125th Problem 2016

We can use the formula $\boxed{d=Vt}$ where: $d$ =distance, $V$ =speed and $t$ =time

Considering the first train:

The distance traveled is

$d_1=V_1t_1=80(t+2)=80t+160$ $\implies$ note that $2$ was added because it started $2$ hours earlier than the second train

Considering the second train:

The distance traveled is

$d_2=V_2t_2=100t$

Now, we know that $d_1$ must be equal to $d_2$ , so

$d_1=d_2$

$80t+160=100t$

$20t=160$

$\large\boxed{\color{#20A900}t=8~hours}$ $\color{#EC7300}\boxed{\text{answer}}$

$\large \text{Distance = Rate}\times \text{Time}$

$\large { r }_{ 1 }{ t }_{ 1 }={ r }_{ 2 }{ t }_{ 2 }$

The distance traveled by both trains must be the same when the second train overtakes the first.

$\large 80(2+{ t }_{ 2 })=100{ t }_{ 2 }\\ \large 160+{ 80 }t_{ 2 }=100{ t }_{ 2 }\\ \large 160=20{ t }_{ 2 }\\ \large \boxed { 8={ t }_{ 2 } }$

$\large \therefore \text{8 hours after the second train leaves it will overtake the first train.}$