Rational Eqns: We hiked 6 miles from point A to point B. Our speed walking back was 1 mph faster. The round trip took 5 hours.

[imjsutkay19]

We hiked 6 miles from point A to point B. Our speed walking back was 1 mph faster. The round trip took 5 hours. What was our walking speed from A to B?

I just don't understand how to solve based off what my teacher taught. How would you explain the steps to this problem?

 

[MarkFL]

Hello, and welcome to FMH!

Using the relationship between distance, average speed and time (where distances are in miles and time is in hours), we may write for the first leg of the journey:

[MATH]6=vt_1[/MATH]
And for the second leg:

[MATH]6=(v+1)t_2[/MATH]
We are also given:

[MATH]t_1+t_2=5[/MATH]
We know have 3 equations in 3 unknowns. Can you proceed?

 

[MarkFL]

To follow up:

The first equation implies:

[MATH]t_1=\frac{6}{v}[/MATH]
And the second equation implies:

[MATH]t_2=\frac{6}{v+1}[/MATH]
Adding these results, we find:

[MATH]t_1+t_2=\frac{6}{v}+\frac{6}{v+1}[/MATH]
Now, we know the LHS is equal to 5:

[MATH]5=\frac{6}{v}+\frac{6}{v+1}[/MATH]
Multiply through by \(v(v+1)\):

[MATH]5v(v+1)=6(v+1)+6v[/MATH]
[MATH]5v^2+5v=12v+6[/MATH]
[MATH]5v^2-7v-6=0[/MATH]
[MATH](5v+3)(v-2)=0[/MATH]
Discarding the negative root, we are left with:

[MATH]v=2\text{ mph}[/MATH]