​Connie's boat travels at 5 mph. Find the rate of the current of the river if she can travel 12 miles upstream in the same time she can travel 28 miles downstream.

Answer 1

I think what you mean is that the boat will travel 5 mph *in still water*.

Problems like this seem complicated, so it best to draw a chart, using the standard d = rt formula.

Since your unknown is the speed of the current, let c = current speed

Upstream values: d=12 miles r=5-c (that is 5 mph minus the speed of the current). t is the same for both up and downstream.

Downstream values: d = 28 miles r=5+c, t is, as above, unknown, but the same as the time upstream.

Since t is the same for both trips, manipulate the basic formula to read t = d/r Then set the upstream d/r equal to the downstream d/r. This would give you:

12/(5-c) = 28/(5+c)

Solve by the usual method. Cross products gives you

60+12c=140-28c

12c= 80-28c (subtracting 60 from both sides)

40c=80 ( adding 28c to both sides)

c=2 (dividing both sides by 40)

The current is 2 mph.

Answer 2

Hello Isabella,

Let x = speed of the current.

5 + x = boat's speed downstream

5 - x = boat's speed upstream

(D/R) = T

(28/(5 + x)) = time for downstream trip

(12/(5 - x)) = time for the upstream trip

Since the times for the two trips is the same,

we can set the two fractional expressions equal to each other.

(28/(5 + x)) = (12/( 5 - x))

Cross-multipy to get: 28(5 - x) = 12(5 + x).

140 - 28x = 60 + 12x

80 = 40x

x = 2 mph for the current.

It helps if you write the calculation steps as you proceed. There's too much mental clutter when you try to hold it all in your head.

You don't have to wait till you know what the answer looks like to pick up your pencil.

Regards,

Anita A.

Answer 3

So, if I can row 5 mph in still water and the rate of the current is 7 mph, I am going to go 5 + 7 = 12 mph going downstream and 5 - 7 = -2 mph trying to go upstream (ending up still going downstream 2 mph).

d = rt, so t = d/r, and the times are equal. Let c = the rate of the current.

12/(5 - c) = 28/(5 + c)

Cross multiply.

12(5 + c) = 28(5 - c)

60 + 12c = 140 - 28c

40c = 80

c = 2 mph

Answer 4

Let's first assign variables and come up with equations to represent traveling upstream and downstream.

velocity of boat v₁ = 5 miles/hour; current of river = v₂ (this is what we are trying to find); x = time in hours

Here is the first equation: x = [12(miles upstream)] / (v₁ - v₂)

Here is the second equation: x = [28(miles downstream)] / (v₁ + v₂)

Now, we need to substitute the first equation for the value of x into the second equation. For simplification, I will not use any units in these next few steps.

12 / (v₁ - v₂) = 28 / (v₁ + v₂) ---> now multiply both sides by (v₁ + v₂) ; you get

[12*(v₁ + v₂)] / (v₁ - v₂) = 28 ---> now divide both sides by 12 ; you get

(v₁ + v₂) / (v₁ - v₂) = 2.333 / 1 ---> now cross multiply ; you get

(v₁ + v₂) = (v₁ - v₂) * 2.333 ---> now distribute the 2.333 ; you get

(v₁ + v₂) = 2.333v₁ - 2.333v₂ ---> now subtract v₁ from both sides ; you get

v₂ = 1.333v₁ - 2.333v₂ ---> now add 2.333v₂ to both sides ; you get

3.333v₂ = 1.333v₁ ---> now divide both sides by 3.333 ; you get

v₂ = 0.39994v₁ ---> we know v₁ = 5 miles/hour; so now

v₂ = 0.39994 * 5 miles/hour

v₂ = 2 miles/hour (this is our answer). you may need to convert it to different units depending on what type of velocity that you need.

Now to check our work with the equations we created:

x = [12(miles upstream)] / (v₁ - v₂)

x = 12 / (5 - 2) ; the amount of time is 4 hours

x = [28(miles downstream)] / (v₁ + v₂)

x = 28 / (5 + 2) ; the amount of time is 4 hours

Now that our answer for v₂ is confirmed with the original equations, you are done with this problem!

Answer 5

You would solve this with a variation of the distance = rate × time formula where you are solving for time instead:

time = distance / rate

Upstream's time = 12 / (x–5)

Downstream's time = 28 / (x+5)

It says the "same time" for upstream and downstream so you would set the their times equal to each. From there, you would cross multiply in order to solve for x.

I hope this helps you solve.

Answer 6

A good place to start is writing out the equation for determine distance traveled, which is:

speed*time = distance

In this problem Connie is encountering two situations, one where her speed and the speed of the river sum and one where the subtract. Since she is "going with the flow" in one instance and upstream in the other.

We will use B=rate of the boat, R=rate of the river, and t=time

(B+R)*t = 28

(B-R)*t = 12

Then, set each equation equal to t:

t = 28/(B+R)

t = 12/(B-R)

Now, the two equations are equivalent:

28/(B+R) = 12/(B-R)

Next, substitute in B=5 and simplify:

28/(5+R) = 12/(5-R)

28*(5-R) = 12*(5+R)

140 - 28R = 60 +12R

80 = 40R

R = 2