N1-G123 Numbers and Orders of Operations

Recap

1.  Positive number are numbers greater than 0 

       Example: 1, 50, 1238

2.  Negative numbers are less than 0

       Example: -1 , -85, -454

3.  Zero 0 is neither positive or negative

The four number operations are 

    Symbol.         Some Words/phrases

        +            add, sum, plus, total, more than, increased by

        -       subtract, difference, minus,less than, decreased

        x           multiply, product, times 

      /, ÷                divide, quotient

Also,

1.  When there is no sign ’attach’ to a number => the number is positive   

                        2 + 3 = 5; 

            2, 3 and 5 are all positive numbers

2.  The sign + , - ‘belongs’ to the number on the right

         -2 (negative 2)

3.  The sign + , - is also add and minus

* NEGATIVE number is different from – MINUS OR SUBTRACT 

           5 + (-3) = 2   is positive 5 add negative 3 

               5 – 3 = 2   is positive 5 minus positive 3   

             5 + (-3) = 5 - 3 

<< Know the difference of the '-' Negative number and Minus Subtract >>

         5 - 3 = 5 + (-3) = -3 + 5 = 2

Addition                                                        

 (1) (+) + (+)              a + b = b + a                                                                                          

                          8 + 5 = 5 + 8 = 13                                         

            

(2) (+) + (-)                a + (-b) = a – b

                          8 + (-5) = 8 – 5 = 3

(3) (-) + (+)                -a + b =  b – a

                                  -8 + 5 = 5 – 8 = -3

(4) (-) + (-)                 -a + (-b) = -a – b                                                                                  

                                  -8 + (-5)= -8 – 5 = -13                                              

Subtraction

(1) (+) – (+)                a – (+b) = a – b                                                                                 

                          8 – (+5) = 8 – 5 = 3                                      

(2) (+) – (-)                 a – (-b) = a + b

                           8 – (-5) = 8 + 5 = 13

(3) (-) – (+)                 -a – (+b) = -a - b                                                                                  

                                       -8 – 5 = -13             [ -(8 + 5) = -13]                                  

(4) (-) – (-)                 -a – (-b) = -a + b = b – a

                                  -8 – (-5) = -8 + 5 = -3   [-5 + 8 = -3]

Multiplication

(1) (+) x (+)                a x b = ab                                                       

                                   6 x 5 = 30                                                      

(2) (+) x (-)                 +a x -b  = -ab

                                      6 x -5 = -30  

(3) (-) x (+)                 -a x +b = -ab       

                                     -6 x 5 = -30

(4) (-) x (-)                  -a x -b  = +ab                                                              

                                     -6 x -5 = 30                                                                                       

Division

(1) (+) x (+)                a ÷ +b = a/b                                                  

                                     6 ÷ 2 = 3                                                     

(2) (+) x (-)                 +a ÷ -b  = -a/6

                                    -6 ÷ -2 = -6/2 = 3

(3) (-) x (+)                 -a ÷ b = -a/b  

                                    6 ÷ -2 = -6/2 = -3

(4) (-) x (-)                    -a ÷ -b  = ab                                                   

                                     -6 ÷ -2 = 6/2 = -3                                                    

Explanation for positive and negative numbers multiplications

                   a x -b = -ab 

               5 x -2 = -2 + (-2) + (-2) + (-2) + (-2)

                             = -10

    => + x - = -

Negative number and negative number multiplications

-5(0) = 0

             -5 [ 3 + (-3) ]    = 0

             -5 x 3 + -5 x -3 = 0

             -15 + (-5)x(-3)  = 0

       (-5) x (-3) must be equal to 15 for -15 + (-5) x (-3) = 0

           => -5 x -3 = 15

           => - x - = +

Order Of Operations

  First Order : (  )  ^e      Bracket  order/power Exponential             

  Second Order : ÷ X    Multiplication And Division     

  Third Order : +   -       Addition And Subtraction       

Do the question by: (1) completing all FIRST order, 

                                (2) follow by all SECOND order,

                                (3) and then the THIRD order. 

Step 1: Underline by the ORDER 

Step 2: Do calculation using order

Step 3: Repeat step 1-2 until question is solved. 

** Order within bracket -> must follow order as well

Example

Find the value of:

            9 + 3 x 4 – (3 + 5) ÷ 2

          9 + 3 x 4 – (3 + 5) ÷ 2        (Step 1 : Underline by Order)

       = 9 + 3 x 4 – (8) ÷ 2                (Step 2 : Do calculation) 

       = 9 + 3 x 4 –   8 ÷ 2                 (Step 3 : <<do ÷ and x>, repeat step 1-2> )

       = 9 +12   –    4                         (Step 3 :  <do + and - > )

       = 21 – 4 = 17          

Example:

Calculate       24 ÷ 4 x 5 ÷ 6 

[For computation with only the same order, working from left to right]

  24 ÷ 4 x 5 ÷ 6         [ all are 2^nd order - Do from left to right ]

                =    6 24 x 5              (Step 2)

                          4     6              [ Write ÷ as a denominator]

                        = 5

Example:

Calculate       25 + 144 – 2 x 5 x 12                      [15/1/5a/1]

                            25 + 144 – 2 x 5 x 12   (Step 1)

                         = 25 + 144 – 120     (Step 2)

                         = 25 + 144 – 120              (Step 3)

                         = 169 – 120                      (Step 3)

                         = 49

Practice

1.    55 – (-4) – 26 = 

2.    35 – (-24) + (-43) =

3.    24 ÷ 8 x 7 = 

4.    5 x 3 ÷ 8 x 12 =

5.    -5 x 3 x 2 =

6.    5 x -3 ÷ 8 x (-12) =

7.  Find the following without using the calculator.

a.         [-44 – (-23)] x (8 -3)]                             

b.         300 ÷ 4 + [(-3)² + (-24)] 

c.         - [365 + (-217)] x (3 – 7)} + 30              

d.         -64 + 81 + [(-1)100 – (-3)]

e.         -3 – 9 x -3 + 20 x (36 + 2)      

f.          -2 x (-3) – 3 x 5 x (-2)

S0T1 Decimals and Four Operations of Decimals

Decimals - Refresh/Recall

Recurring Decimals

Example           4/9 =0.444…

The ... indicates that 4 repeats indefinitely.

Non-recurring decimals

Example:            ¼ = 0.25

Four Operations of Decimals

Addition and Subtraction

** Align the decimal points (one above another) when adding and subtracting **

Example

Evaluate the following:

            12.2 + 1.43

Step 1: Arrange the numbers, aligning the decimals

            12.2

            + 1.43

Step 2: Do the + /           

            12.20   **Fill 0s to the same position as the other number

            + 1.43

             13.63

Example

Find the value of 28.3 – 1.676 – 12.57

Step 1: Arrange the numbers, aligning the decimals

            28.3

           -   1.676

Step2: Do the + /        

              _{7 12 9}   

            28.30¹0                        **Fill 0s to the same position as the other number

           -   1.67 6

                 ₅ 

            26.6¹24

           - 12.5 70

            14.054      

Multiplication and Division

Example

Find the value of 26.2 x 2.1

Step 1: Arrange the number for multiplication

            26.2

           x  2.1

              262

          + 524

5502

Step 2: Count the decimal place/s in both numbers

            26.2 => 1 decimal place

            2.1 => 1 decimal place

            Total 2 decimal places

Step 3: Place decimal places, counting from the right

            26.2

           x  2.1

              262

          + 524

          55.02

               ^{2 1}

Practice

1.  Arrange the following numbers in ascending order

a.  5.2, 5.204, 5.25

b.  8.91, 8.903, 8.95

2.   Find the value of 

a.   1.23 x 4.5

b.    254.8 + 39.05 - 24.3

c.    2(2.5 + 1.5) - (4.8 - 7)

d.    40.6 ÷ 2.2

S0T1 Four Operations on Fractions

<<Recap/Refresh>>

Equivalent Fractions (Fractions with the same values)

Example

            1 = 2 = 3

            2    4    6

Reducing Fractions to its lowest terms

Example

Reduce 12 / 16 to its lowest term

Step1: Break the numbers to its factors

 15 = 3 x 4

 20 = 4 x 4

Step2: ‘strike’ out the common factors

15 =  3 x 5

20 = 4 x 5

Step3: Answer

15 = 3

20 = 4

Mixed numbers and Improper Fractions

Mixed number is a whole number and a fraction written together.

Eg: 5 ½

Improper Fraction : the numerator is greater than or equal to the denominator

Eg: 8/7    ,     2/2   , 9/4

Converting mixed number and improper fraction

Example

Convert 2  ½ to improper fraction

Step 1 : Multiply the whole number with the fraction's denominator

2 x 2 = 4

Step 2 : Add the number to the numerator

4 + 1 = 5

Step 3 : Write the improper fraction - the added number / the denominator

          5 / 2

Example

Convert 17/5 to mixed number

Step 1 : Divide the improper fraction with remainder

123/ 5 = 3 with remainder 2

Step 2 : Write the proper fraction with the remainder as the numerator

2/5

Step 3 : Write the mixed number 

Comparing and Ordering Fractions

Example

Which fraction is greater, 2/5 or 3/5?

Step 1: Convert denominator to same value. Yes

Step 2: Which has a greater numerator value? 3

Step 3: Answer 

            3/5 > 2/5

Ascending and Descending Order

Example

Arrange the fractions 1/2, 2/5, 3/10 in ascending order.

Step 1: Convert denominator to same value

            Common factor for denominator is 10

            ½ = ½ x 5/5 = 5/10, 2/5 x 2/2 = 4/10, 3/10

Step 2: In ascending order => Arrange numerator value from small to big 

Step 3: Answer 

            3/10, 2/5, ½

(2) Converting Decimals and Fractions

Decimals to Fractions

Example

Convert 0.6 to fractions in the lowest terms

Step: Change to fractional part base on decimal place value, 10^th = /10

0.6 = 6/10

      = 3/5

Fractions to Decimals

Example

Convert 1 12/60 to decimal

Step1: Reduce fraction to lowest term

         1 12₁/60₅

      = 11/5

Step2: Use either long division or convert fraction to 10s base

      = 1 1_x2/5_x2

      = 1 2/10

Step3: Answer in decimal form

      = 1.2

 [** Use the sóD button on calculator to convert between a decimal and a fraction]

Compare Fractions and Decimals

Example

Arrange the following in descending order

       0.6, 4/5, 3/4

Step 1: Convert fractions to decimals (with a calculator)

      0.5, 0.8, 0.75

Step 2: Arrange the numbers using place values position

       0.6

       0.8

       0.75

Step 3: Compare and Answer

[Descending – from big to small]

0.8, 0.75, 0.6

Addition and Subtraction of Fractions

** Always reduce the final answer to the lowest terms (when there are common factors in the numerator and denominators

Example

Evaluate the following.

             2 + 3

             3    4

Step 1: Convert to same denominator. 

            2 _{x 4}=8  + 3_x3=9

            3 _{x 4}=12   4_x3  12

Step 2: Add the numerator

            8 +  9 = 17

            12   12   12      

Step3: Answer and reduce to lowest term

             17/12 = 12/12 + 5/12

                       = 1 5/12

Multiplication of Fractions

            a x c = ac

            b    d    bd

Example

Evaluate    2 x 3

                  5    8    

Step1: Any common factors to reduce

             2₁ x 3

             5     8₄             

Step2: Multiple the numerator and denominator

            = -1x 3             (-ve x +ve = -ve)

                5 x 4

Step3: Answer in lowest term

            = -3/20

To avoid mistake, always change a mixed number to improper fraction, then do the multiplication.

Division of Fractions

When we divide by a fraction, we multiply by the reciprocal of fraction => dividing by 3 = multiply x 1/3

Example

Evaluate -4 ÷ 2

                9    3

Step1: ‘Convert’ ÷ to x and flip fraction to right of ÷

            = -4 x 3

                 9   2

Step2: Reduce to lowest term

            = -4₂ x 3₁

                 9₃    2₁

Step3: Check +/- sign and answer

            ** + x - = -

            = -2/3

<<To avoid mistake, always change a mixed number to improper fraction, then do the multiplication>>

Combined Operations on Fractions

** The same rules of the order of operations apply

For Fractions

Do the numerator and the denominator of a fraction separately first.

                  a + b x c = a + bc                  2 + 3 x 5 = 2 + 15 = 17

                    c x e           ce                        3 x 6        18          18

Example:

Calculate

                  1 + 2  – 5 – 3

                     3        5 x 2

Step 1: Write ORDER and Underline Operation BY ORDER 

              (BEDMAS)

              =  1 + 2  – 5 – 3

                     3        5 x 2

Step 2: Do calculation using operation rules

              =  1 + 2  – 5 – 3

                     3        10

              =  1 + 2  – 2

                     3       10

Step 3: Repeat step 1-2 until question is solved. 

              =   3  –   2

                    3      10 5

              = 1 – 1/5

              = 4/5

Example

Evaluate the following.

            2 - 1/4 ÷ (2 1/3 – 1 5/6)

Step1: Any bracket? First order of operation

               2 – ¼ ÷ (2 1x2 / 3x2 – 1 5/6) [change denominator to same value]

            = 2 – ¼ ÷ (2 2/6 – 1 5/6)

            = 2 – ¼ ÷ (14 /6 – 11/6) [change to improper fraction to - ]

            = 2 – ¼ ÷ (3/6) 

            = 2 – ¼ ÷ ½ [reduce to lowest term]

Step2: Any x ÷ ? Do

            = 2 – 1/4₂ x 2₁ [‘flip’ ÷]

Step3: Any + - ?

            = 2 – ½

            = 1 ½

Practice

1.  Arrange the following fractions in ascending order

a.       1/2,  1/4,  1/5,  1/3

b.       2/5,  5/6,  1/2,  3/4

2.   Arrange the following fractions in descending order

a.      1/10, 3/4, 3/5,  1/2

b.       1/4,  1/6, 1/2,  1/3  

3.   Evaluate the following:

a.   -(-¾)              b. +(-⅓)                c.  ⅚ x (-½)

d.    3/8 ÷ 1/3      e.  (-1/5) x (-10/7)      f.   (1/5) ÷ (-3/10)

4.   6 + (2/5 - 1/10) ÷ (-1/10)

5.   2 + 2/3 x (-1/4 + 1/8)

6.  Calculate 4.53 – 1.18                [12/p2/1b/1]

                        6.13 – 2.39

7.   Calculate 5.63 x 23.8               [09/2/1a/1]

                       12.7 + 3.21