wordpress presents QUANT FOR CAT 2009