(1) solve the equation ((x−49)/(50)) + ((x−50)/(49)) = ((49)/(x−50)) + ((50)/(x−49)) (2) How many numbers from 12 to 12345 inclusive have digits which are consecutive an in increasing order, reading from left to right ?


Question and Answers Forum

All Questions Topic List
Permutation and Combination Questions
Previous in All Question Next in All Question
Previous in Permutation and Combination Next in Permutation and Combination
Question Number 118489 by bramlexs22 last updated on 18/Oct/20

$(1) s o l v e t h e e q u a t i o n \frac{x - 49}{50} + \frac{x - 50}{49} = \frac{49}{x - 50} + \frac{50}{x - 49}$ $(2) H o w m a n y n u m b e r s f r o m 12 t o 12345$ $i n c l u s i v e h a v e d i g i t s w h i c h a r e$ $c o n s e c u t i v e a n i n i n c r e a s i n g o r d e r,$ $r e a d i n g f r o m l e f t t o r i g h t ?$
	Answered by benjo_mathlover last updated on 18/Oct/20
	$Solve the equation ((x−49)/(50)) + ((x−50)/(49)) = ((49)/(x−50)) + ((50)/(x−49)) solution : (•) ((x−49)/(50)) + ((x−50)/(49)) = ((49)/(x−50)) + ((50)/(x−49)) let ((x−49)/(50)) = t →x=50t+49 ⇒ t + ((50t−1)/(49)) = (1/t) + ((50)/(50t)) ⇒ t + ((50t)/(49)) −(1/(49)) = (2/t) ⇒((99t)/(49)) − (2/t) = (1/(49)) ; 99t^2 −98 = t ⇒ 99t^2 −t−98 = 0 ; (99t +98)(t−1) = 0 { ((t=1 ⇒x = 99)),((t=−((98)/(99)) ⇒x = 50(−((98)/(99)))+49 = −((49)/(99)))) :}$
	$S o l v e t h e e q u a t i o n \frac{x - 49}{50} + \frac{x - 50}{49} = \frac{49}{x - 50} + \frac{50}{x - 49}$ $s o l u t i o n :$ $(∙) \frac{x - 49}{50} + \frac{x - 50}{49} = \frac{49}{x - 50} + \frac{50}{x - 49}$ $l e t \frac{x - 49}{50} = t \to x = 50 t + 49$ $\Rightarrow t + \frac{50 t - 1}{49} = \frac{1}{t} + \frac{50}{50 t}$ $\Rightarrow t + \frac{50 t}{49} - \frac{1}{49} = \frac{2}{t}$ $\Rightarrow \frac{99 t}{49} - \frac{2}{t} = \frac{1}{49}; 99 t^{2} - 98 = t$ $\Rightarrow 99 t^{2} - t - 98 = 0; (99 t + 98) (t - 1) = 0$ ${\begin{cases} t = 1 \Rightarrow x = 99 \\ t = - \frac{98}{99} \Rightarrow x = 50 (- \frac{98}{99}) + 49 = - \frac{49}{99} \end{cases}$
	Answered by som(math1967) last updated on 18/Oct/20

	$(2) 2 digit . . . . . 8 (last number 89)$ $3 digit . . . . 7 (last number 789)$ $4 digit . . . . 6 (last number 6789)$ $5 digit . . . . . 1 (last number 12345)$ $total 8 + 7 + 6 + 1 = 22$
	Commented by bramlexs22 last updated on 18/Oct/20

	$t h a n k y o u s i r$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum