10. Time Intervals - Repeated Addition or Multiplication by Numbers
First Example:
On three different days this week, Thomas worked for 7 hours and 50
minutes. How many hours and minutes in total did he work?
hours mins
7 50
7 50
7 50 +
23 30
2
He worked for 23 hours and 30 minutes.
Note: 3 × 7 = 21 while 3 × 50 = 150 = 2 × 60 + 30.
If we did not do the conversion while adding, we would have obtained 21
hours and 150 minutes. The latter is 2 hours and 30 minutes (or 2½
hours)
Second Example:
Ten people all work for 8 days, 6 hours and 35 minutes on a project.
Each day is a work day of length 7 hours. What is the total
number of days, hours and minute each work.
|
Days Hours Minutes
|
|
1
2
3
4
5
6
7
8
9
10
|
8 6 35
8 6 35
8 6 35
8 6 35
8 6 35
8 6 35
8 6 35
8 6 35
8 6 35
8 6 35
89 2 50
9 5
|
Calculations:
Minute Count: 10 × 35 = 350 = 5 × 60 + 50 minutes
Conversion gives a +5 carry in the hours column.
Hour Count: 10 × 6 + 5 = 65 = 9 × 7 + 2
The conversion gives a +9 carry in the work day column.
Day Count: 10 × 8 + 9 = 80 +9 =
|
Thus a total of 89 days (7 hours per day), 2 hours and 50 minutes were
worked.
Towards a better format:
Now instead of using the addition column methods with 10 rows, we may use
a multiplication column methods. We carry units in it.
First we write the following without doing any conversions:
8 days, 6 hours, 35 minutes
× 10
350 minutes
60 hours
+ 80 days
Next in place we write the conversions on the right, and align the units
in columns
8 days, 6 hours, 35 minutes
× 10
350 minutes = 5 hours
+ 50 minutes
60 hours = 8 days + 4
hours
+ 80 days = 80
days
Next, we continue the in place writing and add to obtain the following.
t
8 days, 6 hours, 35 minutes
× 10
350 minutes = 5 hours
+ 50 minutes
60 hours = 8 days + 4
hours
+ 80 days = 80
days
80 days + 60 days + 350 minutes = 89 days + 2 hour + 50
minutes
1
<--- multiplication on left ---> <- ----
conversions on right -- >
Here the conversion 5 hours + 4 hours = 9 hours = 1 day + 2 hours gives
the +1 carry.
Thus the result is 89 days (7 hours per day), 2 hours and 50 minute as
before.
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A more compact format
Days Hrs Min Days Hrs Min
8 6 35
× 10
350
= 5 50
60 = 8 4
80 =
80 +
80 60 350 = 89 1 50 <===
The result.
Multiplication on Left and Conversions on the Right.
|
Question: How many hours are in their in 89 work-days where each
is 7 hour long.
Remarks.
- Two column formats appear above. The first carries units through
calculations (and so aid comprhension) while the second one place units
at the top of each column, and so requires less writing.
- Above we see that multiplication by a whole number is just compact
notation for repeated addition. Multiplication by fractions will be
consider in webpages on division.
Another Example:
Multiply 10 hours 33 minutes and 40 seconds by 53.
Solution - using the compact format.
Hrs Mins Secs Hrs Mins Secs
10 33 40
× 53
2200
= 36 40
1749 = 29 9
+ 530 = 530
+
530 1815 2200 = 559 45 40 <=== The result.
Multiplication on Left and Conversions on the Right.
Notes: With the aid of a calculator (or by hand)
40 × 53 = 2200 = 36 × 60 + 40
33 × 53 = 1749 = 29 × 60 + 9
Thus 10 hours, 33 minutes and 40 seconds times 53 is 559 hours, 45
minutes and 40 seconds.
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Learning to do and high marks if it comes to easy is often
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