The numerator and denominator of these rational expressions contain fractions and look very intimidating. We will outline two methods for simplifying them.
Method 1: Obtain a common denominator for the numerator and denominator, multiply by the reciprocal of the denominator, then factor and cancel if possible.
Method 2: Multiply the numerator and denominator of the complex fraction by the LCD of all the simple fractions then factor and cancel if possible.
To illustrate what happened after we multiplied by the LCD we could add an extra step.
For the following solved problems, both methods are used. Choose whichever method feels most comfortable for you.
Given a rational expression, the quotient of two polynomials, we will factor the numerator and denominator if we can and then cancel factors that are exactly the same.
When evaluating rational expressions, plug in the appropriate values either before simplifying or after, the result will be the same. Although, it is more efficient to simplify first then evaluate.
We can see that when evaluating, the result will be the same whether or not we simplify first. It turns out that not all numbers can be used when we evaluate.
The point is that not all real numbers will be defined in the above rational expression. In fact, there are two restrictions to the domain, -2 and 3/5. These values, when plugged in, will result in zero in the denominator. Another way to say this is that the domain consists of all real numbers except for −2 and 3/5.
Tip: To find the restrictions, set each factor in the denominator equal to zero and solve. The factors in the numerator do not contribute to the list of restrictions.
Simplify and state the restrictions to the domain.
Even if the factor cancels it still contributes to the list of restrictions. Basically, it is important to remember the domain of the original expression when simplifying. Also, we must use caution when simplifying, please do not try to take obviously incorrect shortcuts like this:
Since subtraction is not commutative, we must be alert to opposite binomial factors. For example, 5 − 3 = 2 and 3 − 5 = −2. In general,
Simplify and state the restrictions to the domain.
At this point, we evaluate using function notation.
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?
