SOLUTION: Abigail Nightwind has planned a bike trip with some friends from her village. They will bike for 8 hours, then hike for 4 hours. If they can bike on the average 10mph fater than th

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Question 65498This question is from textbook Essential Algebra
: Abigail Nightwind has planned a bike trip with some friends from her village. They will bike for 8 hours, then hike for 4 hours. If they can bike on the average 10mph fater than they can hike and if they will cover a total of 122 miles, how fast can Abigail and her friends bike? This question is from textbook Essential Algebra

Found 2 solutions by funmath, Edwin McCravy:
Answer by funmath(2933)   (Show Source): You can put this solution on YOUR website!
Abigail Nightwind has planned a bike trip with some friends from her village. They will bike for 8 hours, then hike for 4 hours. If they can bike on the average 10mph faster than they can hike and if they will cover a total of 122 miles, how fast can Abigail and her friends bike?
The formula for distance is: , where d=distance, r=rate(or speed), and t=time
Givens for the hike protion:
t=4
r=r (we don't know it)
d=4r
Givens for the bike protion:
t=8
r=r+10 (10 mph faster than they can hike.)
d=8(r+10)
The total of the trip= 122
Problem to solve:
4r+8(r+10)=122
4r+8r+80=122
12r+80=122
12r+80-80=122-80
12r=42
12r/12=42/12
r=3.5
Therefore the rate that they can bike: r+10=3.5+10=13.5 mph
:
Sanity check:
If they hiked for 4 hrs at 3.5 mph and biked for 8 hrs at 13.5 mph, would they travel 122 mi?
4(3.5)+8(13.5)=122 ?
14+108=122
122=122 It appears we are sane.
Happy Calculating!!!!
Answer by Edwin McCravy(20060)   (Show Source): You can put this solution on YOUR website!
Abigail Nightwind has planned a bike trip with 
some friends from her village. They will bike 
for 8 hours, then hike for 4 hours. If they 
can bike on the average 10mph fater than they 
can hike and if they will cover a total of 122 
miles, how fast can Abigail and her friends
bike?

Funmath above let the unknown represent
the hiking rate.  To be a little different,
I'm going to let the unknown represent the
biking rate instead.  After all that's what
we're asked for in the question.  I also
think it makes it easier to see if you use
a DRT chart.

Let x = the biking rate.

Make this DRT-chart

             D        R     T
biking                         
hiking   

and fill in x for the biking rate:

             D        R     T
biking                x      
hiking                                             

>>...They can bike on the average 10mph faster
than they can hike...<<

This means that they can hike on the average
10mph slower than they can bike. So their hiking
rate is 10mph slower or x-10. So fill that in
for the hiking rate 

             D        R     T
biking                x      
hiking               x-10    

>>..They will bike for 8 hours, then hike for 
4 hours...<<

So fill in 8 for the biking time, and 4 for
the hiking time: 

             D        R     T
biking                x     8
hiking               x-10   4

Now we fill in the distances using D = RT

             D        R     T
biking      8x        x     8
hiking    4(x-10)    x-10   4

Now that the chart is filled in, we return
to the word problem to see what we have not 
used in filling in the chart.  It is these
words:

>>...They will cover a total of 122 miles...<<

So,

distance biking + distance hiking = 122 miles

             8x + 4(x-10) = 122

             8x + 4x - 40 = 122

                 12x - 40 = 122

                      12x = 162

                        x = 162/12

                        x = 27/2

                        x = 13.5 mph

Checking: They bike at 13.5mph and hike at 3.5mph

They bike at 13.5mph for 8 hours and that's a distance
of 108 miles they bike.

They hike at 3.5mph for 4 hours and that's a distance
of 14 miles they hike.

108 miles biked + 14 miles hiked = 122 miles total.

Edwin
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