SOLUTION: In a 200 mile bike ride, Duva realized that if she could average 1 mile per hour faster than she did on her last 200 mile bike ride, she would finish 1 hour sooner. How long did sh

Question 935388: In a 200 mile bike ride, Duva realized that if she could average 1 mile per hour faster than she did on her last 200 mile bike ride, she would finish 1 hour sooner. How long did she spend on her last 200 mile ride?
*Write the answer in minutes, and round to the nearest 10 minutes*
Found 2 solutions by josgarithmetic, TimothyLamb:
Answer by josgarithmetic(39620) (Show Source): You can put this solution on YOUR website!
r, speed of the last bike ride;
d, each distance of bike rides;
t, time of the last bike ride;
k, time difference, also given equal to rate difference;
d=200;
k=1.

______________________rate____________time___________distance
LAST__________________r_______________t_______________d
THIS__________________r+k_____________t-k_____________d

The unknown variables are r and t.
The system equations are .
Work with this system as thoroughly as you can completely in symbols to simplify and
to solve for r and t. You will obtain a quadratic equation through the process.

, and already knowing that k=1,

-
Using other equation, r=d/t.
Substituting,

-

, which is in HOURS.

Answer by TimothyLamb(4379) (Show Source): You can put this solution on YOUR website!
x = speed on previous ride
y = speed on current ride
a = time on previous ride
b = time on current ride
---
s = d/t
t = d/s
---
y = x + 1
b = a - 1
---
time on current ride:
b = 200/y
a - 1 = 200/(x + 1)
a = 200/(x + 1) + 1
---
time on previous ride:
a = 200/x
---
equate "a" times:
200/x = 200/(x + 1) + 1
200/x - 200/(x + 1) = 1
200(x + 1)/x(x + 1) - 200x/x(x + 1) = 1
200(x + 1) - 200x = x(x + 1)
200x + 200 - 200x = xx + x
xx + x - 200 = 0
---
the above quadratic equation is in standard form, with a=1, b=1 and c=-200
---
to solve the quadratic equation, by using the quadratic formula, copy and paste this:
1 1 -200
into this solver: https://sooeet.com/math/quadratic-equation-solver.php
---
the quadratic has two real roots at:
---
x = 13.6509717
x = -14.6509717
---
the negative root doesn't fit the problem statement, so use the positive root:
---
x = 13.6509717
a = 200/(x + 1) + 1
a = 200/(13.6509717 + 1) + 1
a = 14.6509716963
---
answer:
a = 14.65 hours
---
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