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You commute to work at a distance of 40 miles and return on the same route at the end of the day.
Your average speed on the return trip is 30 miles per hour faster than your average speed on the morning’s outgoing trip.
If the round trip takes 2 hours, what are your average speeds each way? Define each variable(s).
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Let x be the rate commuting to work (in miles per hour).
Then the rate commuting back is (x+30) mph.
The time commuting to work is hours.
The time commuting back is hours.
The total time commuting to work and back is 2 hours, which gives you the "time" equation
+ = 2 hours. (1)
It is your basic equation. The setup is just completed.
To solve the equation, multiply both sides by x*(x+30). You will get
40*(x+30) + 40x = 2*x*(x+30).
Simplify
40x + 1200 + 40x = 2x^2 + 60x
2x^2 - 20x - 1200 = 0
x^2 - 10x - 600 = 0
Factor left side
(x-30)*(x+20) = 0.
Of the two roots, x= 30 and x= -20, only positive root x= 30 makes sense.
Hence, the ANSWER is: the rate commuting to work is 30 mph; the rate commuting back is 30+30 = 60 mph.
CHECK. I will check if the equation (1) is valid: + = + = 2. ! Correct !
The problem is just solved.
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The method of solution is using the "time" equation.
MEMORIZE this method (!)