AFCAT Numerical Ability · Clock

When the Hands Meet, Point Opposite or Form an Angle

Turn the angle formula round: the hands are θ degrees apart at 2/11 of (30h ± θ) minutes past h, and in 12 hours each position comes round a fixed number of times.

All Clock notes

Why this matters

This is the hardest page of the chapter: a third of its questions are HARD. The method is one formula with a ± sign, and marks go when one of its two answers is dropped, or kept when it falls outside the hour, and when the 11 meetings in 12 hours are taken to be 12.

Concept 1 of 3: When the hands coincide or point opposite

h

o'clock the minute hand is

30h

degrees behind the hour hand and gains

5.5°

a minute. It catches the hour hand once it has closed that gap, and points opposite once it is

180°

ahead, or

180°

behind.

Definition

Coincide between $h$ and $h + 1$ : $m = \dfrac{30h}{5.5} = \dfrac{60h}{11}$ minutes past $h$ .
Opposite: $5.5m = 30h \pm 180$ . Keep the value that lies between 0 and 60.
'In a straight line' covers both coinciding and opposite. 'In a straight line but not together' means opposite.
Between 11 and 12 the hands meet only at 12:00 itself, so there are 11 meetings in 12 hours, not 12.

Coinciding and opposite hands

m = \dfrac{60h}{11} \;\text{(together)}, \qquad m = \dfrac{2}{11}\left(30h \pm 180\right) \;\text{(opposite)}

Worked example

When do the hands coincide between 5 and 6 o'clock, and when are they opposite between 2 and 3?

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 1 · Clock · Hands Coinciding, Opposite or at an Angle

At what time between 8 and 9 o'clock will the hands of a clock be in the same straight line but not together?

Eleven meetings, not twelve

The hands meet once in each hour except between 11 and 12, where the meeting is 12:00 itself. That is why a meeting comes every

65\tfrac{5}{11}

minutes.

A straight line includes together

If a question asks only for 'in a straight line', the moment the hands coincide counts too. Between 1 and 2 that is about 1:05 as well as about 1:38.

Drill 7 more on when the hands coincide or point opposite

Concept 2 of 3: When the hands are at a given angle

Set the angle formula equal to the angle you want:

|30h - 5.5m| = \theta

. The absolute value gives two equations, one before the minute hand passes the hour hand and one after.

Definition

$m = \dfrac{2}{11}(30h \pm \theta)$ minutes past $h$ .
Keep each value from 0 up to 60. Usually both lie inside the hour; near 3 and 9 one of them lands on the hour itself or past it.
Right angles: $\theta = 90$ . Hands ' $k$ minute spaces apart' are $6k$ degrees apart.
Options are often rounded down to the minute, so $27\tfrac{3}{11}$ minutes past 2 shows as 2:27.

Time for a given angle

m = \dfrac{2}{11}\left(30h \pm \theta\right)

Worked example

At what times between 5 and 6 o'clock are the hands at right angles?

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Clock · Hands Coinciding, Opposite or at an Angle

At what time between 1:00 and 2:00 will the angle between the hands of the clock be 72°?

Drop a value outside the hour

Between 10 and 11,

\tfrac{2}{11}(300 + 90) = 70\tfrac{10}{11}

is past the hour, so only

\tfrac{2}{11}(300 - 90) = 38\tfrac{2}{11}

counts.

Minute spaces are not degrees

Hands 5 minute spaces apart are

30°

apart. Putting

\theta = 5

into the formula answers a different question.

Drill 7 more on when the hands are at a given angle

Concept 3 of 3: How many times a position comes round

The minute hand gains one full lap on the hour hand every

65\tfrac{5}{11}

minutes, which makes 11 laps in 12 hours. Each lap brings the hands together once and opposite once, and brings every other angle round twice.

Definition

Multiply by 11 laps in 12 hours, or 22 in a day. For a stretch shorter than 12 hours, list the actual times and count the ones inside it, deciding whether the end times are included.

Position	In 12 hours	In 24 hours	How often
Coinciding	11	22	every $65\tfrac{5}{11}$ min Not 12 in 12 hours: the meeting between 11 and 12 is 12:00 itself.
Opposite ( $180°$ )	11	22	every $65\tfrac{5}{11}$ min
In a straight line (together or opposite)	22	44	every $32\tfrac{8}{11}$ min
At right angles	22	44	every $32\tfrac{8}{11}$ min Not 48 in a day: 3:00 and 9:00 are each shared by two neighbouring hours.
Any other angle between $0°$ and $180°$	22	44	twice in each lap

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Clock · Hands Coinciding, Opposite or at an Angle

How many times in a day do the hands of the clock form a right angle?

Count a stretch by listing

From 2:00 to 5:00 is 3 hours, but

3 \times \tfrac{11}{12} = 2.75

is not a count. List the meetings, about 2:11, 3:16 and 4:22, and there are 3.

44 right angles a day, not 48

Two right angles an hour would make 48. Around 3 and 9 the right angles of neighbouring hours fall on the same moment, 3:00 or 9:00, so 12 hours hold 22 and a day 44.

Drill 13 more on how many times a position comes round

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

When the hands coincide or point opposite
Coinciding and opposite hands
$m = \dfrac{60h}{11} \;\text{(together)}, \qquad m = \dfrac{2}{11}\left(30h \pm 180\right) \;\text{(opposite)}$
When the hands are at a given angle
Time for a given angle
$m = \dfrac{2}{11}\left(30h \pm \theta\right)$

Reference tables (1)

How many times a position comes round5 rows

Position	In 12 hours	In 24 hours	How often
Coinciding	11	22	every $65\tfrac{5}{11}$ min Not 12 in 12 hours: the meeting between 11 and 12 is 12:00 itself.
Opposite ( $180°$ )	11	22	every $65\tfrac{5}{11}$ min
In a straight line (together or opposite)	22	44	every $32\tfrac{8}{11}$ min
At right angles	22	44	every $32\tfrac{8}{11}$ min Not 48 in a day: 3:00 and 9:00 are each shared by two neighbouring hours.
Any other angle between $0°$ and $180°$	22	44	twice in each lap

Watch out for (6)

Eleven meetings, not twelve
→ When the hands coincide or point opposite
A straight line includes together
→ When the hands coincide or point opposite
Drop a value outside the hour
→ When the hands are at a given angle
Minute spaces are not degrees
→ When the hands are at a given angle
Count a stretch by listing
→ How many times a position comes round
44 right angles a day, not 48
→ How many times a position comes round

Drill the questions on this topic

30 questions from the bank, with cart and Word export.

Drill all 30 questions