Clock Reasoning (Angle Problems)

Clock Reasoning Concept


Angle Between Hands Formula           Find the Angle of a Clock Time 04.10 & 02.20


Angle = ∣30H−5.5M∣


Where: H = Hour, M = Minutes, Hour hand moves = 0.5° per minute, Minute hand moves = 6° per minute


Concept (Detailed & Clear): Clock questions test your understanding of angles and time difference between hour and minute hands.


Formulas:


1. Angle between hands = |(30×H-11/2×M)|(H = hour, M = minutes)


2. Each hour = 30° difference


3. Each minute = 6° movement of minute hand


Ex: Find angle at 3:30


= |(30×3-11/2×30)| = |90-165| = 75°


Quick Notes: Hands coincide every 65 5/11 minutes Hands are opposite every 180°



Ex 1: Angle at 5:40


Angle = ∣30(5) - 5.5(40)∣ = ∣150 - 220∣ = 70° Ans: 70°


Ex 2: Angle at 3:30


Angle = ∣30(3) - 5.5(30)∣ = ∣90 - 165∣ = 75° Ans: 75°


Ex 3: Angle at 7:20


Angle = ∣30(7) - 5.5(20)∣ = ∣210 - 110∣ = 100° Ans: 100°


Ex 4: Angle at 9:45


Angle = ∣30(9) - 5.5(45)∣ = ∣270 - 247.5∣ = 22.5° Ans: 22.5°


Ex 5: Angle at 1:50


Angle = ∣30(1)−5.5(50)∣ = ∣30−275∣ = 245°


But the smaller angle = 360 - 245 = 115°


Ans : 115°


Ex 6: An accurate clock shows 8 o'clock in the morning. Through how many degrees will the hour hand rotate when the clock shows 2 o'clock in the afternoon ?


Soln:



  1. 1 hour = 30° (360° ÷ 12 hours)

  2. Count hours from 8 to 2: 8 → 9 → 10 → 11 → 12 → 1 → 2 → 6 hours

  3. Degree moved: 6 × 30° = 180°


Ans: 180°



Q. A clock is started at noon. By 10 minutes past 5, the hour hand has turned through how many degrees ?


Hour hand movement: Hour hand moves 30° per hour. Hour hand also moves 0.5° per minute


Time passed: From 12:00 to 5:10 → 5 hours and 10 minutes


Degree moved: For hours: 5 × 30° = 150°, For minutes: 10 × 0.5° = 5°


Total = 150 + 5 = 155°


Ans: 155°


Short Trick (For Quick Solving)


1st : Multiply hour by 30, 2nd : Multiply minutes by 5.5, 3rd : Subtract, 4th : Take absolute value, 5th : If answer >180°, subtract from 360