A tugboat goes upstream 120 miles in 20 hours. The return trip downstream takes 10 hours. Find the speed of the tugboat without the current and the speed of the current.

Set-up:

Let v=speed of boat
c=speed of current
Distance = speed*time
Upstream:
120=(v-c)*20 ...(1)
Downstream:
120=(v+c)*10 ...(2)

Solve for v and c.

Okay, I got:

v+6-c and c=c-6 for 120=(v-c)*20

AND:

c=12-v and v=6/5-c for 120=(v+c)*10

Now what do I do?

What you really want to do is to reduce the variables in simpler terms by dividing both sides by 20:

v-c=120/20=6 ...(1a)
v+c=120/10=12...(1b)

This is a sum and difference form where the solution can be done mentally:
v=(sum+difference)/2=9
c=(sum-difference)/2=3

To find the speed of the tugboat without the current and the speed of the current, we can use a system of equations and solve them simultaneously.

Let's assume the speed of the tugboat in still water (without the current) is 'x' miles per hour, and the speed of the current is 'y' miles per hour.

For the upstream trip, the effective speed of the tugboat against the current is (x - y) miles per hour. We know that it travels 120 miles in 20 hours, so we can write the equation:

(x - y) * 20 = 120

For the downstream trip, the effective speed of the tugboat with the current is (x + y) miles per hour. We know that it travels 120 miles in 10 hours, so we can write the equation:

(x + y) * 10 = 120

Now we have a system of equations:

Equation 1: (x - y) * 20 = 120

Equation 2: (x + y) * 10 = 120

Let's solve these equations to find the values of 'x' and 'y'.

From Equation 1, we can simplify it to get:

20x - 20y = 120

Divide both sides of the equation by 20:

x - y = 6 (Equation 3)

From Equation 2, we can simplify it to get:

10x + 10y = 120

Divide both sides of the equation by 10:

x + y = 12 (Equation 4)

To solve this system of equations, we can add Equation 3 and Equation 4:

(x - y) + (x + y) = 6 + 12

2x = 18

Divide both sides by 2:

x = 9

Now, substitute the value of x into Equation 4:

9 + y = 12

Subtract 9 from both sides:

y = 3

Therefore, the speed of the tugboat without the current (x) is 9 miles per hour, and the speed of the current (y) is 3 miles per hour.