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Answer :
We are given an initial time on the clock of
[tex]$$
3\!:\!46\!:\!17.
$$[/tex]
Below is a step-by-step explanation of how the other times are obtained.
1. \textbf{Time after 1 hour and 22 minutes:}
Start with the original time, [tex]$3\!:\!46\!:\!17$[/tex].
- First, add 1 hour:
[tex]$$3\!:\!46\!:\!17 \to 4\!:\!46\!:\!17.$$[/tex]
- Next, add 22 minutes:
Adding 22 minutes to [tex]$4\!:\!46\!:\!17$[/tex], the minutes become [tex]$46 + 22 = 68$[/tex]. Since 68 minutes is 1 hour and 8 minutes, we adjust by converting 60 minutes to 1 hour:
[tex]$$4\!:\!46\!:\!17 + 22 \text{ minutes} = 5\!:\!08\!:\!17.$$[/tex]
Thus, in 1 hour and 22 minutes the time will be
[tex]$$
5\!:\!08\!:\!17.
$$[/tex]
2. \textbf{Time 2 hours and 53 minutes ago:}
Again, start with the current time [tex]$3\!:\!46\!:\!17$[/tex].
- Subtract 2 hours:
[tex]$$3\!:\!46\!:\!17 - 2\ \text{hours} = 1\!:\!46\!:\!17.$$[/tex]
- Now subtract 53 minutes:
Subtracting 53 minutes from [tex]$1\!:\!46\!:\!17$[/tex], notice that [tex]$46$[/tex] minutes is insufficient. We need to borrow 1 hour (which is [tex]$60$[/tex] minutes) so that:
[tex]$$1\!:\!46\!:\!17 = 0\!:\!(60+46)\!:\!17 = 0\!:\!106\!:\!17.$$[/tex]
Then subtract 53 minutes:
[tex]$$106 - 53 = 53 \text{ minutes},$$[/tex]
leaving:
[tex]$$
0\!:\!53\!:\!17.
$$[/tex]
Therefore, the time 2 hours and 53 minutes ago was
[tex]$$
0\!:\!53\!:\!17.
$$[/tex]
3. \textbf{Time 4 hours and 14 minutes ago:}
Once more, start with [tex]$3\!:\!46\!:\!17$[/tex].
- First, subtract 4 hours:
[tex]$$3\!:\!46\!:\!17 - 4\ \text{hours} = -1\!:\!46\!:\!17.$$[/tex]
A negative hour indicates that we have gone back past midnight. To adjust for this, we add 24 hours (since the clock cycles every 24 hours):
[tex]$$-1 + 24 = 23,$$[/tex]
so the intermediate time becomes:
[tex]$$23\!:\!46\!:\!17.$$[/tex]
- Next, subtract 14 minutes:
Subtracting 14 minutes from [tex]$46$[/tex] minutes:
[tex]$$46 - 14 = 32 \text{ minutes}.$$[/tex]
The seconds remain unchanged at [tex]$17$[/tex] seconds.
Hence, the time is:
[tex]$$
23\!:\!32\!:\!17.
$$[/tex]
Thus, summarizing all the results:
[tex]$$
\textbf{Current time: }3\!:\!46\!:\!17, \quad \textbf{Time in 1 hour 22 minutes: }5\!:\!08\!:\!17, \quad \textbf{Time 2 hours 53 minutes ago: }0\!:\!53\!:\!17, \quad \textbf{Time 4 hours 14 minutes ago: }23\!:\!32\!:\!17.
$$[/tex]
[tex]$$
3\!:\!46\!:\!17.
$$[/tex]
Below is a step-by-step explanation of how the other times are obtained.
1. \textbf{Time after 1 hour and 22 minutes:}
Start with the original time, [tex]$3\!:\!46\!:\!17$[/tex].
- First, add 1 hour:
[tex]$$3\!:\!46\!:\!17 \to 4\!:\!46\!:\!17.$$[/tex]
- Next, add 22 minutes:
Adding 22 minutes to [tex]$4\!:\!46\!:\!17$[/tex], the minutes become [tex]$46 + 22 = 68$[/tex]. Since 68 minutes is 1 hour and 8 minutes, we adjust by converting 60 minutes to 1 hour:
[tex]$$4\!:\!46\!:\!17 + 22 \text{ minutes} = 5\!:\!08\!:\!17.$$[/tex]
Thus, in 1 hour and 22 minutes the time will be
[tex]$$
5\!:\!08\!:\!17.
$$[/tex]
2. \textbf{Time 2 hours and 53 minutes ago:}
Again, start with the current time [tex]$3\!:\!46\!:\!17$[/tex].
- Subtract 2 hours:
[tex]$$3\!:\!46\!:\!17 - 2\ \text{hours} = 1\!:\!46\!:\!17.$$[/tex]
- Now subtract 53 minutes:
Subtracting 53 minutes from [tex]$1\!:\!46\!:\!17$[/tex], notice that [tex]$46$[/tex] minutes is insufficient. We need to borrow 1 hour (which is [tex]$60$[/tex] minutes) so that:
[tex]$$1\!:\!46\!:\!17 = 0\!:\!(60+46)\!:\!17 = 0\!:\!106\!:\!17.$$[/tex]
Then subtract 53 minutes:
[tex]$$106 - 53 = 53 \text{ minutes},$$[/tex]
leaving:
[tex]$$
0\!:\!53\!:\!17.
$$[/tex]
Therefore, the time 2 hours and 53 minutes ago was
[tex]$$
0\!:\!53\!:\!17.
$$[/tex]
3. \textbf{Time 4 hours and 14 minutes ago:}
Once more, start with [tex]$3\!:\!46\!:\!17$[/tex].
- First, subtract 4 hours:
[tex]$$3\!:\!46\!:\!17 - 4\ \text{hours} = -1\!:\!46\!:\!17.$$[/tex]
A negative hour indicates that we have gone back past midnight. To adjust for this, we add 24 hours (since the clock cycles every 24 hours):
[tex]$$-1 + 24 = 23,$$[/tex]
so the intermediate time becomes:
[tex]$$23\!:\!46\!:\!17.$$[/tex]
- Next, subtract 14 minutes:
Subtracting 14 minutes from [tex]$46$[/tex] minutes:
[tex]$$46 - 14 = 32 \text{ minutes}.$$[/tex]
The seconds remain unchanged at [tex]$17$[/tex] seconds.
Hence, the time is:
[tex]$$
23\!:\!32\!:\!17.
$$[/tex]
Thus, summarizing all the results:
[tex]$$
\textbf{Current time: }3\!:\!46\!:\!17, \quad \textbf{Time in 1 hour 22 minutes: }5\!:\!08\!:\!17, \quad \textbf{Time 2 hours 53 minutes ago: }0\!:\!53\!:\!17, \quad \textbf{Time 4 hours 14 minutes ago: }23\!:\!32\!:\!17.
$$[/tex]
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