Sets

Pratyush @ Toughest Questions

Sets

What is a set?


25 people are enrolled in history, maths, or both. If 20 are in history and 18

are in maths, how many in both history and maths?


Let those in both be n

only H = 20 - n; only M = 18 - n

only H + only M + Both = 25

(20 -n) + (18 - n) + n = 25.

So n = 13


In a certain production lot, 40% of the toys are red and the remaining are

green. Half the toys are small and half are large.


If 10% of the toys are red and small, and 40 toys are

green and large, how many of the toys are red

and large?


Make a table for the data

Red Green TotalSmall 10% 50%Large 50%Total 40% 60% 100%


The completed table is -

Red Green TotalSmall 10% 40% 50%Large 30% 20% 50%Total 40% 60% 100%


Since 20% of the number of toys (let it be n) are green and

large, 0.20n = 40 (40 toys are green and large),

n = 200. Therefore, 30% of the 200 toys,

or 60 toys are red and large


Set Jargon

Empty Set

Contains 0 elements

A B (Subset)

Every member of A part of B


A B (Union)

Set containing all of A & B

A B (Intersection)

Set containing only the common elements of A & B

Ac : Set containing all except A


A A = ? (Union)A

A A = ? (Intersection)A

(Ac)c = ?A


A B = B A ?

Yes - Cumulative Law

A (B C) = (A B) C ?

Yes - Associative Law

(AB) C = (AC) ( BC)?

Yes - Distributive Law


= (A C) ( B C)

What will (A B) C be?


Express in terms of Ac and Bc

(A B)c =

Ac Bc

(A B)c =

Ac Bc


Quiz

Is A B = Ac Bc

No

Is (Ac Bc)c = A B?

Yes


250 people use Colgate and 120 use both Colgate and Close Up. If 330 use only Close Up, how many

use only Colgate?250 - 120 = 130


In a class of 120, 55% have taken Economics and 65% have taken

Mathematics. How many students have taken both

the subjects?


Common =

55 + 65 - 100 = 20%

20% of 120 = 24


Out of 100 families, 50 own radios, 75 have TVs, and 25 have VCRs. Only 10 families have all three and each VCR owner also

has a TV.


If 25 families have only radio, how many have

only TV?


TV = 75. TV & VCR = 25. Only VCR = 0

VCR-Radio but not TV = 0. All 3 = 10.

So, Radio & TV but not VCR = 50 - 10 - 25 = 15.

Only TV = 75 - 15 - 10 - 15 = 35.


With standard weights of 0.5, 1, 2, 4, 8 & 16 kg,

how many different weights can be

measured?


6C1 = 6C5 = 66C2 = 6C4 = 156C3 = 20 & 6C6 = 1

Add all; answer is 63.


There are 6 boxes numbered 1 to 6. Each

box is to be filled up with a white/black ball.

Conditions:1) at least one box should

contain a black ball


2) the boxes containing black balls are adjacent.

What is the total number of ways in which this can be

done?


The answer is 1 + 2 + 3 + 4 + 5 + 6.

(All black balls can be arranged in one way. 5 black can be

arranged in 2 ways. 4 black can be arranged in 3 ways and so

on.)


The population of a town is 35,000. If the number of men by 6% and that of women by 4%, the population would become 36,800. How many men are there ?


M + W = 35000(1.06)M + (1.04)W = 36800. So, M = 20000.


A group of students decided to contribute and buy a tape recorder costing $1800. At the last moment, two of them backed out.


As a result, the remaining students had to add another $10 each. How much did each student plan to contribute originally?


10 + 1800/x = 1800/(x - 2) Solve for x,

x= 20Initial contribution

1800/20 = 90


You have reached Utopia where you find 2 stones, rubies and emeralds, worth $4 m and $5 m respectively and their weights are 0.3 kg and 0.4 kg respectively.


You can carry maximum 12 kg. How many rubies and emeralds would you carry to maximize the profit?


The value per unit weight is 4/0.3 and 5/0.4; that is, 13.33

and 12.5 for rubies and emeralds respectively. So to

maximize total value carry only rubies.


2 friends are going down the descending escalator. Sam walked down, taking 50 steps to reach the bottom; Spade ran down, taking 75.


If Spade ran 3 steps in the time it took Sam to walk 1, how many steps are visible when the escalator is stopped?


Let no. of steps = xWhen Sam walked 50 steps, the escalator had covered x-50 steps. Ratio of elevator speed to Sam’s speed is x-50 : 50Similarly ratio of elevator speed to Spade’s speed is x-75 : 75


Combining the two above, we get ratio of Sam’s to Spade’s speed as x-50 : 50 x-75 : 75= [(x-50) x 75] / [(x-75) x 50]But we know that this is = 3:1Combining both, we get x = 100


You are in a band of 1,000 prisoners to be executed. The prisoners are standing in a single line.


The first one will be shot, followed by every alternate one until one remains and will be freed. Which number would you choose?


In the 1st lot, all the odd numbered prisoners go . In the 2nd lot, 2 and all the multiples

of 2 but not of 4 go . In the next lot, those who remain are

the ones whose numbers are divisible by 8.


In the next lot, the ones who remain are 16 and multiples of 16. This goes on till the last

one remains, who has a number divisible by 256. The

last 3 will be 256, 512 and 768. So, choose 512.

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