A ceiling and floor equation

You have by far one of the best math channels!
I love love love how you don't edit out mistakes. Watching them occur and (sometimes but not always) catching those mistakes really makes me engage with the video more

Lots of love and blessings from India

Best maths teacher ever i have seen ❤ love from india 🇮🇳

You are a very good teacher 😇

Alternative way to do this:
Since floor and ceiling functions always give an integer, the decimal part (0.7) must come from x, so we can easily convert ceiling part to x+0.3 and floor part to x-0.7, and the rest is straightforward

I love your teaching style, sir

How can we know that this is the only solution (among the naturals at least)?

Excellent explanation Sir. Thanks

Love ur vids!

Haven't watched but it looks easy:
Fractional part of both sides, gives fractional part is 0.7, which means the ceiling is I + 1, and floor is I, where I is the integer part.
Now, integer part of both sides: 2(I + 1) + I + I = 10, so 4I = 8 and I=2
So x = 2.7

You are a 💎

Well done ❤

Best math channel ! I found x = 2.07 at 1.50 min
Thank for you explanation

I’m really loving how you’re building off of your previous videos to slowly get more complicated with your subjects. Makes it really satisfying and interesting to watch!

On quick inspection looks like x = 2.7. Let's see if I missed anything.

Awesome we enjoyed a lot

It was very easy ngl. Just by inspection you can tell it's 2.7 in like 10 seconds but nice explanation nonetheless

By observation, knowing Dat floor and ceil only generate integers, x must have a decimal part of 0.7

May Allah bless you my brother

Thank you young man I never knew how to solve ceiling and floor functions

This is My first time coming across this type of problem

Why 2(k+1) and not 2k?!

I've never seen floor of ceiling equations so I don't know if my solution is correct but there it is.
2ceil(x) + floor(x) + x = 10.7
2ceil(x) + floor(x) + (floor(x) + frac(x)) = 10.7
2(ceil(x) + floor(x)) + frac(x) = 10.7 => frac(x) = 0.7
ceil(x) + floor(x) = 5
2floor(x) + 1 = 5
floor(x) = 2
x = floor(x) + frac(x) = 2 + 0.7 = 2.7

I loved the smile when he lead everyone to d=0.7

my thought process was this. at first glance I realized that x has to be some number ending with .7 as the floor or ceiling functions make whatever's inside to an integer. Then after knowing x ends in a .7 I was stuck but then pretended that the ceiling and floor functions didn't exist and it was basically 5x = 10 giving me x = 2 and hence 2.7

Awesome! Could you do derivative of sqrt x using first principles?

2X+X+X = 4X

Quick glance my mind says 2.7

he is so calming when speaking it has no sense

I love you primenewton

Nice question...❤❤❤.

Solution:



Since you add the floor and ceiling of x, which are - by definition - whole numbers and you add x, the ".7" part of the right side has to be the decimal part of x.
As such, the equation becomes:
2 * (x + 0.3) + (x - 0.7) + x = 10.7
2x + 0.6 + x - 0.7 + x = 10.7
4x - 0.1 = 10.7 |+0.1
4x = 10.8 |:4
x = 2.7

Test:
2 * ceil(2.7) + floor(2.7) + 2.7 = 10.7
2 * 3 + 2 + 2.7 = 10.7
6 + 4.7 = 10.7
10.7 = 10.7

Outside in the street 😅 love these videos.

Thank you very much for your excellent explanation!!

Before watching the video: The floor and ceiling functions always generate integers by definition, so x must be an integer + 0.7 to get 10.7 as the end result. Set x=n+0.7 with n being some integer. Then ceiling(x)=ceiling(n+0.7)=n+1 since n+1 is smallest integer greater than n+0.7 and floor(x)=floor(n+0.7)=n as n is greatest integer less than n+0.7. Substituting into the original equation of 2ceiling(x)+floor(x)+x=10.7 gives 2(n+1)+n+(n+0.7)=10.7 or combining like terms on LHS gives 4n+2.7=10.7 or 4n=8 implies n=2 and since x=n+0.7, x=2.7 is the only solution to this equation.

Thank you for teaching me new information

Nice. I like the equation y=x^(1/x) which has no name. It’s the xth root of x. One should try to visualise the graph. If you do, you will find it has a max value exactly where you should expect it to be

Damn smart guy .

I just guessed 2.7 lol, time to watch the video to find out how to actually do it :)

t = flour(x), x = t + 0.7

2(t+1) + t + t = 10
4t = 8
t = 2
x= 2.7

What did you do for 8 minutes?

asnwer=5 isit

Looking forward to more of those...ive never seen an equation like that before

2*ceil(x) + floor(x) + x = 10.7

Taking ceil from both sides:
ceil(2*ceil(x) + floor(x) + x) = ceil(10.7)
3*ceil(x) + floor(x) = 11

So, floor(x) = 11 - 3*ceil(x)

Taking floor from both sides:
floor(2*ceil(x) + floor(x) + x) = floor(10.7)
2*ceil(x) + 2*floor(x) = 10

So, 2*ceil(x) + 2*(11 - 3*ceil(x)) = 10
12 - 4*ceil(x) = 0
ceil(x) = 3
floor(x) = 11 - 3*ceil(x) = 2

From original equation
x = 10.7 - 2*ceil(x) - floor(x) = 2.7

❤️🙏

This man is good at what he does.

first 🎉

❤❤❤

Nice class sir 🎉