SOLUTION: A cyclist travels 60 km. If he reduces his speed by 2km/h, he will take one hour longer. Find the original speed of the cyclist.

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Question 1158453: A cyclist travels 60 km. If he reduces his speed by 2km/h, he will take one hour longer. Find the original speed of the cyclist.
Found 3 solutions by ikleyn, Shin123, MathTherapy:
Answer by ikleyn(52903) About Me  (Show Source):
You can put this solution on YOUR website!
.

            Similar problems were just solved by me SEVERAL times today.

            Therefore, this time I will omit all the words. 


The "time" equation is


    60%2F%28x-2%29 - 60%2Fx = 1    hour longer.


Solve by reducing to a quadratic equation.


Or guess the answer MENTALLY :  x = 12 km/h.


CHECK.  60%2F%2812-2%29 - 60%2F12 = 60%2F10 - 60%2F12 = 6 - 5 = 1 hour.   ! Correct !

Solved.


Answer by Shin123(626) About Me  (Show Source):
You can put this solution on YOUR website!
The cyclist originally traveled x km/h. It takes him 60%2Fx hours to travel 60 km. If he travels (x-2) km/h, it will take him 60%2F%28x-2%29 hours, which is equal to 60%2Fx%2B1. 60%2Fx%2B1=60%2F%28x-2%29 Multiplying both sides by x%28x-2%29, we get 60%28x-2%29%2Bx%28x-2%29=60x.
60x-120%2Bx%5E2-2x=60x. x%5E2-2x-120=0
Solved by pluggable solver: COMPLETING THE SQUARE solver for quadratics
Read this lesson on completing the square by prince_abubu, if you do not know how to complete the square.
Let's convert 1x%5E2%2B-2x%2B-120=0 to standard form by dividing both sides by 1:
We have: 1x%5E2%2B-2x%2B-120=0. What we want to do now is to change this equation to a complete square %28x%2Bsomenumber%29%5E2+%2B+othernumber. How can we find out values of somenumber and othernumber that would make it work?
Look at %28x%2Bsomenumber%29%5E2: %28x%2Bsomenumber%29%5E2+=+x%5E2%2B2%2Asomenumber%2Ax+%2B+somenumber%5E2. Since the coefficient in our equation 1x%5E2%2Bhighlight_red%28+-2%29+%2A+x%2B-120=0 that goes in front of x is -2, we know that -2=2*somenumber, or somenumber+=+-2%2F2. So, we know that our equation can be rewritten as %28x%2B-2%2F2%29%5E2+%2B+othernumber, and we do not yet know the other number.
We are almost there. Finding the other number is simply a matter of not making too many mistakes. We need to find 'other number' such that %28x%2B-2%2F2%29%5E2+%2B+othernumber is equivalent to our original equation 1x%5E2%2B-2x%2Bhighlight_green%28+-120+%29=0.


The highlighted red part must be equal to -120 (highlighted green part).

-2%5E2%2F4+%2B+othernumber+=+-120, or othernumber+=+-120--2%5E2%2F4+=+-121.
So, the equation converts to %28x%2B-2%2F2%29%5E2+%2B+-121+=+0, or %28x%2B-2%2F2%29%5E2+=+121.

Our equation converted to a square %28x%2B-2%2F2%29%5E2, equated to a number (121).

Since the right part 121 is greater than zero, there are two solutions:

system%28+%28x%2B-2%2F2%29+=+%2Bsqrt%28+121+%29%2C+%28x%2B-2%2F2%29+=+-sqrt%28+121+%29+%29
, or

system%28+%28x%2B-2%2F2%29+=+11%2C+%28x%2B-2%2F2%29+=+-11+%29
system%28+x%2B-2%2F2+=+11%2C+x%2B-2%2F2+=+-11+%29
system%28+x+=+11--2%2F2%2C+x+=+-11--2%2F2+%29

system%28+x+=+12%2C+x+=+-10+%29
Answer: x=12, -10.

A speed can't be negative, so his original speed is 12 km/hr.

Answer by MathTherapy(10557) About Me  (Show Source):
You can put this solution on YOUR website!
A cyclist travels 60 km. If he reduces his speed by 2km/h, he will take one hour longer. Find the original speed of the cyclist.
I hope for your sanity, you didn't look at what the other person presented. 

I don't know why some people feel that they can help another person by showing he/she one of the most complex and time-consuming solutions to a math problem.

Is this REALLY help? I think NOT!! 



Let original speed, be S

Then his hypothetical speed is S - 2

We then get the following TIME equation: matrix%281%2C3%2C+60%2FS%2C+%22=%22%2C+60%2F%28S+-+2%29+-+1%29

60(S - 2) = 60S - S(S - 2) ------ Multiplying by LCD, S(S - 2)



(S - 12)(S + 10) = 0

Original speed, or highlight_green%28matrix%281%2C4%2C+S%2C+%22=%22%2C+12%2C+%22km%2Fh%22%29%29           OR              S = - 10 (ignore)