Download Topic 3: Adding and subtracting

Topic 3: Adding and subtracting
Adding and subtracting whole numbers
There are four basic operations in mathematics; addition, subtraction, multiplication and division. When
adding or subtracting numbers, the numbers of the same place value must be lined up before the operation
can be performed. To achieve this, the numbers are placed vertically.
For example:
Add 345, 57 and 224.
Arrange numbers vertically, lining up numbers of the same place value.
Add the numbers in the units column (=16), put the 6 in the units column and the 1
(carry) at the top of the tens column.
Add the numbers in the tens column and the carry(=12), put the 2 in the tens column
and the new carry at the top of the hundreds column.
Add the numbers in the hundreds column and the carry (=6), put the 6 is the
hundreds column, there is no carry.
1 1
345
57
+ 224
626
There are different ways of doing subtraction. The method below is based on borrowing, other methods
work equally well.
For example:
434 take 172.
Arrange numbers vertically, lining up numbers of the same place value.
In the units column, 4-2=2
In the tens column, 3-7 cannot be done, so borrow 1 from the next column and pay
back 10. Now 13-7=6
In the hundreds column, it is now 3-1=2
3 13
434
−1 7 2
262
A harder example:
1024 – 687
Arrange numbers vertically, lining up numbers of the same place value.
In the units column, 4-7 cannot be done, so borrow 1 from the next column and
payback 10. Now 14-7=7.
In the tens column, 1-8 cannot be done, so borrow 1 from the next column, the
hundreds (0). Because borrowing can’t take place, borrow from the next place value,
the thousands, (1) and pay back 10 to the hundreds. Now borrow 1 from the
hundreds to pay back 10 to the tens, which is now 11-8=3
In the hundreds column, it is now 9-6=3
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0
9
10
11
1
14
10 2 4
− 6 8 7
3 3 7
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These examples, performed on a calculator, are below:
345+57+224=626
434p172=262
1024p687=337
Make sure you press the clear key (C), between examples.
When reading questions involving operations, phrases can give cues for selecting the correct operation. In
addition, possible phrases are:
6 add 7
the total of 6 and 7
I have 6 and gain 7
6 added to 7
6 plus 7
7 more than 6
the sum of 6 and 7
increase 6 by 7
13
All of these phrases translate to the mathematical expression 6 + 7 =
In subtraction, possible phrases are:
9 minus 5
9 with a reduction of 5
Discount $9 by $5
9 subtract 5
9 decreased by 5
difference between 9 and 5
5 less than 9
9 less 5
Subtract 5 from 9
All of these phrases translate to the mathematical expression 9 − 5 =
4
For example:
The 125 car parking spaces will increase by 59 when the new car park opens. What is the new number of
parking spaces?
This translates to the mathematical expression 125 + 59 , which can be solved by pen and paper methods
or by calculator.
Calculator
125+59=184
There will be a total of 184 parking spaces available.
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Pen and Paper
Method
1
12 5
+ 59
18 4
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Tom is 184 cm tall and his father is only 176 cm tall, what is the difference between their heights?
This translates to the mathematical expression 184 − 176 , which can be solved by pen and paper methods
or by calculator.
Calculator
184p176=8
There is an 8 cm height difference
Pen and Paper
Method
7
14
18 4
−1 7 6
0 0 8
 Video ‘Adding and Subtracting Whole Numbers’
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Adding and subtracting integers
Integers are all the whole numbers plus the negative numbers, and are represented as
..... − 5, −4, −3, −2, −1,0,1,2,3,4,5,....... . When writing integers, there are two ways of doing this.
For example: negative five could be written as -5 or (-5). You will see the bracketed form when using your
calculator.
Normally positive numbers are written with no sign because the assumption is that they are positive,
however, there are times they are written with a positive sign. So 8, + 8 and (+8) are exactly the same
number.
The expression: positive nine add negative seven could be written in two ways:
these will be solved later.
The expression: negative four is increased by positive one can be written as:
−
+
9 + − 7 or ( +9) + ( −7)
4 + + 1 or ( −4) + ( +1)
The expression: negative six is subtracted from negative four can be written as: − 4 − − 6 or
−
4 − ( −6)
The first step in solving these problems is to replace the double signs between the numbers with a single
sign. The rule here is:
Same signs + + , + ( + ) or − − , − ( − ) are replaced with a + (plus)
−
becomes
4 − −6
−
4+6

with +

Replace − −
Different signs + − , +( − ) or − + , −( + ) are replaced with a – (minus)
+
becomes
9 + −7
+
9−7

with -

Replace + −
Let’s follow these two examples through. So far − 4 − − 6 has been simplified to − 4 + 6 . The solution to
this can be found using a number line. Start at − 4 and move 6 units to the right (adding).
for Subtracting
-10 -9
-8
-7
-6
-5
for Adding
+6
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
Start at -4
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With setting out:
=
−
4 − −6
−
4+6
=2
Using a number line, the second example is:
+
=
9 + −7
-7
+
9−7
=2
-10 -9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8 9 10
Start at 9
-4
-3 -2 -1
Start at -2
0
1
2
3
4
5
6
7
8
Example: − 2 − + 5
−
2 − +5
−
=
-5
2−5
-10 -9
= −7
-7
-8
-6
-5
9
10
Obviously drawing a number line for every example is difficult so from the examples above some
generalisations can be drawn.
Looking at the second step in the solution, the following is noticed:
−
4+6
=2
+
9 −7
=2
−
2−5
= −7
Look at the signs in front of each number
•
If the signs are the same, add the numbers
•
If the signs are different, subtract the numbers
•
The sign of the answer is the sign of the largest number.
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For example: − 8 + − 7
Step 1
Replace the double sign + − with -
−
−
=
Step 2
The signs in front of both numbers
are the same, so add the numbers
(8+7=15)
The largest number has a – sign in
front of it, so the answer has a – sign
in front.
8 + −7
8−7
−
= 15
For example: −15 + + 17
Step 1
Replace the double sign + + with +
−
=
Step 2
The signs in front of both numbers
are different, so subtract the
numbers (17-15=2)
The largest number has a + sign in
front of it, so the answer has a + sign
in front.
15 + + 17
−
15 + 17
= 2
=2
+
When subtracting in step 2 always do: Largest number – Smallest number
For example: 105 + − 217
Step 1
Replace the double sign + − with -
Step 2
The signs in front of both numbers
are different , so subtract the
numbers (217-105=112)
The largest number has a - sign in
front of it, so the answer has a - sign
in front.
105 + − 217
= 105 − 217
= −112
(105 is also + 105 )
For example: − 34 + − 28 − − 57
Step 1
Step 2
Replace the double signs
+ − becomes -, − − becomes +
−
=
34 + − 28 − − 57
−
34 − 28 + 57
Working from left to right, the signs in front of the first = 62 + 57
two numbers are the same, so add the numbers
= −5
(34+28=62). The largest number has a - sign in front of
it, so the answer has a - sign in front ( − 62 ). Now
consider the remaining numbers, the signs in front of
the numbers are different, so subtract (62-57=5). The
largest number has a – sign in front of it, so the answer
has a – sign in front.
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−
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These can be performed on a calculator. Entering a negative sign is achieved by pressing the z key. This
key may appear on other calculators as Y. Calculators work on the principle that positive numbers are
entered without a sign.
Example
−
8 + −7
Calculator sequence of keys
z8+z7=-15
15 + 17
z15+17=2
−
105 + 217
105+z217= -112
−
−
−
34 + 28 − 57 z34+z28pz57=-5
−
+
If many numbers are being added and subtracted, working left to right will obtain an answer. There is
another method that reduces the amount and difficulty of calculations. This method is based on the
analogy of a bank account where the balance of the account is equal to deposits take withdrawals.
Balance = Sum of Deposits – Sum of Withdrawals
Answer = Sum of Positive numbers – Sum of Negative Numbers
For example: −105 − 203 − − 64 + − 81 + 44
Step 1
Step 2
−
Replace the double signs
105 − 203 + 64 − 81 + 44
Think of +numbers as deposits and – Deposits – withdrawals
numbers as withdrawals. The
(64 + 44) − (105 + 203 + 81)
balance of the account is the sum of
= 108 − 389
the deposits – the sum of the
= − 281
withdrawals.
For example: 73 − + 29 − −121 − 43 + − 28
Step 1
Step 2
Replace the double signs
73 − 29 + 121 − 43 − 28
Think of +numbers as deposits and – Deposits – withdrawals
numbers as withdrawals. The
(73 + 121) − (29 + 43 + 28)
balance of the account is the sum of = 194 − 100
the deposits – the sum of the
= 94
withdrawals.
 Video ‘Adding and Subtracting Integers’
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Activity
1.
(a)
(b)
(c)
(d)
Perform the operation indicated by both pen and paper and calculator methods.
Add 4709, 864 and 21 109
Take 672 from 2709
Discount $37 606 by $488
Increase $294 529 by $57 294
2.
Perform the operation indicated by both pen and paper and calculator methods.
−
(b)
46 − 56
3 + −4
(a)
(c)
(e)
(g)
27 − − 36
(d)
411 − + 276
(f)
11 + − 8 − −15 + 12 − + 9
(h)
+
−
−
−
37 + + 16
123 + − 204
−
234 + − 76 − −127
3.
A pilot is flying at an altitude of 3680 metres when she is instructed by the control tower to
increase her altitude by 550m. What is her altitude after the increase?
4.
Write a mathematical expression for this scenario and solve. Represent deposits as positive values
and withdrawals as negative. On Monday, Jim opens a bank account with a deposit of $500, on
Tuesday he withdraws $137, on Wednesday he withdraws $67, on Thursday he deposits $36 and on
Friday he withdraws $247. What is the balance at the end of the week?
5.
Joe is reading his superannuation statements for the past 3 years. In 2007 he had a balance of
$127 456.28, in 2008 a balance of $153 291.74 and in 2009 a balance of $ 151 875.28.
(a)
How much did his superannuation increase from 2007 to 2008?
(b)
How much did his superannuation increase from 2008 to 2009?
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