Question

Tina bicycles 160 miles at the rate of r mph. The same trip would have taken two hours longer if she had decreased her speed by 4 mph. Find r.

Answer 1

So we now that 'r' is the speed in mph and 't' is the time in hours Tina took to run the 160 miles. So the ecuation is
\[r \times t=160\]
On the other hand, going 4 miles slowler would take two hours more

4 mph slowler is \[r -4\] and two hours longer is \[t +2\]
So the second ecuation is
\[\left( r-4 \right) \times \left( t +2 \right)=160\]
So you have two ecuations with two unknown variables

To solve it just equalize t like this
\[t = 160/r\]and
\[t= \left( 160/\left( r-4 \right) \right)-2\]
So you have
\[160/r=(160/(r-4))-2\]
get the r from there

Answer 2

Tina bicycles 160/r hours.
r-4... Tinas speed if she decreased her first speed by 4 mph
(160/r) +2... time, if Tina decreases her speed by 2 mph
(r-4)*((160/r)+2)=160... distance is the same, you can make an equation which to solve

160+2r-640/r-8=160
2r-640/r-8=0
r-320/r-4= |*r
\[r^2-4r-320=0\]

\[r_{1,2}=\frac{4\pm \sqrt{16+1280}}{2}=\frac{4\pm 36}{2}=20;-16\]
You are interested in positive values as a positive speed.
so
\[r=20mph\]

Answer 3

* (160/r) +2... time, if Tina decreases her speed by "4" mph