Sanjana jogged uphill for a while at an average speed of 3 miles per hour, then jogged downhill for a while at an average speed of 8 miles per hour. If Sanjana jogged the uphill and downhill stretches in a total of 40 minutes at an average speed of 4 miles per hour, how far did she jog uphill?
A. 1 mile
B. 1 1/3 miles
C. 1 1/2 miles
D. 1 3/5 miles
E. 1 2/3 miles
Can this question be solved using any other approach apart from algebra?
This is a MIXTURE problem.
Two speeds (3 mph and 8 mph) are combined to form a mixture with an average speed of 4 mph.
To determine how much time must be spent at each speed, we can use ALLIGATION.
Let U = the uphill speed and D = the downhill speed.
Step 1: Plot the 3 speeds on a number line, with the speeds for U and D on the ends and the speed for the mixture in the middle.
U 3---------4----------8 D
Step 2: Calculate the distances between the values on the number line.
U 3----
1----4----
4----8 D
Step 3: Determine the time ratio for the two speeds.
The time ratio for U and D is equal to the RECIPROCAL of the distances in red.
U : D = 4:1.
Implication:
Of every 5 minutes of travel time, 4 minutes are spent at the uphill speed, while 1 minute is spent at the downhill speed.
Thus, 4/5 of the total travel time is spent at the uphill speed:
(4/5)(40 minutes) = 32 minutes = 32/60 hour = 8/15 hour.
Since 8/15 hour is spent at the uphill speed of 3 mph, we get:
Distance uphill = rt = (3)(8/15) = 8/5 = 1 3/5 miles.
The correct answer is
D.
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