The hare leaps 500 centimeters every 20 seconds. the tortoise crawls 250 centimeters, but gets a 1000-centimeter head start. use any method you know to determine how long it takes the hare to catch up to the tortoise

It takes 80 seconds.

Since the hare runs at a speed of 500 cm in 20 seconds, we can write this as a unit rate by dividing:
500/20 = 25 cm/sec

We will do the same thing for the speed of the tortoise:
250/20 = 12.5 cm/sec

The distance the hare travels can be written as 25x, where x is the number of seconds.
The distance the tortoise travels can be written as 12.5x, where x is the number of seconds; but he also has a 1000 cm head start, so his expression is 12.5x+1000.
We know that at some point, these two will be at the same place, so we set the expressions equal to each other.

25x = 12.5x+1000

We want all of the variables on one side, so we subtract 12.5x from both sides:
25x-12.5x = 12.5x+1000-12.5x
12.5x = 1000

Divide both sides by 12.5:
12.5x/12.5 = 1000/12.5
x = 80