Lesson Uniform Rates, Travel: same direction but different travel times meet when?

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About josgarithmetic: Academic and job experience with beginning & intermediate Algebra. Tutorial help mostly for Basic Math and up through intermediate algebra.

Different Speeds and Departure Times but Same Direction Meet When?

Here is a common kind of uniform rates, travel problem exercise.

A specific example.
A train leaves a station and travels north at 70km/hr. Two hours later, a second train leaves on a parallel track and travels north at 90km/hr. How far from the station will they meet?

The basic uniform rate equation for travel is {{{R*T=D}}}, and in this example, R is speed in kilometers per hour, T is time in hours, and D is distance in kilometers.    The question asks, what will the two trains equal distance be.  One train travels slow and spends more travel time and the  other train goes faster but starts the trip later, but travels less time.

Assign variables this way:
d = distance that each train travels when both are the same distance from starting point
t = travel time of the faster, later starting train

A data table helps to understand the problem analysis:

Train_________speed_________time_________distance
SLOW________70_____________t+2__________d=70(t+2)
FAST_________90_____________t____________d=90t

The expressions for d are equal when the fast train reaches the slow train.
{{{d=highlight_green(70(t+2)=90t)}}}
Solve for t, but then use the value to answer the question for d.

{{{70t+140=90t}}}
{{{70t+140-70t=90t-70t}}}
{{{140=20t}}}
{{{20t=140}}}
{{{highlight(t=7)}}}, seven hours after the fast train started, it meets with the slower train.

Either distance equation can be used for computing the distance, the {{{90t=d}}} equation being simpler to use:   {{{d=90*t}}}, {{{d=90*7}}}
{{{highlight(d=630)}}} kilometers

This example and any which fit this form can be solved completely in symbols when one wants to.

GENERALIZATION OF THIS EXAMPLE
A traveler departs in one direction  at  R km/hr. k hours later, a second traveler departs from the same starting point and travels in the same direction  at r km/hr. How far from the departure point  will they meet?  
{{{R>r}}} and {{{k>0}}}

______________speed_______time_________distance
SLOW EARLY____r___________t+k__________r(t+k)
FAST LATE______R___________t____________Rt

Solve first for t.
The distances will be equal.
{{{highlight_green(r(t+k)=Rt)}}}
{{{rt+rk=Rt}}}
{{{Rt-rt=rk}}}
{{{(R-r)t=rk}}}
{{{highlight(t=(rk)/(R-r))}}}

Form a formula solution for d, the distance when the travelers meet, using the now solved t.
{{{d=Rt}}}
{{{d=R(rk/(R-r))}}}
{{{highlight(d=(Rrk)/(R-r))}}}

Actual variable choices can vary, but must be maintained consistantly within each problem being analyzed and solved.