Kate can row a boat 10 miles per hour in still water. in a river where the current is 5 miler per hour, it take her 4 hour longer to row a given distance upstream than to travel the same distance downstream. Find how long it takes her to row upstream, how long to row downstream, and how many miles she rows.

please help/ thank you.

rate against the current = 10-5 = 5 mph

rate with the current = 10+5 = 15 mph

let the distance be x

time against the current = x/5
time with the current = x/15

x/5 - x/15 = 4
times 15
3x - x = 60
2x = 60
x = 30

each way is 30 miles
time upstream = 30/5 = 6 hrs
time downstream = 30/15 = 2 hrs
(note the difference is 4 hrs, as it should be)

To solve this problem, we can use the concept of relative speed. Let's break down the given information and find the solution step by step:

1. Kate's rowing speed in still water is 10 miles per hour.
2. The current in the river is 5 miles per hour.

Let's assume the distance Kate rows is 'd' miles.

1. Speed downstream: When Kate rows downstream, the speed of the river current adds up to her rowing speed. The speed downstream will be the sum of Kate's rowing speed and the current's speed:
Speed downstream = Kate's rowing speed + Current's speed = 10 mph + 5 mph = 15 mph.

2. Speed upstream: When Kate rows upstream, the speed of the river current subtracts from her rowing speed. The speed upstream will be the difference between Kate's rowing speed and the current's speed:
Speed upstream = Kate's rowing speed - Current's speed = 10 mph - 5 mph = 5 mph.

3. Time taken downstream: We can use the formula: Time = Distance / Speed. Since the speed downstream is 15 mph and the distance is 'd', the time taken downstream can be calculated as:
Time downstream = d / 15 hours.

4. Time taken upstream: Similarly, since the speed upstream is 5 mph and the distance is 'd', the time taken upstream can be calculated as:
Time upstream = d / 5 hours.

According to the given information, it takes Kate 4 hours longer to row upstream than to travel downstream. So, we can create an equation using the information above:

Time upstream = Time downstream + 4.

Now, let's substitute the equations for time downstream and time upstream:

d / 5 = d / 15 + 4.

To solve this equation, we start by getting rid of the fractions:

1. Multiply through by the least common multiple of 5 and 15, which is 15:

3d = d + 60.

2. Simplify the equation:

3d - d = 60,
2d = 60,
d = 30.

So, the distance Kate rows is 30 miles.

Now, we can substitute the value of 'd' in the time equations:

Time downstream = 30 / 15 = 2 hours.

Time upstream = 30 / 5 = 6 hours.

Thus, it takes Kate 6 hours to row upstream, 2 hours to row downstream, and she rows a total distance of 30 miles.

Hope this explanation helps you understand how to solve the problem!