Evaluating Expressions Involving Fractions

This short note is intended to remind you of principles already described under the heading “Order of Operations,” and to illustrate their application to arithmetic expressions involving fractions.

In evaluating an arithmetic expression, the order in which operations are done is:

(1) bracketed expressions first, starting with the innermost pair of brackets

(2) powers second

(3) multiplications and divisions third

(4) additions and subtractions last Within each priority level, operations are done from left to right.

These rules apply to all arithmetic expressions, including those which involve fractions.

Example:

Simplify

solution:

The expression in brackets gets evaluated first:

thus

Now, the multiplication in the second term gets done, because it has a higher priority than the subtraction:

Finally, we do the subtraction:

Thus, our final answer is

A common error is to start by doing the subtraction:

This is an error, because it carries out the subtraction ahead of the higher priority brackets and multiplication. If carried to completion this will give an incorrect final answer.

Example:

Simplify

solution:

The expression in brackets has the highest priority, so do it first:

So, the original expression becomes

Of the remaining operations, the multiplication has the highest priority, so do it next:

Now, we’re left with subtraction and addition. Both of these operations have the same priority, so we work from left to right. First, subtract 5 / 16 from 7 / 8 :

so that

Since 67 is a prime number, this fraction cannot be simplified further, so the final answer here is

Note that if the addition and subtraction were done in the reverse order (thus violating the priority rules), we would get

and then

which is very different from the correct answer. So, to get the correct answer, the conventional priority rules must be followed very carefully.