Distance, Rate, and Time

Question

Jack leaves Kingstown at 1:00 P.M. and drives to Queensville, 160 mile distant, at 45 miles per hour. At 1:15 P.M. Jill leaves Queensville and drives to Kingstown at 40 miles per hour. At what time do they pass each other on the road?

Answer 1

The best way I’ve seen to solve these was in a sheltered math class I observed during my first year as a math teacher. It involved setting up a distance-rate-time chart, and using it to write and solve the equations which result from it.

The easy part is plugging in what you are given.

For Jack, at any given time, his rate is 45 miles/hour, and his time traveled is t:

	d (distance)	r (rate)	t (elapsed time)
Jack		45	t
Jill

And since

d = rt

The distance traveled is 45t:

	d (distance)	r (rate)	t (elapsed time)
Jack	45t	45	t
Jill

For Jill, her rate is 40 miles/hour:

	d (distance)	r (rate)	t (elapsed time)
Jack	45t	45	t
Jill		40

Also, since she is starting 15 minutes, or ¼ hour after Jack did, her time doesn’t start counting until ¼ hour after Jack’s time started. It helps me to think that after ¼ hour (“Jack time”; this whole problem is in “Jack time”) Jill’s time is zero.

So Jill’s time travel is:

t – ¼

	d (distance)	r (rate)	t (elapsed time)
Jack	45t	45	t
Jill		40	t – ¼

Now we have to be careful.

Jill is traveling backwards, from the destination toward the origin, She is starting 160 miles from the origin, so this r * t progress has to be subtracted from 160.

So

Distance = 160 - rate * (time – ¼)

= 160 – 40(t – ¼)

= 160 – 40t + 10

= 170 – 40t

	d (distance)	r (rate)	t (elapsed time)
Jack	45t	45	t
Jill	170 – 40t	40	t – ¼

Now, we have enough information to confront what the problem is asking us: namely, when do they pass each other?

In other words: when are their distances from the origin equal to each other?

With this chart, it’s easy to write an equation just by looking at it.

When Jack and Jill’s distances from Queensville are the same, the chart above tells us:

45t = 170 – 40t

Solving,

85t = 170

t = 2

So, after two hours “Jack time”, they pass each other…

… but… is that what we were asked to find?

NO.

It’s interesting, and important, but NOT the result we were asked for.

We were asked what time it was when they passed each other.

Getting that is pretty easy. Just add the 2 hours to Jack’s start time:

1:00 P.M. + 2 hours = 3:00 P.M.

They’ll pass each other at 3:00 P.M.

Answer 2

15 minutes is 1/4 hour. Jack will have traveled 45/4 = 11.25 miles before Jill starts.

So the distance is now 160 - 11.25 = 148.75 miles.

The question now is how long will it take them to travel a combined total of 148.75 miles.

Their combined speed is 45 + 40 = 85 miles per hour.

148.75 / 85 = 1.75 hours, or 1 hour, 45 minutes.

Adding 1 hour, 45 minutes to 1:15 pm results in 3 pm exactly.