Clock Gain Calculation Explained

This problem requires us to calculate the actual time when a clock that runs faster than normal (a gaining clock) shows a specific time. We are given how much the clock gains and the time it indicates.

Calculate Clock Indicated Time

First, we need to determine the total duration shown by the faulty clock since it was set correctly.

• The clock was initially set correctly at 8 a.m.
• The clock indicates 1 p.m. on the following day.
• To find the total time shown by the clock:
• From 8 a.m. one day to 8 a.m. the next day is 24 hours.
• From 8 a.m. the next day to 1 p.m. the next day is 5 hours (9 a.m., 10 a.m., 11 a.m., 12 p.m., 1 p.m.).
• Therefore, the total time indicated by the clock is 24 hours + 5 hours = 29 hours.

Determine the Clock's Gain Rate

The question states that the clock gains 10 minutes in every 24 hours. This means that for every 24 hours of actual, true time that passes, the faulty clock advances by 24 hours and 10 minutes.

Establish the True Time Ratio

We can establish a relationship between the time shown by the faulty clock and the actual true time elapsed. Since the clock is gaining time, the true time elapsed will always be less than the time shown by the clock.

The ratio is formulated based on the gain:

When the clock shows 24 hours and 10 minutes, the true time elapsed is 24 hours.

$ \frac{\text{True Time}}{\text{Clock Time}} = \frac{24 \text{ hours}}{24 \text{ hours } 10 \text{ minutes}} $

To simplify calculations, let's convert both values to minutes:

$ 24 \text{ hours} = 24 \times 60 \text{ minutes} = 1440 \text{ minutes} $

$ 24 \text{ hours } 10 \text{ minutes} = (24 \times 60) + 10 \text{ minutes} = 1440 + 10 \text{ minutes} = 1450 \text{ minutes} $

Now, the ratio becomes:

$ \frac{\text{True Time}}{\text{Clock Time}} = \frac{1440 \text{ minutes}}{1450 \text{ minutes}} $

Simplifying the fraction by dividing both the numerator and denominator by 10:

$ \frac{\text{True Time}}{\text{Clock Time}} = \frac{144}{145} $

Calculate the True Elapsed Time

We know that the clock indicated a total of 29 hours. Using the ratio derived above, we can calculate the actual true time that has passed.

$ \text{True Time} = \text{Clock Time} \times \frac{144}{145} $

Substitute the value of Clock Time (29 hours):

$ \text{True Time} = 29 \text{ hours} \times \frac{144}{145} $

$ \text{True Time} = \frac{29 \times 144}{145} \text{ hours} $

Calculate the product in the numerator:

$ 29 \times 144 = 4176 $

So, the true time elapsed is:

$ \text{True Time} = \frac{4176}{145} \text{ hours} $

Convert True Time to Hours and Minutes

To find the exact time, we need to convert this fraction of hours into hours and minutes.

Divide 4176 by 145:

$ \frac{4176}{145} \approx 28.8 $

Performing the division more precisely:

$ 4176 \div 145 = 28 \text{ with a remainder of } 116 $

This means the true time elapsed is $28 \frac{116}{145}$ hours.

Now, convert the fractional part of the hour ($\frac{116}{145}$) into minutes by multiplying by 60:

$ \text{Minutes} = \frac{116}{145} \times 60 $

Simplify the expression:

$ \text{Minutes} = \frac{116 \times 60}{145} $

Divide 60 and 145 by their greatest common divisor, which is 5:

$ \text{Minutes} = \frac{116 \times 12}{29} $

Notice that $116$ is a multiple of $29$ ($116 = 4 \times 29$):

$ \text{Minutes} = \frac{(4 \times 29) \times 12}{29} $

Cancel out the $29$ from the numerator and denominator:

$ \text{Minutes} = 4 \times 12 = 48 \text{ minutes} $

So, the true time elapsed is 28 hours and 48 minutes.

Determine the Final True Time

Finally, we add the calculated true elapsed time to the original starting time.

• Starting Time: 8 a.m.
• True Time Elapsed: 28 hours 48 minutes.
• True Time = 8 a.m. + 28 hours 48 minutes
• First, add 24 hours to the starting time: 8 a.m. + 24 hours = 8 a.m. the next day.
• Now, add the remaining time (28 hours 48 minutes - 24 hours = 4 hours 48 minutes) to 8 a.m. the next day:
• 8:00 a.m. + 4 hours 48 minutes = 12:48 p.m.

Therefore, when the clock indicates 1 p.m. on the following day, the true time is 48 minutes past 12 p.m.