Subtract Fractions (mixed numbers)

$\require{cancel}$

Just as you sometimes have to borrow when you subtract whole numbers, you sometimes have to borrow when you subtract mixed number fractions. Essentially, when you're working in the realm of positive numbers, you can subtract from any number a number that is smaller. BUT, even when an entire number is smaller, every digit is not smaller, so we have to borrow from the digit to the left.  

When working with fractions, you can subtract a mixed number from another mixed number if it's smaller, but sometimes even if the entire mixed number is smaller, the fraction of the number you are subtracting is bigger than the fraction you are subtracting from.  In that case, you need to borrow from the whole number in order to complete the problem.  When you borrow from a whole number you borrow a 1.  Because it's easiest to combine fractions with fractions of the same denominator, when you borrow that one, you borrow it in the form of a fraction with the same denominator as the rest of your subtraction problem (so if you are subtracting $\dfrac{1}{4}-\dfrac{3}{4}$, borrow the one in the form of $\dfrac{4}{4}$.

Although students often find this process complicated, it's exactly the same as borrowing with whole numbers, it just uses fractions!  For more on the process of subtracting fractions see lessons: Subtract Fractions (like denominators) and Subtract Fractions (unlike denominators).

Example: 

$$\eqalign{2\dfrac{1}{6}-1\dfrac{3}{8}\\ \text{Write it vertically  }& \left\{ \begin{array}{rcl} 2 \dfrac{1}{6}&\\ -1\dfrac{3}{8}& \\ \hline \qquad \qquad\\ \qquad  \end{array}\right.  \\ \text{Find the common denominator: 24}& \left\{ \begin{array}{rcl}   2\dfrac{1}{6}&=&&\dfrac{4}{24}\\ -1\dfrac{3}{8}&=&&\dfrac{9}{24}\\ \hline \qquad \qquad \\ \qquad \end{array}\right.\\ \text{Try to subtract, but bottom fraction is larger} & \left\{ \begin{array}{rcl} 2\dfrac{4}{24}\\ -1\dfrac{9}{24}\\ \hline \qquad \qquad \end{array}\right.\\ \text{Borrow a whole from the 2 and add it to the } \dfrac{4}{24}& \left\{ \begin{array}{rcl}   \overset{1}{\bcancel{2}}\dfrac{4}{24}+\dfrac{24}{24}=&\dfrac{28}{24}\\\qquad -1\qquad\quad\qquad &\dfrac{9}{24}\\ \hline \qquad \qquad \\ \qquad \end{array}\right.\\ \text{Subtract the fractions and the whole numbers } \dfrac{4}{24}& \left\{ \begin{array}{rcl}   \overset{1}{\bcancel{2}}\dfrac{4}{24}+\dfrac{24}{24}=&\dfrac{28}{24}\\ \qquad-1\qquad \quad  \qquad &\dfrac{9}{24}\\ \hline \qquad \qquad \\ \qquad \end{array}\right.\\ &=\begin{array}&\quad && \;\; 0 &&& \quad &\dfrac{19}{24}\end {array} }$$

When you finish the subtraction, you're finished: $2\dfrac{1}{6}-1\dfrac{3}{8}=\dfrac{19}{24}$

Borrowing is the most efficient way to subtraction fractions with mixed numbers.  See the alternate method below for another common method.