Clock Reasoning Questions: Angles and Calendars

📑 On this page

⚡ The short answer

The angle between the hands is |30H − 5.5M| degrees; if it's over 180, take 360 minus it. The hands overlap 11 times in 12 hours, every 720/11 ≈ 65.45 minutes. Calendars run on odd days: an ordinary year moves a date's weekday by 1, a leap year by 2. Number puzzles: find one rule that fits every line.

🎬 The 30-second story

The samosa bet

“Hands overlap once an hour. So 12 times in 12 hours. Samosa says I'm right.”

🕐

Aarav

Aarav, betting on a 2018 CLAT question.

“Watch from 11 to 1. They meet at exactly 12:00, and that's it.”

🕚

Meher

Meher points at the hostel clock.

“So the 11-to-12 overlap and the 12-to-1 overlap are the same one!”

💡

Zoya

Zoya gets there first.

“Fine. Eleven. Take the samosa.”

🥟

Aarav

The samosa changes hands.

Tara's verdict: The minute hand laps the hour hand, but the hour hand keeps creeping forward, so each lap takes 720/11 ≈ 65.45 minutes, not 60. That's 11 overlaps in 12 hours and 22 in a day. Count with the formula, not with the clock face.

How does CLAT word clock and number puzzles?

Stem	What it really asks	First move
“From 11 o'clock to 7 o'clock, how many times will the hands overlap?” (CLAT 2018)	Count events in a window	List overlap times, every 65 5/11 min from 12:00
“If 1 □ 7 = 8, 2 □ 7 = 16… what is 9 □ 7?” (CLAT 2019)	Find the hidden operation	Test one rule on every line
“Another bag has 1/3rd extra coins… the difference is 5” (CLAT 2019)	Words into a fraction	Call the first bag x
“What is the angle between the hands at 3:40?” (common pattern)	The clock formula	\|30H − 5.5M\|
“What day of the week was 26 January 1950?” (common pattern)	Odd days	Remainders after dividing by 7

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How do the hands of a clock actually move?

The minute hand covers 360° in 60 minutes: 6° a minute. The hour hand covers 30° in 60 minutes: 0.5° a minute. So every minute the minute hand gains 5.5°. Everything else is bookkeeping.

🧮 The formula

The clock angle formula

🕐

30 × H

the hour hand's head start

−

⏱️

5.5 × M

what the minute hand has gained

📐

θ = |30H − 5.5M|

over 180°? use 360° − θ

Hand	Per hour	Per minute
Minute hand	360°	6°
Hour hand	30°	0.5°
Minute hand gains	330°	5.5°

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📌 Why 11 overlaps, not 12

To lap the hour hand, the minute hand must gain 360° at 5.5° a minute: 360 ÷ 5.5 = 720/11 = 65 5/11 minutes. In 720 minutes that's exactly 11 laps. The 11-to-12 and 12-to-1 hours share one overlap: 12:00.

How do you solve any clock or calendar question in four steps?

1
Fix the reference
Clocks: the 12 o'clock line. Calendars: a day you know (a date in the question, or a year's 1 January).
2
Convert to one unit
Clocks in minutes and degrees; calendars in days. 3:40 is H = 3, M = 40.
3
Apply the one tool
Angle: |30H − 5.5M|. A given angle or overlap: M = (2/11)(30H ± θ). Calendar: the remainder after dividing by 7.
4
Sanity-check
Picture the face. At 3:40 the minute hand is on 8 and the hour hand is just past 3: a bit under five numbers apart, so a bit under 150°.

Which clock questions can CLAT ask?

🧩 At a glance

Six clock families

📐Angle at a time

|30H − 5.5M|. At 3:40: |90 − 220| = 130°.

🤝Hands together

Between H and H+1, at 60H/11 min past H. Between 4 and 5: 21 9/11 min past 4.

📏Right angles

22 times in 12 hours, 44 in a day (not 48).

↔️Opposite (180°)

11 times in 12 hours, 22 in a day.

🪞Mirror image

Subtract from 11:60. 3:40 looks like 8:20.

🐢Fast or slow clock

Use the ratio of clock time to real time.

Finding the time for a given angle

Rearrange the formula. Between H and H+1, the hands are θ apart at M = (2/11)(30H ± θ) minutes past H. Try both signs and keep answers between 0 and 60. With θ = 0 you get the overlap: M = 60H/11.

Between	Overlap at	Between	Overlap at
12 and 1	12:00	6 and 7	6:32:44
1 and 2	1:05:27	7 and 8	7:38:11
2 and 3	2:10:55	8 and 9	8:43:38
3 and 4	3:16:22	9 and 10	9:49:05
4 and 5	4:21:49	10 and 11	10:54:33
5 and 6	5:27:16	11 and 12	none: shared 12:00

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⚠️ Nothing on a clock happens once an hour

In 12 hours: 11 overlaps, 11 opposites, 22 right angles. The missing ones are shared: the overlap at 12:00, the opposite at 6:00, and the right angles at 3:00 and 9:00.

Mirror images and slow clocks

For a mirror image, subtract from 11:60: 3:40 reflects as 8:20 and 10:25 as 1:35. For a slow clock, set up a ratio: a clock losing 5 minutes an hour shows 55 minutes for every 60 real ones.

How do odd days crack calendar questions?

Weekdays repeat every 7 days, so only the remainder after dividing by 7 matters: the odd days. Count odd days from a date you know and move the weekday forward by that many. Number them 0 = Sunday, 1 = Monday, up to 6 = Saturday.

Span	Days	Odd days
Ordinary year	365	1
Leap year	366	2
31-day month	31	3
30-day month	30	2
February	28 / 29	0 / 1
100 years	76 ordinary + 24 leap	5
200 / 300 / 400 years		3 / 1 / 0

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Leap years divide by 4, but century years only if they divide by 400: 2000 was a leap year, 1900 wasn't. Same date next year? The weekday moves 1, or 2 if a 29 February falls between. CLAT 2026 is on Sunday 6 December; 6 December 2027 is a Monday and 6 December 2028 a Wednesday. 2026's whole calendar returns in 2037.

How do you crack a number puzzle in 30 seconds?

A number puzzle hides one rule behind a few examples. Propose a rule and test it on every line, not just the first. A rule that fits one line and fails the next is the trap the examiner built.

🔢 Series breakdown

The 2019 operator puzzle, lined up

8162432

1st differences

+8+8+8

Rule: Each step of 1 in n adds 8, so n □ 7 = 8n, which is n × 7 + n. Don't read on to the 5th term (40): jump straight to n = 9, giving 72.

💡 Fractions in words, and a divisibility shortcut

'1/3 more than the first' means second = first + first/3, so the difference is a third of the first. And a quantity split into thirds must be a multiple of 3: in the 2019 coin puzzle only 9 and 15 survive before you write any algebra.

Worked examples: clocks, calendars and a slow clock

✍️ Worked example Warm-up ⏱ 20 sec

The angle at 3:40

What is the angle between the hands of a clock at 3:40?

1
Reference: 12 o'clock. Units: H = 3, M = 40.
2
Formula: 30H = 90 and 5.5M = 220, so |90 − 220| = 130°.
3
Under 180°, so keep it (the reflex angle is 230°). Sanity: under 150°, as pictured.

Answer: 130°

⚡ The exam-hall shortcut

Minute hand on 8, hour hand 20° past 3 (40 × 0.5°). From 3 to 8 is 150°; minus 20° = 130°.

✍️ Worked example CLAT level ⏱ 45 sec

Right angles between 4 and 5

At what times between 4 and 5 o'clock are the hands at right angles?

1
Reference: 4 o'clock, so H = 4 and θ = 90.
2
Minus sign: (2/11)(120 − 90) = 60/11 = 5 5/11 min past 4.
3
Plus sign: (2/11)(120 + 90) = 420/11 = 38 2/11 min past 4.
4
Both are between 0 and 60, so both count. Check: at 4:38 2/11 the minute hand is at 229.1° and the hour hand at 139.1°. Gap 90°.

Answer: 4:05 5/11 and 4:38 2/11

⚡ The exam-hall shortcut

Two right angles in almost every hour; the hours 2–3, 3–4, 8–9 and 9–10 share theirs at 3:00 and 9:00.

✍️ Worked example CLAT level ⏱ 60 sec

Republic Day 1950

On what day of the week was 26 January 1950?

1
Split it: 1,949 complete years, plus 26 days of January 1950.
2
1,600 years → 0 odd days. The next 300 years (to 1900) → 1.
3
1901–1949: 49 years, 12 of them leap (1904 to 1948). 49 + 12 = 61 → remainder 5.
4
26 days → remainder 5. Total 0 + 1 + 5 + 5 = 11 → remainder 4 = Thursday.

Answer: Thursday

⚡ The exam-hall shortcut

Anchor on a date you know: 26 January 2026 was a Monday. Going back 76 years with 19 leap days = 95 odd days, remainder 4. Monday minus 4 = Thursday.

✍️ Worked example Tricky ⏱ 75 sec

Kabir's slow café clock

Kabir's café clock loses 5 minutes every hour. He sets it right at 6:00 a.m. That evening it shows 5:00 p.m. (a) What is the real time? (b) A customer is told her table is ready 'when your clock shows 8 p.m.'. What will the real time be then?

1
Reference: 6:00 a.m., correct. The clock shows 55 minutes for every 60 real minutes, so real = clock × 60/55 = clock × 12/11.
2
(a) Clock time passed: 11 hours. Real = 11 × 12/11 = 12 hours, so 6:00 p.m.
3
(b) Clock time passed: 14 hours = 840 min. Real = 840 × 12/11 = 916 4/11 min = 15 h 16 4/11 min, so about 9:16 p.m.

Answer: (a) 6:00 p.m. (b) about 9:16 p.m.

⚡ The exam-hall shortcut

Every 11 clock-hours are 12 real hours. Spot that and part (a) needs no arithmetic.

How has CLAT asked clocks and number puzzles?

📝 Asked in CLAT 2018 · Q120 Overlaps in a window

From morning 11 O'clock to Evening 7 O'clock, how many times the hour hand and minutes hand will overlap each other?

Reveal answer + reasoning

Answer: C

Overlaps come every 720/11 minutes, starting at 12:00. Inside the window: 12:00 noon, about 1:05, 2:11, 3:16, 4:22, 5:27 and 6:33. The one before (10:54:33) is too early and the next (7:38) too late. That's 7. Cross-check: 8 hours × 11/12 = 7.33, so 7 whole overlaps. Answer C.

Why the others fail

A — Counts overlaps at 11:00 and 7:00 too. At 11 the hands are 30° apart; at 7 they're 150° apart.
B — Assumes one overlap per hour across 8 hours: Aarav's samosa mistake.
D — Forgets the overlap at exactly 12:00 noon.

🕒 Timeline

11 a.m. to 7 p.m.: every overlap

▶️
11:00

Window opens

Hands 30° apart
1
12:00

Overlap 1

Exactly noon
2
1:05

Overlap 2

1:05:27
3
2:11

Overlap 3

2:10:55
4
3:16

Overlap 4

3:16:22
5
4:22

Overlap 5

4:21:49
6
5:27

Overlap 6

5:27:16
7
6:33

Overlap 7

6:32:44
⏹️
7:00

Window closes

Next overlap 7:38, outside

📝 Asked in CLAT 2019 · Q11 Fraction puzzle

In a bag there are some gold coins. In another bag there are 1/3rd extra gold coins as compared to first bag. If the difference in the number of gold coins in first and second bag is 5, then how many coins are there in the first bag?

C13

D15

Reveal answer + reasoning

Answer: D

Let the first bag hold x. '1/3rd extra' means the second holds x + x/3, so the difference is x/3 = 5, giving x = 15. Check: 15 and 20 differ by 5. Answer D (our solution; CLAT 2019 has no official key).

Why the others fail

A — Not a multiple of 3, so a third of it isn't a whole coin.
B — Second bag 12, difference 3, not 5.
C — Not a multiple of 3.

📝 Asked in CLAT 2019 · Q16 Hidden operator

If 1 □ 7 = 8, 2 □ 7 = 16, 3 □ 7 = 24, 4 □ 7 = 32, then what is the value of 9 □ 7?

A63

B72

C81

D90

Reveal answer + reasoning

Answer: B

(The paper prints a symbol between the numbers; we show it as □.) Test a rule on every line. 1 + 7 = 8 fits line one, but 2 + 7 = 9, not 16. The answers are 8 × 1, 8 × 2, 8 × 3, 8 × 4, so n □ 7 = 8n = n × 7 + n. Then 9 □ 7 = 63 + 9 = 72. Answer B (our solution).

Why the others fail

A — 9 × 7 drops the + n that every line has.
C — 9 × 9: no line multiplies n by itself.
D — 9 × 10 assumes the multiplier grows.

Which clock and calendar myths cost marks?

💥 Myth vs fact

Traps students fall for

🙅 Myth

At 3:15 the hands are together.

Tap to bust it →

✅ Fact

The hour hand has crept 7.5° past 3, so the angle is 7.5°. They meet at 3:16:22.

🙅 Myth

Every century year is a leap year.

Tap to bust it →

✅ Fact

Only if it divides by 400. 2000 was; 1900 and 2100 aren't.

🙅 Myth

If a rule fits the first line, it's the rule.

Tap to bust it →

✅ Fact

Test every line. 1 + 7 = 8 works once and fails at 2 □ 7 = 16.

🎯 Quick quiz

Five quick ones

0 / 5

Q1. What is the angle between the hands at 7:20?

Meher's mock-test series starts on Saturday 3 October. A mock is held every 9 days after that, so the second one is on Monday 12 October. Her counsellor wants to know which weekday the fifth mock falls on, so that Meher can keep the day before free for revision. Meher insists every mock will be on a Saturday because the first one was.

Q2. On which day is the fifth mock?

Q3. How many times are the hands of a clock at right angles in a day?

Kabir has locked the café's tip jar and left a note for his staff: 'If 2 ◆ 3 = 13, 3 ◆ 4 = 25 and 4 ◆ 5 = 41, the code is 5 ◆ 6.' Aarav guesses the rule is 'multiply, then add 7', because 2 × 3 + 7 = 13, and he punches in his answer. Zoya stops him and tells him to test the rule on the other two lines before he locks the jar for good.

Q4. What is the code?

Q5. A clock seen in a mirror shows 4:15. What is the real time?

🎯 Clocks, calendars and puzzles in a nutshell

Angle = |30H − 5.5M|; over 180°, use 360° − θ.
Hands meet every 720/11 min: 11 times in 12 hours, 22 a day.
Given an angle: M = (2/11)(30H ± θ).
Calendars: odd days. Ordinary year +1, leap year +2.
Number puzzles: one rule, tested on every line.

📎 Cheat sheet

“angle between the hands at…” → clock angle → |30H − 5.5M|.
“how many times… overlap / right angle / opposite” → event count → 11 / 22 / 11 per 12 hours, then check the window's edges.
“at what time between H and H+1…” → reverse formula → M = (2/11)(30H ± θ).
“mirror shows…” → mirror time → subtract from 11:60.
“what day of the week…” → odd days → remainder after dividing by 7.
“if a □ b = …” → hidden operator → test one rule on every line.

Frequently asked questions

How many times do the hands of a clock overlap in a day?

22 times. The minute hand gains 5.5° a minute, so it laps the hour hand every 720/11 minutes, about 65.45 minutes. That gives 11 overlaps in 12 hours, not 12, because the overlaps in the 11-to-12 and 12-to-1 hours are the same one, at 12:00.

What is the formula for the angle between clock hands?

Angle = |30H − 5.5M| degrees, where H is the hour and M the minutes. If the result is more than 180°, subtract it from 360° for the smaller angle. At 7:20 it is |210 − 110| = 100°. To find when the hands are θ apart, use M = (2/11)(30H ± θ).

What are odd days in calendar questions?

Odd days are the days left over after removing complete weeks. An ordinary year has 1 odd day and a leap year 2, so a date's weekday moves forward by 1 or 2 each year. Add up the odd days from a date you know, divide by 7, and move the weekday forward by the remainder.

Do clock and calendar questions still appear in CLAT?

They appeared in 2018 and 2019, when Logical Reasoning had standalone puzzles. Since 2020 the section has been mainly passage-based, but 2025 brought back a seating set, so analytical questions can return. With the formulas on this page, a clock or calendar question takes under a minute.