# Emily rows six miles downstream in 1 hour and her friend Ashley, rowing 1 mile per hour faster, completes the return trip in 2 hours. If Emily and Ash ley were rowing separately, who would complete...

Emily rows six miles downstream in 1 hour and her friend Ashley, rowing 1 mile per hour faster, completes the return trip in 2 hours. If Emily and Ash ley were rowing separately, who would complete their trip first and by how long? Round to the nearest hundredth, if necessary.

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Let Emily's speed be r in mph. Then Asley's speed will be (r+1) in mph.

We use distance equals rate times time or d=rt:

For Emiy's trip downstream she rows with the current; if the speed of the current is c then her rate is r+c.

So 6=(r+c)(1) 88 time is 1 hour ==> c=6-r

For Ashley's trip upstream she rows against the current; again with the speed of the current c we have her rate as r+1-c. (1 mph faster rowing than Emily, but against the current so subtract.)

So 6=(r+1-c)(2) **time here is 2 hours ==> c=r-2

Then 6-r=r-2 ==> r=4.

Thus Emily's rowing speed is 4mph, the current's speed is 2mph, and Ashley's rowing speed is 5mph.

(Check: rowing with the current Emily goes 4+2=6mph; so she takes 1 hour to row 6 miles. Ashley rows against the current so she goes 5-2=3mph and will take 2 hours to row 6 miles.)

If Emily rowed down and back at 4 mph she would travel 6 miles at 6 mph and 6 miles at 4-2=2mph. It would take 1 hour for the downstream leg and 3 hours for the upstream leg for a total of 4 hours.

If Ashley rows down and back at 5mph she will travel 6 miles at 5+2=7mph downstream and 6 miles at 5-2=3mph upstream. The downstream leg will take `6/7` of an hour while the upstream leg will take 2 hours for a total of `2 6/7` hours.

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The current flows at 2mph. Emily's rowing speed is 4mph (without considering current) while Ashley's rowing speed (again without considering current) is 5mph.

Emily's round-trip time is 4 hours.

Ashley's round-trip time is `2 6/7` hours.

Ashley finishes first by `4-2 6/7=1 1/7` hours or 1.14hours.

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Let the speed of rowing of Emily be E mph.

So, the speed of rowing of Ashley is (E+1) mph per hour. Also assume that the speed of current in the river is C.

For the downstream trip of Emily, `6/(E+C)=1`

`rArr (E+C)=6` ,

and `E=6-C` ------ (i)

For the return trip of Ashley, `6/(E+C+1)+6/(E-C+1)=2`

Plugging in the values from eqn. (i),

`6/(6+1) + 6/(6-C-C+1)=2`

`rArr 6/7 + 6/(7-2C) = 2`

`rArr C=7/8` mph.

and E = (6-7/8)=41/8 mph.

Total time required by Emily for her return trip

`=6/(E+C)+6/(E-C)=1+6/(34/8) = 2 7/17` hrs.

Total time required by Ashley for her return trip = 2 hrs (given).

Therefore, Ashley would complete the return trip faster by 7/17 hrs. i.e. by **0.41** hrs.