Question 212330
HOPE THIS HELPS---ON THESE PROBLEMS YOU EITHER HAVE TO DEAL IN MINUTES OR HOURS---YOU CAN'T MIX. I WILL DEAL IN HOURS:
Distance(d) equals Rate(r) times Time(t) or d=rt;r=d/t and t=d/r
Let t1=time to the lake
Let t2 equal time from the lake
Now we are told that t1+t2=32/60 =8/15 hr---------------eq1
distance to the lake =8t1
distance from the lake =6t2
We know that distance to lake=distance from lake, so:

8t1=6t2------------------------------------------eq2
From eq2, t2=(4/3)t1.  Substitute this for t2 in eq1:
t1+(4/3)t1=8/15
(7/3)t1=8/15  multiply each side by 15
35t1=8
t1=8/35 hr--------------------------Time for Joan to run from home to the lake

ANOTHER WAY::
Let d=the distance to/from home to lake

Time to run from home to the lake=d/8
Time to run back from the lake to home=d/6
And we are told that the above two times add up to 32/60 or 8/15 hr, so:
d/8 +d/6 = 8/15  multply each term by 120
15d+ 20d=64 
35d=64
d=64/35 mi
So time for Joan to run from her home to the lake=(64/35)/8=8/35 hr
And time for Joan to run back home=(64/35)/6=64/210 hr
CK
Total time then is: 8/35 + 64/210=48/210 + 64/210=112/210=8/15 hr

Hope this helps---ptaylor