PYQ Vault

CDS Mathematics · Time, Speed and Distance

Clocks and Angles

The minute hand turns 6° a minute and the hour hand 0.5°, so the minute hand gains 5.5° every minute.

All Time, Speed and Distance notesCDS Mathematics strategy

Why this matters

Five PYQs. Two numbers do it all: the angle between the hands at h:m is |30h − 5.5m|, and the hands coincide every 720/11 minutes. The second hand gains 354° a minute on the minute hand.

Concept 1 of 1: Angles between the hands

Treat the hands as runners on a circular track. The hour hand starts 30h30h degrees ahead at hh o'clock, and the minute hand closes that gap at 5.5∘5.5^\circ a minute.

Definition

  • Minute hand: 6∘6^\circ/min. Hour hand: 30∘30^\circ/hour =0.5∘= 0.5^\circ/min. Second hand: 360∘360^\circ/min.
  • Angle at h:mh{:}m: ∣30h−5.5m∣|30h - 5.5m| (take 360∘360^\circ minus it if over 180∘180^\circ and the smaller angle is asked).
  • Hands coincide after hh o'clock at 60h11\dfrac{60h}{11} minutes past: 72011\dfrac{720}{11} minutes apart, 1111 times in 1212 hours.
  • Second and minute hands coincide every 6059\dfrac{60}{59} minutes.
  • Radians: multiply degrees by π180\dfrac{\pi}{180}.

Angle between the hands

θ=∣30h−5.5m∣\theta = |30h - 5.5m|

Worked example

Find the angle between the hands at 7:307{:}30.
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2022 · CDS (II) 2022 — Elementary Mathematics · Q23Easy

Example 1 · Time, Speed and Distance · Clocks and Angles

What is the angle between the hour hand and the minute hand of a clock when the clock shows 4 hours 40 minutes ?

The hour hand moves too

At 8:208{:}20 the hour hand is not on the 88; it has moved 10∘10^\circ toward the 99. Ignoring that gives 120∘120^\circ instead of 130∘130^\circ.
Drill 4 more on angles between the hands

Summary — formulas & gotchas at a glance

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Formulas (1)

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Test yourself on Time, Speed and Distance

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