Question

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

Answer 1

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