\[t'=\frac{(1-\frac{v}{c})t}{\sqrt{1-\frac{v^{2}}{c^{2}}}}=\frac{t-\frac{vt}{c}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}=\frac{t-t_v}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\]