A Sprinter Thief and a Sprinter Dog !!......

At the time the police and dog start chasing the thief they are a distance of $40*3 = 120$ km behind. As the police catch up to the thief at a relative speed of $50 - 40 = 10$ km/hr, we know that both the police and the dog will be on the move for $\frac{120}{10} = 12$ hours. This means that the dog will have run a total of $12*60 = 720$ km by the time the police catch up to the thief.

Now since the dog catches up to the thief at a relative speed of $60 - 40 = 20$ km/hr, it will first reach the thief after $\frac{120}{20} = 6$ hours. At this point, since the police move at a speed $10$ km/hr less than the dog, the distance between the police and the dog will be $6*10 = 60$ km. Once the dog turns around and returns to the police, the dog will approach the police at a relative speed of $50 + 60 = 110$ km/hr, so the dog will return to the police after $\frac{60}{110}$ hours. For this one 'cycle', the dog will have spent a fraction

$\dfrac{6}{6 + \frac{60}{110}} = \dfrac{660}{720}$

of the time traveling in the direction of the thief. For each subsequent 'cycle' the dog runs, (i.e., from the police to the thief and then back again), the fraction of time the dog spends running towards the thief will be the same. So since the dog runs a total of $720$ km, the distance it travels in the direction of the thief will be

$\dfrac{660}{720} * 720 = \boxed{660}$ km.

• 120 + 40t = 50t $\Rightarrow$ (50 - 40)t = 120 $\Rightarrow$ t = 12 hr
• Distance cut by the Police = 50 t = 50 * 12 = 600 Km
• Total distance cut by the Dog = 60 t = 60 * 12 = 720 Km
• The new distance cut by the Dog = Distance cut by the Police = 600 Km
• Repeated back and forth distance = 720 - 600 = 120 Km

As the repeated back and forth are the same because the dog returns distance (r) to the police and run the same distance (r) towards the thief + new distance (n) for the new position of the thief which computed in (The new distance cut by the dog = 600 Km) .

• Then the return distance towards the police = $\frac {120}{2}$ = 60 Km

$\Rightarrow$ The total distance cut by the dog towards the thief = 720 - 60 = 660 Km.

If you confused with the last paragraph let's write it in Series.

• let the new distance cut by the dog = $\displaystyle \sum_{i=1}^kn_{i}$ = $n_{1}$ + $n_{2}$ + $n_{3}$ + $\ldots$ + $n_{k}$ = 600 Km
• and the return distance towards the police = $\displaystyle \sum_{i=1}^{k-1}r_{i}$ = $r_{1}$ + $r_{2}$ + $r_{3}$ + $\ldots$ + $r_{k-1}$

then the total distance cut by the dog will be:

$n_{1}$ + $r_{1}$ + ( $r_{1}$ + $n_{2}$ ) + $r_{2}$ + ( $r_{2}$ + $n_{3}$ ) + $\ldots$ + $r_{k-1}$ + ( $r_{k-1}$ + $n_{k}$ ) = 720 Km

then ( $n_{1}$ + $n_{2}$ + $n_{3}$ + $\ldots$ + $n_{k}$ ) + ( $2r_{1}$ + $2r_{2}$ + $2r_{3}$ + $\ldots$ + $2r_{k-1}$ ) = 720 Km

$\Rightarrow$ $\displaystyle \sum_{i=1}^kn_{i}$ + $2\displaystyle \sum_{i=1}^{k-1}r_{i}$ = 720 Km $\Rightarrow$ $\displaystyle \sum_{i=1}^{k-1}r_{i}$ = $\frac {720 - 600}{2}$ = 60 Km

$\Rightarrow$ The total distance cut by the dog towards the thief = 720 - 60 = 660 Km.