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PROFIT AND LOSS PROBLEMS

Aptitude

CLOCKS -> IMPORTANT FORMULAE

The face or dial of a watch is a circle whose circumference is divided into 60 equal parts, called minute spaces.
A clock has two hands, the smaller one is called the hour hand or short hand while the larger one is called the minute hand or long hand.
I. In 60 minutes, the minute hand gains 55 minutes on the hour hand.
II. In every hour, both the hands coincide onece.
III. The hands are in the same straight line when they are coincident or opposite to each other.
IV. When the two hands are at right angles, they are 15 minute spaces apart.
V. When the hands are in opposite directions, they are are 30 minute spaces apart.
VI. Angle traced by hour hand in 12 hrs = 360°.
VII. Angle traced by munute hand in 60 min. = 360°.
Too Fast and Too Slow : If a watch or a clock indicates 8.15, when the correct time is 8, it is said to be 15 minutes too fast.
On the other hand, if it indicates 7.45, when the correct time is 8, it is said to be 15 minutes too slow.

CLOCKS -> SOLVED EXAMPLES

1. 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.
  Sol. At 8 o’clock, the hour hand is at 8 and the minute hand is at 12, i.e. the two hands are 20 min. spaces apart.
To be i the same straight line but not together they will be 30 minute spaces apart. So, the minute hand will have to gain (30 - 20) = 10 minute spaces over the hour hand.
55 minute spaces are gained in 60 min.
10 minute spaces will be gained in [60/55 * 10] min. = 10 10/11 min.
∴ The hands will be in the same straight line but not together at 10 10/11 min. past 8.
2. Find the angle between the hour hand and the minute hand of a clock when the time is 3.25.
  Sol. Angle traced by the hour hand in 12 hours = 360°.
Angle traced by it in 3 hrs 25 min. i.e. 41/12 hrs =
[360 / 12 * 41/12] = 102½°
Angle traced by minute hand in 60 min. = 360°.
Angle traced by it in 25 min. = [360 / 60 * 25]° = 150°.
∴ Required angle = [150° - 102½°] = 47½.
3. A watch which gains uniformly, is 5 min. slow at 8 o’clock in the morning on Sunday and it is 5 min. 48 sec. fast at 8 p.m. on following Sunday. When was it correct?
  Sol.
Time from 8 a.m. on Sunday to 8 p.m. on following Sunday = 7 days 12 hours = 180 hours.
∴ The watch gains [5+5 4/5] min. or 54/5 min. in 180 hrs.
Now 54/5 min. are gained in 180 hrs.
∴ 5 min. are gained in [180 * 5/54 * 5] hrs. = 83 hrs 20 min. = 3 days 11 hrs 20 min.
∴ It will be correct at 20 min. past 7 p.m. on Wednesday.

CLOCKS -> Exercise

4. At what time between 4 and 5 o’clock will the hands of a watch point in opposite directions?
 
  • A. 54 6/11 min. past 4
  • B. 50 min. past 4
  • C. 45 min past 4
  • D. 40 min past 4
Ans: A.
Sol.
At 4 o’clock, the hands of the watch are 20 min. spaces apart.
To be in opposite directions, they must be 30 min. spaces apart.
∴ Minute hand will have to gain 50 min. spaces.
55 min. spaces are gined in 60 min.
50 min. spaces are gained in [60/55 * 50] min. or 54 6/11 min.
∴ Required time = 54 6/11 min. past 4.
 
5. At what time between 9 and 10 o’clock will the hands of a watch be together?
 
  • A. 49 1/11 min. past 9
  • B. 40 min. past 9
  • C. 45 min. past 9
  • D. 50 min. past 9
Ans: A.
Sol.
To be together between 9 and 10 o’clock, the minute hand has to gain 45 min. spaces. 55 min. spaces gined in 60 min.
45 min. spaces are gained in [60 / 55 * 45] min. or 49 1/11 min.
∴ The hands are together at 49 1/11 min. past 9.
 
 
6. How many times in a day, are the hands of a clock in straight line but opposite in direction?
 
  • A. 22
  • B. 23
  • C. 24
  • D. 25
Ans: A.
Sol.
Tha hands of a clock point in opposite directions (in the same straight line) 11 times in every 12 hours (Because between 5 and 7 they point in opposite directions at 6 o’clock only). So, in a day, the hands point in the opposite directions 22 times.