Question 255014
On the 42 km go kart course Arshia drives .4 km/hr faster than Sarah, but she has engine trouble part way around the track and has to stop to get the cart fixed.
 This stop costs Arshia one half hour, and so she arrives 15 minutes after Sarah at the end of the course.
 How fast did each girl drive and how long did each girl take to finish the course? 
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I am not sure how you did this, but here's my take on it:
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Change 15 min to .25 hr
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Let s = Sarah's speed
then
(s+.4) = Arshia's speed
:
{{{42/s}}} = S's travel time
then
{{{42/((s+.4))}} + .5 = A's travel time
:
S's travel time + 15 min = A's travel time + half hr
{{{42/s}}} + .25 = {{{42/((s+.4))}}} + .5 
:
{{{42/s}}} = {{{42/((s+.4))}}} + .5 - .25
:
{{{42/s}}} = {{{42/((s+.4))}}} + .25
Multiply by s(s+.4), results
42(s+.4) = 42s + .25s(s+.4)
:
42s + 16.8 = 42s + .25s^2 + .1s
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Arrange as a quadratic equation
.25s^2 + .1s + 42s - 42s - 16.8 = 0
.25s^2 + .1s - 16.8 = 0
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Multiply by 20, get rid of those decimals
5s^2 + 2s - 336 = 0
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Factor this to
(5s + 42)(s-8) = 0
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positive solution
s = 8 km/h is Sarah's speed
and
8.4 km/h is Arshia's speed
;
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Find the times of each:
42/8 = 5.25 hrs 
42/8.4 = 5 hrs, + .5  = 5.5 hrs (checks out, 15 min after Sarah)