Clock and Calendar

Competitive Mathematics · Time, Work and Distance

5 Part8 worked examples69 practice questions

A clock is really two runners in a race. The minute hand makes one full turn in 60 minutes, the hour hand in 12 hours. So every hour the minute hand gets 55 minute-spaces ahead — and almost every clock question is answered from this “gain”. The calendar is the opposite: days repeat every 7, so you only need the remainder on dividing by 7 — these are called odd days.

मिनट की सुई : 6^{\circ} / मिनट, घंटे की सुई : 0. 5^{\circ} / मिनट, विषम दिन = दिन mod 7

The key to both halves

The hands of a clock

Angles, meetings and the gain

The angle between the hands, when they meet, are perpendicular or opposite, and fast and slow clocks.

Angles and speeds

6^{\circ}

and

0. 5^{\circ}

per minute

Each minute the minute hand turns $6^{\circ}$ and the hour hand $0. 5^{\circ}$ ; the angle is their difference.

θ = 30 h - \frac{11}{2} m (यदि > 18 0^{\circ}, 36 0^{\circ} - θ)

Angle between the hands at

h

hours

m

minutes

The dial has 12 hour marks and 60 minute marks. One hour mark is $3 0^{\circ}$ and one minute mark is $6^{\circ}$ .
Hour hand from 12: $30 h + 0.5 m$ degrees. Minute hand from 12: $6 m$ degrees.
Take the difference; if it exceeds $18 0^{\circ}$ subtract from $36 0^{\circ}$ (we want the smaller angle).
In 12 hours the hands: meet 11 times, are at right angles 22 times and opposite 11 times; double in a day.

Example 1 · Basic

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

Solution, step by step

1Hour hand $= 30 \times 7 + 0.5 \times 20 = 22 0^{\circ}$ ; minute hand $= 6 \times 20 = 12 0^{\circ}$ .
2Difference $= 10 0^{\circ}$

Answer 100°

Example 2 · Standard

What is the smaller angle between the hands at 3:50?

Solution, step by step

1Hour hand $= 90 + 25 = 11 5^{\circ}$ ; minute hand $= 30 0^{\circ}$ ; difference $= 18 5^{\circ}$ .
2 $185 > 180$ so $360 - 185 = 17 5^{\circ}$

Answer 175°

BasicTry it yourself first

What is the angle between the hour hand and the minute hand of a clock at 7:20?

BasicTry it yourself first

What is the angle between the hour hand and the minute hand of a clock at 2:40?

StandardTry it yourself first

What is the angle between the hour hand and the minute hand of a clock at 5:45?

StandardTry it yourself first

What is the angle between the hour hand and the minute hand of a clock at 10:10?

StandardTry it yourself first

What is the angle between the hour hand and the minute hand of a clock at 1:35?

AdvancedTry it yourself first

What is the angle between the hour hand and the minute hand of a clock at 8:16?

AdvancedTry it yourself first

What is the angle between the hour hand and the minute hand of a clock at 11:25?

AdvancedTry it yourself first

What is the angle between the hour hand and the minute hand of a clock at 6:05?

AdvancedTry it yourself first

What is the angle between the hour hand and the minute hand of a clock at 3:50?

BasicTry it yourself first

Through how many degrees does the minute hand of a clock turn in 10 minutes?

BasicTry it yourself first

Through how many degrees does the hour hand of a clock turn in 40 minutes?

StandardTry it yourself first

Through how many degrees does the minute hand of a clock turn in 25 minutes?

StandardTry it yourself first

Through how many degrees does the hour hand of a clock turn in 90 minutes?

AdvancedTry it yourself first

Through how many degrees does the hour hand of a clock turn in 48 minutes?

BasicTry it yourself first

How much time passes from 7:25 am to 7:48 pm on the same day?

StandardTry it yourself first

How much time passes from 5:50 am to 5:15 pm on the same day?

StandardTry it yourself first

How much time passes from 9:08 am to 9:56 pm on the same day?

AdvancedTry it yourself first

How much time passes from 4:35 am to 1:10 pm on the same day?

AdvancedTry it yourself first

How much time passes from 10:05 am to 10:50 pm on the same day?

When will the hands meet

Between

n

and

n + 1

o’clock

Meeting, right angle, opposite and “ $x$ spaces apart” are all solved by the “gain”.

मेल : \frac{60 n}{11} मिनट बाद, बढ़त = \frac{60}{55} \times (खाने)

After

n

o’clock

At $n$ o’clock the hour hand is at $5 n$ spaces and the minute hand at 12 — a gap of $5 n$ spaces.
For the target gap (together: 0, right angle: 15, opposite: 30, $x$ apart: $x$ ) find the number of spaces the minute hand must gain.
The minute hand gains 55 spaces in 60 minutes ⟹ for $g$ spaces it takes $\frac{60}{55} \times g = \frac{12 g}{11}$ minutes.
Write the answer as “... minutes past $n$ ”.

Example 3 · Standard

At what time between 4 and 5 o’clock will the hands of a clock be together?

Solution, step by step

1At 4 the gap is $20$ spaces. The minute hand must gain $20$ spaces.
2Time $= \frac{60}{55} \times 20 = \frac{240}{11} = 21 \frac{9}{11}$ minutes ⟹ $21 \frac{9}{11}$ minutes past 4

Answer $21 \frac{9}{11}$ min past 4

Example 4 · Standard

At what time between 2 and 3 o’clock will the hands be in opposite directions?

Solution, step by step

1At 2 the gap is $10$ spaces; for opposite the minute hand must gain $30 + 10 = 40$ spaces.
2Time $= \frac{60}{55} \times 40 = \frac{480}{11} = 43 \frac{7}{11}$ minutes ⟹ $43 \frac{7}{11}$ minutes past 2

Answer $43 \frac{7}{11}$ min past 2

Example 5 · Standard

At what time between 6 and 7 o’clock are the hands first at right angles?

Solution, step by step

1At 6 the minute hand is $30$ spaces behind. For a right angle the gap must be $15$ ⟹ gain $= 30 - 15 = 15$ spaces.
2Time $= \frac{60}{55} \times 15 = 16 \frac{4}{11}$ minutes ⟹ $16 \frac{4}{11}$ minutes past 6

Answer $16 \frac{4}{11}$ min past 6

BasicTry it yourself first

At what time between 2 and 3 o'clock will the two hands of a clock be together?

BasicTry it yourself first

At what time between 5 and 6 o'clock will the two hands of a clock be together?

StandardTry it yourself first

At what time between 6 and 7 o'clock will the two hands of a clock be together?

StandardTry it yourself first

At what time between 7 and 8 o'clock will the two hands of a clock be together?

StandardTry it yourself first

At what time between 8 and 9 o'clock will the two hands of a clock be together?

AdvancedTry it yourself first

At what time between 9 and 10 o'clock will the two hands of a clock be together?

AdvancedTry it yourself first

At what time between 10 and 11 o'clock will the two hands of a clock be together?

BasicTry it yourself first

At what time between 1 and 2 o'clock will the hands of a clock point in opposite directions?

StandardTry it yourself first

At what time between 4 and 5 o'clock will the hands of a clock point in opposite directions?

StandardTry it yourself first

At what time between 7 and 8 o'clock will the hands of a clock point in opposite directions?

AdvancedTry it yourself first

At what time between 8 and 9 o'clock will the hands of a clock point in opposite directions?

AdvancedTry it yourself first

At what time between 9 and 10 o'clock will the hands of a clock point in opposite directions?

AdvancedTry it yourself first

At what time between 10 and 11 o'clock will the hands of a clock point in opposite directions?

AdvancedTry it yourself first

At what time between 11 and 12 o'clock will the hands of a clock point in opposite directions?

StandardTry it yourself first

At what time between 4 and 5 o'clock are the hands of a clock first at right angles?

StandardTry it yourself first

At what time between 4 and 5 o'clock are the hands of a clock at right angles for the second time?

StandardTry it yourself first

At what time between 5 and 6 o'clock are the hands of a clock first at right angles?

AdvancedTry it yourself first

At what time between 6 and 7 o'clock are the hands of a clock at right angles for the second time?

AdvancedTry it yourself first

At what time between 8 and 9 o'clock are the hands of a clock first at right angles?

AdvancedTry it yourself first

At what time between 9 and 10 o'clock are the hands of a clock first at right angles?

AdvancedTry it yourself first

At what time between 10 and 11 o'clock are the hands of a clock first at right angles?

StandardTry it yourself first

At what time between 2 and 3 o'clock are the hands first 3 minute-spaces apart?

StandardTry it yourself first

At what time between 5 and 6 o'clock are the hands first 10 minute-spaces apart?

StandardTry it yourself first

At what time between 6 and 7 o'clock are the hands first 5 minute-spaces apart?

AdvancedTry it yourself first

At what time between 8 and 9 o'clock are the hands first 12 minute-spaces apart?

AdvancedTry it yourself first

At what time between 9 and 10 o'clock are the hands first 8 minute-spaces apart?

AdvancedTry it yourself first

At what time between 10 and 11 o'clock are the hands first 20 minute-spaces apart?

AdvancedTry it yourself first

At what time between 4 and 5 o'clock are the hands first 6 minute-spaces apart?

StandardTry it yourself first

Between 2:00 am and 8:00 am (excluding the ends) how many times do the two hands of a clock coincide?

StandardTry it yourself first

Between 4:00 am and 10:00 am (excluding the ends) how many times do the two hands of a clock are at right angles?

AdvancedTry it yourself first

Between 1:00 am and 7:00 am (excluding the ends) how many times do the two hands of a clock point in opposite directions?

AdvancedTry it yourself first

Between 1:00 am and 11:00 am (excluding the ends) how many times do the two hands of a clock coincide?

AdvancedTry it yourself first

Between 5:00 am and 11:00 am (excluding the ends) how many times do the two hands of a clock are at right angles?

Fast and slow clocks

Clock minutes versus true minutes

In a correct clock the hands meet every $65 \frac{5}{11}$ minutes; this tells how much a clock gains or loses.

A correct clock: the hands meet every $\frac{720}{11} = 65 \frac{5}{11}$ minutes. In another clock, if they meet every $x$ minutes of true time, the daily gain / loss is $\frac{1440 ( 720/11 - x )}{x}$ minutes; $x < 65 \frac{5}{11}$ ⟹ gains, $x >$ ⟹ loses.
**A clock gaining $g$ minutes in 24 hours:** $1440 + g$ clock-minutes $= 1440$ true minutes. When it shows $D$ minutes, true time $= D \times \frac{1440}{1440 + g}$ .
**A watch $a$ minutes slow that is $b$ minutes fast after $T$ hours:** total gain $a + b$ ; correct after $T \times \frac{a}{a + b}$ .

Example 6 · Standard

The minute hand of a clock overtakes the hour hand every 66 minutes of true time. How much does the clock gain or lose in a day?

Solution, step by step

1 $66 > 65 \frac{5}{11}$ ⟹ the clock loses. A correct clock takes $\frac{720}{11}$ , here $66$ .
2Daily loss $= 1440 \times \frac{66 - 720/11}{66} = 1440 \times \frac{6/11}{66} = \frac{1440}{121} = 11 \frac{109}{121}$ minutes

Answer loses $11 \frac{109}{121}$ min

Example 7 · Standard

A clock is set right at 6:00 am. It gains 60 minutes in 24 hours. What is the true time when the clock shows 7:00 am the next day?

Solution, step by step

1 $1500$ clock-minutes $= 1440$ true minutes. The clock showed $25$ hours $= 1500$ minutes.
2True time passed $= 1440$ minutes $= 24$ hours ⟹ 6:00 am the next day

Answer 6:00 am

Example 8 · Advanced

A watch gains uniformly. On Monday at 8:00 am it was 4 minutes slow and 6 days later at the same time it was 8 minutes fast. When was it correct?

Solution, step by step

1Gain in 6 days $= 4 + 8 = 12$ minutes. Time to gain 4 minutes $= 144 \times \frac{4}{12} = 48$ hours.
2Monday 8:00 am $+$ 2 days $=$ Wednesday 8:00 am

Answer Wednesday, 8:00 am

StandardTry it yourself first

The minute hand of a clock overtakes the hour hand at intervals of 60 minutes of correct time. How much does the clock gain or lose in a day?

StandardTry it yourself first

The minute hand of a clock overtakes the hour hand at intervals of 80 minutes of correct time. How much does the clock gain or lose in a day?

StandardTry it yourself first

The minute hand of a clock overtakes the hour hand at intervals of 90 minutes of correct time. How much does the clock gain or lose in a day?

AdvancedTry it yourself first

The minute hand of a clock overtakes the hour hand at intervals of 100 minutes of correct time. How much does the clock gain or lose in a day?

AdvancedTry it yourself first

The minute hand of a clock overtakes the hour hand at intervals of 50 minutes of correct time. How much does the clock gain or lose in a day?

AdvancedTry it yourself first

The minute hand of a clock overtakes the hour hand at intervals of 72 minutes of correct time. How much does the clock gain or lose in a day?

StandardTry it yourself first

A clock is set right at 6:00 am. It gains 60 minutes in 24 hours. What is the true time when the clock shows 7:00 am?

StandardTry it yourself first

A clock is set right at 7:00 am. It loses 40 minutes in 24 hours. What is the true time when the clock shows 6:00 pm?

AdvancedTry it yourself first

A clock is set right at 8:00 am. It gains 120 minutes in 24 hours. What is the true time when the clock shows 9:00 pm?

AdvancedTry it yourself first

A clock is set right at 9:00 am. It loses 120 minutes in 24 hours. What is the true time when the clock shows 8:00 pm?

AdvancedTry it yourself first

A clock is set right at 10:00 am. It gains 40 minutes in 24 hours. What is the true time when the clock shows 11:00 pm?

AdvancedTry it yourself first

A clock is set right at 5:00 am. It gains 10 minutes in 24 hours. What is the true time when the clock shows 3:00 pm?

StandardTry it yourself first

A watch gains uniformly. It was 4 minutes slow on Monday at 8:00 am and 8 minutes fast 6 days later at the same time. When was it correct?

StandardTry it yourself first

A watch gains uniformly. It was 5 minutes slow on Wednesday at 7:00 am and 15 minutes fast 6 days later at the same time. When was it correct?

AdvancedTry it yourself first

A watch gains uniformly. It was 6 minutes slow on Friday at 9:00 am and 12 minutes fast 7 days later at the same time. When was it correct?

AdvancedTry it yourself first

A watch gains uniformly. It was 3 minutes slow on Sunday at 6:00 am and 9 minutes fast 4 days later at the same time. When was it correct?

AdvancedTry it yourself first

A watch gains uniformly. It was 2 minutes slow on Tuesday at 10:00 am and 10 minutes fast 5 days later at the same time. When was it correct?

The calendar

Leap years, odd days and weekdays

The rule for leap years, odd days, the weekday of any date and when a calendar repeats.

Leap years and odd days

The remainder on dividing by 7

1 odd day in an ordinary year, 2 in a leap year; 5 in 100 years, 0 in 400 years.

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The weekday of a date

Count the days between, take the remainder mod 7

The weekday of another date from a known one; and the weekday of a date with nothing given.

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Traps and fast tricks

Slips · tricks

Where marks go in clock-and-calendar questions.

The commonest mistakes

Six traps

The slips that cost marks.

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Fast tricks

Rules and checks

Habits for speed.

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The practice ladder

Basic → standard → advanced

New questions of every type, in three levels.

Level 1 — Basic

Direct rules

One-rule questions.

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Level 2 — Standard

Two ideas combined

Exam-level questions.

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Level 3 — Advanced

Many steps and clever thinking

Questions at the hard end of competitive exams.

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Revision — formula cards

Flip each card and test yourself

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Clock and Calendar

The hands of a clock

Angles and speeds

When will the hands meet

Fast and slow clocks

The calendar

Leap years and odd days

The weekday of a date

Traps and fast tricks

The commonest mistakes

Fast tricks

The practice ladder

Level 1 — Basic

Level 2 — Standard

Level 3 — Advanced

Revision — formula cards

Free All Access Passevery mind map · Hindi + English · 12 months

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