Question

A jogger ran 8 miles and then walked 6 miles. The jogger's running speed was 5 miles per hour faster than her walking speed. The total time for jogging and walking was 2 hours. What was the jogger's walking speed and jogging speed?

Answer 1

`S_r=10`mph
`S_w=5`mph

Explanation:

Let 'r' denote running and 'w' denote walking.
Let D be distance travelled. So,
`(D_r)=8`mi
`(D_w)=6`mi

Let "S" be the speed. So,
And since speed of running is 5mph faster than walking. So,
`(S_r)=S_w + 5mph`

Let T be the time.So,
Total time taken is 2 hrs.
Since the time includes `T_r` and `T_w`. We could say,
`(T_r)+(T_w)=2`
`or, (T_r)=2-(T_w)`

We know,
`S = (DeltaD)/(Deltat)`
`(S_r)=8/(2-T_w)` and
`(S_w)=6/(T_w)`

As mentioned
`(S_r)=(S_w)+5mph`
`or, 8/(2-T_w)=5+6/(T_w)`
For sake of ease let's suppose `(T_w)=x`
So,
`8/(2-x)=5+6/x`
`or, 8/(2-x)=(5x+6)/x`
Cross multiplying,
`or, 8x=(5x+6)(2-x)`
`or, 8x=4x-(5x^2)+12`
`or, 8x-4x+(5x^2)-12=0`
`or, (5x^2)+4x-12=0`
Factorization by splitting the middle term,
`or, (5x^2)+(10-6)x-12=0`
`or, (5x^2)+10x-6x-12=0`
`or, 5x (x+2)-6 (x+2)=0`
Use reverse of distributive property.
`or, (5x-6)(x+2)=0`

Since time is positive we must choose such binomial from above that yields positive value.
Choose
`(5x-6)=0`
`x=6/5`
`x=(T_w)=6/5`hrs

So,
`S_w=(deltaD_w)/(deltaT_w)`
`=6/(6/5)`
`=30/6`
`=5` mph

Again `S_r=5mph+S_w`
`=5mph+5mph`
`=10mph`

`:.S_r=10`mph
`:.S_w=5`mph