Basic  Refresher

Place Value

T H U t h th

 

We use a decimal  number system – it has a base of 10.

The decimal point is always next to the units part of the number and only needs to be shown if there is a decimal part, or if the number is represented to a given number of decimal places.

Every time a digit moves to the left, it increases in size by  a factor of 10.

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102 = One hundred, no tens, two units.
                This is read one hundred and two point .

3300 = three thousands, three hundreds, no tens, no units.
              This is read three thousand, three hundred.    
              Zeroes are put in as place holders.

 

Every time a digit moves to the right, it idecreases in size by  a factor of 10.

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102.35 = One hundred, no tens, two units, 3 tenths, 5 hundredths.
                This is read one hundred and two point three five.

3020.020 = three thousands, no hundreds, two tens, no units, no tenths and two hundreths.
This is read three thousand and twenty point zero two.    

Note: 3020.020 = 3020.02 = 3020.0200000

The zeroes at the end , after the decimal point, do not matter.


x 10   x100  x1000

 

Examples

102.35 x 10 = 1023.5           10352.3577x 10 =103523.577      30 x 10 = 300   50230 x 10 = 502300

 

e.g    102.35 x 100 = 10235          10352.3577x 100 =1035235.77      

  30 x 100 = 3000         50230 x 100 = 5023000

 

e.g    102.35 x 1000 = 102350          10352.3577x 1000 =10352357.7      30 x 1000 = 30000      50230 x 1000 = 50230000


÷  10   ÷ 100  ÷ 1000

e.g  102.35 ÷ 10 = 10.235          10352.3577 ÷ 10  =1035.23577    

       30 ÷ 10  = 3                        50230÷ 10  = 5023

e.g    102.35 ÷ 100 = 1.0235          10352.3577 ÷ 100  =103.523577   

           30 ÷ 100  = 0.3                        50230÷ 100  = 502.3

e.g    102.35 ÷ 1000 =0.10235          10352.3577 ÷ 1000  =10.3523577   

           30 ÷ 1000  = 0.03                        50230÷ 1000  = 50.23


+ Addition

When we add numbers, we are adding up the totals in each of the units, tens, hundreds etc columns.
Since base 10 only allows the single digits 0 to 9, any extras are carried forward to the next column.

Examples

113 +65 =178  224 +102=326  7+6=13 27+6=313=33

To add numbers together, it is easier to write them out underneath each other.
Start from the units column.  Make sure that the numbers stay in line !

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- Subtraction

            
To subtract two numbers,

To check your answer mentally  add it to the bottom number.
 You should then find the top number.

Examples

Calculate a) 178 – 65             b) 326 -102

8 9

  check 113 +65 =178 5    224 +102=326 5

 

Order is important.

123-89 is not the same as 89-123 !!!

Problems often occur when the bottom digits are bigger than the top digits.
The temptation then is to switch to bottom take top.
This is wrong.

Example

The solution to the sum "Calculate 123 -89"

is often given as

10

This is clearly incorrect, since adding the  answer to the bottom number gives
166 +89 =255   

There are many methods available to find the correct solution.

Decomposition

 

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check 34 +89 =123    5

 

Subtraction by addition

 

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Equivalence

 As long as the same thing is done to both numbers, the original sum can be changed into an easier one.

123 – 89   is the same as 124 – 90, since 1 has been added to both numbers

124 -90 = 34 so 123 – 89 =34

 

Borrow and payback

 

 This is based on a combination of equivalence
and decomposition.

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X Multiplication

 

Twenty times table

10

12

18 x 17 = 306

5 x 30 = 5 x (3 x 10) = (5 x 3) x 10 = 15 x10 = 150

 

Then

 

 Examples

115

 

Decimal Multiplication

To multiply decimals,  write them down underneath each other.

Then 
   *   Ignore the decimal point  .

      Starting  at right hand end of  answer, count that number of digits to the left.
        Place decimal point

Example
 

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Egyptian multiplication

With this method, you only need to be able to double numbers and add.

Example
Calculate 17 x 27

First, set up a table

To complete the table,  double each entry

1

This table will now allow us to find our answer by adding the relevant entries together.

27 = 16 +8 +2 +1
so adding these entries will give

17 x 27 = 272 +136+34 +17 = 459

 

Example
Calculate 45 x 36

 

2

45 x 36 = 1440 +180 =1620

Grid Multiplication

This method is often used in primary school.
It basically breaks the numbers down into a grid, then the individual components are multiplied and finally added together.

Example 45 x 36

2

1

 

Gelosia Multiplication

This is a method from the middle ages.
Draw a grid, complete with diagonals.
Multiply the individual components, putting any tens above the diagonal.
Add up each diagonal
Read off the answer.

 

Example 45 x 36

12 Answer 45 x 36 = 1620

4 x 3 = 12, so write 1 above the diagonal and 2 below.

Complete for the rest.
Add up each diagonal.
Don’t forget to carry if necessary!

Sometimes this method is written the other way:

45

 


÷ Division

 

Short Division

 

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    Tip : First write out the ten times table  for the number you are dividing by, or use the Egyptian multiplication method shown above.

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Dividing  a decimal

Divide as normal. Instead of writing down a remainder, put a decimal point in the answer ( in line with the question) and put a zero after your remainder.
 Then keep on going  until you have no remainders left, or have reached a suitable point.

e.g. For an answer to 3 decimal places, work it out to 4 decimal places first.

 

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Dividing  by a decimal

164

 

166

 

168 169

 

 Long Division

 

174

    

© Alexander Forrest