Section 5.3 The 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.

Rational Numbers All 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.

Rational Numbers When the numerator of a fraction has a smaller absolute value than the denominator, we call the fraction 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:

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

EXAMPLE 2 Writing a Mixed Number as an Improper Fraction Write as an improper fraction. SOLUTION Step 1 Multiply the whole number part (7 in this case) by the denominator (5), then add the numerator (2): = = 37 Step 2 Write as a fraction with the result of step 1 as numerator, and the original denominator of the fractional part.

Facts About Fractions Since 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. Different Signs Negative Value 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. SOLUTION Both 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.

EXAMPLE 3 Reducing a Fraction to Lowest Terms SOLUTION We 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.

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.

Multiplying and Dividing Fractions Multiplying Fractions To multiply two fractions, multiply the numerators and the denominators separately. That is, Dividing Fractions To 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 Fractions Adding/Subtracting with Common Denominators To 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 a calculator. 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

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 ….

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

Fractions and Decimals Writing 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. SOLUTION Step 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 99

Change 0.63 to a fraction. EXAMPLE 13 Writing a Repeating Decimal as a Fraction SOLUTION Step 1 Write n = … and multiply both sides of the equation by 100 to get 100n = … Step 2 Subtract the first equation from the second one as shown. 100n = … – n = … 99n = 63 Step 3 Divide both sides by n = 63, so n =63 =

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?

SOLUTION The 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