242 Chapter 3 Fractions and Mixed Numbers

Similar objects

Because the denominators of 35 and 15 are the same, we say that they have a common denominator. Since the fractions have a common denominator, we can add them. The following figure explains the addition process.

We can make some observations about the addition shown in the figure.

The sum of the numerators is the numerator of the answer.

The answer is a fraction that has the same denominator as the two fractions that were added.

These observations illustrate the following rule.

Adding and Subtracting Fractions That Have the Same Denominator

To add (or subtract) fractions that have the same denominator, add (or subtract) their numerators and write the sum (or difference) over the common denominator. Simplify the result, if possible.

Caution! We do not add fractions by adding the numerators and adding the denominators!

The same caution applies when subtracting fractions.

Solution

We read 73 1 23 2as “negative seven-thirds minus negative two-thirds.” Thus, the number to be subtracted is 23 . Subtracting 23 is the same as adding its opposite, 23 .

2 Add and subtract fractions that have different denominators.

Now we consider the problem 35 13 . Since the denominators are different, we cannot add these fractions in their present form.

To add (or subtract) fractions with different denominators, we express them as equivalent fractions that have a common denominator. The smallest common denominator, called the least or lowest common denominator, is usually the easiest common denominator to use.

Least Common Denominator

The least common denominator (LCD) for a set of fractions is the smallest number each denominator will divide exactly (divide with no remainder).

The denominators of 35 and 13 are 5 and 3. The numbers 5 and 3 divide many numbers exactly (30, 45, and 60, to name a few), but the smallest number that they divide exactly is 15. Thus, 15 is the LCD for 35 and 13 .

To find 35 13 , we build equivalent fractions that have denominators of 15. (This procedure was introduced in Section 3.1.) Then we use the rule for adding fractions

WHY To add (or subtract) fractions, the fractions must have like denominators.

Solution

Since the smallest number the denominators 2 and 3 divide exactly is 6, the LCD is 6.

This fraction is in simplest form.

1

Success Tip In Example 6, did you notice that the denominator 5 is a factor of the denominator 15, and that the LCD is 15. In general, when adding (or subtracting) two fractions with different denominators, if the smaller denominator is a factor of the larger denominator, the larger denominator is the LCD.

Caution! You might not have to build each fraction when adding or subtracting fractions with different denominators. For instance, the step in blue shown below is unnecessary when solving Example 6.

25 1115 25 33 1115 11

3 Find the LCD to add and subtract fractions.

When we add or subtract fractions that have different denominators, the least common denominator is not always obvious. We can use a concept studied earlier to determine the LCD for more difficult problems that involve larger denominators. To illustrate this, let’s find the least common denominator of 38 and 101 . (Note, the LCD is not 80.)

We have learned that both 8 and 10 must divide the LCD exactly. This divisibility requirement should sound familiar. Recall the following fact from Section 1.8.

The Least Common Multiple (LCM)

The least common multiple (LCM) of two whole numbers is the smallest whole number that is divisible by both of those numbers.

Thus, the least common denominator of 38 and 101 is simply the least common multiple of 8 and 10.

We can find the LCM of 8 and 10 by listing multiples of the larger number, 10, until we find one that is divisible by the smaller number, 8. (This method is explained in Example 2 of Section 1.8.)

Multiples of 10: 10, 20, 30, 40, 50, 60, . . .

This is the first multiple of 10 that is divisible by 8 (no remainder).

Since the LCM of 8 and 10 is 40, it follows that the LCD of 38 and 101 is 40.

We can also find the LCM of 8 and 10 using prime factorization. We begin by prime factoring 8 and 10. (This method is explained in Example 4 of Section 1.8.)

8 2 2 2

10 2 ~5

The LCD for 1328 and 211 is 84.

We will compare the prime factorizations of 28, 21, and the prime factorization of the LCD, 84, to determine what forms of 1 to use to build equivalent fractions for 1328 and 211 with a denominator of 84.

4 Identify the greater of two fractions.

If two fractions have the same denominator, the fraction with the greater numerator is the greater fraction.

For example,

If the denominators of two fractions are different, we need to write the fractions with a common denominator (preferably the LCD) before we can make a comparison.

Solution

Since the smallest number the denominators will divide exactly is 24, the LCD for 56 and 78 is 24.

Next, we compare the numerators. Since 21 20, it follows that 2124 is greater than 2024 . Thus, 78 56 .

5Solve application problems by adding and subtracting fractions.