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1. At what time between 4 and 5 o'clock are the hands of the clock together?
  A.  21
9 / 11
min past 4
  B.  22
3 / 7
min past 7
  C.  21
8 / 3
min past 3
  D.  20
5 / 11
min past 5
     
   
View Answer

Shortcut:
Between z and (z + 1) o'clock, the two hands will be together at 5z
(
12 / 11
)
minutes past z.
Here, z = 4
Using the above value in the shortcut, we get:
Required value = 5 x 4
(
12 / 11
)

=
(
12 x 20 / 11
)
=
(
240 / 11
)

= 21
(
9 / 11
)
minutes past 4.

2. At what time between 3 and 4 o'clock are the hands of a clock together?
  A.  15
4 / 13
min past 5
  B.  16
4 / 11
min past 3
  C.  14
5 / 9
min past 7
  D.  16
5 / 3
min past 3
     
   
View Answer

Shortcut:
Between z and (z + 1) o'clock, the two hands will be together at 5z
(
12 / 11
)
minutes past z.
Here, z = 3
Using the above value in the shortcut, we get:
Required value = 5 x 3
(
12 / 11
)

=
12 x 15 / 11
=
180 / 11
= 16
4 / 11
minutes past 3.

3. At what time between 5 and 6 are the hands of a clock coincident?
  A.  25
5 / 11
minutes past 3
  B.  27
3 / 11
minutes past 2
  C.  26
7 / 11
minutes past 5
  D.  27
9 / 2
minutes past 4
     
   
View Answer

Shortcut:
Between z and (z + 1) o'clock, the two hands will be together at 5z
(
12 / 11
)
minutes past z.
Here, z = 5
Using the above value in the shortcut, we get:
Required value = 5 x 5
(
12 / 11
)

=
12 x 25 / 11
=
300 / 11
= 27
3 / 11
minutes past 5.

4. At what time between 5 and 6 o'clock will the hand of clock be at right angle?
  A.  10
10 / 11
min past 4
  B.  12
11 / 5
min past 2
  C.  11
11 / 10
min past 5
  D.  13
11 / 12
min past 1
     
   
View Answer

Shortcut:
Between z and (z + 1) o'clock, the two hands will be at right angled at (5z ± 15)
(
12 / 11
)
minutes past z.
Here, z = 5
Using the above value in the shortcut, we get:
Required value = (5 x 5 ± 15)
(
12 / 11
)

= (25 − 15)
12 / 11
and (25 + 15)
12 / 11

= 10 x
12 / 11
and 40 x
12 / 11

=
120 / 11
and
480 / 11

= 10
10 / 11
min past 4.

5. At what times are the hands of a clock at right angles between 7 amd and 8 am?
  A.  53
9 / 10
min past 5, 22
8 / 12
min past 3
  B.  54
6 / 11
min past 7, 21
9 / 11
min past 7
  C.  54
2 / 11
min past 8, 20
8 / 9
min past 9
  D.  56
8 / 12
min past 5, 23
8 / 5
min past 2
     
   
View Answer

Shortcut:
Between z and (z + 1) o'clock, the two hands will be at right angled at (5z ± 15)
(
12 / 11
)
minutes past z.
Here, z = 7
Using the above value in the shortcut, we get:
Required value = (5 x 7 ± 15)
(
12 / 11
)

= (35 − 15)
12 / 11
and (35 + 15)
12 / 11

= 20 x
12 / 11
and 50 x
12 / 11

=
240 / 11
and
600 / 11

= 21
9 / 11
and 54
6 / 11

Hence, the hands of a clock will at at right angles 21
9 / 11
minutes past 7 and 54
6 / 11
minutes past 7.

6. At what time between 5:30 amd 6 will the hands of a clock be at right angles?
  A.  43
7 / 11
minutes past 5
  B.  41
7 / 12
minutes past 7
  C.  39
8 / 9
minutes past 2
  D.  43
13 / 5
minutes past 3
     
   
View Answer

Shortcut:
Between z and (z + 1) o'clock, the two hands will be at right angled at (5z ± 15)
(
12 / 11
)
minutes past z.
Here, In this case direct formula is not applicable because, given data is not in the form of z and z + 1. At 5 o'clock, the hands are 25 minutes spaces apart. To be at right angles and that too between 5:30 and 6, the minute hand has to gain (25 + 15) or 40 min spaces.
Now, by using the above value in the shortcut, we get:
Required value = 40 x
12 / 11
=
480 / 11

= 43
7 / 11
minutes past 5.

7. At what time between 10 and 11 o'clock will the hand of clock be at right angle?
  A.  41
5 / 13
min past 9
  B.  40
3 / 11
min past 13
  C.  39
3 / 8
min past 11
  D.  38
2 / 11
min past 10
     
   
View Answer

Shortcut:
Between z and (z + 1) o'clock, the two hands will be at right angled at (5z ± 15)
(
12 / 11
)
minutes past z.
Here, z = 10
Using the above value in the shortcut, we get:
Required value = (5 x 10 ± 15)
(
12 / 11
)

= (50 − 15)
12 / 11
and (50 + 15)
12 / 11

= 35 x
12 / 11
and 65 x
12 / 11

=
420 / 11
and
780 / 11

= 38
2 / 11
and 70
10 / 11

Hence, the hands of a clock will at right angles 38
2 / 11
minutes past 10.

8. At what time between 4 and 5 will the hands of a watch point in opposite direction?
  A.  55
7 / 9
min past 9
  B.  56
7 / 11
min past 6
  C.  54
6 / 11
min past 4
  D.  55
5 / 9
min past 5
     
   
View Answer

Shortcut:
Between z and (z + 1) o'clock, the two hands are in the same straight line
Case I: (a) When they are in opposite directions i.e, 30 minutes spaces apart, at (5z − 30)
(
12 / 11
)
minutes past z. [where z > 6]
(b) When z< 6, the above formula will become as (5z + 30)
(
12 / 11
)
minutes past z.
Note: At 6 o'clock two hands will be in opposite direction.
Case II: When they coincide (or come together), i.e 0 minutes spaces apart at 5z x
12 / 11
minutes past z.
Here, z = 4
Using the above value in the shortcut, we get:
Required value = (5 x 4 + 30) x
12 / 11
minutes past 4
= (20 + 30) x
12 / 11
minutes past 4
= 50 x
12 / 11
minutes past 4
=
600 / 11
minutes past 4
= 54
6 / 11
minutes past 4

9. Find at what time between 8 and 9 o'clock will the hands of a clock be in the same straight line but not together.
  A.  11
12 / 11
min past 5
  B.  10
10 / 11
min past 8
  C.  9
11 / 10
min past 2
  D.  8
9 / 10
min past 3
     
   
View Answer

Shortcut:
Between z and (z + 1) o'clock, the two hands are in the same straight line
Case I: (a) When they are in opposite directions i.e, 30 minutes spaces apart, at (5z − 30)
(
12 / 11
)
minutes past z. [where z > 6]
(b) When z< 6, the above formula will become as (5z + 30)
(
12 / 11
)
minutes past z.
Note: At 6 o'clock two hands will be in opposite direction.
Case II: When they coincide (or come together), i.e 0 minutes spaces apart at 5z x
12 / 11
minutes past z.
Here, z = 8
Using the above value in the shortcut, we get:
Required value = (5 x 8 − 30) x
12 / 11
minutes past 8
= (40 − 30) x
12 / 11
minutes past 8
= 10 x
12 / 11
minutes past 8
=
120 / 11
minutes past 8
= 10
10 / 11
minutes past 8

10. Find at what time between 9 and 10 o'clock will the hands of a colck be in the same straight line but not together.
  A.  16
4 / 11
min past 9
  B.  15
8 / 9
min past 8
  C.  17
5 / 10
min past 5
  D.  15
7 / 11
min past 3
     
   
View Answer

Shortcut:
Between z and (z + 1) o'clock, the two hands are in the same straight line
Case I: (a) When they are in opposite directions i.e, 30 minutes spaces apart, at (5z − 30)
(
12 / 11
)
minutes past z. [where z > 6]
(b) When z< 6, the above formula will become as (5z + 30)
(
12 / 11
)
minutes past z.
Note: At 6 o'clock two hands will be in opposite direction.
Case II: When they coincide (or come together), i.e 0 minutes spaces apart at 5z x
12 / 11
minutes past z.
Here, z = 9
Using the above value in the shortcut, we get:
Required value = (5 x 9 − 30) x
12 / 11
minutes past 9
= (45 − 30) x
12 / 11
minutes past 9
= 15 x
12 / 11
minutes past 9
=
180 / 11
minutes past 9
= 16
4 / 11
minutes past 9

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