Fraction to Integer Relation

Algebra Level 3

Let $\{t\}$ and $\lfloor t \rfloor$ denote the fractional and integral part of a positive real number $t$ respectively.

If $4\{t\} = t + \lfloor t \rfloor$ , what is the value of $3t$ ?

2 solutions

Chew-Seong Cheong
Jan 29, 2015

$4\{t\} = t + [t] \quad \Rightarrow 4\{t\} = [t] + \{t\} + [t] \quad \Rightarrow 4\{t\} - \{t\} = [t] + [t]$

$\Rightarrow 3\{t\} = 2[t] \quad \Rightarrow \{t\} = \frac {2}{3}[t]$

Since $\{t\} < 1\quad \Rightarrow [t] = 1\quad \Rightarrow t = 1 + \frac {2}{3} = \frac {5}{3}\quad \Rightarrow 3t = \boxed {5}$

Absolutely Right Sir

Mayank Srivastava - 6 years, 4 months ago

Soham Shanbhag
Jan 29, 2015

4{t} = t + [t] => 3{t} = 2[t]. Thus [t] = 1 and so {t} = 2/3. Thus 3t = 5 .