Perform all calculations within the innermost Parentheses first.
Evaluate Exponent expressions.
Apply the Multiplication and Division operations from left to right.
Lastly, work all Addition and Subtraction operations from left to right.
Caution: Please do not dismiss the fact that multiplication and division should be worked from left to right. Many standardized exams will test you on this fact. The following example illustrates the problem.
Order of Operations - The Basics
Simplify.
Order of operation problems get a bit more tedious when fractions are involved. Remember that when adding or subtracting fractions you need to first find the equivalent fractions with a common denominator. Multiplication does not require a common denominator.
Simplify.
The Order of Operations
We will see that some of the problems have different looking parentheses { [ ( ) ] }, treat them the same and just remember to perform the innermost parentheses first. Some problems may involve an absolute value, in which case you will need to apply the innermost absolute value first as you would if it were a parentheses.
To avoid these unnecessary mistakes, work one operation at a time and for each step rewrite everything. This may seem like a lot of work but it really helps avoid errors.
Fractions can be a barrier to beginning algebra students. Also referred to as rational numbers, fractions are simply real numbers that can be written as a quotient, or ratio, of two integers.
Equivalent fractions can be expressed with different numerators and denominators. For example,
If you eat 4 out of 8 slices of pizza, that is the same as eating one-half of the pie. Usually we will be required to reduce fractions to lowest terms. Fractions in lowest terms have no common factors in the numerator and denominator other than 1.
An alternative and more common method of reducing is to identify the greatest common factor (GCF) of the numerator and denominator and then divide both by that number.
An improper fraction is one where the numerator is larger than the denominator. To convert an improper fraction to a mixed number, simply divide. The quotient is the whole number part and the remainder is the new numerator.
Reduce.
To multiply fractions, you multiply the numerators and the denominators. NO COMMON DENOMINATOR is required.
To divide fractions, you multiply the numerator by the reciprocal of the divisor.
When multiplying, look for factors to cancel before actually multiplying the numerators and denominators. This will eliminate the need to reduce the end result.
Multiply or Divide.
Convert mixed numbers to improper fractions before you multiply or divide. To do this, multiply the denominator with the whole number then add the numerator, this will be the new numerator.
Divide.
Sometimes mixed numbers are confused with multiplication. Be sure to remember that 4 2/3 does not imply multiplication, it represents addition.
The difficulty with fractions is usually caused by addition and subtraction. These operations require a common denominator. If I were to say that I ate two pieces of pizza, you would want to know what size each piece was. To say that I ate 2 pieces out of a pizza cut into 4 big slices is not the same as eating 2 pieces of a pizza cut into 8 slices!
If fractions have a common denominator, simply add or subtract the values found in the numerator and write the result over the common denominator and reduce if necessary.
Add or subtract.
Many of the problems that you are likely to encounter will have different denominators. You will have to find the equivalent fractions with a common denominator before you can add or subtract them.
Step 1: Determine the least common multiple of the denominators (LCD). Step 2: Multiply numerator and denominator by what you need to obtain equivalent fractions with that LCD. Step 3: Add the numerators and write the result over the common denominator. Step 4: Reduce if necessary.
Add or subtract.
Division Word Problem: A board that is 5 1/4 feet long is cut into 7 pieces of equal length. What is the length of each piece?
Multiplication Word Problem: A muffin recipe calls for 1 2/3 cups of flour to make 4 muffins. How many cups of flower does the recipe require if you wish to make 16 muffins?
Order of operation problems get a bit more tedious when fractions are involved. Remember that when adding or subtracting fractions you need to first find the equivalent fractions with a common denominator. Multiplication does not require a common denominator.
