On a city road, the last traffic light glows green for 60 s and red for 120 s. The range of speeds of vehicles is from $\frac{50}{3} {ms}^{- 1} to \frac{200}{9} {ms}^{- 1}$ in a group. The speed of each vehicle is constant. It is found that at a distance x from traffic light, then the successive groups pushing through traffic light place will become indistinguishable. The value of x (in km) is_________

Moderate

Solution

Multiples of 60: 60,120,180,240, etc
Multiples of 120: 120,240,480 etc
Multiples of 24: 24,48,72,96,120,144 etc
common multiples of 60,120 and 24
LCM of 60,120,24 is 120
$\begin{array}{l} x = \frac{200}{9} t = \frac{50}{3} (t + 120) \\ or \frac{200 t}{9} = \frac{50 t}{3} + \frac{50}{3} \times 120 \\ or \frac{200 t - 150 t}{9} = 2000 or 50 t = 2000 \times 9 \\ ∴ t = \frac{2000 \times 9}{50} = 360 s \\ ∴ x = \frac{200}{9} \times 360 = 200 \times 40 = 8000 m = 8 km \end{array}$

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