Understanding Subtraction of Fractions:

Under the Addition of Fractions, we saw how placing the fractions on number lines necessitated determining an equal number of subdivisions for both lines. That same approach applies when working with subtraction.

On the "thirds" line, if we cut each "third" into two equal sections we will get 6 in total.
On the "halves" line, if we cut each "half" into three equal sections we will get 6 in total.
(This number 6 will be called our common denominator.)

 Subtract:

Now that both lines contain subdivisions of the same size (both have 6 sections), we can express our fractions using this new subdivision size of 1/6. Then we can subtract the fractions.

Let's combine our number lines onto one line to demonstrate this difference.

 Same Denominators:
When the fractions have the same denominators, adding and subtracting are easy.
Think of it as already dealing with equal sections on the number line.

When adding (or subtracting) fractions with the same denominators,
just add (or subtract) the numerators.

Since the denominators are the same, the answer can be found by subtracting the numerators (tops).
 This answer can be simplified (reduced).

 Using the "separate parts" method:       Notice the parentheses in the first line. You must subtract the ENTIRE amount of the second mixed number. When the parentheses are removed, notice that 5/11 is also being subtracted. (The subtraction has been distributed across the parentheses.)
 Sometimes, during subtraction, you need to borrow, so you will have a sufficient amount from which to subtract. Consider the following: We just borrowed a "1" from 7, changed it to "5/5" and created a fraction with a larger numerator. A fraction with a larger numerator may be useful when subtrating.
 Using the "line up" method: You can immediately see a problem: you cannot subtract 4/5 from 2/5. We need to borrow a "1" or "5/5" from the 7, and add that amount to the 2/5.
 Visualize:

 Different Denominators:
When fractions have different denominators, you must work with common denominators. Think of it as finding the needed number of subdivisions for the number line.

When adding (or subtracting) fractions with different denominators,
you must
find a common denominator before adding (or subtracting).

 Least common denominator of 6 and 9 is 18.

 Least common denominator of 4 and 6 is 12. Borrowing will be needed.

This type of problem can be confusing. Think carefully before writing your final answer.

First, let's LOOK at what the answer should be.

"Separate parts horizontally method":
For this question, this method can be a bit confusing. Be careful!

"Line up vertically":
This method gives you a bit of a hint that something is missing in this problem. Be careful not to simply "drop" the 2/3 down into the answer.
 You could think of the empty oval as . We need to "borrow" 1 from the 4: