Answer:
1.98 miles per hour.
Step-by-step explanation:
Let x miles per hour be the average rate on the return trip,
∵ The speed on round trip is 5 miles per hour faster than on return trip,
So, the speed in round trip = (x+5) miles per hour,
Now, distance in one sided trip = 6 miles,
Since,
![Speed = \frac{Distance}{Time}\implies Time = \frac{Distance}{Speed}](https://tex.z-dn.net/?f=Speed%20%3D%20%5Cfrac%7BDistance%7D%7BTime%7D%5Cimplies%20Time%20%3D%20%5Cfrac%7BDistance%7D%7BSpeed%7D)
According to the question,
Time taken in return trip - Time taken in round trip =
hours,
![\implies \frac{6}{x}-\frac{6}{x+5}=\frac{13}{6}](https://tex.z-dn.net/?f=%5Cimplies%20%5Cfrac%7B6%7D%7Bx%7D-%5Cfrac%7B6%7D%7Bx%2B5%7D%3D%5Cfrac%7B13%7D%7B6%7D)
![\frac{6x+30-6x}{x(x+5)}=\frac{13}{6}](https://tex.z-dn.net/?f=%5Cfrac%7B6x%2B30-6x%7D%7Bx%28x%2B5%29%7D%3D%5Cfrac%7B13%7D%7B6%7D)
![\frac{30}{x^2 + 5x}=\frac{13}{6}](https://tex.z-dn.net/?f=%5Cfrac%7B30%7D%7Bx%5E2%20%2B%205x%7D%3D%5Cfrac%7B13%7D%7B6%7D)
![180 = 13x^2 + 65x](https://tex.z-dn.net/?f=180%20%3D%2013x%5E2%20%2B%2065x)
![13x^2 + 65x - 180 = 0](https://tex.z-dn.net/?f=13x%5E2%20%2B%2065x%20-%20180%20%3D%200)
By quadratic formula,
![x=\frac{-65\pm \sqrt{65^2 - 4\times 13\times -180}}{26}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B-65%5Cpm%20%5Csqrt%7B65%5E2%20-%204%5Ctimes%2013%5Ctimes%20-180%7D%7D%7B26%7D)
![x=\frac{-65\pm \sqrt{4225 + 9360}}{26}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B-65%5Cpm%20%5Csqrt%7B4225%20%2B%209360%7D%7D%7B26%7D)
![x=\frac{-65\pm \sqrt{13585}}{26}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B-65%5Cpm%20%5Csqrt%7B13585%7D%7D%7B26%7D)
![x\approx 1.98\text{ or }x=-6.98](https://tex.z-dn.net/?f=x%5Capprox%201.98%5Ctext%7B%20or%20%7Dx%3D-6.98)
∵ speed can not be negative,
Hence, average rate on the return trip would be 1.98 miles per hour.