A certain preliminary science class contains 50 students all of whom take Mathematics. 18 study Chemistry, 17 study Biology, 24 study Physics. Of those taking three subjects, 5 study Physics and Chemistry, 7 study Physics and Biology and 6 study Chemistry and Biology while 2 take all four subjects. How many students study only Mathematics?


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Question Number 198321 by necx122 last updated on 17/Oct/23

$A c e r t a i n p r e l i m i n a r y s c i e n c e c l a s s$ $c o n t a i n s 50 s t u d e n t s a l l o f w h o m t a k e$ $M a t h e m a t i c s . 18 s t u d y C h e m i s t r y, 17$ $s t u d y B i o l o g y, 24 s t u d y P h y s i c s . O f$ $t h o s e t a k i n g t h r e e s u b j e c t s, 5 s t u d y$ $P h y s i c s a n d C h e m i s t r y, 7 s t u d y P h y s i c s$ $a n d B i o l o g y a n d 6 s t u d y C h e m i s t r y$ $a n d B i o l o g y w h i l e 2 t a k e a l l f o u r s u b j e c t s .$ $H o w m a n y s t u d e n t s s t u d y o n l y M a t h e m a t i c s ?$
Commented by necx122 last updated on 17/Oct/23
This is another question that troubled me. please help me with the Venn diagram and the solution . Thanks in advance.
	Answered by mr W last updated on 17/Oct/23

	Commented by mr W last updated on 17/Oct/23

	$13 s t u d y o n l y m a t h e m a t i c s .$
	Commented by necx122 last updated on 17/Oct/23
	Thank you sir. I really understand this but the textbook says the answer is 7 students.
	Commented by mr W last updated on 17/Oct/23

	$t h e n t e x t b o o k i s w r o n g .$
	Commented by necx122 last updated on 17/Oct/23
	Ok sir. I actually believe your solving than my textbooks however. After seeing your Venn diagram and the answer you got I resubstituted the values and got 7. I substituted it such that the intersection between each of biology and physics only, biology and chemistry only and chemistry and physics only to reduce by 2. Thus giving us 5 , 4 and 3. Yet, I'll want to agree with you sir. Thank you.
	Commented by deleteduser1 last updated on 18/Oct/23

	$T h e a n s w e r s h o u l d b e 7 .$
	Commented by mr W last updated on 18/Oct/23

	$c a n y o u p l e a s e s h o w t h e c o r r e s p o n d i n g$ $v e n n d i a g r a m ?$
	Commented by deleteduser1 last updated on 18/Oct/23

	$W e c a n a l s o u s e t h e f o r m u l a i n t h e a n s w e r$ $b e l o w .$
	Commented by mr W last updated on 18/Oct/23

	$a c c o r d i n g t o q u e s t i o n :$ $\underset{―}{o f t h o s e t a k i n g t h r e e s u b j e c t s}, 5 s t u d y$ $p h y s i c s a n d c h e m i s t r y .$ $b u t a c c o r d i n g t o y o u r v e n n d i a g r a m$ $o n l y 3 s t u d y p h y s i c s a n d c h e m i s t r y .$
	Commented by deleteduser1 last updated on 18/Oct/23

	$3 + 2 = 5, w e a d d t h e 2 f r o m P, C a n d B .$ $3 t a k e P h y s i c s a n d C h e m i s t r y o n l y, b u t 3 + 2 = 5$ $t a k e P h y s i c s a n d C h e m i s t r y .$
	Commented by mr W last updated on 18/Oct/23

	$b u t t h e 2 s t u d e n t s t a k e a l l f o u r s u b j e c t s,$ $s o t h e y a r e n o t t h o s e t a k i n g t h r e e$ $s u b j e c t s .$
	Commented by mr W last updated on 18/Oct/23

	$i o n l y a c c e p t t h e a n s w e r o f t h e t e x t b o o k,$ $i f i n t h e q u e s t i o n t h e u n d e r l i n e d$ $s e n t e n c e {d o e s n}^{'} t e x i s t, o r i f i t i s$ $\underset{―}{o f t h o s e t a k i n g a t l e a s t t h r e e s u b j e c t s .}$
	Commented by deleteduser1 last updated on 18/Oct/23

	$M y s o l u t i o n b e l o w i s a l s o a n o t h e r w a y t o l o o k$ $a t i t .$
	Commented by mr W last updated on 18/Oct/23

	$a t f i r s t i s o l v e d e x a c t l y l i k e y o u, t h e n$ $i c h a n g e d m y m i n d d u e t o t h e$ $s e n t e n c e ‘ ‘ \underset{―}{o f t h o s e t a k i n g t h r e e}$ $\underset{―}{s u b j e c t s}, . . . .^{″} .$ $a n y w a y i t h i n k t h e a n s w e r 7 i n t e x t b o o k$ $i s w r o n g, i f w e s t r i c t l y f o l l o w t h e$ $t e x t i n t h e q u e s t i o n .$
	Commented by deleteduser1 last updated on 18/Oct/23

	$2 t a k e a l l s u b j e c t s i m p l y 2 t a k e P h y s i c s,$ $C h e m i s t r y a n d B i o l o g y .$
	Commented by mr W last updated on 18/Oct/23

	$a n d m a t h e m a t i c s a s w e l l, s i n c e a l l$ $s t u d e n t s i n t h e c l a s s s t u d y$ $m a t h e m a t i c s .$
	Commented by deleteduser1 last updated on 18/Oct/23

	$Y e s, b u t w e a r e c o n s i d e r i n g P, B a n d C l i k e i n$ $t h e a n s w e r b e l o w .$ $W e f i n d t h e t o t a l n u m b e r t h a t t a k e P, B o r C,$ $w h i c h i s 43 . T h i s m e a n s t h e r e a r e 50 - 43 = 7 t h a t$ $d o n o t a k e a n y o f P, B o r C . S o, t h e r e m a i n i n g$ $t a k e M a t h s o n l y .$
	Commented by necx122 last updated on 18/Oct/23
	Thank you very much sir. These comments have helped me understand the question properly. Thank you.
	Answered by deleteduser1 last updated on 18/Oct/23

	$∣ P u B u C ∣=∣ P ∣ + ∣ B ∣ + ∣ C ∣ - ∣ P n B ∣ - ∣ B n C ∣ - ∣ P n C ∣$ $+ ∣ P n B n C ∣\Rightarrow∣ P u B u C ∣= 18 + 17 + 24 - 5 - 7 - 6 + 2$ $= 43$ $\Rightarrow∣ M a t h s o n l y ∣= 50 - 43 = 7$
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