Copyright © Cengage Learning. All rights reserved. Functions 1 Basic Concepts

Copyright © Cengage Learning. All rights reserved. Unit 1B Review of Operations with Fractions

Copyright © Cengage Learning. All rights reserved. Addition and Subtraction of Fractions 1.7

4 Addition and Subtraction of Fractions Adding Fractions That is, to add two or more fractions with the same denominator, first add their numerators. Then place this sum over the common denominator and simplify.

5 Example 1 Add: Add the numerators and simplify.

6 Addition and Subtraction of Fractions To add fractions with different denominators, we first need to find a common denominator. When reducing fractions to lowest terms, we divide both numerator and denominator by the same nonzero number, which does not change the value of the fraction. Similarly, we can multiply both numerator and denominator by the same nonzero number without changing the value of the fraction.

7 Addition and Subtraction of Fractions It is customary to find the least common denominator (LCD) for fractions with unlike denominators. The LCD is the smallest positive integer that has all the denominators as divisors. Then, multiply both numerator and denominator of each fraction by a number that makes the denominator of the given fraction the same as the LCD.

8 Addition and Subtraction of Fractions To find the least common denominator (LCD) of a set of fractions: 1. Factor each denominator into its prime factors. 2. Write each prime factor the number of times it appears most in any one denominator in Step 1. The LCD is the product of these prime factors.

9 Example 4 Find the LCD of the following fractions: and Step 1: Factor each denominator into prime factors. 6 = 2  3 8 = 2  2  2 18 = 2  3  3 Step 2: Write each prime factor the number of times it appears most in any one denominator in Step 1. The LCD is the product of these prime factors.

10 Example 4 Here, 2 appears once as a factor of 6, three times as a factor of 8, and once as a factor of 18. So 2 appears at most three times in any one denominator. Therefore, you have 2  2  2 as factors of the LCD. The factor 3 appears at most twice in any one denominator, so you have 3  3 as factors of the LCD. Now 2 and 3 are the only factors of the three given denominators. The LCD for and must be 2  2  2  3  3 = 72. Note that 72 does have divisors 6, 8, and 18. This procedure is shown in Table 1.1. cont’d

11 Example 4 From the table, we see that the LCD contains the factor 2 three times and the factor 3 two times. Thus, LCD = 2  2  2  3  3 = 72. cont’d

12 Addition and Subtraction of Fractions After finding the LCD of the fractions you wish to add, change each of the original fractions to a fraction of equal value, with the LCD as its denominator.

13 Addition and Subtraction of Fractions Subtracting Fractions To subtract two (or more) fractions with the same denominator, first subtract their numerators. Then place the difference over the common denominator and simplify.

14 Example 10 Subtract: Subtract the numerators.

15 Addition and Subtraction of Fractions To subtract two fractions that have different denominators, first find the LCD. Then express each fraction as an equivalent fraction using the LCD, and subtract the numerators. Adding Mixed Numbers To add mixed numbers, find the LCD of the fractions. Add the fractions, then add the whole numbers. Finally, add these two results and simplify.

16 Example 13 Add: and First change the proper fractions to the LCD, 10.

17 Addition and Subtraction of Fractions Subtracting Mixed Numbers To subtract mixed numbers, find the LCD of the fractions. Subtract the fractions, then subtract the whole numbers and simplify.

18 Example 14 Subtract:from First change the proper fractions to the LCD, 12.

19 Addition and Subtraction of Fractions If the larger of the two mixed numbers does not also have the larger proper fraction, borrow 1 from the whole number. Then add it to the proper fraction before subtracting.

20 Applications Involving Addition and Subtraction of Fractions

21 Applications Involving Addition and Subtraction of Fractions An electrical circuit with more than one path for the current to flow is called a parallel circuit. See Figure Parallel circuit Figure 1.25

22 Applications Involving Addition and Subtraction of Fractions The current in a parallel circuit is divided among the branches in the circuit. How it is divided depends on the resistance in each branch. Since the current is divided among the branches, the total current (I T ) of the circuit is the same as the current from the source. This equals the sum of the currents through the individual branches of the circuit. That is, I T = I 1 + I 2 + I

23 Example 18 Find the total current in the parallel circuit in Figure Figure 1.26

24 Example 18 I T = I 1 + I 2 + I 3 + I 4 + I 5 First change the proper fractions to the LCD, 8. cont’d