9. Comparison and Subtraction of Time Intervals
The Easy Case - Conversions not needed
Here are two time intervals in cosmetic normal formal
T1 = 6 hours 35 minutes and 15 seconds
T2 = 8 hours 45 minutes and 30 seconds
The second time interval T2 has more hours, more minutes
and seconds than the first T1. So it is larger.
The subtraction
T2 = 8 hours 45
minutes and 30 seconds
T1 = 6 hours 35
minutes and 15 seconds __
T2 - T1 = 2 hours 10 minutes and 15
seconds
suggests T2 is 2 hours 10 minutes and
15 seconds more than T1.
Let us check that by an addition:
2 hours 10
minutes and 15 seconds
T1 = 6 hours 35 minutes and 15
seconds +
8 hours
45 minutes and 30 seconds = T2
Harder Cases - Conversions Needed
Here are two more time intervals
T1 = 7 hours 35 minutes and 15 seconds
T2 = 5 hours 45 minutes and 30 seconds
The first time interval T1 has 2 more hours than the
second T2. Just one of those extra hours by itself is larger
than the minute and second part of the other time interval
T2. So in this case, the first time interval is
more than (larger than) the second. Subtraction will say how
much more.
The first time interval T1 in its minutes and second portion
has fewer minutes and fewer seconds than the other time interval
T2 . So the first column method for subtraction
introduced above would have us write:
The subtraction
T1 = 7 hours 35
minutes and 15 seconds
T2 = 5 hours 45
minutes and 30 seconds __
T2 - T1 = 2 hours ?? minutes and ??
seconds
If we avoid negative numbers or counts, we will have difficulty
with the seconds and minute entries in this subtraction. But the first
time interval T1 has 2 more hours than the second
T2. Just one of those extra hours by itself is larger than the
minute and second part of the other time interval
T2. That suggests a
conversion:
1 hour = 59 minutes and 60 seconds and so 7 hours = 6 hours, 59
minutes and 60 seconds.
Now T1 = 7 hours 35 minutes and 15
seconds
= 6 hours, 59 minutes and 60 seconds + 35 minutes and
15 seconds
We want to subtract T2 = 5 hours 45
minutes and 30 seconds from the latter.
T1 = 6 hours, 59
minutes and 60 seconds + 35 minutes and 15 seconds
T2 = 5
hours 45 minutes and 30
seconds
__
T1 - T2 = 1 hour 14 minutes
and 30 seconds + 35 minutes and 15 seconds
That latter gives
T1 - T2 = 1 hour 49 minutes
and 45 seconds
The latter gives how much, the first time length T1 is more
than the other time interval T2.
Let us check that:
1 hour 49 minutes and 45
seconds
T2 = 5
hours 45 minutes and 30 seconds +
7 hours, 35 minutes and 15 seconds =
T1
1
1
Observe the presence of carries in the minute and hours columns.
The foregoing calculations suggest a column method for subtraction when
conversions are needed.
First, write
T1 = 7 hours 35
minutes and 15 seconds
<======= The extra line.
T2 = 5 hours 45
minutes and 30 seconds __
as before, but with an extra line. The extra line is needed.
We will cross-out the 7 hours on the top line, and replace its by 6
hours, 59 minutes and 60 seconds
on the second line. That gives
T1 = 7
hours 35 minutes and 15 seconds
6 hours, 59 minutes and 60
seconds <======= The
extra line.
T2 = 5 hours 45
minutes and 30 seconds __
The task now is to subtract the third line from the top two
lines.
T1 = 7
hours 35 minutes and 15 seconds
6 hours, 59 minutes and 60
seconds
T2 = 5 hours 45
minutes and 30 seconds
__
T1 - T2 = 1 hour
49 minutes and 45
seconds
as before. Here 15 + 60 - 30 = 15 + 30 = 45
gives the second count
35 + 59 - 45 = 35 + 14 = 49 gives the minute count
6 - 5 = 1 gives the hour count
The result is the same as before. Note the extra line is present if
a conversion is needed.
Worked Examples:
Take the shorter time interval away from the longer time interval.
Example A.
23 hours 18 minutes 17 seconds versus 34 hours 45 minutes 25
seconds
Solution: The second time interval is larger as it has more
hours than than first. The difference is
give by the calculation:
hours mins
secs
34
45 25
23
18 17 -
9
27 8
No extra line is needed here because no conversions are
needed. The second time interval is 9 hours, 27
minutes and 8 seconds more than the first. Check
that in place by seeing i if the sum of the bottom two rows in the above
column method for subtraction equals the first.
Example B.
50 hours 23 minutes 17 seconds versus 34 hours 25 minutes 25 seconds.
Solution: The first time interval is larger as it has more
hours than than first. The difference is
give by first writing
hours mins
secs
50
23 17
23
25 25 -
with an extra line since conversions will be needed in the minute and
seconds column. Now in place, cross-out the 50 and replace it by an
entry of 49 hours, 59 minutes and 60 seconds in line 2. That should give
you
hours mins
secs
50
23 17
49
59 60
23
25 25 -
Now continue the calculation in place to obtain the
following:
hours mins
secs
50
23
17 (line 1)
49
59
60 (line 2)
23
25 25
- (line 3)
26
57
52 (line
4)
Work: 17 +
60 - 25 = 17 + 35 = 52
23+ 59 - 25 = 23 + 37 = 57
49 - 26 = 23
The longer time interval should be 26 hours, 57 minutes
and 52 seconds more than the first. Finally, check that in
place by verifying the original line 1 (no cross-out) is the sume of
lines 3 and 4
Example C.
30 hours 35 minutes 13 seconds versus 30 hours 20 minutes 25 seconds.
Solution: Again, the first time interval is larger as it has
more minutes than than first. The difference is calculated by first
writing
hours mins
secs
30 35
13
30
20 25 -
with an extra line since conversions will be needed in the minute and
seconds column. Now in place, cross-out the 35 in the minute column
and replace it by an entry of 34 minutes and 60 seconds in line 2. That
should give you
hours mins
secs
30 35
13
34 60
30 20
25 -
Now continue the calculation in place to obtain the
following:
hours mins
secs
30 35
13 (line 1)
34
60 (line 2)
30 20
25 - (line 3)
14
48 (line
4)
Work: 13 + 60 - 25 = 13 + 35 = 48
34 - 20 = 14
The longer time interval should be 14 minutes and 48 seconds
more than the first. Finally, check that in place by
verifying the original line 1 (no cross-out) is the sum of lines 3 and 4
Example D.
55 hours 43 minutes 35 seconds versus 35 hours 13 minutes 55 seconds.
Solution: Again, the first time interval is larger as it has
more hours than than first. The difference is calculated by first
writing
hours mins
secs
55
43 35
35
13 55 -
with an extra line since conversions will be needed in the seconds
column. No conversions are needed elsewhere. The first line
has more minutes than the last line. Now in place,
cross-out the 43 in the minute column and replace it by an entry of 42
minutes and 60 seconds in line 2. That should give you
hours mins secs
55
43 35
42 60
35
13 55 -
Now continue the calculation in place to obtain the
following:
hours mins
secs
55
43 35
42 60
35
13 55 -
20
29 40
Work: 35 + 60 - 55 = 35 + 5 = 40
42 - 13 = 29
The longer time interval should be 20 hours, 29 minutes and
40 seconds more than the first. Finally, check that in place
by verifying the original line 1 (no cross-out) is the sum of the last
two lines.
Example D - Revisited
55 hours 43 minutes 35 seconds versus 35 hours 13 minutes 55 seconds.
Solution: Again, the first time interval is larger as
it has more hours than than first. The difference is
calculated by first writing
hours mins
secs
55 43
35
35
13 55 -
with an extra line since conversions will be needed in the seconds
column. No conversions are needed elsewhere.
In this second version of Example D, we will convert 55 hours
into 54 hours, 59 minutes and 60 seconds. That is not
necessary because 43 minutes and 35 seconds is longer than 13
minutes and 55 seconds, and hence conversion of 43 minutes into
42 minutes and 60 seconds was good enough (and quicker). We
doing this to compare and contrast the two routes for obtaining a
solution. You need the experience.
The first line has more hours than the last line. Now in
place, cross-out the 55 in the hour column and replace it by an
entry of 54 hours, 59 minutes and 60 seconds in line 2. That should
give you
hours mins secs
55 43
35
54 59
60
35
13 55 -
Now continue the calculation in place to obtain the
following:
hours
mins secs
55
43 35
54
59 60
35
13 55 -
20
29 40
1
Work: 35 + 60 - 55 = 35 + 5 = 40 as
before
43 + 59 -13 = 43 + 46 = 89 or 1 hour and 29 minutes
(that gives a carry of 1 hour to add into the hours column
1+ 54 - 35 = 1 + 19 = 20
Again, the time interval should be 20 hours, 29 minutes and
40 seconds more than the first.
In this second approach, the conversion of 55 hours into 54 hours,
59 minutes and 60 seconds was not wrong, but it the effect of
introducing the conversion of 89 minutes into the answer.
Without the conversion of 89 minutes into 1 hour and 29 minutes,
the above calculation would have given
hours
mins secs
55
43 35
54
59 60
35
13 55 -
19
89 40
or 19 hours, 89 minutes and 40 seconds as a result, correct
except it is not in cosmetic normal form.
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Example E.
25 hours 45 minutes 10 seconds versus 40 hours 18 minutes 55
seconds versus .
Solution: The second time interval has more hours than the
first. So it is larger.
The difference is calculated by first writing
hours mins
secs
40 18
55
25 45
10 -
with an extra line since conversions will be needed. Now in place, 55
secs - 10 secs gives 45 secs.
In place that gives
hours mins
secs
40 18
55
25
45 10 -
45
Now in place, cross-out the 40 in the hour column and replace it by
an entry of 39 hours and 60 minutes in the extra line.
hours mins
secs
40
18 55
39 60
25
45 10 -
45
Now continue the calculation in place to obtain the following:
hours mins
secs
40
18 55
39 60
25
45 10 -
14
33
45
Work: 18 + 60 - 45 = 18 + 15 = 33
39 - 25 = 14
The longer time interval should be 14 hours, 33 minutes and
45 seconds more than the other. Finally, check that in place
by verifying the original line 1 (no cross-out) is the sum of the last
two.
Remark: We could use negative numbers here.
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