To travel 100 miles, it takes sue, riding a moped, 3 hours less time than it takes doreen to travel 42 miles riding a bicycle. sue travels 19 miles per hour faster than doreen. find the times and rates of both girls.

Let us assume Doreen rate = x miles per hour.

Sue travels 19 miles per hour faster than Doreen.

Therefore,

Rate of Sue = (x+19) miles per hour.

We know time, rate and distance relation as

Time = Distance / rate.

Therefore, time taken by Doreen to travel  42 miles at the rate x miles per hour =

.

And time taken by Sue to travel  100 miles at the rate (x+19) miles per hour =

.

Sue takes 3 hours less time than it takes Doreen.

Therefore,

We need to solve the equatuion for x now.

We can't take rates as negative numbers.

So, the rate of Doreen (x) = 6 miles per hour.

Rate of Sue = x+19 = 6+19 = 25 miles per hour.

Time taken by Doreen @ 6 miles per hour to cover 42 miles = 42/6 = 7 hours.

Time taken by Sue @ the rate 25 miles per hour to cover 100 miles = 100/25 = 4 hours.