N2-G123 Map Scale

Map Scale (G1-S3)

Scale of Drawing = Drawing length : Actual length

          Map Scale = Map distance : Actual Distance

If the scale of a map is given as 1 : 200 000

=> 1cm represents 200 000 cm    (200 000 cm = 2000m = 2km)

=> 1cm represents 2 km

1cm : 2km

1cm x 1cm = 1cm²: 2km x 2 km = 4km²

1cm²: 4 km²  

* Area calculation: (first calculate 1cm² represent how many km²)

Example

The scale of a map is given as 1 : 200 000             

  

Scale representation and conversion 

a. Complete the following

                        Scale of map is 1cm to       km

Step 1: Write the conversion 

               1km = 100 x 1000 

                        = 100 000 cm      

Step 2: Find the conversion scale

            Divide given scale by conversion

           200 000 / 1001000 = 2km

Step 3: Solve

           Scale of map is 1cm to   2   km

To Find Distance on the map ( Given actual distance)

b.     Find the distance on the map between two towns that are actually 30 km apart.

Step1: Write the scale

              1 cm : 2 km   [[  MUST write this scale FIRST when doing ALL questions  ]]

Step2: ‘line’ the number and compute

              1cm :  2km

                 ?   :  30km

              30 km apart is 30/2 = 15 cm on map

              Distance on the map is 15 cm. 

To find Actual Distance (Given distance on the map)

c. Find the actual distance of two towns are that 10 cm apart on the map.

            Working:

                1 cm : 2 km

              10 cm = 10 x 2 = 20

                 Actual distance = 20 km

To Find Actual Area (Given area of map)

d.  A forest has an area of 5 cm² on the map. Find the actual area of the forest in square kilometres.

Step 1: Write the scale and compute to area mapping

              1 cm : 2 km   

             1cm² => 2km x 2km = 4 km²

Step 2: ‘line’ the number and compute (multiply to find actual value)

              1cm² :  4 km²

               5 cm² :    ?

              5cm² => 4 x 5 = 20

        Actual area of forest = 20 km²

To find Area on Map (Given actual area)

e.  The actual area of a town is 120 km2. What is the area on the map?

Step 1: Write the scale and compute to area mapping

              1 cm : 2 km  

             1cm² => 2km x 2km = 4 km²

Step 2: ‘line’ the number and compute (divide to find map value)

              1cm² :  4 km²

                    ? : 120 km²            

         120 km² => 120 / 4 = 30cm²

Area of town on map is 30cm²        

Practice

1.  Write 480 : 400 in its simplest form.

2.  The scale of the map in given as 1cm : 2 km

a.   Scale of the map is 1 : ___________ 

b.   The distance between two towns is 30km. What is this distance, in cm on the map?

c.   A forest has an area of 8cm² on the map. What is the actual area of the forest in km².   

3a.  Write 50 000 cm in kilometres.

       A map is drawn to a scale of 1 : 50 000

b.   The distance between two stations on the map is 12cm

      Find the real distance between the stations in kilometres

c.   The distance between two towns is 35 km. How far is this distance on the map?

S3T3 Index Notation, Standard Form

Index/Power

The small, raised number next to a normal letter or number, to be multiplied by itself

Example

b²= b x b

4³ = 4 x 4 x 4

 

Square power 2 : ²

- multiplying by itself

Example

(1)        Square of 4 = 4² = 4 x 4 = 16                    

(2)        3² = 3 x 3 = 9

(3)        Square of (-5)² = (-5) x (-5) = 25

Perfect Square are the squares of whole numbers: 

             (2 x 2) 4, (3 x 3) 9, (4 x 4) 16, …

Square Root - the Opposite of Square

            A number that when multiplied by itself, gives the number

                                    Symbol: √ 

       

5 -->         square 5²       --> 25

                        5 <--    Square root √25  <-- 25

Example

(1) 16 = 4 x 4

     √16 = 4

(2) a² = 25        

      a = + √5 x 5 =    = +5             

Why +5 when a number is square-root?

   (-5) x (-5) = 25 ( -ve x -ve = + ve)

    5 x 5 = 25

  => 25 = (-5)² = 5² = 25

       a2 =  + √a x a = -a or a

Cube - Power of 3 : ³

- Multiplying the number by 3 times

Example:

(1)        Cube of 3 = 33 = 3 x 3 x 3 = 27                                

(2)        4³ = 4 x 4 x 4 = 64

(3)        cube of (-5) ³ = (-5) x (-5) x (-5) = = -125                             

Cube Root

            A value that when ‘cubed’ gives the original number

                        Symbol: ³√

4 -->        cube 4³         --> 64

                4 <--  cube root ³√64    <-- 64

Example: 

                  3√8 = 3√2 x 2 x 2 = 2

              ³√216 = ³√6 x 6 x 6 = 6

Index Notation

• Representing number/letters that multiplied themselves a number of time

Example 

Write 18 in index notation

Using LCM : 

                     2   |  18  (smallest divisible no=2, 18/2=9 , place 9 below)

                     3   |    9  (next smallest no to divide:3)

                     3   |    3  (divide by 3, till = 1)

                              1

 The index notation of 18 = 2 x 3 x 3     

                                        = 2 x 3²

Example

Write 40 in index notation

Using LCM : 

                     2   |  40  (smallest divisible no=2, 40/2=20 , place 20 below)

                     2   |  20  (next smallest no to divide:2)

                     2   |  10  (next smallest = 2)

                     5   |  5     (next smallest = 5, till = 1)

                             1

 The index notation of 40 = 2 x 2 x 2 x 5    

                                        = 2³ x 5

STANDARD FORM  (S3/NA/T)                                                                            

            K x 10^m   where K is between 1 to 9      1 <= K < 10

Example: 

Change 123.5 to standard form

Step 1: Change number to 1 <= k < 10                         

            123.5 = 1.235 x 100 

Step 2: Change to 10ⁿ 

            100 = 10²

Step 3: Answer in standard form

            123.5 = 1.235 x 10²

Example: Change 0.0012 to standard form

                       

            0.0012 = 1.2 x 0.001 (Step 1: Change number to 1 <= k < 10)

   

               0.001 = 10^-3       (Step 2: Change to 10ⁿ )

         0.0012 = 1.2 x 10^-3 (Step 3: Answer in standard form)

Practice

1. Calculate (6.2 x 10³) x (1.5 x 10⁶). Give your answer in standard form.

2.   Find the value of 

      (a) 8³

      (b) 5^-2

3.  Write 0.0000567 in standard form.

4.   Calculate 6.1 x 10⁶ + 1.4 x 10⁷. Give your answer in standard form.

5.   Find the value of 2⁴ + 5²

6.   Write the number 315.17

     (a)  in standard form

     (b)  correct to one decimal place

     (c)  correct to the nearest 10.

7. Write 4.67 x 10⁶ as an ordinary number

8. Calculate (6.2 x 10³) x (1.5 x 10⁶). Give your answer in standard form.

9.  Write the number 3.6148 correct to

      a.  4 significant figures

      b.  2 decimal places

N1-G23 Numbers, HCF, LCM, power of

Number definition

Integers

Whole numbers including negative whole number

Example: …,-4, -3, -2, -1, 0, 1, 2, 3, 4,…

A positive number is any number greater than zero.

Example: 1, 2, ½, 6.4

Negative Numbers

We read -1 as negative one.

A negative number is any number less than zero.

Example: -1, -2.5, -4/7

More Numbers Defintions                                                                                   

Natural numbers are whole numbers equal of greater than 0.

Example:        1 ,2 ,3, 4,… 

Rational numbers: a/b where a and b are integers and b is not 0, and terminating 

Example: 2/3, 6/7, -4/5, 1.2, -10.8, 200, -430

Irrational numbers: Real numbers that cannot be written as a/b

Example: π, √2

Real numbers: The limit of convergent sequence of rational numbers; rational and irrational numbers.

Prime Number

Greater than 1 that cannot be formed by multiplying 2 smaller natural number.

Example: 2, 3, 5, 7, …

2 = 1 x 2, 13 = 1 x 13

* 1 is not a prime number because prime numbers are greater than 1

* All even numbers are not prime number except 2.

Example

List all the prime number that are greater than 10 and less than 20

Step 1: List all the numbers 

 11, 12 , 13 , 14, 15, 16, 17, 18, 19

Step 2: Strike out the non-prime number

            11, 12 , 13 , 14, 15, 16, 17, 18, 19

Step 3: List out the prime number

  11, 13, 17, 19

Lowest Common Factor

The smallest common multiples of two or more numbers

Example

   Find the LCM of 8 and 20

                        2  | 8  20

                        2  | 4  10  

                              2   5

            LCM is 2 x 2 x 2 x 5 = 40

Highest Common Factor (for NA/O)

Largest factor that divides 2 numbers without remainder.

Example 

Find the HCF of 12 and 32 

                 2  |   12    32   (smallest common divisible no=2 for 12,32, place result below)

                 2  |     6    16   (  " " " " = 2 for  6, 16, place result below)

                           3      8   (no more common divisible number)

            HCF = 2 x 2 = 4

Example

Find the HCF of 24 and 30 (Multiplying all factors that appear in both list)

            24 = 2 x 2 x 2 x 2 x 3,          30 = 2 x 3 x 5

            HCF is 2 x 3 = 6

Lowest Common Factor
The smallest common multiples of two or more numbers
Example
Find the LCM of 8 and 20
   2 | 8    20
   2 | 4    10
         2   5

LCM is 2 x 2 x 2 x 5 = 40

Prime factorisation (for NA/O)

Numbers in the form of product of prime numbers

Example:         6 = 2 x 3, 

                      18 = 2 x 3 x 3

 Index/Power

The small, raised number next to a normal letter or number, to be multiplied by itself

Example

b²= b x b

4³ = 4 x 4 x 4

 

Square power 2 : ²

- multiplying by itself

Example

(1)        Square of 4 = 4² = 4 x 4 = 16                    

(2)        3² = 3 x 3 = 9

(3)        Square of (-5)² = (-5) x (-5) = 25

Perfect Square are the squares of whole numbers: 

             (2 x 2) 4, (3 x 3) 9, (4 x 4) 16, …

Square Root - the Opposite of Square

            A number that when multiplied by itself, gives the number

                                    Symbol: √ 

        

5 -->         square 5²       --> 25

                        5 <--    Square root √25  <-- 25

Example

(1) 16 = 4 x 4

     √16 = 4

(2) a² = 25        

      a = + √5 x 5 =    = +5             

Why +5 when a number is square-root?

   (-5) x (-5) = 25 ( -ve x -ve = + ve)

    5 x 5 = 25

  => 25 = (-5)² = 5² = 25

       a2 =  + √a x a = -a or a

Cube - Power of 3 : ³

- Multiplying the number by 3 times

Example:

(1)        Cube of 3 = 33 = 3 x 3 x 3 = 27                                

(2)        4³ = 4 x 4 x 4 = 64

(3)        cube of (-5) ³ = (-5) x (-5) x (-5) = = -125                            

Cube Root

            A value that when ‘cubed’ gives the original number

                        Symbol: ³√

4 -->        cube 4³         --> 64

                4 <--  cube root ³√64    <-- 64

Example: 

                  3√8 = 3√2 x 2 x 2 = 2

             ³√216 = ³√6 x 6 x 6 = 6

Practice

1. Write all the factors of the following numbers.

(a) 120 (b) 324 (c) 112

2. Find the prime factorization of the following numbers

(a) 180 (b) 1560 (c) 214

3. Write the prime factorization of 180 x 214

4. Find the HCF of the following pairs of numbers

(a) 114 and 270 (b) 45 and 135

5. Find the LCM of the following numbers.

(a) 55 and 88 (b) 420 and 245

6. The LCM of x and 22 x 33 x 7 is 22 x 34 x 5 x 72 x 11. Find the smallest possible

value of x.