70% of the employees in a multinational corporation have VCD players, 75% have microwave ovens, 80% have ACs and 85% have washing machines. At least what percentage of employees has all four gadgets?


Question and Answers Forum

All Questions Topic List
Set Theory Questions
Previous in All Question Next in All Question
Previous in Set Theory Next in Set Theory
Question Number 114236 by Aina Samuel Temidayo last updated on 18/Sep/20

$70 % of the employees in a$ $multinational corporation have VCD$ $players, 75 % have microwave ovens,$ $80 % have ACs and 85 % have washing$ $machines . At least what percentage of$ $employees has all four gadgets ?$
	Answered by Dat_Das last updated on 18/Sep/20

	$10 %$
	Answered by 1549442205PVT last updated on 18/Sep/20

	$Denote VCD players unit byV$ $M - the microwave ovens$ $A - the ACs$ $W - the washing machines$ $m - number of the employees have$ $only V$ $n - number of the employees have$ $only M$ $p - number of the employees have$ $only A$ $w - number of the employees have$ $only W$ $a - number of the employees have$ $only two gadgets VM$ $b - number of the employees have$ $only two gadgets VA$ $c - number of the employees have$ $only two gadgets VW$ $d - number of the employees have only two gadgets$ $MA$ $e - number of the employees have only two gadgets$ $MW$ $f - number of the employees have only two gadgets$ $AW$ $x - number of the employees have only$ $three gadgets MAV$ $y - number of the employees have only$ $three gadgets VAW$ $z - number of the employees have only$ $three gadgets VMW$ $t - number of the employees have only$ $three gadgets VMA$ $T - number of the employees have$ $all four gadgets .$ $Then from the hypothesis we have$ $m + n + p + q + a + b + c + d + e + f$ $+ x + y + z + t + T = 100 (1)$ $m + a + b + c + z + y + t + T = 70 (2)$ $n + a + d + e + x + z + t + T = 75 (3)$ $p + b + d + f + x + y + t + T = 80 (4)$ $q + c + e + f + x + y + z + T = 85 (5)$ $Adding up four above equalities we get$ $m + n + p + q + 2 (a + b + c + d + e + f)$ $+ 3 (x + y + z + t) + 4 T = 310 (6)$ $Multiplying two sides of (1) by 3 then$ $substracting from (6) we get$ $- 2 (m + n + p + q) - (a + b + c + d + e + f) + T = 10$ $\Rightarrow T = 10 + 2 (m + n + p + q) + (a + b + c + d + e + f)$ $\Rightarrow T ⩾ 10 (%)$ $The equality ocurrs if and only if$ $m = n = p = q = a = b = c = d = e = f = 0$ $which means nobody has only one$ $gadgets and also nobody has only$ $two gadgets all have at least three$ $gadgets up$ $Answer : at least 10 % of the employees$ $have all four gadgets$
	Commented by Aina Samuel Temidayo last updated on 19/Sep/20

	$Thanks .$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum