# Faulty Clocks

The questions framed on this topic will generally ask you to find:

- the time at which the inaccurate clock will show the correct time.
- the total time gained or lost in an inaccurate clock.

## Definition

Faulty clock - A clock which gains or loses time.

If a clock indicates more than the actual time, then the clock is said to be fast or gaining time. E.g. if a clock indicates 10:15 when the correct time is 10, then it is 15 minutes too fast.

If a clock indicates less than the actual time, then the clock is said to be slow or losing time. E.g. if a clock indicates 09:45, when the correct time is 10, it is said to be 15 minutes too slow.

## Coinciding time

Whenever a clock is too fast or too slow, then both the hands of the clock will not coincide at intervals of 66 $\frac{5}{11}$ min

If coinciding time > 65$\frac{5}{11}$ then the clock is going slower than normal (i.e. clock is loosing time)

And if coinciding time < 65$\frac{5}{11}$ then the clock is going faster than normal (i.e. clock is gaining time).

##### Q. How much time is gained/lost by a clock in 11 hours, if minute and hour hands of the clock overlap every 66 minutes?

(a) 6 $\frac{5}{11}$ minutes (b) 5 $\frac{5}{11}$ minutes

(c) 5 $\frac{9}{11}$ minutes (d) 6 $\frac{5}{7}$ minutes

Explanations :

Both hands of a clock overlap every 65$\frac{5}{11}$ 𝑚𝑖𝑛 when the clock is working perfectly.

But the minute and hour hands of the clock in question overlap every 66 minutes.

Thus in 66 minutes, the time lost = 66 𝑚𝑖𝑛 - 65$\frac{5}{11}$ 𝑚𝑖𝑛 = $\frac{6}{11}$ minutes.

Then, time lost in 1 minute = $\frac{6}{11}$ x $\frac{1}{66}$ = $\frac{1}{121}$ minutes

So, time lost in 60 minutes (i.e. in 1 hr) = 60 x $\frac{1}{121}$ = $\frac{60}{121}$ minutes

And time lost in 11 hours = 11 x $\frac{1}{121}$ = $\frac{60}{11}$ min = 5$\frac{5}{11}$ minutes

Answer: (b)

If the minute hand of a clock overtakes the hour hand at intervals of x min of the correct time, then the clock loses or gains (5x ± t) $\frac{12}{11}$ minutes in a day.

If the result is (+ ve), then clock gains and if the result is (–ve), then clock loses.

In the given question, x = 66 min.

According to the formula:

Time lost in a day = ($\frac{720}{11}$ - 66) (60 × $\frac{24}{66}$) = (-$\frac{6}{11}$) ($\frac{240}{11}$) minutes

So, time lost in 11 hr = (-$\frac{6}{11}$) ($\frac{240}{11}$) ($\frac{11}{24}$) = -$\frac{60}{11}$ = - 5$\frac{5}{11}$ minutes

(minus sign denotes that the clock loses time)

Answer: (b)