The number of times in a day the Hour-hand and the Minute-hand of a clock at right angles is

1. Basic Geometry of a Clock Dial (basic)

To master clock problems, we must first view the clock face as a geometric circle of 360°. Because the dial is divided into 12 equal hour sections, the angular distance between any two consecutive numbers (like 12 to 1) is 30° (360 / 12). Furthermore, since each hour is divided into 60 minutes, the minute hand moves 6° per minute (360 / 60), while the slower hour hand moves only 0.5° per minute (30 / 60). This basic math is the foundation for understanding how the hands interact over time Physical Geography by PMF IAS, Latitudes and Longitudes, p.243.

A common point of confusion in competitive exams is the frequency of specific hand positions, such as the right angle (90°). Conceptually, a right angle occurs when there is a 15-minute space between the hands. While it appears this should happen twice every hour, the movement is relative. Just as the Earth's rotation of 15° signifies the passage of one hour in local time, the hands of a clock are in constant, repeating motion Certificate Physical and Human Geography, The Earth's Crust, p.11. This periodic repetition is what allows us to measure intervals so precisely Science-Class VII . NCERT, Measurement of Time and Motion, p.111.

In a standard 12-hour cycle, the hands actually form a right angle only 22 times, not 24. This happens because the right angles at exactly 3:00 and 9:00 are unique; they act as the second occurrence for the previous hour and the first occurrence for the following hour simultaneously. Consequently, in a full 24-hour day, the hands are perpendicular exactly 44 times (22 occurrences × 2 cycles).

Sources: Physical Geography by PMF IAS, Latitudes and Longitudes, p.243; Certificate Physical and Human Geography, The Earth's Crust, p.11; Science-Class VII . NCERT, Measurement of Time and Motion, p.111

2. Angular Speed and Relative Velocity of Hands (intermediate)

To master the mechanics of a clock, we must first understand that the hands are not just moving—they are rotating at specific angular speeds. Much like how we calculate linear speed as distance divided by time Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.113, we calculate angular speed by dividing the total degrees in a circle (360°) by the time taken to complete a revolution. The minute hand covers 360° in 60 minutes, giving it a speed of 6° per minute. The hour hand, being much slower, takes 12 hours (720 minutes) to complete the same circle, resulting in a speed of 0.5° per minute.

Because both hands move in the same clockwise direction, we focus on their relative velocity—the rate at which the minute hand "gains" on the hour hand. By subtracting the slower speed from the faster one (6° - 0.5°), we find the relative speed is 5.5° per minute (or 11/2° per minute). This concept is similar to how the Earth’s rotation determines local time, where 360° is covered in 24 hours, meaning the Earth rotates at 15° per hour Certificate Physical and Human Geography , GC Leong, The Earth's Crust, p.11. In a clock, this relative speed is the key to finding when the hands will overlap, be opposite, or form specific angles.

A common point of confusion arises when counting how many times the hands form a right angle (90°). While it seems they should be perpendicular twice every hour (24 × 2 = 48 times a day), the reality is different. Because the hands are both moving, the minute hand only "laps" the hour hand 11 times every 12 hours. During this cycle, the positions at 3:00 and 9:00 are unique; the hands form a right angle exactly on the hour, which "robs" the count of one occurrence in the preceding hour. Therefore, the hands are perpendicular only 22 times in 12 hours, totaling 44 times in a full 24-hour day.

Hand Type	Time for 360°	Angular Speed
Minute Hand	60 Minutes	6° / min
Hour Hand	720 Minutes	0.5° / min
Relative	--	5.5° / min

Sources: Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.113; Certificate Physical and Human Geography , GC Leong, The Earth's Crust, p.11

3. The General Formula for Angles (intermediate)

To master clock-based aptitude questions, we must first view the clock face not just as a timekeeper, but as a geometric circle of 360°. Historically, time was measured using the angle of elevation of the Sun or the flow of water clocks, as documented in early Indian texts like the Arthasastra Science-Class VII, Measurement of Time and Motion, p.108. In modern quantitative problems, we focus on the relative angular speed between the minute hand and the hour hand.

The Minute Hand completes a full circle (360°) in 60 minutes, moving at a rate of 6° per minute. The Hour Hand is much slower, covering only 30° (the space between two numbers) in 60 minutes, which equates to 0.5° per minute. Therefore, the minute hand gains 5.5° over the hour hand every minute. This leads us to the General Formula for the Angle (θ) between the hands at any given time (H hours, M minutes):

One of the most frequent applications of this logic is determining how often the hands form a right angle (90°). While it might seem like this happens twice every hour (e.g., at 24 × 2 = 48 times a day), the reality is slightly different due to the overlap that occurs during specific periods. Just as the angle of deviation in optics describes a specific change in path Science, class X, The Human Eye and the Colourful World, p.166, the positions of the hands "deviate" from a simple count because the hour hand is also moving. Specifically, in every 12-hour cycle, the hands are at right angles only 22 times, not 24. This reduction happens around 3:00 and 9:00, where one of the expected right-angle positions is shared or coincides exactly with the hour mark.

Hand	Speed (per minute)	Speed (per hour)
Minute Hand	6°	360°
Hour Hand	0.5°	30°
Relative Gain	5.5°	330°

Sources: Science-Class VII, Measurement of Time and Motion, p.108; Science, class X, The Human Eye and the Colourful World, p.166

4. Hands Coinciding and Straight Line Positions (intermediate)

To master clock problems, we must first understand that a clock is a periodic system where two hands move at different speeds. Much like the Earth rotates 360° in 24 hours Certificate Physical and Human Geography, GC Leong, The Earth's Crust, p.11, the hands of a clock complete their own circular orbits. The Minute Hand moves 360° in 60 minutes (6° per minute), while the Hour Hand moves only 30° in 60 minutes (0.5° per minute). This difference in speed (5.5° per minute) determines when the hands will meet or form specific shapes. Hands Coinciding (0°): This occurs when the minute hand 'overlaps' the hour hand. While you might expect this to happen 12 times in 12 hours, it actually happens only 11 times. This is because the hands coincide only once between 11:00 and 1:00 (exactly at 12:00). Therefore, in a full 24-hour day, the hands coincide 22 times. Modern clocks use quartz or atomic vibrations to ensure these intervals remain precise Science-Class VII . NCERT, Measurement of Time and Motion, p.111, but the geometric logic remains constant. Straight Line Positions: In aptitude tests, the term 'straight line' covers two distinct scenarios: hands coinciding (0°) and hands pointing in opposite directions (180°). Just like the coinciding position, the 'opposite' position occurs 11 times in 12 hours (the exception being between 5:00 and 7:00, where they are opposite only at exactly 6:00).

Position Type	Angle	Occurrences (12 Hours)	Occurrences (24 Hours)
Coinciding	0°	11 times	22 times
Opposite	180°	11 times	22 times
Straight Line (Total)	0° or 180°	22 times	44 times

Sources: Certificate Physical and Human Geography, GC Leong (Oxford University press 3rd ed.), The Earth's Crust, p.11; Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.111

5. Clock Reflections and Faulty Timing (exam-level)

In the study of Quantitative Aptitude, mastering clocks requires understanding how they behave under non-ideal conditions, such as lateral inversion (reflections) or mechanical inaccuracies (faulty timing). When we observe a clock in a plane mirror, the image undergoes lateral inversion—the left side appears on the right and vice versa Science, class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.141. This means that a clock showing 3:00 will appear as 9:00 in the reflection. To solve these problems quickly without drawing, we use the Subtraction Rule: subtract the mirror time from 11:60 (which is the same as 12:00) to find the actual time.

Faulty clocks present a different challenge. Historically, even the best early pendulum clocks, like those by Huygens, could gain or lose 10 seconds a day, whereas modern atomic clocks are incredibly precise Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.111. In exam problems, a "faulty" clock is one that is either too fast or too slow. The key is to establish a ratio between Incorrect Time and Correct Time. For instance, if a clock loses 10 minutes every 24 hours, it only covers 23 hours and 50 minutes of "clock time" for every 24 hours of real time.

Finally, we must consider the frequency of hand positions. While it seems intuitive that the hands of a clock would form a right angle (90°) twice every hour, the geometry of the circular face dictates otherwise. Over a 12-hour period, there are two instances—exactly 3:00 and 9:00—where the hands form a right angle that is shared between two different hour slots. This reduces the count from 24 to 22 times per 12-hour cycle. Consequently, in a full 24-hour day, the hands are at right angles exactly 44 times.

Position Type	Occurrences (12 Hrs)	Occurrences (24 Hrs)
Coincidence (0°)	11	22
Straight Line (180°)	11	22
Right Angle (90°)	22	44

Sources: Science, class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.141; Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.111

6. The Right-Angle Frequency Logic (90°) (exam-level)

To master clock problems, we must move beyond simple arithmetic and understand the relative motion of the hands. On a standard clock face, the minute hand moves at 6° per minute, while the hour hand moves at a sluggish 0.5° per minute. A right angle (90°) occurs when the space between these two hands is exactly 15 'minute spaces.' While our intuition suggests this should happen twice every hour (once when the minute hand is 'behind' and once when it is 'ahead'), the reality is slightly different due to the overlapping nature of time cycles. In a standard 12-hour period, a naive calculation would suggest 24 occurrences (12 hours × 2). However, the actual count is 22 times per 12 hours. This happens because the right-angle positions at exactly 3:00 and 9:00 are unique. For instance, in the period from 2:00 to 4:00, you don't see four right angles; you see only three. The right angle at 3:00 PM 'belongs' to both the 2-3 PM hour and the 3-4 PM hour, effectively reducing the count by one. The same logic applies to the 8:00 to 10:00 window, where 9:00 is the shared pivot. Just as the Earth's rotation of 360° in 24 hours defines our global time zones Physical Geography by PMF IAS, Latitudes and Longitudes, p.243, these internal clock mechanics define the frequency of hand alignments. When we scale this up to a full 24-hour day, we simply double the 12-hour result. Since there are two 12-hour cycles in a day, and each cycle 'loses' 2 right angles (one at the 3:00 mark and one at the 9:00 mark), we subtract a total of 4 from the theoretical 48. Therefore, the hands of a clock form a right angle exactly 44 times in a 24-hour period. This is a crucial 'constant' to memorize for competitive exams, as it allows you to bypass complex relative speed equations during the heat of the paper.

Sources: Physical Geography by PMF IAS, Latitudes and Longitudes, p.243

7. Solving the Original PYQ (exam-level)

Now that you have mastered the Relative Speed of clock hands and the concept of Angular Displacement, this question brings those building blocks together. You already know that the minute hand gains 5.5 degrees per minute over the hour hand. To form a right angle, the hands must have a relative separation of 15-minute spaces (or 90 degrees). While your first instinct might be to simply double the hours in a day, this question is a classic test of your ability to identify mathematical exceptions within a repeating cycle.

To solve this like a pro, start with the baseline: in a standard hour, the hands reach a 90-degree position twice. In a 24-hour day, a simple calculation suggests 48 times (24 × 2). However, you must account for the overlap periods. Around 3:00 and 9:00, the hands reach exactly 90 degrees at the start/end of the hour, meaning one instance is shared between two hour-long windows. This reduces the count by 2 in every 12-hour cycle. Therefore, in a full 24-hour day, we subtract 4 from the naive total, arriving at the correct answer: (A) 44. This logical nuance is a staple in Quantitative Aptitude for Competitive Examinations by R.S. Aggarwal.

UPSC designed the options to catch common errors in reasoning. Option (B) 48 is the "naive trap" for those who forget the overlap at the 3 and 9 pivots. Option (C) 24 is a trap for students who mistakenly believe hands only reach a right angle once per hour, similar to the 0-degree overlap. Finally, (D) 12 is a distractor that ignores the 24-hour requirement of the question. Remember, in CSAT, the edge cases—those moments where the standard rule breaks—are exactly where the marks are hidden.