7. Addition of Time Intervals - Column Methods
Compare with column methods for decimal addition.
Three people John, Sally and Jane work for the following time intervals:
John: 14 hours, 15 minutes
Sally: 16 hours, 50 minutes
Jane: 16 hours, 35 minutes
How many hours and minutes did they work altogether?
Alternative Terms: The phrases time interval, time length and
time period all have the same meaning in this lesson.
Solution: The sum or total of these time lengths is
wanted.
John:
14 hours, 15 minutes
Sally:
16 hours, 50 minutes
Jane:
16 hours, 35 minutes +
Altogether: 46 hours, 100 minutes
But as a matter of convention, each multiple of 60 minutes in the
minute count 100 is converted (changed) into hours.
Here 100 minutes = 60 minutes + 40 minutes = 1
hour + 40 minutes.
So altogether, they work 46 hours + 100 minutes = 46 hours + 1 hour
+ 40 minutes = 47 hours + 40 minutes. So the total is 47
hours, 40 minutes with the comma playing the role of a plus or addition
sign.
Briefer form of Solution (with carries or conversions)
Solution: The sum or total of these time lengths is
wanted.
John:
14 hours, 15 minutes
Sally:
16 hours, 50 minutes
Jane:
16 hours, 35 minutes +
Altogether: 47 hours, 40 minutes
1 <----------------------------------- Here addition of
minutes column
gives 100 minutes, which in turn gives 40 minutes
and a carry of 1 hour in the hour column.
Now the hours column addition gives 14 + 16 + 16 + 1 =
47
Compare and contrast the foregoing with a decimal sum that gives 23
tens and 45 ones. Conversions would rewrite the sum as 2
hundreds, 7 tens and 5 ones or 275 in brief.
Conversion into Cosmetic Normal Form
The time length 8 hours, 25 minutes and 54 seconds is in cosmetic
normal form. But the time length 10 hours, 59 minutes and 193
seconds is not. The latter after one conversion equals 10 hours,
(59 + 3) minutes and 13 seconds, or 11 hours, 2 minutes and 13
seconds as (59+3) minutes = 62 minutes = 1 hour and 2 minutes.
Here 10 hours, 59 minutes and 193 seconds measures or gives the
time length 11 hours, 2 minutes and 13 seconds.
More on the addition of time intervals - Alternate Notation
8 hours, 26
minutes
6 hours, 51
minutes +
15 hours, 17
minutes
1
The carry of one hour in the hour columns follow from the addition of the
minutes column.
That is, 26 minutes + 51 minutes = 77 minutes = 1 hour and 17
minutes.
Here is the same calculation is a different form.
hrs mins
8 26
6
51 +
15
17
1
The identification of units of time measure at the top of each column
leads to the units being written fewer times.
We will stop with a last example where 4 time intervals are added.
hrs mins
secs
Work & Comments
8
50
30
The seconds sum to 170 = 2 × 60 + 50
45
45
50
That gives the carry of 2 in the minutes column
10
55 35
12
50 55
+
With the carry of 2, the minutes add to
78
22
50
202 = 3 × 60 + 22
3
2
That gives the carry of 3 in the hours column.
Now the hours with the carry of 3 total 78
The four lengths of time sum to 78 hours, 22 minutes and 50
seconds.
We could go further and convert the 78 hours into 3 twenty-four
hour calendar days and 6 hours. Or, we might convert the 78 hours
into 9 eight-hour work days and 6 hours left-over. These
further conversion all depend on how many hours there are in the kind
of day in question.
General Rule
Addition of lengths of time, lengths expressed with mixed units, will
gives another length of time in seconds, minutes, hours and days in which
the number of seconds or minutes may be more than 60, the number of hours
may be more than 24, and so on. Conversions or carries may be done
to rewrite or measure the time interval (length) in question, so the
number of seconds is between 0 and 59, the number of minutes is between 0
and 59, the number of hours is between 0 and 23 - assuming 23+1 = 24
hours in a day. In the latter case, we have expressed the time
length in cosmetic normal form.
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Secondary
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