Math word problem plz help me

Question

Sarah wanted to catch Jim. However, although they started at the same time, Jim traveled at 80 km/h and Sarah traveled at 120 km/h. How much of a head start did Jim have if it took three hours for Sarah to catch him? 

Answer 1

Hello Caleb.

If one person tries to catch another, you need to find difference of speeds first

120 - 80 = 40; in one hour they close 40 miles.

Since it took 3 hours;

40*3 = 120 miles of head start

Answer 2

Let's call h the head start (in kilometers) that Jim had.

At the time when Sarah catches Jim, they are at the same location. Let's say that Jim traveled a distance of d (kilometers) from where he started. Sarah traveled a bit further, since Jim had a head start, so she traveled a total of d + h.

But they traveled for the exact same amount of time, from the time they started (they started at the same time) to the time when she caught him. We can express that by saying T_S = T_J (equality of duration, time for Sarah equals time for Jim). That also happens to be exactly 3 hours.

We also know the general equation that speed = distance / time, or time = distance / speed. If we apply this equation to both Sarah and Jim, and we take into account the fact that they both traveled for the same amount of time, we get:

distance_S / speed_S = distance_J / speed_J

We also know the distance traveled and the speed for both of them, so:

(d + h) / 120 = d / 80

If we multiply both sides by 120, we get:

d + h = (3 d) / 2

or:

h = d / 2

We also know d since that's the distance traveled by Jim, for three hours, at a speed of 80 km / h, so d must be 3 * 80 = 240 km. h must be half of that, or 120 km.

In short, Jim had a head start of 120 kilometers.