Section 5.3The Rational Numbers

Math in Our World

Learning Objectives

Define rational numbers. Convert between improper fractions and mixed

numbers. Reduce fractions to lowest terms. Multiply and divide fractions. Add and subtract fractions. Write fractions in decimal form. Write terminating and repeating decimals in

fraction form.

Rational Numbers

The word “ratio” in math refers to a comparison of the sizes of two different quantities. Ratios are often written as fractions. For that reason, numbers that can be written as fractions are called rational numbers.

A rational number is any number that can be written as a fraction in the form ,

where a and b are both integers (and b is not zero). The integer a is called the numerator of the fraction, and b is called the denominator.

ab

Rational NumbersAll of the integers we’ve studied are also rational numbers because they can be written as fractions with denominator 1. For example, 3 = , which fits

the definition of rational number. This means that a number like 3 is a natural number, a whole number, an integer, and a rational number.Every rational number can also be written in decimal form.

31

Rational NumbersWhen the numerator of a fraction has a smallerabsolute value than the denominator, we call thefraction a proper fraction. (i.e. 2/3)

When the absolute value of the numerator is larger than or equal to that of the denominator, we call the fraction an improper fraction. (i.e. – 11/2)

A mixed number consists of a whole number and a fraction. It’s important to understand that a mixed number is really an addition without the + sign:

1 13 32 2

EXAMPLE 1 Writing an Improper Fraction as a Mixed Number

Write as a mixed number.

SOLUTION

First, divide 17 by 6:

The quotient is 2, with remainder 5.

That tells us that

In mixed number form we write

176

17 52

6 6

17 52

6 6

EXAMPLE 2 Writing a Mixed Number as an Improper Fraction

Write as an improper fraction.

SOLUTIONStep 1 Multiply the whole number part (7 in this case) by

the denominator (5), then add the numerator (2):

7 • 5 + 2 = 35 + 2 = 37

Step 2 Write as a fraction with the result of step 1 as numerator, and the original denominator of the fractional part.

725

2 3775 5

Facts About FractionsSince fractions can be thought of as the division of two integers, we can take advantage of rules for division of signed numbers to address the sign of a fraction.

You can reduce a fraction to lowest terms by dividing the numerator and denominator by the same number.

3 3 3

5 5 5

Different Signs Negative Value

4 4 4 1

8 8 4 2

We can also multiply the numerator and denominator by a number to

raise it to higher terms.

EXAMPLE 3 Reducing a Fraction to Lowest Terms

Reduce to lowest terms.

SOLUTIONBoth the numerator and denominator can be divided by 2 with

no remainder:

Before we congratulate ourselves, notice that there’s still more that can be done: each of 9 and 12 is divisible by 3!

Now the fraction is in lowest terms because 3 and 4 have no common divisors.

1824

18 18 2 9

24 24 2 12

9 9 3 3

12 12 3 4

EXAMPLE 3 Reducing a Fraction to Lowest Terms

SOLUTIONWe could have accomplished the previous answer in one

step by finding the greatest common factor (GCF) of 18 and 24, which is 6, then dividing the numerator and denominator by it:

Either method is acceptable.

18 18 6 3

24 24 6 4

EXAMPLE 4 Rewriting a Fraction with a Larger Denominator

Change each fraction to an equivalent fraction with the indicated denominator.

(a) (b)

EXAMPLE 4 Rewriting a Fraction with a Larger Denominator

SOLUTION(a) To change the denominator from 8 to 32, we have to multiply by 4. So to make the fractions equivalent, we also need to multiply the numerator by 4. The equivalent fraction is

(b) It may be a bit harder to notice what number we need to multiply 4 by to get 56, so we divide 56 ÷ 4 = 14. The number we need to multiply the numerator and denominator by is 14.

3 4 12

8 4 32

5 14 70

4 14 56

Multiplying and Dividing Fractions

Multiplying FractionsTo multiply two fractions, multiply the numerators and the denominators separately. That is,

Dividing FractionsTo divide two fractions, multiply the first by the reciprocal of the second. That is,

EXAMPLE 5 Multiplying Fractions

Find each product, and write the answer in lowest terms.

(a) Rather than actually multiply out the numerator and denominator, we’ll write as 5 • 3 and 8 • 5, which allows us to reduce easily.

(b) The product will be negative since the two fractions have opposite signs.

EXAMPLE 5 Multiplying Fractions

SOLUTION

(c) Our multiplication rule doesn’t apply to mixed numbers, so we should first rewrite each as an improper fraction, then multiply.

EXAMPLE 6 Dividing Fractions

Find each quotient, and write the answer in lowest terms.

(a) Multiply 3/4 by the reciprocal of – 5/8 . The two fractions have opposite signs, so the quotient will be negative.

(b) Again, multiply the first fraction by the reciprocal of the second. This time, the quotient cannot be reduced.

EXAMPLE 6 Dividing Fractions

SOLUTION

Adding & Subtracting FractionsAdding/Subtracting with Common DenominatorsTo add or subtract two fractions with the same denominator, add or subtract the numerators, and keep the common denominator the same in your answer.

Adding/Subtracting with Different Denominators

Step 1 Find the least common multiple of the denominators. (This is usually called the least common denominator, or LCD.)

Step 2 Rewrite each fraction as an equivalent fraction with denominator equal to the LCD.

Step 3 Add or subtract.

EXAMPLE 7 Adding & Subtracting Fractions with a Common Denominator

Find each sum or difference.

EXAMPLE 7 Adding & Subtracting Fractions with a Common Denominator

SOLUTION

EXAMPLE 8 Adding & Subtracting Fractions

Find each sum or difference.

EXAMPLE 8 Adding & Subtracting Fractions

SOLUTION(a) The LCD of 4 and 6 is 12, so we rewrite each fraction with denominator 12, then add.

(b) The LCD of 9 and 5 is 45. Rewrite each fraction with denominator 45 and subtract:

(c) First, we need to rewrite the mixed numbers as improper fractions. The LCD of 2 and 4 is 4, so we rewrite 5/2 as 10/4 , then add.

Fractions and Decimals

You might be familiar with the decimal form of fractions, especially when using acalculator. Any fraction can be written in decimal form. (The opposite is not true—many decimals can be written as fractions, but not all). In order to work with numbers in decimal form, you need to be familiar with place value, which is described on the next slide.

Place Values

To change a fraction to decimal form, we use long division.

EXAMPLE 9 Writing a Fraction in Decimal Form

Write as a decimal.

SOLUTION

Divide 5 by 8, as shown.

Since we got a remainder of zero, we’re done. The answer is 0.625.

58

EXAMPLE 10 Writing a Fraction in Decimal Form

Write as a decimal.

SOLUTION

Divide 5 by 6, as shown.

Notice the pattern will keep repeating, so the answer is 0.8333….

56

Repeating Decimals

The decimal 0.8333 . . . , is called a repeating decimal. Repeating decimals can be written by placing a line over the digits that repeat.

0.8333 . . . is written as0.626262 . . . is written as

Any terminating decimal can be written in fraction form, using the following procedure.

Fractions and Decimals

Writing a Terminating Decimal as a Fraction

Step 1 Drop the decimal point and place the resulting number in the numerator of a fraction.

Step 2 Use a denominator of 10 if there was one digit to the right of the decimal point, a denominator of 100 if there were two digits to the right of the decimal point, a denominator of 1,000 if there were three digits to the right of the decimal point, and so on.

Step 3 Reduce the fraction if possible.

EXAMPLE 11 Writing a Termination Decimal as a Fraction

Write each decimal as a fraction.

(a) 0.8 (b) 0.65 (c) 0.024

SOLUTION

Fractions and DecimalsWriting a Repeating Decimal as a Fraction Step 1 Write n = the repeating decimal, and multiply both sides of that equation by 10 if one digit repeats, 100 if two digits repeat, etc.

Step 2 Now you will have two equations: subtract the first equation from the second. The repeating part of the decimal will subtract away.

Step 3 Divide both sides of the resulting equation by the number in front of n. This will be the fractional equivalent of the repeating decimal. (Reduce if necessary.)

EXAMPLE 12 Writing a Repeating Decimal as a Fraction

Change 0.8 to a fraction.

SOLUTIONStep 1 Write n = 0.888… and multiply both sides of the

equation by 10 to get 10n = 8.888…Step 2 Subtract the first equation from the second one as

shown.10n = 8.888… – n = 0.888… 9n = 8

Step 3 Divide both sides by 9. 9n = 8 , so n =

8 9 9

9

Change 0.63 to a fraction.

EXAMPLE 13 Writing a Repeating Decimal as a Fraction

SOLUTIONStep 1 Write n = 0.636363… and multiply both sides of the

equation by 100 to get 100n = 63.636363…Step 2 Subtract the first equation from the second one as

shown.100n = 63.636363… – n = 0.636363… 99n = 63

Step 3 Divide both sides by 99. 99n = 63 , so n =

63 = 7 99 99

99 11

EXAMPLE 14 Applying Rational Numbers to Fitness Training

In her final 2 days of preparation for a triathlon, Cat hopes to swim, run, and bike a total of 50 miles. She works out at a nearby state park with a swimming quarry and running/biking trail. One lap in the quarry is 1/4 mile and the trail is 3-1/4 miles. The first day, Cat swims six laps, runs the trail twice, and bikes it five times. How many more miles does she need to cover the second day?

SOLUTIONThe total distance she covers on the first day is

The mixed number 3-1/4 is 13/4 as an improper fraction. Using order of operations, we perform the multiplications first, then add.

Now we subtract from 50 to get the distance Cat needs to cover the second day.

EXAMPLE 14 Applying Rational Numbers to Fitness Training