Math word problem plz help me out

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

distance = rate * time

1. We are measuring our speed in kilometers per hour
2. Jim traveled at a rate of 180 kilometers every 1 hour
3. Sarah traveled at a rate of 120 kilometers every one hour
4. it took Sarah 3 hours to catch up to Jim
5. Jim had an unknown head start

distance = rate * time

Sarah's rate is 120 km/hr

Sarah traveled for 3 hours;

Sarah's distance:

120km * 3hrs = 360 kilometers

Now we know the distance both Sarah and Jim went, Jim just had a head start in time by 3 hours.

so, in Jim's equation based on his rate;

d = rate * time

and we have defined that

t = sarah's travel time

Josh's time must be equivalent to (t + 3)

and his d = 360 km

360 = 80 * (t +3)

4.5 = t + 3

1.5 = t

Jim had a head start of 1.5 hours

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.

Answer 3

Sarah will catch Jim when the total distances match.

120(3) = 80(3) + d

d = 360-240 = 120 km

Jim's head start time = 120 km/80km/hr= 1.5 hour head start