On an up hill hike ted climbs at 3 mph. going back down, he runs at 5 mph. if it takes him forty minutes longer to climb up than run down, then how is the long is the hike?

If the amount of time it takes to climb the hill is x hours, then 3x is the distance of the hill. In addition, if the amount of time it takes to go back down is y, then 5y is the distance of the hill too, so 5y=3x. Since it takes 40 minutes longer (2/3rds of an hour) to climb up the hill, y+2/3=x, making it 5y=3(y+2/3)=3y+6/3=2+3y using the distributive property. Subtracting 3y from both sides, we get 2y=2. Dividing both sides by 2, we get y=1. Multiplying that by 5, we get that 5*1=the hike (1 way)=5 miles.