## Easy Mathematics Step-by-Step (2012)

### Chapter 5. Fractions

In this chapter, you learn how to work with fractions.

**Fraction Concepts**

As greater precision in measurement was needed, the concept of fractions (rational numbers) was invented. If a *unit* is broken into equal parts, then a fraction represents one or more of those equal parts. For example, if an inch is broken into 10 equal parts, then represents seven of those equal parts. When the *unit* (the distance between 0 and 1) is divided into 10 equal parts, the fraction is a point on the number line, as shown in __Figure 5.1__.

**Figure 5.1** The fraction

The division line in a fraction is the *fraction bar*. The number above the fraction bar is the *numerator*, and the number below the fraction bar is the *denominator*. For example, in the fraction , 7 is the numerator, and 10 is the denominator.

The denominator of a fraction cannot be 0.

**Reducing Fractions to Lowest Form**

A fundamental principle of fractions is the *cancellation law*.

**Cancellation Law**

If both the numerator and the denominator of a fraction are multiplied by the same nonzero number, then the value of the fraction is unchanged. That is, provided that and/or .

The cancellation law is not valid if the operation is addition instead of multiplication. That is, . This is a common error observed in the arithmetic of fractions.

A fraction is in *lowest form* when all the common factors of the numerator and denominator have been cancelled. The process of cancelling the common factors is called *reducing the fraction*.

**Problem** Reduce to lowest form.

**Solution**

*Step 1*. Write the numerator and denominator in a factored form.

*Step 2*. Cancel the common factor.

*Step 3*. If possible, factor again.

*Step 4*. Cancel the common factor.

*Step 5*. If there are no more common factors, state the lowest form.

You can simplify the reduction process considerably if you factor by using the *greatest common factor* (GCF). As implied by its name, the GCF of two numbers is the largest factor that is common to the two numbers. For example, the GCF of 30 and 36 is 6. The factoring in the previous problem could have been , and then the 6 canceled to get the reduced form of in just two steps. You can determine the GCF of two numbers by listing all the factors of the two numbers and selecting the largest factor that is common to both.

**Problem** Find the GCF of 30 and 36.

**Solution**

*Step 1*. List all the factors of 30 and all the factors of 36.

The factors of 30 are 1, 2, 3, 5, 6, 10, 15, 30, and the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36.

*Step 2*. Examine the two lists and select the greatest factor common to both.

6 is the greatest factor that is common to both lists. Thus, the GCF of 30 and 36 is 6.

Another way to accomplish the reduction process is to *divide* the numerator and denominator by the same number and write the resulting quotients as the equivalent fraction. The common way of writing the reduction is

, where you divide both the numerator and the denominator by 6.

**Problem** Write in lowest form.

**Solution**

*Step 1*. Divide the numerator and denominator by 22.

*Step 2*. If there are no more common factors, state the lowest form.

The process of reduction is very useful in operations involving fractions, especially in the area of algebra. Also, is much easier to locate or visualize on the number line than is . Reduction is a simple way of writing a fraction in a more recognizable form to help determine its relative size.

**Equivalent Fractions**

Fractions are the only real numbers that use two numerical components (a numerator and a denominator) to express the number. Consequently, a concept known as *equivalent fractions* is peculiar to fractions. There are several ways to express this idea.

**Equivalent Fractions**

Two fractions, and , are *equivalent* if and only if

(a) they locate the same point on the number line;

(b) one can be reduced to the other; or

(c) *ad = bc*.

Each of the ways of expressing equivalence has its use, but the method in (c) above is of special significance when doing arithmetic with fractions.

**Adding and Subtracting Fractions**

**Addition and Subtraction of Fractions**

**Like denominators:** If and are fractions with *like* denominators, then

**Unlike denominators:** If and are any two fractions, then

**Problem** Perform the indicated operation.

**Solution**

*Step 1*. The denominators are alike, so apply and reduce, if needed.

*Step 1*. The denominators are not alike, so apply and reduce, if needed.

*Step 1*. The denominators are not alike, so apply and reduce, if needed.

*Step 1*. The denominators are not alike, so apply and reduce, if needed.

Because adding or subtracting fractions is very simple when they have the same denominator, a third method of combining any two fractions is to write the two fractions as equivalent fractions with like denominators and then add (or subtract). This method, referred to as *finding a common denominator*, is used often.

For example, and . Hence, the cancellation law ensures both can be written as equivalent fractions with the same denominator. This is not a unique process because and . Again, both can be written as equivalent fractions with like denominators. However, 28 is the least common denominator and is preferred in most cases.

Here is an example of finding a common denominator to subtract two fractions.

**Problem** Compute as indicated: .

**Solution**

*Step 1*. Find the common denominator.

The common denominator of 5 and 11 is 55.

*Step 2*. Write and as equivalent fractions that have the denominator 55.

*Step 3*. Subtract the equivalent fractions and reduce, if needed.

The method of combining fractions by finding a common denominator is used by many, but using or as shown above for combining two fractions can be faster; it also automatically produces a common denominator and transfers easily to algebra. Nevertheless, you should choose the method that works best for you.

**Multiplying and Dividing Fractions**

**Multiplication of Fractions**

If and are two fractions, then .

To multiply two fractions, multiply the two numerators and the two denominators and then reduce, if needed.

**Problem** Multiply: .

**Solution**

*Step 1*. Apply and reduce, if needed.

Common denominators are not needed when multiplying fractions.

The fraction is the reciprocal of the fraction provided . This concept is used when you divide fractions.

**Division of Fractions**

If and are two fractions and , then

To divide two fractions, multiply the first fraction by the reciprocal of the divisor fraction.

**Problem** Divide: .

**Solution**

*Step 1*. Apply and reduce, if needed.

**Working with Mixed Numbers and Improper Fractions**

A *mixed number* is a whole number in combination with a fraction such as (read as “two and five-sevenths”). This mixed number is an alternate representation of the *improper fraction* . Any mixed number can be converted to its fractional form by the method . Also, the added form is .

One way to add or subtract with mixed numbers is to change the mixed number to an improper fraction before adding or subtracting, as shown in the following problem.

**Problem** Compute as indicated. Write the answer as a mixed number.

__a____.__ Add and .

__b____.__ Subtract from .

**Solution**

** a**. Add and .

*Step 1*. Convert the mixed number to an improper fraction.

*Step 2*. Add and reduce to lowest form, if needed.

*Step 3*. Write as a mixed number.

** b**. Subtract from .

*Step 1*. Write both fractions as improper fractions.

*Step 2*. Subtract and reduce to lowest form, if needed.

*Step 3*. Write as a mixed number.

Another way to add or subtract with mixed numbers is to work with the whole number parts and fractional parts separately, as shown in the following problem.

**Problem** Add and .

**Solution**

*Step 1*. Write both fractions in added form.

*Step 2*. Add the whole numbers and then the fractions.

When you multiply or divide mixed numbers, change mixed numbers to improper fractions before multiplying or dividing.

**Problem** Compute as indicated.

**Solution**

*Step 1*. Write both fractions as improper fractions.

*Step 2*. Do the division and reduce to lowest form, if needed.

*Note:* You can divide out common factors before multiplying.

*Step 1*. Write both fractions as improper fractions.

*Step 2*. Do the multiplication and reduce to lowest form, if needed.

**Exercise 5**

*Compute as indicated. Reduce to lowest form and write as a mixed number, when possible*.