David started across a lake 15 miles wide in his fishing boat at 10 miles per hour. He had to row the rest of the way at only 5 miles per hour after his motor went out. If he was rowing for half the time that the trip took, how long did the total trip take

Total time = 2hrs

Step-by-step explanation:

Firstly, to outline some variables, with which we can construct some equations:

x = the time taken to cross the river at 10mph (in hrs)

y = the time taken to cross the river at 5mph (in hrs)

t = total time taken to cross the river (in hrs)

a = distance travelled at speed 10mph (in miles)

b = distance travelled at speed 5mph (in miles)

Then,

a + b = 15

t = x + y

y = ¹/₂ (t) ∴ t = 2y, so,

x + y = 2y ∴ x = y

Secondly, we know speed = distance/time, so,

10 = a/x ⇒ x = a/10

5 = b/x ⇒ x = b/5

So,

a/10 = b/5 ∴ a = 2b

Thirdly, if a + b = 15, then,

2b + b = 15 ⇒ 3b = 15 ⇒ b = 5

[a = 2b ∴ a = 10]

Fourthly, x = b/5, so,

x = 5/5 = 1,

Since x = y, y = 1 and since t = 2y,

t = 2 (1) = 2

The whole trip, therefore, took 2hrs.