The ceiling of a floor and the floor of a ceiling equation

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@alvesrubtch8636 7 ай бұрын

This saved me big time, I passed my Calculus 1 exam. Thank you Mr Prime Newton. Not only this videos but , the rest too🥺

@PrimeNewtons 7 ай бұрын

Congratulations 🎉

@RickyWallace 11 ай бұрын

Very nice video and audio editing to correct the inequality. The effort is appreciated! Enjoyed the video

@mikefochtman7164 11 ай бұрын

Whew, yes I worked it out before watching, but at 3:22 when you had

@vxllabh09 11 ай бұрын

@Prime Newtons i have a small doubt, how can we substantiate the use of the inequality, k≤x

@CashueTM 11 ай бұрын

Yes, but in this equation x cannot be an integer, because the answer is 37.3, which has a fractional part. An integer plus and integer (5) times an integer + an integer (2) times an integer = an integer. But as seen it does not equal an integer, meaning x has to have a fractional part and his logic works.

@mikefochtman7164 11 ай бұрын

Yes, that caught me up for a moment as well. But I noticed the right-hand side was NOT an integer, so the 2x term had to be a noninteger and therefore x is not an integer. In that case, what he said about k≤x

@V-for-Vendetta01 11 ай бұрын

if 'x' was an integer, then you could re-write the equation as x + 5x + 2x = 37.3; but by doing so, we see that x is not an integer. he did this initially in some of his other videos solving floor/ceiling equations.

@LumberLopper 11 ай бұрын

I actually mucked up my floor and ceiling definitions the first time I tried this, but got 4.15 as my answer. Also technically k

@xyz9250 4 ай бұрын

Alternatively could just write x = k + j where 0

@benjaminvatovez8823 3 ай бұрын

Thank you for your video. I think we could simply say that [x] is an integer already so the extra ceiling/flooring operation is useless. I often prefer to let x = k+{x} where {x} is the decimal part of x we get k+5(k+1)+2(k+{x})=37.3 thus 4k+{x}=16.15 and we immediately get {x}=0.15 and k=4 (which is in Z).

@enxel493 11 ай бұрын

For some reason I always thought ceiling(x) = x+1, and that's why I initially got 3.66 < k

@dan-florinchereches4892 4 ай бұрын

Case 1 : x is an integer then floor and roof are the same . Say x=k then 4k=373 so there are no integer solutions as k=46.625 This lands us a range to look for x in but we'll ignore for now. If x is not an integer Floor(roof(x))=roof(x)=k+1 Roof(floor(x))=floor(x)=k K

@gdbyzantryx810 11 ай бұрын

We note that ceil(floor)=floor and floor(ceil)=ceil since floor and ceil fix the integers. We quickly note that there are no integer solutions since 8 times an integer cannot be 37.3 We now consider each interval (n, n+1) to narrow down where any solutions must be. In (n,n+1), the equation becomes n + 5n + 5 + 2x = 37.3, so 6n + 2x = 32.3. 6n + 2x < 6n + 2n + 2 = 8n + 2. Thus, we get that 30.3 < 8n. That is, n>=4. Similarly, 6n + 2x > 8n. So 32.3 > 8n. That is, n

@alex_paterson 11 ай бұрын

I kinda got there before watching! I somehow didn't get the minimum value of k, so calculated x for k being {0,1,2,3,4} which gave me 4.15 as one of my answers, but redid my calculations halfway through your video once I realised my mistake! Thanks again for the video - I didn't know about the floor and ceiling function before your previous videos, so thank you :)

@Moj94 11 ай бұрын

If you just ignore everything and write it as: 8x=37.3 you'll get x=4.66 . Now we know that x is not actually 4.66 but it should be between 4 and 5. so we say floor is 4 and ceiling is 5. we'll get 29+2x=37.3 which results in x=4.15 (Keep in mind this is not the way to do things and it only works if you know where to use)

@m.h.6470 11 ай бұрын

By replacing ⌊x⌋ with k, ⌈x⌉ with k + 1 and x with k + d, you end up with 8k + 2d = 32.3. Given, that d is the only non-integer and it has to be between 0 and 1, it could either be 2d = 0.3 or 2d = 1.3. Since 32.3 - 1.3 = 31 and 31 is not a multiple of 8 (has to be one, for k to be integer), 2d = 1.3 is incorrect. So we end up with k = 4 and d = 0.15, so x = 4.15.

@m.h.6470 11 ай бұрын

Solution: ⌈⌊x⌋⌉ = ⌊x⌋, as ⌈x⌉ is x, when x is an integer. ⌊⌈x⌉⌋ = ⌈x⌉, as ⌊x⌋ is x, when x is an integer. So the actual equation is: ⌊x⌋ + 5⌈x⌉ + 2x = 37.3 let x = k + d, k is the integer part of x; d is the decimal part of x k + 5(k + 1) + 2(k + d) = 37.3 k + 5k + 5 + 2k + 2d = 37.3 2d = 0.3 → d = 0.15, as "k + 5(k + 1) + 2k" are all integers and 0 < d < 1 Therefore we have: k + 5k + 5 + 2k = 37 8k + 5 = 37 |-5 8k = 32 |:8 k = 4 x = k + d = 4 + 0.15 = 4.15

@m.h.6470 11 ай бұрын

Slight correction: d *COULD* also be 0.65, but that can be disproven, as 32.3 - 2(0.65) = 31, which is not a multiple of 8. Therefore d can't be 0.65.

@nothingbutmathproofs7150 11 ай бұрын

I have two questions for you. Is there a way to tell if there will be only one solution? You made a mistake by writing

@williamperez-hernandez3968 11 ай бұрын

Slight change to the order of the solution for similar problems. Since the ceiling = floor when x is an integer, we should first verify that x cannot be an integer. In simple cases (like here) it can be done by inspection, since the total is not an integer. In other cases we can assume that ceiling = floor, solve, and if the result is not an integer we know ceiling=/floor. If we do get an integer, then we got the correct answer for x.

@alexki3135 11 ай бұрын

So first of all i reduced the formula to (floor x) + 5(ceil x) + 2x = 37.3 Because floor and ceil return integer and so stack them Shouldnt change anything. Then i did the k thing with k 2d = 0.3 d = 0.15 => d > 0 => ceil x = k + 1 => k + 5(k+1) + 2k +2d = 37.3 k +5k + 5 + 2k = 37 k + 5k + 2k = 32 8k =32 k = 4 => x = (floor x) + d = k + d = 4 + 0.15 = 4.15

@mikefochtman7164 11 ай бұрын

I like this. And it includes testing that ceil(x) != floor(x) by virtue that d> 0. I had to just rationalize that by virtue the right-hand side wasn't an integer, but I like your technique instead.

@DiverseDose11 11 ай бұрын

7:08 Bro if you're bad at math then Others:Moye Moye 😂

@PrimeNewtons 11 ай бұрын

You never know 😅😅😅😅

@vitotozzi1972 11 ай бұрын

I like this equation so much!

@antonionavarro1000 11 ай бұрын

Gracias a los videos anteriores pude resolver este nuevo ejercicio. Tu primera intuición al escribir ' 4

@jacobgoldman5780 11 ай бұрын

Am I missing something or is this the same as floor(x)+5ceiling(x)+2x=37.3 since once you take a floor or ceiling of a number any more floors or ceilings don't do anything to the number since it is already an integer?

@martinrosenau478 5 ай бұрын

You asked to try it ourself and leave a comment: The following approach took me about 2 minutes: Because foor(x) and ceil(x) are integers, 2x must be n+0.3 (while n is an integer). This means that x must be either N+0.15 or N+0.65 (while N is also an integer). In both cases, x is not an integer, so floor(x) is N and ceil(x) is (N+1). Now we have two equations with the variable N that may lead to solution(s): N+5(N+1)+2(N+0.15)=37.3 N+5(N+1)+2(N+0.65)=37.3 The second equation leads to an invalid (because it is non-integer) solution for N; the first one leads to N=4 so x=4.15.

@JourneyThroughMath 11 ай бұрын

Okay, So I attempted it and got 4.15. I hypothesized that i can drop the outside ceiling and floor. My only remaining question is is that the only answer. I am confident that it is.

@tomasbeltran04050 11 ай бұрын

Isn't ðe function monotonically increasing? If it is, 4.15 is ðe only answer

@JourneyThroughMath 11 ай бұрын

@@tomasbeltran04050 yes, but I didnt think of that. To me the detail that stood out was the decimal. I also wanted an excuse to watch the video

@Christian_Martel 3 ай бұрын

Yes, this is the only solution. Since the reminder is contained in the 2x term. The reminder of x is either 0.15 or 0.65. (2*0.15=0.3) or (2*0.65=1.3). I have verify both cases and only 0.15 works with this equation.

@Vi-kun 6 ай бұрын

Sir cant we suppose x is some number as y+ dec where xy is integer and dec is decimal part then the equation will be y +5(y+1)+2y+2dec=37.3 So 8y + 2dec +5 = 37.3 8y+2dec=32.3 Comparing 8y= 32 y=4 2dec=0.3 dec=0.15 so x will be 4.15

@Vi-kun 6 ай бұрын

Nvm I found the problem that it will only work if, 2dec

@therichcircle.8819 11 ай бұрын

This a great video

@surendrakverma555 11 ай бұрын

Thanks Sir

@GreenMeansGOF 11 ай бұрын

I solved this on my own. If k is the floor of x, I deduced that {x}-0.15=16-4k. Since the right hand side is an integer, it follows that {x}-0.15=0 and therefore, k=4. Thus, x=4.15.😊

@loietkenp 11 ай бұрын

what if x equals to to floor of x? is still celling of x k+1?

@PrimeNewtons 11 ай бұрын

That's a smart answer in this case. However, it's impossible for x to be equal to k.

@loietkenp 11 ай бұрын

@@PrimeNewtons that's true... i wonder if there's a case when that is a problem

@Christian_Martel 3 ай бұрын

Looking at the equation, I see the following: integer + integer + 2x = 37.3, therefore x is not an integer and has a reminder of 0.15 or similar. First, I set x as integer + reminder: Let x = a + b, x: R, a: Z, b: R [0, 1). C(): Ceiling(), F(): Floor() C(F(x)) + 5F(C(x)) + 2x = 37.3 C(F(a+b)) + 5F(C(a+b)) + 2(a+b) = 37.3 C(a) + 5F(a+1) + 2a + 2b = 37.3 a + 5(a+1) + 2a + 2b = 37.3 8a + 5 = 37.3 - 2b Note that 2b: R [0, 2). 2b = 1.3 or 2b = 0.3 If 2b = 1.3 -> 8a + 5 = 36 8a = 31 -> a = 31/8: not Z -> not a solution If 2b = 0.3 -> 8a + 5 = 37 8a = 32 -> a = 4 -> a: Z 2b = 0.3 -> b = 0.15 Solution: x = a + b x = 4 + 0.15 x = 4.15

@KannaKamui21000 11 ай бұрын

I did it and found 2 possible answers. . Dots are anti-spoiler for those who don't want to be spoiled. . I've found x=4,15 and x=4,525. But for the second one, when using x=n+d decomposition, n integer and d in [0;1), for 4,15 we find exactly n=4 and d=0,15. But fir x=4,525, we find exactly n=3,875 and d=0,65. For the second one, n has to be an integer, and 3,875 is obviously not, so the answer x=4,525 is rejected. That leaves the only answer x=4,15. Tips : - floor(ceil(x)) = ceil(x) - ceil(floor(x)) = floor(x) Here's a problem for those who wants to solve more equations : find all real numbers x so that floor(x) * ceil(x) = (round(x))². I'll post the solution in answer to this post in one week !