Operations with Decimals

Addition and Subtraction

- Since many decimal numbers can be written as using fractions, we add them and subtract them in a manner similar to adding and subtracting fractions: add/subtract "like" fractions (same denominators and same decimal places)
Example: 
  24.076
+  3.19
=27.266

Subtraction with decimal squares, one square is the unit.

Pencil and paper algorithm is the traditional way of doing addition and subtraction
  2.4      5.6
+3.2    -3.4
=5.6   =2.2

Multiplication
Whole number x decimal
3 x 0.2
= 0.2 + 0.2 + 0.2
= 0.6

Area Model:

Pencil and paper algorithm:

Powers of 10:
Example:
0.3 (or 3/10) x 10^2 = 30

Example:
4.012 x 10^7
[4(1)+0(10^-1)+1(10^-2)+2(10^-3)] 10^7

Example:
28.7 x 10^-3
287/100 x 1/1000 = 287/10000 = 0.0287

Division:
Measurement/Subtractive
.9 / .3 = 3
---I---I---   

Sharing/Partitive
.9 / 3= 3
(---) (---) (---)
This shows that 3 friends are each given 3 peices of candy

Paper and pencil algorithm:

Division and converting decimals to fractions

Division powers of ten

Example: 4.07 / 10^2 = 0.0407
               Check by doing this: 4.07 x 10^-2 [4(1) + 0(10^1) + 7(10^-2)] (10^-2)
Example: 78.392 / 10^7 = 0.0000078392
               A positive exponent moves the decimal to the left that many places, and moves the decimal to the right if the exponent is negative.

Converting Decimals to Fractions

Helpful Hints
-  0.11111... = 1/9
-  0.01010... = 1/99
-  0.001001... = 1/999

A number over the fraction of 9 is repeating.
Examples:
2/9 = 0.2222...
3/9 = 1/3 = 0.3333...
4/9 = 0.4444...
5/9 = 0.5555...
6/9 = 2/3 = 0.6666...
7/9 = 0.7777...
8/9 0.8888...
25/99 = 0.25252525

To write a repeating decimal into a fraction, there are two ways.

One way is:

Another way is:

More ways of converting decimal examples are:
0.47 = 47/100
0.028 = 28/1000
0.777... = 7/9
0.2121... = 21/99