Functions will blow your mind

Algebra Level 5

Find the number of x x satisfying the equation 100 { x } = [ 3 x ] + x 100\{x\}=[3x]+x .

where { x } \{x\} represents the fractional part of x x , and [ x ] [x] is the greatest integer function.


The answer is 27.

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Dhiraj Agarwalla by dhiraj agarwalla , 23, India

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1 solution

Dhiraj Agarwalla
Sep 30, 2014

Here the tricky part is [ 3 x ] [3x] .

The problem can be solved by taking into consideration three cases.

CASE 1: When 0 ≤ { x } < 1 3 0 \le \{x\} < \frac{1}{3} then [ 3 x ] = 3 [ x ] [3x]=3[x] .

CASE 2: When 1 3 ≤ { x } < 2 3 \frac{1}{3} \le \{x\} < \frac{2}{3} then [ 3 x ] = 3 [ x ] + 1 [3x]=3[x]+1 .

CASE 3: When 2 3 ≤ { x } < 1 \frac{2}{3} \le \{x\} < 1 then [ 3 x ] = 3 [ x ] + 2 [3x]=3[x]+2 .

Now using the above three cases we can easily form 3 equations, which will give 27 \fbox{27} solutions.

9 Helpful 0 Interesting 0 Brilliant 0 Confused

I cnt understand abt the tricky part. ...if fract x is .33333... The GIF part vil be ..666 aprox vhich z not posibl it should be a integer part...totly confused

Navii Sheikh - 6 years, 5 months ago

Edited to use L a T e X LaTeX and fixed incorrect statements. You can click the pencil at the top right corner to see what I have changed.

Kenny Lau - 6 years, 8 months ago

Can you show how??

Aakash Khandelwal - 5 years, 6 months ago

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