Question

Ralph is driving from his home to his grandmother’s house at an average rate of 40 miles per hour (mph). He had driven 6 miles when his sister Maria left their house and began driving along the same route. At what average rate, in miles per hour, is Maria driving if she catches up with Ralph in 1 1/2 hours?
A) 60

B) 44

C) 40

D) 46

Answer 1

Let's call Maria's average speed $M$

Imagine for a moment that Maria has a stopwatch, that measures in hours.
She starts her stopwatch when she starts driving.

If Maria looks at her stopwatch at a time $t$, her distance traveled will be $M * t$ miles. (This is true since $distance = speed * time$.)

What about Ralph? If Ralph could look at Maria's stopwatch, his distance traveled would be $40*t + 6$, since his speed is 40mph and he got a six-mile head start. Remember that $t$ is measured from Maria's start, not Ralph's.

So when they meet, after 1.5 hours, their distances are equal. So:

$Distance(Maria) = Distance(Ralph)$

$M*t=40t+6$

And since that time is 1.5, we plug that in for t:

$M*1.5=40*1.5+6$

$M*1.5=60+6$

$M*1.5=66$

Divide both sides by 1.5:

$\frac{M*1.5}{1.5} =\frac{66}{1.5}$

Which cancels on the left:

$M = \frac{66}{1.5} = 44$ miles per hour.