Lesson #3 Fraction Arithmetic

An understanding of basic fraction arithmetic is necessary for further work in Algebra. We will concentrate on the properties of fractions and the rules for addition, subtraction, and multiplication. First and foremost, keep in mind that the rules for positive and negative signs apply for all arithmetic, as does the order of operations.

The Meaning of a Fraction: A fraction can be written as either a/b or .

In both cases, the fraction bar means division: 8/2 means 8 divided by 2, which is 4. The top number (a) is called the numerator. The bottom number (b) is called the denominator. A fraction represents some portion of a whole number: the denominator tells what size portion, or how many pieces in one whole; the numerator tells how many such pieces. So the fraction means 2 parts of 3, where it takes 3 parts to make 1. We will reduce a fraction whenever possible, but if the division is such that you don't get an easier answer, just leave it as a fraction (even if the numerator is larger than the denominator). So we would leave the fraction as is (improper form is okay!), since 4 does not divide evenly into 7.

Decimal Form of a Fraction: Convert a fraction into a decimal number by carrying out the division. means 7 divided by 4, which is 1.75. You can do the division by calculator or by long-division methods. Since calculators are especially good at decimal arithmetic, we can usually rely on the calculator to perform this task.

Reducing a Fraction: Every time you work with a fraction, you should check to see that it is fully reduced. A fraction is fully reduced if the numerator and the denominator have no common factor. is fully reduced because 4 does not divide into 7, and no smaller factor divides both 7 and 4. On the other hand, can be reduced: 4 does not divide into 10, but the smaller factor 2 does. Simply divide both numerator and denominator by 2 to get . The hard part about reducing may be in finding the right common factor to use. Don't worry if you take more than one step to complete the reduction. For example, reduces to since 2 divides both top and bottom. But reduces further to and (reduce by 2 twice more). So . You could have reduced by 8 from the start to complete the reduction in just one step, but it is fine to use more than one step.

Addition and Subtraction of Fractions: To add or subtract two fractions, both fractions must have the same denominator. Since the denominator determines the portion size, we must have the same denominator to compare pieces. Thus is a sensible computation.

 When the denominators are the same, simply add (or subtract) the numerators.

If the denominators are unequal, you must first change each fraction into a new fraction with the Least Common Denominator (LCD). The LCD is the smallest number that both denominators can divide into. For each fraction, build the common denominator by "unreducing" each fraction: multiply both top and bottom by the same factor.

Example 1. Add . The denominators are different, so we first need to find the LCD. 15 works, since both 3 and 5 divide into 15. The first fraction is multiplied by 3 to bring the denominator up to 15, and the second fraction is multiplied by 5. That is, and . So the addition problem becomes (add numerators in the last step).

Example 2. Simplify . Again, we first need the LCD of 28. Change to (multiplying by 4's), and change to (multiplying by 7's). Note that we keep the negative sign with the second fraction. Then we complete the subtraction by combining the numerators: . The answer here is negative, since we are subtracting 21 parts from 8 parts (the larger portion is negative). The negative sign may be written in front or as part of the numerator.

 To add or subtract fractions: 1. Convert each fraction to a new fraction with the LCD; 2. Add/Subtract the numerators as the signs indicate; 3. Reduce the final answer if possible. Your answer may be left as an improper fraction.

Multiplication of Fractions: Multiplication is even easier. No common denominator is necessary to multiply fractions. Simply multiply each of the numerators together, and multiply each of the denominators together. Then reduce the final answer as necessary. Your answer may be left as an improper fraction.

Example 3. Multiply . Do the multiplication of the numerators and denominators separately. The numerator is, and the denominator is. So the product is which reduces to . Alternatively, the reduction may be done first: .

Example 4. Multiply. It's easiest to reduce first. Find any common factor that is used on both the numerator and the denominator. In this case, reduce by 2's to get

.

Division of Fractions: Division is the opposite of multiplication. In the case of fractions, division is relatively simple, because fractions already involve divisions. Perform divisions by a fraction by multiplying instead by the reciprocal. That is, turn the second fraction upside down and multiply as above.

Example 5. Divide:

We first change the division by to multiplication by .

So .

In all fraction problems, the most important thing to remember is to pay close attention to which operation shows between the numbers (as always!). Also, be careful not to lose the positive or negative signs.

Exercises

Simplify each of the following by performing the indicated operations.

 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) (What's first?) 27) 28) 29) 30) 31) 32) 33) 34) 35) 36)

Evaluate each of the following when x = 3, y = -2, and z = 6.

 37) 38) 39) 40) 41) 42)

Simplify each of the following algebraic expressions containing fractions.

 43) 44) 45) 46) 47) 48)

Check your answers to Lesson #3 exercises.

Go on to Lesson #4.

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