Subtracting Integers With Alike and Different Signs

Subtracting Integers With the Same Signs:

                     When subtracting integers with the same signs you simply ignore their signs, find the difference and put in the sign after subtracting. Or you can use integer chips to solve it. After you take away the chips that you needed to take away, what your left with is your answer.

                                                                   
                                                                     ^                                                          
Examples:                                                  /                                                                 

                                                                  /                                                                 

                                                               /                                                                 

              (+6) - (+2) =       O O O O O O                                                                                                                                          

              6 - 2 = 4                                             

              (+6) - (+2) = (+4)                                  ^

                                                                            /

                                                                          /
              (-9) - (-4) =       O O O O O O O O O  

               9 - 4 = 5

              (-9) - (-4) = (-5)



Subtracting Integers With Different Signs:

                     When subtracting integers with different signs using integer chips you have to create zero pairs in order take away what is needed. (Zero pairs will not affect the number you are subtracting from). If you are not using integer chips you can take the second number and make it its opposite [example: opposite of (-2) is (+2)]. After doing that, you change the subtraction statement to an addition statement, then you add (see Adding Integers by RJ Tabelina).        

Example:

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

       O O O O O O O O    /

                           O O O  /

                                |                                     

                            Zero Pair

         
           (+7) - (-9) =                                                                                                              + (+9) = (+16)   



                            

                                              

Adding Integers

When adding integers that have the same signs you just have to ignore the sign and then add what you have after ignoring the signs then when you get your answers just add the signs.

(+3)+(+7)=(+10)
    |     |
     3 + 7=10

How to Add and Subtract Fractions with Unlike Denominators

When you're adding fractions with an unlike denominator you make a common denominator by multiplying it with each other or if one denominator is the answer to the other denominator you multiply the factor and don't change the other fraction it stays like that until you add. So when you multiply the denominator you multiply the numerator with that multiple. Then you add it but you don't add your denominator because that tells you how many equal pieces make a whole. You only add the numerator. After you put it into lowest terms, if there are any. When you're subtracting its the exact same but you just have to subtract instead of add.

Examples: 

2  x4 +  1 x3
3 x4      4 x3

 8    +    3      =   11
12         12          12


7       +      7 x3   
12             4 x3

7       +       21      =   28
12              12            12

Adding fractions and Subtracting fractions with like denominators

Adding fractions and subtracting fractions with like denominators (bottom number) is easy, because all you have to do is add or subtract the numerators together (top number) . You only add or subtract the numerators because the denominators have the same number, which means they have the same number to make a whole. So you don't worry about adding the denominators. Lastly, if you fraction isn't in lowest terms (simplified) then make it in lowest terms. But how? Divide the fractions with it's biggest common multiple.

Example:

21/30 - 11/30 = 10/30 divided by 10 (biggest common multiple) = 1/3

Adding fractions & subtracting fractions (without lowest terms):

1/4 + 2/4 = 3/4    (1 + 2 = 3 --> then just put the denominator in the fraction.)
                                                                      ^ (3/4)
3/4 - 2/4 = 1/4    (3 - 2 = 1 --> then just put the denominator in the fraction.)
                                                                      ^ (1/4)

Adding And Subtracting Fractions With Like And Unlike Denominators

Addition:

Like Denominators: To add with like denominators, you add the numerators only. 

For example, 
5/12 + 2/12 = 7/12
Notice how I only added 5+2? That's because you do not add denominators. It's the equal pieces the whole is cut into, you do not change the whole.

Try these:
6/10 + 2/10, 3/8 + 1/8, 3/9 + 4/9.

Unlike Denominators: To add with unlike denominators, you find a common denominator.
Once you find one, simply add the numerators of both the fractions to get your answer. 

For example,
4/12 + 1/4 = 7/12
1/4 can be changed into 3/12 by multiplying 1/4 by 3/3. 1/4 x 3/3 = 3/12. 
Re-write the question, 4/12 + 3/12 = 7/12. It's easier now, right? 

Try these:
5/7 + 2/14, 6/10 + 1/5, 2/6 + 1/3. 


Subtraction:

Like Denominators: To subtract fractions with like denominators, all you need to do is subtract the numerators. Then figure out your answer. It's the same with adding fractions with like denominators, except you subtract. 

For example, 
7/12 - 4/12 = 3/12  
Do not subtract denominators. (Because those are the equal pieces your whole is cut into, your whole does not change! As said before!)  

Try these:
14/20 - 8/20, 9/13 - 7/13, 4/5 - 3/5.

Unlike Denominators: To subtract fractions with unlike denominators, you have to find a common denominator. You can do it without finding a common denominator, but that would take more work. (E.g. Diagrams, etc.) 

For example, 
9/12 -  2/6 = 5/12 
You can change 2/6 into 4/12 by multiplying the fractions, 2/6 x 2/2 = 4/12. 
Then, re-write the question and simply subtract the numerators again for your answer. 
9/12 - 4/12 = 5/12 

Try these:
4/10 - 1/5, 7/16 - 2/8, 6/14 - 2/7. 

(Put answers into lowest terms if asked to. This applies to both adding and subtracting.) 

Adding & subtracting fractions(Like & unlike denominators)

Adding Alike: When adding fractions with the same denominators you add the numerators for example if it's 1/6 + 3/6 adding the numerators and keep the denominators 1+3=4 don`t add the denominator. Answer then is 4/6
Try these:  5/12 + 3/12  3/10 + 5/10

Adding Unalike: When adding fractions that have different denominators you have to determine a common denominator for example if it's 1/3 + 2/6 you can either multiply the 1/3 by 2 = 2/6 or multiply both fractions by the other fractions denominator 1/3x6=6/18 2/6x3=6/18 then add the numerators and keep the denominator
Try these: 3/12 + 2/6 5/10 + 2/5

Subtracting with Alike: When subtracting with alike you do the same thing for adding but subtract the 2 numbers for example if it's 8/10 - 3/10 do 8-3=5 so the answer is 5/10
Try these: 9/10 - 4/10  5/6 - 3/6

Subtracting with Unalike: When subtracting with unalike denominators you still have to find a common denominator like this: 6/10 - 2/5 you can either multiply that 2/5 by 2 and get 4/10 and subtract then answer is 2/10 or 6/10 x5 = 30/50 and 2/5x 10 = 20/50 and subtract then answer is 10/50
Try these: 7/12 - 3/6  7/8 - 1/4

Adding and Subtracting Fractions with Unlike Denominators

Adding:
When adding you first have to multiply the first denominator by the second one and the second one by the first denominator or if one of the denominators can equally be multiplied to turn into the same denominator as the other, you can just multiply the fraction by that number which will make the numerator change also. By doing this both fractions will have like denominators now and you can add the numerators together to get the answer. If the answer is not yet in lowest terms you just have to divide.
5/12 + 2/6 = 3/4
2/6 x 2/2 = 4/12
4/12 + 5/12 = 9/12 ÷ 3 = 3/4

Subtracting:
When subtracting (just like adding) you first have to change the unlike denominators to like denominators before starting to actually subtract. So I just multiplied 2/3 by 5/5 to make the denominator the same because 3 can be multiplied 5 times to get 15. After subtracting you need to put the answer in lowest terms if it is not in lowest terms.
2/3 - 7/15 = 1/5
2/3 x 5/5 = 10/15
10/15 - 7/15 = 3/15 ÷ 3 = 1/5