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3.04 Adding and Subtracting Fractions from Basic Algebra: One Step at a Time © 2002 P. 265-280 Dr. Robert J. Rapalje Seminole State College of Florida
ANSWERS TO ALL EXERCISES ARE INCLUDED AT THE END OF THIS PAGE
Before beginning the algebra of adding and subtracting fractions, it might be helpful to do a few “simple” fractions, especially since you have a calculator to help out. As always, when using the calculator, it is good to begin with simple exercises to check it out first. Solve each of the following by calculator (especially if you have fractions capabilities) and check by regular LCD methods.
EXAMPLE 1.
CALCULATOR: If you have a b/c button, type [1] [a b/c] [8] [+] [3] [a b/c] [8] [=] _____ Your calculator should say “1 ┘ 2", which means 1/2.
Without the fractions key, you can enter [1] [÷] [8] [+] [3] [÷] [8] [=] _____ The calculator gives the decimal 0.5 which is 1/2. In your head: you add to get 4/8, which reduces to 1/2.
EXAMPLE 2.
CALCULATOR: If you have [a b/c] button, type [5] [a b/c] [12] [+] [3] [a b/c] [16] [=] _____ Did you get “29 ┘ 48" (i.e. 29/48)?
Without fractions key, you get a decimal your calculator can probably convert to a fraction. In your head: It is much harder. Find LCD = 48. We’ll save the rest of the check for later.
EXAMPLE 3.
CALCULATOR: If you have a b/c button, type [3] [a b/c] [4] [+] [1] [a b/c] [2] [=] _____ The calculator says “1 1 ┘ 4", which means a mixed fraction “1 1/4", which is 5/4.
Without the fractions key, you can enter [3] [÷] [4] [+] [1] [÷] [2] [=] _____ The calculator gives the decimal 1.25 which is 1 1/4 or 5/4. In your head: you must find LCD of 4. Then 3/4 + 2/4 gives you 5/4, or 1 1/4.
EXAMPLE 4.
CALCULATOR: If you have a b/c button, you can easily enter the numbers and get "1 2 ┘ 2", which means 1 ½ or 3/2.
Even without the fractions key, you should easily get the decimal value 1.5 which is 1 ½ or 3/2. In your head: The LCD = 24, and with some work (later!) the answer reduces to 3/2.
EXERCISES. Use a calculator to add or subtract the following fractions:
1.
5.
The first question that must be answered when adding and subtracting fractions without a calculator is this: “Is there a common denominator?” If the fractions already have a common denominator, then just put down the common denominator as THE denominator of the answer. To get the numerator of the answer just add (or subtract) the numerators. Then, of course, try to reduce the fractions, if possible, as you did in an earlier section.
9.
= = ________(Reduce!) =
13.
15. Factor: = Reduce: =
17.
=
=
= Notice that in the next few exercises, the primary concept in adding or subtracting fractions is putting down the LCD and then adding or subtracting numerators. Whether the numerators can be factored or not is usually irrelevant. Don’t forget that if you were multiplying or dividing the fractions, the first step would be to factor everything, both numerators and denominators. However, when adding or subtracting fractions, there is usually no need to factor numerators! (Even if they do factor, it is usually not a good idea!)
19.
=
=
=
= In the next exercises, be careful of the signs:
21.
=
=
23.
NOW, if the fractions do not have a common denominator, then finding the common denominator is your first priority! If the denominator is not in factored form, then you must factor each denominator completely and find the common denominator, as in the previous section. The next step will require a few examples to understand. You must compare each denominator of the problem to the LCD and in each case decide “WHAT’S MISSING?” You must multiply the numerator and denominator of each fraction by the “missing factors.” Don’t forget! Always multiply the numerator AND denominator! Then combine numerators adding or subtracting like terms, as illustrated in the next examples. On the next page is a summary in outline form. This will be a very helpful reference as you study the examples and do the exercises that follow.
EXAMPLE 5. Solution. Step I (Find the LCD): The LCD is 20. Step II (What’s Missing?): 1st denominator has 4, missing: 5 2nd denominator has 5, missing: 4 Multiply numerator and denominator of each fraction by “What’s Missing”:
=
=
Step III (Add or Subtract): Add the numerators and place over the LCD.
=
EXAMPLE 6. Solution. Step I (Find the LCD): The LCD is 20x. Step II (What’s Missing?): 1st denominator has 4 x, missing: 5 2nd denominator has 5 x, missing: 4 Multiply numerator and denominator of each fraction by “What’s Missing”:
=
= Step III (Add or Subtract): Add the numerators and place over the LCD.
=
EXAMPLE 7. Solution. Step I (Find the LCD): The LCD is 40xy. Step II (What’s Missing?): 1st denominator has 8x, missing: 5y 2nd denominator has 10y, missing: 4x Multiply numerator and denominator of each fraction by “What’s Missing”:
=
=
Step III (Add or Subtract): Add the numerators and place over the LCD.
=
=
EXAMPLE 8. Solution. Step I (Find the LCD): The LCD is 12x2y3. Step II (What’s Missing?): 1st denominator has 4x2y, missing: 3y2 2nd denominator has 3xy3, missing: 4x Multiply numerator and denominator of each fraction by “What’s Missing”:
=
=
Step III (Add or Subtract): Add the numerators and place over the LCD.
=
=
EXAMPLE 9.
Solution. Step I (Find the LCD): The LCD is 24x3y.
Step II (What’s Missing?): 1st denominator has 8x2y, missing: 3x 2nd denominator has 6x3, missing: 4y
Multiply numerator and denominator of each fraction by “What’s Missing”:
=
=
Step III (Add or Subtract): Add the numerators and place over the LCD.
=
EXERCISES. Add or subtract the fractions as indicated.
25.
Step I (Find the LCD): The LCD is _____.
Step II (What’s Missing?): 1st denominator has 3, missing: _____ 2nd denominator has 4, missing: _____ Multiply numerator and denominator of each fraction by “What’s Missing”:
=
Step III (Add or Subtract): Add the numerators and place over the LCD.
=
26.
Step I (Find the LCD): The LCD is _____.
Step II (What’s Missing?): 1st denominator has 8, missing: _____ 2nd denominator has 6, missing: _____ Multiply numerator and denominator of each fraction by “What’s Missing”:
=
Step III (Add or Subtract): Add the numerators and place over the LCD.
=
27.
29.
31. Step I (Find the LCD): The LCD is _______.
Step II (What’s Missing?): 1st denominator has 5x, missing: _____ 2nd denominator has 15y, missing: _____ Multiply numerator and denominator of each fraction by “What’s Missing”:
=
Step III (Add or Subtract): Add the numerators and place over the LCD.
=
32. Step I (Find the LCD): The LCD is _______.
Step II (What’s Missing?): 1st denominator has 7x2, missing: _____ 2nd denominator has 28xy2, missing: _____ Multiply numerator and denominator of each fraction by “What’s Missing”:
=
Step III (Add or Subtract): Add the numerators and place over the LCD.
=
33.
35.
37.
39.
41.
EXAMPLE 10. Solution. Step I (Find the LCD): The first step is to factor the denominators.
=
Step II (What’s Missing?): 1st Denom missing: (x+1) 2nd Denom missing: (x) Multiply numerator and denominator of each fraction by “What’s Missing”.
=
=
Step III (Add or Subtract): Add numerators and place over LCD.
=
=
=
43. Step I (Find the LCD): Factor the denominators
Step II (What’s Missing?):
Step III (Add or Subtract): Add numerators and place over LCD.
=
44.
=
=
=
=
45.
47.
EXAMPLE 11. Solution. Step I (Find the LCD): The first step is to factor the denominators.
= Step II (What’s Missing?): 1st Denom missing: (x+5) 2nd Denom missing: (x-3) Multiply numerator and denominator of each fraction by “What’s Missing”.
=
=
Step III (Add or Subtract): Add numerators and place over LCD.
=
49.
=
=
50.
52.
In #54 - 56, factoring the numerators at the beginning of the problem, even if possible, is usually irrelevant! However, after you finish the problem, you may be able to reduce the fractions by factoring.
54.
56.
NOTE: Problems #57 - 59 are from New School Algebra (1898), by G.A. Wentworth.
57.
58.
59.
FACTORS of “x - y” and “y- x” (Optional--if time permits!)
As in the first two sections of this chapter, it is
frequently necessary to work with factors and their negatives. This may
occur in reducing fractions, multiplying/ dividing fractions, or
adding/subtracting fractions. Sometimes it is helpful to factor a “-1" from
one of the factors, in order to make them “match up.” Another helpful hint
is to remember that any number (except zero, of course!) divided by its
negative is “-1". Consider:
Likewise, since “4-x” is the negative of “x-4",
and “y-x” is the negative of “x-y”:
EXERCISES.
60.
64. = -1(x+ 4) or - x - 4
When adding and subtracting fractions such as
EXAMPLE 12:
Solution. =
=
=
=
=
66.
69.
72.
ANSWERS 3.04 p. 266 - 280:
1. 19/15;
2. 29/24; 3. 1/2; 4.
1/12; 5. 99/221; 6. 25/828; 7.
16/9; 8. 7/12; 9.
65.
-(X+6) or
‑X‑6; 66. 0; 67. Return to main page Math in Living C O L O R !! Return to Basic Algebra page Return to Intermediate Algebra page Return to College Algebra page
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