A bicycle trip of 120 miles would have taken 3 hours less if

A bicycle trip of 120 miles would have taken 3 hours less if the average speed had been increased by 2 miles per hour. Find the average speed of the bicycle.

Solution

A bicycle trip of 120 miles would have taken 3 hours less if the average speed had been increased by 2 miles per hour. Find the average speed of the bicycle.
x= avg speed of bicycle
The distance is the same (120) before and after the increase
The Rate changes from x to x+2
Note that in each case the time is Distance/Rate

Rate x Time = Distance
Before increase
x....120/x.....120
After increase
x+2...120/x+2..120

New Time is 3 less
120/(x+2) = 120/x -3
Multiply each term by LCM=x(x+2)
x(x+2)(120)/(x+2) = x(x+2)(120)/x -3(x)(x+2)
Cancel denominators
120x=120(x+2)-3x(x+2)
120x=120x+240-3x^2-6x
0=-3x^2-6x+240
divide by -3
0=x^2+2x-80
Factor right side
0=(x-8)(x+10)
x-8=0 .... x=8 avg speed of bicycle
x+10=0 ... x=-10 (reject, speed can\'t be negative!)