1.4 Multiplying Whole Numbers
Multiplication
Other terms for multiplication :
times, multiplied by, of.
Multiplication Properties of 0 and 1:
a ٠ 0 = 0 Example: 4 ٠ 0 = 0 + 0 + 0 + 0 = 0
a ٠ 1 = a Example: 4 ٠ 1 = 1 + 1 + 1 + 1 = 4
Multiplication by Multiples of 10
When one or more factors is a multiple of 10, such as 20, 80, or 300, you may
temporarily ignore such ending zeros and include them together in the final
product.
30·90 Count the ending zeros…two. Calculate the product of
the
leading digits : 3·9 = 27. Attach the two zeros and the final
answer is 2700.
5,010·200 There are 3 ending zeroes, the zero between 5
and 1
doesn’t count because it’s in the middle. 501·2 = 1002, attach
the three zeros and the final answer is 1002000 = 1,002,000
Commutative Property: a ٠ b = b ٠ a
5 ٠ 4 = 4 ٠ 5
Associative Property: (a ٠ b) ٠ c = a ٠ (b ٠ c)
21 ٠ 6 = 21 ٠ (3 ٠ 2)= (21 ٠ 3) ٠ 2 = 63 ٠ 2 = 126
IMPORTANT!! The commutative and associative properties are
only in effect when an expression is either ALL addition or
ALL multiplication .
Distributive Property :
a ٠ (b+ c) = a ٠ b + a ٠ c
2 ٠ (4+ 5) = 2 ٠ 4 + 2 ٠ 5
= 8 + 10
= 18
Relevance: If you don’t know a product through mental
math, you can
break up one of the factors into a sum that will allow you to do
mental math .
Example 6 p.43
a) Multiply 6 ٠ 14
If you don’t now what 6 times 14 is , you can break up 14 into a sum of
10 and 4, then use the distributive property.
6 ٠ 14 =6 ٠ (10+ 4) = 6 ٠ 10 + 6 ٠ 4 = 60 + 24 = 84
b) Multiply 7 ٠ 59
Example: Mileage
Specifications for a Ford Explorer 4x4 are shown in the table below. How far can
it
travel on a tank of gas?
| |
Engine |
4.0 L V6 |
|
| |
Fuel Capacity |
21 gal ← |
Number
of gallons
in 1 tank |
mpg means
Miles per Gal
= miles/gal |
→ Fuel economy
(mpg) |
15 city/ 19 hwy |
|
We are being asked to find “how far”, which is a distance.
Distance is measured in
miles. We will assume this distance is traveled in the city so we’ll use city
mileage
(15 mpg).
Dimensional analysis is used to convert units . We are
given miles per gal and gallons
per tank. We want to know how many miles can be traveled on 1 tank.
Note: if you don’t know what 15 x 21 is doing mental math,
you can do it using the
distributive property:
15 x 21 = 15(20+1) = 15(20) + 15(1) = 300+15=315
Example: Calculating production
The labor force of an electronics firm works two 8-hour shifts each day and
manufactures 53 TV sets each hour. Find how many sets will be manufactured in 5
days.
We are looking for the number of TV sets manufactured in 5
days.
Use Dimensional Analysis. We are given TV sets manufactured per hour, and hours
worked per shift, and shifts worked per day. If you included all the pertinent
information, you should have cancelled out all the unnecessary units ( like units
on top
cancel out like units on the bottom), and the units left should be “TV sets”,
which is
what we want.
Rectangular Patterns
If you have a rectangular pattern objects, such as rows
and
columns, you can determine the total number of objects by
multiplying the number of rows times the number of columns (or
objects per row).
Example: Our classroom has 7 rows with 5 seats in each
row.
Therefore our classroom can hold 7x5 = 35 people.
Area: The area of a rectangle is length X width= lw
The is the amount of area covered within the rectangle.
Note: Perimeter is the distance around the rectangle.
If length is in feet, and width is in feet,
Area = length X width means the units of the Area are feet2, or
“square feet”
p.49 #51
The parking lot at the Hillside Christmas Craft Fair has 14 rows of
parking with 12 parking spaces in each row. How many cars can
fit in the parking lot?
All these operation symbols mean the same thing: “10
divided by 2 is 5”
Division Properties:
Division is the “inverse” of multiplication. That is, does the opposite of what
multiplying does.
a(b) = c
|
If a times b is c, then c/b answers the question,
“c times what equals b?” |
| a = c/b |
and c/a answers the question, |
| b = c/a |
“c times what equals a?” |
Division with 0
If a represents any nonzero number, 0/a = 0
If a represents any nonzero number, a/0 is undefined.
0/0 is undetermined.
More Division Properties
a/1 = a
a/a = 1 (provided that a ≠ 0) |
Checking Division:
Quotient X Divisor + Remainder = Dividend
Translation: The answer to the division problem times the number you divided
by, plus
the remainder, should equal the number you divided into.
What about 16/3?
 |
Check:
Quotient X Divisor + Remainder = Dividend
5 x 3 + 1 = 15+1=16 yes |
Example
A total of 216 girls tried out for a city volleyball
program. How many girls
should be put on the team roster if the following requirements must be
met?
1)All the teams must have the same number of players.
(find a number that goes exactly into 216, so there is no remainder)
2) A reasonable number of players on a team is 7 to 10
(divide 216 players by
7 players per team, then 8, then 9, then 10). But don’t bother with 10
because we know 10 doesn’t go exactly into 216.
3) There must be an even number of teams. (The quotient
must be EVEN).
