5, 25, 50 'Trick'

The key to this trick is remembering that all numbers can be expressed in countless ways. When multiplying by 5, (you should know your multiplication tables, but sometimes we multiply by large numbers or numbers with decimal places), think of 5 as 10*(1/2).

For example if multiplying 5 by 365.24, you can divide 365.24 by 2 then multiply by ten by simply moving the decimal place one to the right, or by moving the decimal place first then dividing by 2. This will yield 1826.2

So, to re-express 50 to make multiplying easier, think of it as 100*(1/2). Move the decimal place right two and divide by two, or take half the number you’re multiplying by then multiply by 100. For example, multiplying 50 by 365.24 will yield 18, 262.

Now the number 25 can be re-expressed as 100*(1/4). Multiplying 25 by 365.25 will then yield 9131, which is half of 18, 262.

Multiplying a Fraction by the Numerator

When you have a fraction like (19/20) and multiply it by its denominator (19). The answer is the denominator subtracted from the numerator (usually a negative number) added to the numerator. The numerator of the remainder is difference between the two squared and the denominator is the denominator from the original fraction.
E.x.
(19/20) * 19 = 18(1/20)
19-20 = -1 + 19 = 18
then 19-20=-1, -1^2 = 1, over 20.

Dividing by 9 into a mixed number

1.  First, add all the digits together and divide by 9, keeping in mind the whole number and the remainder.

2.  Write the remainder over 9, this is the fraction part of the answer.  (Make sure the fraction is in simplest form.)

3.  Add all the digits together again, except for the last one.  If there was a whole number from step 1, add this too.  Write the number down, carrying if need be.

4.  Continue to add all the digits together, subtracting off one digit each time, until you are left with the first digit. Write these numbers down, carry if need be.

5.   Write the last number down.  (Remember to add any carried numbers.)

  

Ex   8380 ÷ 9 = __________(mixed number).

a.  Add all the digits together and divide by 9: 8+3+8+0=19. 19 divided by 9 is 2 with 1 remainder.

b.  Write ¹/₉, this is the fraction of the answer.

c.  Add 8+3+8+2 = 21.  Write 1. (remember the 2)

d.  Add 8+3+2 = 13.  Write 3. (remember the 1)

e.  The last number is 8, 8+1=9 so write 9.

f.  The answer is 931 ¹/₉.

Adding a sequence (1+2+3...+n)

1+2+3+4+...+10
An easy was to do this is (n*(n+1))/2 where n is the last number (so 10 in the above example)
so 1+2+3+4+...+10=55 or (10*11)/2
An easy trick (inside the trick) is that one of the multiplied numbers will be even...
Ex. 1+2+3+...+49 is the same as 49*25 or 49*(50/2)

Happy Numbers

A happy number is as follows. Starting with any positive integer, replace that number by the sum of the squares of its digits, and repeat the process until the number equals 1 (where it will stay), or until you determine that it loops endlessly in a cycle which does not include 1. Those numbers for which this process ends in 1 are happy numbers, while those that do not end in 1 are unhappy numbers.

The first 20 happy numbers are:

1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, 97, 100

LOIF

We should be familiar with multiplying two binomials using the FOIL method
(X+4)*(X+5)
First: X*X
Outside: 5*X
Inside: 4*X
Last: 4*5
Answer = X^2 + 9X + 20
In Number Sense, we can do the same when multiplying two 2-digit numbers using a similar method, LOIF.
*Example*
43x27
Last: Multiply the last 2 digits and write down a single digit in the one's place(Carry if necessary)
          3x7 = 21
     Write down the 1 and remember the 2
Outside: Multiply the outside digits but don't write anything else down yet.
          4x7 = 28
     Remember the 28
Inside: Multiply the inside digits and then add to what you got from the outside digits (add any numbers carried from the last digits). Write down a single digit in the ten's place and carry if necessary.
          3x2 = 6
          6+28 = 34
          34+2 = 36
     Write down the 6, carry the 3  
 First: Multiply the first digits and then add any numbers that were carried from the previous step. Write down the number in the hundred's place.
          4x2 = 8
          8+3 = 11
     Write down the 11
Answer = 1161

The 11 Trick!

The 11 Trick is one of the easiest tricks out there, but it can be INCREDIBLY useful on the number sense test. When multiplying any number greater than 10 by 11, there's a very easy process to do that gets you the right answer with practically no effort. For example:

65 x 11

The number in the units place of the answer will be 5. For the middle number, add 6 and 5 together, to get 11. Place a 1 in the tens place of the answer and carry the one over to the 6, which becomes a 7, for the hundreds place.

To make it more simple, when multiplying 11 by the number A'B, you will end up with answer A'A+B'B. This works with longer numbers as well - simply continue adding the middle two numbers, starting from the right, and carry over the remainders. Here are some other examples:

43 x 11 = 473
89 x 11 = 979
129 x 11 = 1419
462 x 11 = 5082

This might help in giving you a clearer picture on how the questions on the test are spread out. I personally would only focus on 1-20 until you master it, then move on to the rest.
http://www.uiltexas.org/files/academics/number-sense-sequencing.pdf

Roman Numerals

First off what are Roman Numerals and why are they still relevant?
  Roman Numerals were the symbols that utilized letters from the Roman alphabet, collectively known as an ancient numerical system used in Ancient Rome. Today we use Roman Numerals for copyright credits at the end of films, organization of outlines (i) (ii) (iii) (iv), numbering Super Bowls, names of Monarchs and royalty.

What are our Roman Numerals?
  I=1
  V=5
  X=10
  L=50
  C=100
  D=500
  M=1,000
 

 The following were introduced later to represent larger, more complex numbers. They should be represented with a Bar over their tops. These however are highly unlikely to be on any Number Sense tests.

*V=5,000
*X=10,000
*L=50,000
*C=100,000
*D=500,000
*M=1,000,000