126th Problem 2016

We can use the formula,

$\large\boxed{d=Vt}$ where: $d$ =distance, $V$ = speed and $t$ =time

Considering the train that leaves at 3:00 pm:

The total time it travels from $3:00~pm$ to $6:00~pm$ is $3$ hours. So we have

$d_1=V_1t_1=50(3)=150~miles$

Considering the train that leaves at 4:00 pm:

The total time it travels from $4:00~pm$ to $6:00~pm$ is $2$ hours. So we have

$d_2=V_2t_2=60(2)=120~miles$

Finally, the distance between them at exactly $6:00~pm$ is

$d=d_1+d_2=150+120=$ $\large\boxed{\color{#D61F06}270~miles}$ $\color{#3D99F6}\boxed{\large\text{answer}}$

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

$\large { d }_{ total }={ r }_{ 1 }{ t }_{ 1 }+{ r }_{ 2 }{ t }_{ 2 }$

Since both trains are travelling in opposite directions, their total distance apart is the sum of the distances they each traveled.

$$

Time between:

${ t }_{ 1 }: \text{ 3:00 AM and 6:00 AM}$

${ t }_{ 2 }: \text{ 4:00 AM and 6:00 AM}$

$\large { t }_{ 1 } = 3$

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

$$

$\large { d }_{ total }={ (50) }{ (3) }+{ (60) }{ (2) }$

$\large { d }_{ total }={ 270 }$

$\large \therefore \text{The trains are 270 miles apart.}$