A boat traveled 45 miles downstream (w/ the current) in 3 hours and made the return trop (against the current) in 5 hours. What was the speed of the boat and what was the speed of the current?

b=boat (mph)

c=current (mph)

Set up equations using distance=speed*time
3(b+c)=45 ...(1)
5(b-c)=45 ...(2)

Multiply (1) by 5 and (2) by 3:
15b+15c = 225 ...(1a)
15b-15c = 135 ...(2a)

Add (1a) and (2a)
30b+0 = 360
b = 360/30 = 12

Subtract (2a) from (1a)
0+30c = 90
c = 90/30 = 3

Boat speed is 12 mph,
current speed is 3 mph

speed of boat ---- x mph

speed of current -- y mph

then :
1. 3(x+y) = 45 or x+y = 15
2. 5(x-y) = 45 or x-y = 9

add them:
2x = 24
x = 12

from one of the others, y = 3

etc.

Well, let's put on our detective hats and solve this mystery! We'll call the speed of the boat "B" and the speed of the current "C".

When the boat is going downstream, its effective speed increases because it gets a helpful push from the current. So, the boat's speed can be represented as B + C.

When the boat is going upstream, it has to fight against the current, so its effective speed decreases. Thus, the boat's speed can be represented as B - C.

Now, we know that the boat traveled 45 miles downstream in 3 hours. So the equation becomes:

(B + C) × 3 = 45

Similarly, when the boat made the return trip (45 miles), it took 5 hours. So the equation becomes:

(B - C) × 5 = 45

Now, let's solve this riddle! Starting with the first equation:

(B + C) × 3 = 45

We can simplify it by dividing both sides by 3:

B + C = 15

Now, let's focus on the second equation:

(B - C) × 5 = 45

Again, we can simplify it by dividing both sides by 5:

B - C = 9

Now, we have a system of two equations:

B + C = 15
B - C = 9

If we add these equations together, the "C" variables will cancel out:

(B + C) + (B - C) = 15 + 9

This simplifies to:

2B = 24

Now, divide both sides by 2:

B = 12

So the speed of the boat is 12 miles per hour. But wait, there's more!

Now that we know the boat's speed, let's find the speed of the current by substituting the value of B into one of the original equations. Let's use the first one:

(B + C) × 3 = 45

(12 + C) × 3 = 45

Now, let's divide both sides by 3:

12 + C = 15

Subtract 12 from both sides:

C = 3

Voila! The speed of the current is 3 miles per hour. So, the boat's speed is 12 mph, and the current's speed is 3 mph.

Now that our detective work is done, it's time to set sail with that newfound knowledge!

To find the speed of the boat and the speed of the current, we can use the formula:

Boat speed + Current speed = Downstream speed
Boat speed - Current speed = Upstream speed

Let's set up the equations using the given information:
Equation 1: Boat speed + Current speed = Downstream speed
Equation 2: Boat speed - Current speed = Upstream speed

Given:
Downstream distance = 45 miles
Downstream time = 3 hours
Upstream time = 5 hours

First, we need to find the downstream speed:
Downstream speed = Downstream distance / Downstream time
Downstream speed = 45 miles / 3 hours
Downstream speed = 15 miles per hour

Using Equation 1, we can substitute the downstream speed into the equation:
Boat speed + Current speed = 15 miles per hour

Next, we need to find the upstream speed:
Upstream speed = Downstream distance / Upstream time
Upstream speed = 45 miles / 5 hours
Upstream speed = 9 miles per hour

Using Equation 2, we can substitute the upstream speed into the equation:
Boat speed - Current speed = 9 miles per hour

We now have a system of equations:
Equation 1: Boat speed + Current speed = 15 miles per hour
Equation 2: Boat speed - Current speed = 9 miles per hour

To solve these equations, we can use the method of elimination. By adding the two equations together, we can eliminate the "Current speed" variable:
(Boat speed + Current speed) + (Boat speed - Current speed) = 15 miles per hour + 9 miles per hour

Simplifying the equation gives us:
2 * Boat speed = 24 miles per hour

Dividing both sides by 2 gives us the boat speed:
Boat speed = 24 miles per hour / 2
Boat speed = 12 miles per hour

To find the current speed, we can substitute the boat speed into either Equation 1 or Equation 2. Let's substitute it into Equation 1:
12 miles per hour + Current speed = 15 miles per hour

Subtracting 12 miles per hour from both sides gives us:
Current speed = 15 miles per hour - 12 miles per hour
Current speed = 3 miles per hour

Therefore, the speed of the boat is 12 miles per hour and the speed of the current is 3 miles per hour.

thank you guys so much