A floor value equation

[@mjones207]

I just used straight, old-fashioned algebra all the way: x = ⌊x⌋ + {x}. ⌊x⌋ ∈ ℤ and 0 < {x} < 1. {x} ≠ 0 because it's in the denominator of one of the original fractions. 8/{x} = 9/(⌊x⌋ + {x}) + 10/⌊x⌋. Subtracting, 8/{x} - 9/(⌊x⌋ + {x}) - 10/⌊x⌋ = 0. Multiplying by ⌊x⌋{x}(⌊x⌋ + {x}) to rid fractions, 8⌊x⌋(⌊x⌋ + {x}) - 9⌊x⌋{x} - 10{x}(⌊x⌋ + {x}) = 0. Distributing, 8⌊x⌋² + 8⌊x⌋{x} - 9⌊x⌋{x} - 10⌊x⌋{x} - 10{x}² = 0. Combining like terms, 8⌊x⌋² - 11⌊x⌋{x} - 10{x}² = 0. Factoring, (8⌊x⌋ + 5{x})(⌊x⌋ - 2{x}) = 0. If 8⌊x⌋ + 5{x} = 0, then {x} = -(8/5)⌊x⌋, and because 0 < {x} < 1, 0 < -(8/5)⌊x⌋ < 1, hence -5/8 < ⌊x⌋ < 0. This is a contradiction because there are no integers between -5/8 and 0. Therefore, ⌊x⌋ - 2{x} = 0, and {x} = ⌊x⌋/2. Because 0 < {x} < 1, 0 < ⌊x⌋/2 < 1, hence 0 < ⌊x⌋ < 2. The only integer between 0 and 2 is one, so ⌊x⌋ = 1, {x} = ⌊x⌋/2 = 1/2, and x = ⌊x⌋ + {x} = 3/2.

[@SyberMath]

Very good!

[@Germankacyhay]

👍👍👍👍❤❤❤❤❤

[@srijanbhowmick9570]

ahh finally ! Satisfaction ! I LOVE THSE TYPE OF PROBLEMS SO MUCCCCHHH ! Thanks man !!

[@SyberMath]

Happy to help!

[@ramaprasadghosh717]

Let n be the integral part of x and f be the fractional.part of x so 8/f = 9/(n+f) + 10/ n or 8/f - 9/(n+f) = 10/n or (8n - f )/(f*n + f*f) = 10/ n or 8n*n - f*n = 10 *f*n + 10"f*f or 8 n*n -11 f*n - 10f*f = 0 or 8 n*n -16 n*f + 5n*f -10n*f = 0 or (n -2f) ( 8n +5f) = 0 since neither n nor f is negative n = 2f also f < 1 or 2f < 2 i e n < 2 or n = 1 so x = 1 +1/2

[@CriticSimon]

Wow! This problem looks really cool

[@SyberMath]

Thank you! 🥰

[@CriticSimon]

@@SyberMath Oh sure

[@michaelempeigne3519]

x = floor ( x ) + { x }

[@SyberMath]

Yessir!

[@fasihullisan3066]

very nice explanation

[@SyberMath]

Thank you!

[@elvistvaz]

CASE 1: X = 0 X can't be 0 because 9/x. X is diferent than zero. CASE 2:X Difer than zero {X} = A; ==> 0 B E Z (and B cant be Zero) X=A+B; Then 8/A = 9/(A+B) + 10/B; Lets put all in the same base [8(A+B)B]/(A(A+B)B) = [9AB+10A(A+B)]/(A(A+B)B); lets multiply both (A(A+B)B) 8(A+B)B = 9AB+10A(A+B); Distribuiting 8AB + 8B^2 = 9AB+10A^2 +10AB; lets puting all in the same side 8B^2 - 11AB - 10A^2 = 0 Using the the second degree equation we have: B = 2A or B = -5A/8 SUB CASE 2.1: B = -5A/8 0 = -5A/8 > -5/8 ==> Using B = -5A/8 and we know that: -5/8 > -8/8 = -1 0 >= B > -5/8 > -1 0 >= B > -1 ==> B Pertence to Z and cant be Zero, then this solution is inposible. SUB CASE 2.2: B = 2A 0

[@asimsinha2531]

Amazing!

[@SyberMath]

Thank you! Cheers!

[@tonyhaddad1394]

Great( floor/fractionl part)problem!!!!

[@SyberMath]

Thank you!!!

[@nicogehren6566]

great solution sir thanks

[@SyberMath]

No problem! You're very welcome!

[@piyushdaga357]

Most Beautiful use of inequality I have ever seen 😍

[@SyberMath]

Thank you!

[@זאבגלברד]

When I look at the graph of these 3 function I can write 8/(x-n) = 9/x + 10/n where n=0,1,2,3,4, ... (obviously we cancel n=0 , x=0 , x=n). I get x= 1.5n (x>0 must be). n=1 gives a solution. n can not be even since fractional of an integer is zero. If n is 3,5,7,... left side of the equation is always 16 but right side decreases.

[@SyberMath]

Good thinking! 🤩

[@skylardeslypere9909]

What I did was: Let a < x < a+1 for a an integer (nonzero) Substitute the equation, solving for x gives us x = 3a/2. Why doesn't this work with every integer? So floor(x) = a and frac(x)=x-a

[@SyberMath]

What do you mean?

[@haricharanbalasundaram3124]

@@SyberMathI think it is right, actually... that's the way I did it. The greatest integer of x is some integer 'n', the fractional part of x is x - n. I can just substitute them in the equation, and get x in terms of n, then using n < x < n + 1 gives the result

[@mcwulf25]

Put n = [x] and {x} = x -n Multiply through by xn(x-n) and you get 10x^2 - 9xn - 9n^2 = 0 (2x-3n)(5x+3n) = 0 (1) x = (3/2)n which is only possible if x = 3/2 as per the video. (2) x = (-3/5)n which has no solutions.

[@ramaprasadghosh717]

Let the fractional part of x be f and integral part of x be n. Hereby 8 /f = 9/(n+f) + 10/n ( 8f + 8n - 9f)/(f*(n+f))= 10/n 8n*n -f*n = 10*f*(f+n) 8n*n -11f*n -10f*f = 0 8n*n -16*f*n + 5f*n -10f*f = 0 or (n-2f)(8n +5f) = 0 n = 2f is the only feasible solution f = 1/2 , n = 1 now 8/f = 16 9/(n+f) = 6 10/ n = 10

[@vuyyurisatyasrinivasarao3140]

Gòod solution nice explanation

[@SyberMath]

Thanks

[@scottleung9587]

Nice!

[@mab9316]

Genius 🎉

[@nawusayipsunam1643]

Great

[@SyberMath]

Thank you!

[@barakathaider6333]

👍👍

[@Germankacyhay]

👍

[@ganda3454]

8/x=19/x 8x=19x x=0

[@gamepplayker-of6in7do8to9nice]

*7*

[@SyberMath]

What do you mean?

[@gamepplayker-of6in7do8to9nice]

@@SyberMath 142857

[@kimsanov]

Still confused about {x} for x < 0

[@SyberMath]

Think of a negative number like -2.5. The goal is to write it as an integer plus a decimal between 0 and 1. -2-0.5 won't work because the decimal needs to be added to the integer. The correct way to do it is -3+0.5 So if x=-2.5, ⌊x⌋ = -3 and {x} = 0.5 x = ⌊x⌋ + {x}

[@kimsanov]

@@SyberMath Feels counter intuitive. If 3.14 = 3 + 0.14 Why mathematicians don't use e.g. - 3.14 = - (3 + 0.14) Kind a rhetorical question :)

[@razor-xn2ve]

@@kimsanov Definition of floor value function is all that you need to consider floor(x) is the greatest integer less than or equal to x Example: floor(3.4) Is an integer smaller than 3.4 So you may guess 1,2,3 But we are looking for greatest among these so we say Floor(3.4)=3 Now consider floor(-3.4) Seek integers *smaller* than - 3.4 You should know that -3 is greater than -3.4, so is -2 , Those smaller than -3.4 include -4,-5,-6... Now pick up the largest among these so it is -4 I hope it clears

[@georget8008]

The {x} = x-floor(x) even for x

[@SyberMath]

What is x?

[@georget8008]

@@SyberMath I m referring to the definition of {x}. The fractional part of a number x.

[@SyberMath]

@@georget8008 -3.14=-4+0.86=floor(-3.14)+{-3.14}

[@hoangnguyennguyen6445]

i let x = [x] + {x} then i let [x] = a;{x} = b and i just solve it

[@SyberMath]

Nice!

[@hoangnguyennguyen6445]

@@SyberMath thanks

[@Mik1604]

X can not be an integer