Download Section 1.7 - Shelton State

Concepts
1 BasicFunctions
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Unit 1B
Review of Operations with
Fractions
Copyright © Cengage Learning. All rights reserved.
1.7 Addition and Subtraction of Fractions
Copyright © Cengage Learning. All rights reserved.
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.
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Example 1
Add:
Add the numerators and simplify.
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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.
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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.
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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.
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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.
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Example 4
cont’d
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.
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Example 4
cont’d
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.
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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.
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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.
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Example 10
Subtract:
Subtract the numerators.
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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.
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Example 13
Add:
and
First change the proper fractions to the LCD, 10.
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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.
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Example 14
Subtract:
from
First change the proper fractions to the LCD, 12.
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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.
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Applications Involving Addition and
Subtraction of Fractions
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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 1.25.
Parallel circuit
Figure 1.25
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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 (IT) 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 3 + . . ..
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Example 18
Find the total current in the parallel circuit in Figure 1.26.
Figure 1.26
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Example 18
cont’d
IT = I1 + I2 + I3 + I4 + I5
First change the proper fractions to the LCD, 8.
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