Download Multiplying and dividing

Multiplication is really just repeated addition, so 3  3  3  3  3  5 lots of 3  5  3 . When
multiplying numbers, knowledge of multiplication tables up to 10x will make calculations much faster.
Firstly, multiplication by a single digit number will be covered.
For example: 317  6
Arrange numbers vertically, lining up numbers of the same place value.
Multiply the units number by 6. 6x7=42, so the 2 will go in the units column and the 4
will be carried over to the tens column.
Multiply the tens number by 6. 6x1=6, then the carry from the units must be added
6+4=10. The 0 will go in the tens column and the 1 is carried over to the hundreds
column.
Multiply the hundreds number by 6. 6x3=18 plus the carry from the tens18+1=19.
The 9 will go in the hundreds column and the 1 must go in the thousands column
1 4
31 7
6
1 902
For example: 2407  5
Arrange numbers vertically, lining up numbers of the same place value.
Multiply the units number by 5. 5x7=35, so the 5 will go in the units column and the 3
will be carried over to the tens column.
Multiply the tens number by 5. 5x0=0, then the carry from the units must be added
0+3=3. The 3will go in the tens column and there is no carry over to the hundreds
column.
Multiply the hundreds number by 5. 5x4=20 plus no carry equals 20. The 0 will go in
the hundreds column and the 2 must go in the thousands column
Multiply the thousands number by 5. 5x2=10 plus the carry from the hundreds must
be added 10+2=12. The 2 will go in the thousands column and the 1 must go in the
ten thousands column
2
3
2407
5
12 0 3 5
Before multiplying by numbers containing 2 or more digits, knowledge of multiplying by multiples of 10
must be covered.
For example: 7  10,100,1000,10 000 etc.
Question
7  10
7  100
7  1000
7  10 000
Mental calculation
7 x1 attach 0
7 x1 attach 00
7 x1 attach 000
7x1 attach 0 000
Answer
70
700
7000
70 000
Extending this idea:
Question
3  30
20  50
70  300
400  20 000
Mental calculation
3x3 attach 0
2x5 attach 00
7x3 attach 000
4x2 attach 000 000
Answer
90
1000
21 000
8 000 000
Video ‘Multiplying by Multiples of 10’
When multiplying by a two digit number, the method used consists of two multiplications. If the question
is 34 x 26, first 34 is multiplied by 6 then 34 is multiplied by 20. We know already that when a number is
multiplied by a multiple of ten, the number will have a 0 in the units place. The method for doing this is
often called Long Multiplication.
Example: 34 x 26
Arrange numbers vertically, lining up numbers of the same
place value.
6 x 4 = 24 put down the 4 and carry the 2
6 x 3 = 18 plus carry of 2 gives 20, put down the 0 and the 2
must
go in the next place.
Before multiplying by 20, go and cross out the carry numbers
to
2
34
2 6
This is the answer to 34 x 6
204
680
This is the answer to 34 x 20
884
This is the answer to 34 x 26
avoid confusion.
Put a 0 in the units column from multiplying by a multiple of
ten..
2x4 = 8 put this in the next place value
2x3=6 put this in the next place value
Add the two part answers
Example: 619 x 45
Arrange numbers vertically, lining up numbers of the
same place value.
5 x 9 = 45 put down the 5 and carry the 4
5 x 1 = 5 plus carry of 4 gives 9, put down the 9
5 x 6 = 30 put down the 0 and the 3 in the next place value
Before multiplying by 40, go and cross out the carry
numbers to avoid confusion. Put a 0 in the units column
from multiplying by a multiple of ten.
4 x 9 = 36 put down the 6 and carry the 3
4 x 1 =4 plus the carry 3 gives 7 put this in the next place
value
4 x 6 = 24 put down the 4 and the 2 in the next place value
Add the two part answers
3
4
6 19
 45
30 95
 2 4 71 6 0
This is the answer to 619 x 40
2 7 855
This is the answer to 619 x 45
These examples, performed on a calculator, are below.
This is the answer to 619 x 5
317O6=1902
2407O5=12035
34O26= 884
619O45=27855
When reading questions involving operations, phrases can give cues for selecting the correct operation. In
multiplication, possible phrases are:
the product of 6 and 7
6 lots of 7
6 times 7
of
6 multiplied by 7
6 groups of 7
These phrases translate to 6  7  42 .
For example:
There were 7 groups of 23 students through the visitor information centre today. How many is this
altogether?
Calculator
Pen and Paper
Method
23O7=161
2
23
7
16 1
There were 161 students through the centre.
Video ‘Multiplying Whole Numbers’
If the signs of the integers are the same, the answer is positive.
If the signs of the integers of are different, the answer is negative.
Example
positive x positive  positive
6  4  24
negative x negative  positive


Same signs

6  4  24


positive x negative  negative
6  4  24
negative x positive  negative

For example:

3813  8

6  4  24
Same signs
Different signs
Different signs
Do the multiplication, ignoring signs ie: 3813 x 8
Now think: a negative x positive  negative (different signs)
The answer is -30504
6 1 2
3 8 13
8
3 0 5 04
This example, performed on a calculator, is below.
z3813O8=-30504
When multiplying more than two numbers, the first step is to multiply the numbers ignoring signs and then
work out the sign of the answer.
For example:

5   4   2  3
Multiply, ignoring signs 5  4  2  3  120
Sign: working left to right      then     then    
The answer is 120
A quicker way to work out the signs is:
An odd number of negatives  negative
An even number of negatives  positive
For example: (1)  6  (2)  (4)  2
Multiply, ignoring signs 1 6  2  4  2  96
There are an odd number of negative signs, overall negative.
The answer is  96
These examples, performed on a calculator, are below.
z5O4Oz2Oz3=-120
z1O6Oz2Oz4O2=-96
Video ‘Multiplying Integers’
Division is best described as a sharing process. Suppose the cost of dinner is $228 and is to be shared
between 4 people, each person would pay $228 ÷ 4 which is $57 each. If 12 people share a lotto win of $1
200 000, each person would receive $1 200 000 ÷ 12 which is $100 000 each.
In this module, answers to division questions will be whole numbers. In the decimal module, remainders
will be dealt with. There are two types of division. In this module the focus is to do short division only.
Traditionally, dividing by a single digit number was answered by short division and dividing by a two digit
number was answered by long division. The thought processes in each are identical, the difference being
that these are part of the setting out in long division.
The question 371 divided by 7 can be written as 371  7 or as 371/ 7 which should be written as 371 .
7
The setting out for this question is:
3 divided by 7 cannot be done, so think 37 divided by 7 = 5 and 2 left
over.
5
7 3 7 21
53
21 divided by 7 = 3
7 3 7 21
The answer is 53.
For example: 6075  5
6 divided by 5 = 1 and 1 left over
1
5 6 10 7 5
10 divided by 5 = 2
12
5 6 10 7 5
7 divided by 5 = 1 and 2 left over
1 21
5 6 10 7 2 5
25 divided by 5 = 5
1 21 5
5 6 10 7 2 5
The answer is 1215.
For example: 90702  3
9 divided by 3 = 3
3
3 90702
0 divided by 3 = 0
30
3 90702
302
7 divided by 3 = 2 and 1 left over
3 9 0 7 10 2
302 3
10 divided by 3 = 3 and 1 left over
3 9 0 7 10 12
302 34
12 divided by 3 = 4
3 9 0 7 10 12
Answer is 30 234
These examples, performed on a calculator, are below:
371P7=53
6075P5=12035
90702P3=30234
For example: 7 000  10,100, etc.
Question
7000  10
7000  100
7000  1000
Extending this idea:
Question
900  30
1000  50
80000  200
24 000 000
Mental calculation
Remove a 0
Remove 00
Remove 000
Answer
700
70
7
Mental calculation
First 10 , which gives 90, then 3
First 10 , which gives 100, then 5
First 100 , which gives 800, then 2
First 10 000 , which gives 2400, then 2
Answer
30
20
400
1200
20 000
Video ‘Dividing by Multiples of 10’
When reading questions involving operations, phrases can give cues for selecting the correct operation. In
division, possible phrases are:
40 divided by 5
quotient of 40 and 5
5 divided into 40
per
5 goes into 40
40 shared between 5
For example:
Jayne drove 5 journeys of 34km and 3 journeys of 57km. She used 31 litres of fuel to cover this distance,
how many km per litre did she obtain?
In the first sentence, ‘of’ means multiply and ‘and’ means add.
The first sentence translates to 5  34  3  57 . The two multiplications must be performed first (See Topic
5 – Order of Operations).
Calculator
Pen and Paper Method
34O5=170
2
34
5
17 0
A distance of 170 km was covered.
Calculator
Pen and Paper Method
57O3= 171
2
57
3
17 1
A distance of 171 km was covered.
Calculator
Pen and Paper Method
170+171=341
The total distance travelled was 341 km.
1
17 0
17 1
3 41
The word ‘per’ in the second sentence is a cue for division.
Calculator
Pen and Paper Method
1 1
341P31= 11
The answer is 11. The car covered 11km per litre
Video ‘Dividing Whole Numbers’
31 3 4 3 1
The same rules apply here as for multiplication.
If the signs are the same, the answer is positive.
If the signs are different, the answer is negative.
Example
positive ÷ positive  positive
negative ÷ negative  positive
positive ÷ negative  negative
negative ÷ positive  negative
24  4  6

24   4   6
24   4   6

24  4   6
Same signs
Same signs
Different signs
Different signs
24  4   6 as a
multiplication is 4   6   24 , the sign is correct here, so the division answer sign must also be correct.
Remember, these can always be checked by turning the question into a multiplication.

For example:  945   7
Do the division, ignoring the signs ie:945÷7
1
7 924 5
Now think: a negative ÷ a negative  positive
13
7 9 2 4 35
The answer is 135 or 135
13 5
7 9 2 4 35
For example: 1233   9
Do the division, ignoring the signs ie:1233÷9
1
9 12 3 3 3
Now think: a positive ÷ a negative  negative
13
9 12 3 3 6 3
The answer is 137
These examples, performed on a calculator, are below.
z945Pz7=135
1233Pz9=-137
Video ‘Dividing Integers’
137
9 12 3 3 6 3
1.
(a)
(d)
(g)
Perform these multiple of ten multiplications mentally.
9 x 100
(b)
90 x 10
40 x 50
(e)
220 x 20
5 x 900
(h)
6000 x 50
(c)
(f)
(i)
770 x 1000
410 x 1000
40 x 25
2.
(a)
(b)
(c)
(d)
Perform the operation indicated by both pen and paper and calculator methods.
Find 12 lots of $512
Find the quotient of 50 274 and 7
Find the product of 479 and 7
Share 1686 smarties between 6 people
3.
Perform the operation indicated by both pen and paper and calculator methods.


(b)
(c)
132   5
12  11
121 11
(a)
316   24
(d)
4.

(e)
96  6

(f)
3807   9
Evaluate the following.
(a)

3  4  5  2
(b)

40   5   2
(c)

36  6   2   3
5.
One codeine tablet contains 30mg of active ingredient. How much active ingredient will 12 tablets
contain?
6.
From a roll of material of length 100m, 16 lengths of 6 metres have been cut off and sold. How
much should be left on the roll?
7.
An athlete runs a marathon (42km) in 3hrs 30 mins. What was his average time per km? (Hint:
convert the time to minutes)
8.
John can travel 630 km on a tank of fuel. How many journeys of 35 km can he do?