A cruise boat travels 48 miles downstream in 3 hours and returns upstream in 6 hours. Find the rate of the stream

THANK YOU

To find the rate of the stream, we can use the concept of relative motion. The speed of the boat in still water is the average of the speeds downstream and upstream.

Let's call the speed of the boat in still water "b" and the rate of the stream "s".

Given:
Distance downstream = 48 miles
Time downstream = 3 hours

The speed of the boat downstream is given by:
Speed downstream = b + s

Using the formula: Speed = Distance / Time, we can write the equation:
48 / 3 = (b + s)

Simplifying, we have:
16 = b + s ---(1)

Given:
Distance upstream = 48 miles
Time upstream = 6 hours

The speed of the boat upstream is given by:
Speed upstream = b - s

Using the formula: Speed = Distance / Time, we can write the equation:
48 / 6 = (b - s)

Simplifying, we have:
8 = b - s ---(2)

Now, we have a system of linear equations (1) and (2). We can solve this system to find the values of "b" and "s".

Adding equations (1) and (2), we get:
16 + 8 = (b + s) + (b - s)

Simplifying, we have:
24 = 2b

Dividing both sides by 2, we get:
b = 12 mph

Now, substitute the value of "b" into equation (1):
16 = 12 + s

Subtracting 12 from both sides, we get:
s = 4 mph

Therefore, the rate of the stream is 4 mph.

To find the rate of the stream, we need to use the concept of relative speed.

Let's assume that the speed of the boat is B mph, and the speed of the stream is S mph.

When the boat is moving downstream (i.e., with the stream), the combined speed of the boat and the stream will be (B + S) mph. Similarly, when the boat is moving upstream (i.e., against the stream), the combined speed will be (B - S) mph.

Given that the boat travels 48 miles downstream in 3 hours, we can use the formula: Distance = Speed × Time. So, 48 = (B + S) × 3.

Similarly, when the boat returns upstream and travels 48 miles in 6 hours, we can use the formula: 48 = (B - S) × 6.

Now we have a system of two equations:
1. 48 = (B + S) × 3
2. 48 = (B - S) × 6

Let's solve these equations to find the values of B and S:

Expanding equation 1, we get:
48 = 3B + 3S

Similarly, expanding equation 2, we get:
48 = 6B - 6S

Rearranging equation 1, we have:
3B + 3S = 48

Next, dividing equation 2 by 6, we get:
8 = B - S

We can now solve these two equations simultaneously by substitution or elimination. For simplicity, let's use the elimination method.

Multiplying equation 1 by 2, we have:
6B + 6S = 96

Now, adding this equation to equation 2 (B - S = 8), we get:
6B + 6S + B - S = 96 + 8

Simplifying the equation, we get:
7B = 104

Dividing both sides by 7, we find:
B = 104 / 7

Therefore, the speed of the boat is approximately 14.86 mph.

To find the speed of the stream, substitute the value of B back into equation 2:
8 = B - S
8 = 14.86 - S

Rearranging the equation to solve for S, we get:
S = 14.86 - 8

Calculating this, we find that the rate of the stream is approximately 6.86 mph.

Thus, the rate of the stream is approximately 6.86 mph.

distance = rate x time

downstream 48 mi in 3 hours so rate = 16 mi/hr.
Let B = boat speeed and S = stream speed so rate = 16 mi/hr = B + S
Upstream is 49 mi in 6 hours or 48/6 8 mi/hr
or B-S = 8
Two equation in two unknowns.
B+S = 16
B-S = 8
Solve for B and S (although S is what you want). Check my work.