SOLUTION: Miguel went on a bike ride of 24 miles. He realized that if he had gone 6 mph faster, he would have arrived 9 hours sooner. How fast did he actually ride?

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Question 1150319: Miguel went on a bike ride of 24 miles. He realized that if he had gone 6 mph faster, he would have arrived 9 hours sooner. How fast did he actually ride?
Found 3 solutions by josmiceli, ikleyn, MathTherapy:
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Let +r+ = his normal rate in mi/hr
Let +t+ = his normal time in hrs
-------------------------------------------
Normal rate:
(1) +24+=+r%2At+
Faster rate:
(2) +24+=+%28+r+%2B+6+%29%2A%28+t+-+9+%29+
------------------------------------
(1) +t+=+24%2Fr+
(2) +24+=+%28+r+%2B+6+%29%2A%28+24%2Fr+-+9+%29+
(2) +24+=+24+%2B+144%2Fr+-+9r+-+54+
(2) +0+=+144%2Fr+-+9r+-+54+
(2) +144%2Fr+=+9r+%2B+54+
(2) +144+=+9r%5E2+%2B+54r+
(2) +16+=+r%5E2+%2B+6r+
(2) +r%5E2+%2B+6r+-+16+=+0+
(2) +%28+r+%2B+8+%29%2A%28+r+-+2+%29+=+0+
(2) +r+=+2+ ( can't use the negative result )
His normal rate was 2 mi/hr
-------------------------------
check answer:
(1) +24+=+r%2At+
(1) +24+=+2%2At+
(1) +t+=+12+ hrs
and
(2) +24+=+%28+r+%2B+6+%29%2A%28+t+-+9+%29+
(2) +24+=+%28+2+%2B+6+%29%2A%28+t+-+9+%29+
(2) +24+=+8%2A%28+t+-+9+%29+
(2) +t+-+9+=+3+
(2) +t+=+12+
OK

Answer by ikleyn(52810) About Me  (Show Source):
You can put this solution on YOUR website!
.

            The way on how @josmiceli solves the problem,  may scare the reader.

            Meanwhile,  all such problems have very straightforward  SIMPLE  LOGIC  nice solutions using  "time"  equation. 


Let v be the unknown rate of the biker.

Then the time of his biking is  24%2Fv hours.


Had he gone 6 miles faster, his time would be  24%2F%28x%2B6%29 hours.


The time equation is


    48%2Fv - 48%2F%28x%2B6%29 = 9  hours.


To solve it, multiply both sides by x*(x+6).  You will get


    48*(x+6) - 48x = 9x*(x+6)

    48x + 288 - 48x = 9x^2 + 54x

    9x^2 + 54x - 288 = 0

     x^2 +  6x -  32 = 0

     (x+8)*(x-2)     = 0


The only positive root x= 2 makes sense.


It gives the ANSWER :  the rate of biking is  2 miles per hour.


CHECK.  I will check equation (1) by substituting the value x= 2 there


        24%2F2 - 24%2F%282%2B6%29 = 12 - 3 = 9  hours.    ! Correct !

Solved.

--------------------

Using  "time"  equation is the  STANDARD  method of solving such problems.
From this lesson,  learn on how to write,  how to use and how to solve a  "time"  equation.

To see many other similar solved problems,  look into the lessons
    - Had a car move faster it would arrive sooner
    - How far do you live from school?
    - Earthquake waves
    - Time equation: HOW TO use, HOW TO write and HOW TO solve it
in this site.


Answer by MathTherapy(10555) About Me  (Show Source):
You can put this solution on YOUR website!
Miguel went on a bike ride of 24 miles. He realized that if he had gone 6 mph faster, he would have arrived 9 hours sooner. How fast did he actually ride?
This DOESN'T have to be as COMPLEX as the other person presents it to be.



This is the correct way to do the problem:

Let the ACTUAL speed be S

Then the HYPOTHETICAL speed = S + 6

The HYPOTHETICAL trip would've taken 9 hours less, so we get the following TIME equation: matrix%281%2C3%2C+24%2FS%2C+%22=%22%2C+24%2F%28S+%2B+6%29+%2B+9%29 

matrix%281%2C3%2C+8%2FS%2C+%22=%22%2C+8%2F%28S+%2B+6%29+%2B+3%29 ------ Reducing equation by factoring out GCF, 3, in the numerator

8(S + 6) = 8S + 3S(S + 6) ------ Multiplying by LCD, S(S + 6)

matrix%281%2C3%2C+8S+%2B+48%2C+%22=%22%2C++8S+%2B+3S%5E2+%2B+18S%29

matrix%281%2C3%2C+3S%5E2+%2B+18S+%2B+8S+-+8S+-+48%2C+%22=%22%2C+0%29

matrix%281%2C3%2C+3S%5E2+%2B+18S+-+48%2C+%22=%22%2C+0%29



(S - 2)(S + 8) = 0

S, or highlight_green%28matrix%281%2C5%2C+ACTUAL%2C+speed%2C+%22=%22%2C+2%2C+mph%29%29       OR       S = - 8 (ignore)