Uniform Motion Problems

Question

David and Amy live 435 miles apart. They are planning to meet up somewhere between the two locations. If David travels at 60 mph and Amy travels at 45 mph, and Amy is planning on leaving two hours later than David, how many miles will David travel before they meet?

Answer 1

Sketch the problem so David drives to the left and Amy drives to the right.

Initially David and Amy are 435 miles apart. As both drive toward each other, the combined distance traveled will eventually equal to 435 miles at the point where they meet.

Distance = speed x time

The different starting time adds a bit complexity to the problem, so we need to express the amount of time Amy drives as 2 hours less than the amount of time David drives.

So for David, the distance he travels to the meeting point is 60 x T

For Amy, the distance she travels to the meeting point is 45 x (T-2)

Adding these 2 expressions 60T + 45(T-2) = 435. then solve for T ; 105T - 90 = 435 ; T = 525 / 105 = 5

Hence David drives 60x5 = 300 miles to the meeting point (Amy drives 135 miles).

Answer 2

Distance = rate * time or Distance = speed * time

Let Dd = David's distance

Let t = time that David spends driving

Let rD = rate at which David travels (60 mph)

Let Ad = Amy's distance

Let rA = rate at which Amy travels (45 mph)

Amy's time in the car will be David's time (t) less 2 hours because she leaves two hours later than David, so Amy's time driving will be (t - 2)

The total distance is 435 miles = Ad + Dd

(Amy's Distance + David's Distance = total distance between homes

Amy's Distance = Amy's speed * Amy's time so Ad = 45(t-2)

David's Distance = David's speed * Davids time Dd = 60t

Together, their distances will add up to 435 miles. Ad + Dd = 435 45(t-2) + 60t = 435

distribute the 45 into the parenthesis

45t - 90 + 60t = 435

combine like terms

105t - 90 = 435

add 90 to both sides

105t = 525

divide both sides by 105

t = 5 hours

David drives for 5 hours at 60 miles per hour, so David's distance is 300 miles

Amy drives for 3 hours at 45 miles per hour, so Amy's distance is 135 miles

Final answer: David drives 300 miles before they meet.