Episode 8 — Aptitude and Reasoning / 8.19 — Clocks

8.19.b Clocks - Tips, Tricks and Shortcuts

Trick 1: The One Formula You Must Know

Angle between hands at H:M = |30H - 5.5M| degrees

If the result > 180, subtract from 360.

This single formula solves 80% of all clock problems. Memorize it cold.

Trick 2: Quick Angle Calculation

Instead of computing |30H - 5.5M| directly, think of it as:

Step 1: At H:00, the angle between hands = 30 * H degrees
Step 2: From H:00 to H:M, the minute hand gains 5.5 * M degrees on the hour hand
Step 3: Combine: start angle +/- gain

Example: Angle at 5:20

At 5:00: angle = 150 degrees
In 20 min, minute hand gains: 5.5 * 20 = 110 degrees
New angle = 150 - 110 = 40 degrees

Example: Angle at 2:40

At 2:00: angle = 60 degrees
In 40 min, minute hand gains: 5.5 * 40 = 220 degrees
New angle = |60 - 220| = 160 degrees

Trick 3: Quick Reference - Angles at Exact Hours

TimeAngle
12:000 degrees
1:0030 degrees
2:0060 degrees
3:0090 degrees
4:00120 degrees
5:00150 degrees
6:00180 degrees
7:00150 degrees
8:00120 degrees
9:0090 degrees
10:0060 degrees
11:0030 degrees

Use these as starting points and adjust for minutes.

Trick 4: Overlap Time Formula

Between H and (H+1), hands overlap at:

M = 60H / 11 minutes past H

Quick calculation: 60/11 = 5.4545... = 5 + 5/11 minutes
So each overlap is approximately 5 min 27.3 sec past the "expected" position.

Quick overlap times (approximate):

~12:00, ~1:05, ~2:11, ~3:16, ~4:22, ~5:27,
~6:33, ~7:38, ~8:44, ~9:49, ~10:55, then 12:00 again

Trick 5: The 11 and 22 Rule

Memorize these counts:

In 12 hours:
  Overlaps (0 degrees): 11 times
  Opposite (180 degrees): 11 times
  Right angles (90 degrees): 22 times
  Straight lines (0 or 180): 22 times

In 24 hours: Double all the above.

Why 11 and not 12? Between 11 and 1 (passing through 12), the hands overlap only once (at 12:00), not twice.

Trick 6: Time Between Events

Between consecutive overlaps: 720/11 = 65 + 5/11 minutes (~65 min 27 sec)
Between consecutive right angles: 360/11 = 32 + 8/11 minutes (~32 min 44 sec)
Between consecutive 180-degree positions: 720/11 = 65 + 5/11 minutes

Trick 7: Mirror Image Instant Calculation

Mirror time = 11:60 - Actual time
Actual time = 11:60 - Mirror time

Quick examples:

Mirror shows 1:50 -> Actual = 11:60 - 1:50 = 10:10
Mirror shows 4:15 -> Actual = 11:60 - 4:15 = 7:45
Mirror shows 9:30 -> Actual = 11:60 - 9:30 = 2:30
Mirror shows 6:00 -> Actual = 11:60 - 6:00 = 5:60 = 6:00 (same!)

When mirror time is past 11:60 (i.e., for times between 12:00 and 12:59):

Use: Actual time = 23:60 - Mirror time

Mirror shows 12:20 -> Actual = 23:60 - 12:20 = 11:40

Trick 8: Faulty Clock Quick Formula

Clock gains g minutes per hour:

In 1 real hour, clock shows: 60 + g minutes

Real time when clock shows T hours:
Real time = T * 60 / (60 + g) hours

Clock loses l minutes per hour:

In 1 real hour, clock shows: 60 - l minutes

Real time when clock shows T hours:
Real time = T * 60 / (60 - l) hours

When does a faulty clock show correct time again?

Clock gains g min/hour:
  Time = 720 / (g * 24) days = 30/g days

Clock loses l min/hour:
  Time = 720 / (l * 24) days = 30/l days

Trick 9: Angle at Half-Hour Times

At H:30, the formula simplifies nicely:

Angle at H:30 = |30H - 5.5*30| = |30H - 165|

Quick results:
1:30 -> |30 - 165| = 135 degrees
2:30 -> |60 - 165| = 105 degrees
3:30 -> |90 - 165| = 75 degrees
4:30 -> |120 - 165| = 45 degrees
5:30 -> |150 - 165| = 15 degrees
6:30 -> |180 - 165| = 15 degrees
7:30 -> |210 - 165| = 45 degrees
8:30 -> |240 - 165| = 75 degrees
9:30 -> |270 - 165| = 105 degrees
10:30 -> |300 - 165| = 135 degrees
11:30 -> |330 - 165| = 165 degrees
12:30 -> |360 - 165| = 195 -> 360 - 195 = 165 degrees

Notice the symmetry: 5:30 and 6:30 both give 15 degrees.

Trick 10: Working with Fractions of 11

Since 5.5 = 11/2, many clock answers involve fractions of 11. Quick conversion:

1/11 = 0.0909...   ~5.45 seconds
2/11 = 0.1818...   ~10.91 seconds
3/11 = 0.2727...   ~16.36 seconds
4/11 = 0.3636...   ~21.82 seconds
5/11 = 0.4545...   ~27.27 seconds
6/11 = 0.5454...   ~32.73 seconds
7/11 = 0.6363...   ~38.18 seconds
8/11 = 0.7272...   ~43.64 seconds
9/11 = 0.8181...   ~49.09 seconds
10/11 = 0.9090...  ~54.55 seconds

Trick 11: "Between H and (H+1)" Problems

For "At what time between H and (H+1) will the hands...":

For 0 degrees (overlap):   M = 60H/11
For 90 degrees:            M = (60H - 180)/11  or  (60H + 180)/11
For 180 degrees:           M = (60H - 360)/11
For X degrees:             M = (60H - 2X)/11   or  (60H + 2X)/11

(Take values where 0 <= M < 60)

Simplified: M = (30H +/- X) / 5.5 = 2(30H +/- X) / 11

Trick 12: Angle Traced Shortcuts

Minute hand: 6 degrees per minute (easy to compute)
Hour hand: 0.5 degrees per minute = 1 degree every 2 minutes

In t minutes:
  Minute hand traces: 6t degrees
  Hour hand traces: t/2 degrees

For hour hand: Think "half the minutes = degrees."

In 40 minutes, hour hand moves: 40/2 = 20 degrees
In 90 minutes, hour hand moves: 90/2 = 45 degrees

Trick 13: Reflex Angle

If a problem asks for the "reflex angle" between the hands:

Reflex angle = 360 - (smaller angle between hands)

The reflex angle is always > 180 degrees.

Trick 14: Clocks Showing Same Time

Two clocks, one gaining and one losing, show the same time when:

Total time gained + Total time lost = 12 hours = 720 minutes

If clock A gains a min/hr and clock B loses b min/hr:
Hours until same time = 720 / (a + b)

Exam Strategy Tips

Time Management

  1. One formula: Master |30H - 5.5M|. Practice it until it is automatic.
  2. Memorize the 11 times: Overlaps, right angles, and straight lines counts are instant answers.
  3. Mirror image: The 11:60 subtraction is a 5-second calculation.

Common Traps

  1. Angle > 180: Always check and subtract from 360 if needed.
  2. "Between H and (H+1)": Both solutions from the quadratic may be valid - check the range.
  3. 11 overlaps, not 12: The missing overlap is the 11-to-1 crossing at 12:00.
  4. Clock gains vs. loses: Read carefully which clock is fast and which is slow.
  5. "Correct time" for faulty clocks: It means the faulty clock displays the same time as a correct clock - this happens only after gaining/losing exactly 12 hours.

Quick Verification

  • At 12:00, angle should be 0: |30*12 - 5.5*0| = 360 -> 360-360 = 0. Correct.
  • At 6:00, angle should be 180: |30*6 - 5.5*0| = 180. Correct.
  • At 3:00, angle should be 90: |30*3 - 5.5*0| = 90. Correct.

Next: 8.19.c - Solved Examples