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26 Mar 2026, 00:09
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E
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
85%
(hard)
Question Stats:
38% (03:43) correct
63%
(03:49)
wrong
based on 16
sessions
History
Date
Time
Result
Not Attempted Yet
Theofanos and Gretchen cycled from Anton to Beauville along the same route. Theofanos cycled half the time at 12 miles per hour and the other half at 18 miles per hour, whereas Gretchen cycled half the distance at 12 miles per hour and the other half at 18 miles per hour. If Gretchen took 20 minutes longer than Theofanos, how many miles did each person cycle?
A. 60
B. 70
C. 72
D. 90
E. 120
A. 60
B. 70
C. 72
D. 90
E. 120
26 Mar 2026, 02:13
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The sneaky thing about this one is that Theofanos and Gretchen look like they're doing the same thing. They're not. One splits time equally, the other splits distance equally. That's the whole trap.
This is a classic Problem Solving question testing average speed, and I got burned by this exact setup when I didn't read carefully.
1. Theofanos: half the TIME at 12 mph, half the TIME at 18 mph.
When time is split equally, the average speed is a simple arithmetic mean:
Speed = (12 + 18)/2 = 15 mph
Time for Theofanos = D/15
2. Gretchen: half the DISTANCE at 12 mph, half the DISTANCE at 18 mph.
When distance is split equally, you need the harmonic mean approach:
Time = (D/2)/12 + (D/2)/18 = D/24 + D/36
Common denominator: 3D/72 + 2D/72 = 5D/72
Time for Gretchen = 5D/72
3. Gretchen took 20 minutes longer. 20 min = 1/3 hour.
5D/72 - D/15 = 1/3
4. Find common denominator for the left side (LCD = 360):
25D/360 - 24D/360 = 1/3
D/360 = 1/3
D = 120
5. Answer: E (120 miles)
Quick check: Theofanos time = 120/15 = 8 hours. Gretchen time = 5(120)/72 = 600/72 = 25/3 hours. Difference = 25/3 - 8 = 1/3 hour = 20 minutes. Checks out.
The takeaway: "half the time" and "half the distance" give different average speeds. Always pause and ask which one you're dealing with.
This is a classic Problem Solving question testing average speed, and I got burned by this exact setup when I didn't read carefully.
1. Theofanos: half the TIME at 12 mph, half the TIME at 18 mph.
When time is split equally, the average speed is a simple arithmetic mean:
Speed = (12 + 18)/2 = 15 mph
Time for Theofanos = D/15
2. Gretchen: half the DISTANCE at 12 mph, half the DISTANCE at 18 mph.
When distance is split equally, you need the harmonic mean approach:
Time = (D/2)/12 + (D/2)/18 = D/24 + D/36
Common denominator: 3D/72 + 2D/72 = 5D/72
Time for Gretchen = 5D/72
3. Gretchen took 20 minutes longer. 20 min = 1/3 hour.
5D/72 - D/15 = 1/3
4. Find common denominator for the left side (LCD = 360):
25D/360 - 24D/360 = 1/3
D/360 = 1/3
D = 120
5. Answer: E (120 miles)
Quick check: Theofanos time = 120/15 = 8 hours. Gretchen time = 5(120)/72 = 600/72 = 25/3 hours. Difference = 25/3 - 8 = 1/3 hour = 20 minutes. Checks out.
The takeaway: "half the time" and "half the distance" give different average speeds. Always pause and ask which one you're dealing with.
