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# Distance, Rate and Time Problem

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Karen can row a boat 10 kilometers per hour in still water. In a river where the current is 5 kilometers per hour, it takes her 4 hours 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 kilometers she rows?

https://brainmass.com/math/basic-algebra/distance-rate-time-problem-12493

## SOLUTION This solution is FREE courtesy of BrainMass!

These problems, however, can be tricky.

One of the best ways to start a problem like this is to make a table that uses all the information you have been given. Let's make one for the information we have about the distance, rate, and time Karen travels when she is going both upstream and downstream.

We'll call the time it takes to row downstream x, which means that the time it takes to row upstream is x +4.

We'll start by calculating Karen's rates going upstream and downstream. When she is traveling against the current, she won't be able to row 10 kilometers/hour. Her speed relative to the shore will only be 5 kilometers per hour because the force of the current, which is flowing at 5 kilometers/hour, slows her rate by 5 km/hour. When Karen is rowing downstream, however, the current helps her go faster, so she moves 10 + 5 = 15 km/hour.

We can use the formula, written as Rate x Time = Distance:

Therefore,
Downstream = 15 x (x) = 15x ;where 15=rate, x=time, and 15x=distance

And upstream = 5 x (x+4) = 5(x+4) ;where 5=rate, x+4=time, and 5(x+4) = distance

Because Karen goes the same distance upstream and downstream, we know that the two expressions of distance - for upstream and downstream - must be equal; we can set the upstream distance equal to the downstream distance. This produces the following equation, which we solve for x:

Statement of original equation: 15x = 5(x+4)
Distributing on right side: 15x = 5x+20
Subtracting 5x from both sides: 10x = 20
Dividing both sides by 10: x = 2

x equals the time it takes Karen to row downstream, or 2 hours. Since it takes her four hours longer to row upstream, this time will be 2 + 4 = 6 hours.

How many kilometers does she row? Look at the distance column in the table. Since x is in hours, Karen's downstream distance is 15 x 2 = 30 kilometers.

The problem states that Karen rows the same distance upstream as down. Let's check our work... yes, 5(2+4) = 5 x 6 = 30 kilometers.

As is frequently the case with word problems, setting up the equations is the hardest part. Once that's done, the rest is relatively easy. Remember always to answer what the question asks - don't stop once you've solve for x, because that may be only part of what the question asked - and always check your answer.

BEST OF LUCK!

This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!