A clock gains 3 minutes every hour.If the clock is set correctly at midnight on January 1,when will it next show the correct time ?
Hello, symmetry!
A clock gains 3 minutes every hour.
If the clock is set correctly at midnight on January 1,
when will it next show the correct time?
To show the correct time, the clock must gain exactly $\displaystyle 12$ hours.
. . This is $\displaystyle 60 \times 12 \:=\:720$ minutes.
Gaining only $\displaystyle 3$ minutes per hour,
. . it will take: $\displaystyle \frac{720}{3} \:=\:240\text{ hours }\:=\:10\text{ days.}$
It will tell the correct time at midnight of January 11.
I understand exactly what you said.
You said:
"To show the correct time, the clock must gain exactly 12 hours."
My question: What section of the problem indicated that the clock must gain 12 hours? Where is that stated in the question?
Actually that assumption may not be true. A military clock would require 24 hours to show the correct time again. However most clocks are on a 12 hour cycle, so the assumption is probably true.
This kind of assumption is called "common knowledge." The writer assumes you have knowledge of this, as it is supposedly so basic that everyone knows this. I have often found these assumptions to not be quite as well known as the writer assumes.
-Dan
Dan,
I agree with you what you said. Most math teachers, math textbook writers assume that students have enough common sense to figure out word problems by applying what they already know but this is not always the case. I had no idea that the assumption made by Soroban needed to be made in order to find the answer.
May I butt in.
Maybe I should have posted my solution even after I saw Soroban's solution then. I decided not to, because I thought our solutions were almost the same.
Anyway, here's that solution.
(Correct time, Erroneous time)
(0000, 0000) ---starting at midnight, both times are the same.
(0100, 0103) ---after 1 hr, the erroneous time is 3 minutes faster.
(0200, 0206)
(0300, 0309)
.
(1200, 1236) ---after 12 hrs, the erronuous time is 36 minutes faster.
.
(2000, 2060) = (2000, 2100) ---after 20 hrs, the erroneous time is one hour faster.
So, after 40 hrs, the erroneous time is faster by 2 hrs.
After 60 hrs, faster by 3 hrs.
.
After 120 hrs, faster by 6 hrs.
.
After 240 hrs, faster by 12 hrs.
But at this 240 hrs, the correct time is 12 o'clock midningt.
So is the erroneous time. Because the clock is up to 12 hrs only in its dial.
So, (240 hrs)*(1day/24hrs) = 10 days.
Therefore, in 10 days after midnight of January 1, both correct time and erroneous time will show the same time. And that is at midnight of January 11. -----------answer.