a motorboat, which can travel at 20 miles per hour in still water, takes 3/5 as long to travel downstream on a river from a to b, as to return. Find the rate of the rivers current.
speed of current --- x mph
distance ---- d miles
time to go downstream = d/(20+x)
time to go upstream = d/(20-x)
d/(20+x) = (3/5)d /(20-x)
divide both sides by d
1/(20+x) = 3/(5(20-x))
cross-multiply
60+3x = 100 - 5x
8x = 40
x = 5
the speed of the current is 5 mph
check:
suppose the distance is 100 miles
time to go downstream = 100/25 hrs = 4 hrs
time to go upstream = 100/15 = 20/3 hrs or
and (3/5)(20/3) = 4
Wow I wish I really understood this it sounds very interesting I wish I knew what y'all were talking about I wish some body would sit and explain it to me step by step but that will probably never happen
Let's assume the rate of the river's current is "x" miles per hour.
When the motorboat travels downstream with the current, its effective speed would be the sum of its own speed and the speed of the current. So, its effective speed would be (20 + x) miles per hour.
Similarly, when the motorboat travels upstream against the current, its effective speed would be the difference between its own speed and the speed of the current. So, its effective speed would be (20 - x) miles per hour.
Given that the motorboat takes 3/5 as long to travel downstream compared to traveling upstream, we can set up the following equation:
Time downstream = (3/5) * Time upstream
Distance = Speed * Time
Distance downstream = Distance upstream
So, we can write the equation as:
(20 + x) * (3/5) = (20 - x)
Now, let's solve this equation to find the rate of the river's current (x).
(3/5)(20 + x) = (20 - x)
Multiplying both sides by 5 to get rid of the fraction:
3(20 + x) = 5(20 - x)
60 + 3x = 100 - 5x
Combining like terms:
8x = 40
Dividing both sides by 8:
x = 5
Therefore, the rate of the river's current is 5 miles per hour.
To find the rate of the river's current, we can set up a proportion using the given information.
Let's say the rate of the river's current is "r" miles per hour.
When the boat is traveling downstream, the effective speed of the boat (including the current) will be 20 + r miles per hour.
When the boat is traveling upstream (against the current), the effective speed of the boat will be 20 - r miles per hour.
The time taken to travel downstream is given as 3/5 times the time taken to return. So, we can set up the proportion:
(Downstream Time) / (Return Time) = 3/5
To find the downstream time, we can use the formula:
Distance = Speed * Time
Let's say the distance between points A and B is "d" miles.
For the downstream journey, the effective speed is 20 + r miles per hour, and the time taken is Td hours:
d = (20 + r) * Td
For the return journey (upstream), the effective speed is 20 - r miles per hour, and the time taken is Tr hours:
d = (20 - r) * Tr
Now, we need to express the downstream time (Td) in terms of the return time (Tr) using the given proportion:
Td / Tr = 3/5
To solve for Td in terms of Tr, we can multiply both sides of the equation by Tr:
Td = (3/5) * Tr
Now we can substitute this value of Td into the equation for the downstream distance:
d = (20 + r) * (3/5) * Tr
Similarly, substitute the value of Td into the equation for the return distance:
d = (20 - r) * Tr
Since both equations are equal to the same distance, we can equate them:
(20 + r) * (3/5) * Tr = (20 - r) * Tr
Simplifying this equation:
(3/5)(20 + r) = 20 - r
Solving for r, we can start by getting rid of the fractions by multiplying both sides by 5:
3(20 + r) = 5(20 - r)
Now distribute on both sides:
60 + 3r = 100 - 5r
Combine like terms:
8r = 40
Divide both sides by 8:
r = 5
So, the rate of the river's current is 5 miles per hour.