RWM101 Study Guide

Unit 4: Fractions

4a. Identify parts of a fraction

  • What is the number on the top of a fraction called?
  • What is the number on the bottom of the fraction called?

Fractions are made up of three parts: the numerator, the denominator, and the fraction bar.

The fraction bar is the line in the middle, with the numerator on top, and the denominator on the bottom, like so

 \begin{array}{c} \text { numerator } \\ \hline \text { denominator } \end{array}

To review, see Fractions of Whole Numbers.

4b. Recognize fractions in lowest terms

  • How can you tell if a fraction is in lowest terms?
  • How can you rewrite a fraction in lowest terms?

Fractions are most often written in lowest terms. While it is common to say, "I will eat 1/2 of a piece" of something, you would not generally say, "I will eat 3/6ths of a piece", even though ½ = 3/6.

A fraction is in lowest terms if the numerator and denominator have no factors in common. For example, 3/7 is in lowest terms because 3 and 7 have no factors in common besides 1. On the other hand, 4/8 is NOT in lowest terms, because 4 is a factor of both 4 and 8. Rewriting this fraction in lowest terms would give us ½.

To rewrite a fraction in lowest terms, which is also called reducing or simplifying a fraction, find a common factor between the numerator and the denominator. By breaking the numerator and denominator into factors, you can cancel the common factor, and end up with a reduced fraction.

For example:

 \frac{12}{16}=\frac{4 \times 3}{4 \times 4}=\frac{4}{4} \times \frac{3}{4}=\frac{3}{4}

As you can see, we found 4 was a common factor in 12 and 16. Once we rewrite the fraction, we can see that the 4s can be canceled, since 4/4 = 1. Since 3 and 4 have no common factors, we can say the fraction is now written in lowest terms.

To review, see Fractions in Lowest Terms and Reducing Fractions.

4c. Recognize equivalent fractions

  • How can you verify that two fractions are equivalent?

Fractions can be written in many ways and still have the same value. For example, 1/3 and 2/6 are equivalent fractions, because they both represent ⅓ of a whole.

You can verify that fractions are equivalent by writing them in their lowest terms and then checking that the lowest terms are identical, or by dividing the numerator by the denominator and verifying that their decimal equivalents are the same.

To review, see Demonstrating Equivalent Fractions.

4d. Identify improper fractions and mixed numbers

  • What makes a fraction an improper fraction?
  • What is a mixed number?

A fraction with a value of less than 1 can only be written one way, which is called a proper fraction. There are two ways, however, to write fractions with a value greater than 1: as a mixed number, or as an improper fraction.

A fraction with a numerator greater than the denominator is called an improper fraction. Despite its name, improper fractions are commonplace and acceptable answers. For example  \frac{12}{5}, \frac{11}{9}, \text { and } \frac{22}{2} are all improper fractions.

A mixed number is a number consisting of a whole number and a fraction. For example  2 \frac{1}{3} \text { and } 5 \frac{1}{6} are mixed numbers.

To review, see Proper and Improper Fractions, and Proper Fractions, Improper Fractions, and Mixed Numbers.

4e. Convert between mixed numbers and improper fractions

  • How can you convert from a mixed number to an improper fraction?
  • How can you convert from an improper fraction to a mixed number?

To convert a mixed number to an improper fraction, convert the whole number into a fraction with the same denominator as the fraction in the mixed number. Then add these two fractions together, resulting in an improper fraction. For Example:  2 \frac{1}{3}=2+\frac{1}{3}=\frac{6}{3}+\frac{1}{3}=\frac{7}{3}
    . As shown, convert the whole number (2) into a fraction (6/3), and then add it to 1/3 to get the improper fraction 7/3.

To convert an improper fraction to a mixed number, start by dividing the numerator by the denominator. The whole number in the answer will be the whole number in your mixed number, and the remainder will be the numerator in the fraction part. The denominator will always stay the same. For example, when converting  \frac{13}{5} to a mixed number, divide 13 by 5. You will find an answer of 2 with a remainder of 3. This means you were able to divide it 2 whole times, with 3 left over. This can now be written as a mixed number of  2 \frac{3}{5}

To review, see Changing from an Improper Fraction and Visualize Fractions.

4f. Determine the least common denominator between fractions

  • When do you need a common denominator?
  • How do you find the least common denominator?

As you know, the denominator of a fraction is the number on the bottom. So finding a common denominator is making the denominators of two or more fractions the same. This involves finding the least common multiple of the denominators of the fractions. Common denominators are needed when adding or subtracting fractions, and are also helpful if you want to compare two fractions.

For example, consider \frac{1}{6} and \frac{1}{8} . To find a common denominator, think of the least common multiple of 6 and 8, which is 24. 24 is the smallest number that we can divide by both 6 and 8, therefore it is the least common multiple. Since we have to multiply 6 by 4 to get 24, we must also multiply its numerator by 4, giving us the equivalent fraction \frac{4}{24} . Similarly, we multiply 8 by 3 to get 24, so we multiply its numerator by 3 and get the equivalent fraction 8/24. Now, we have found two equivalent fractions with the least common denominator.

To review, see Finding Common Denominators.

4g. Use equivalent fractions to add and subtract fractions and mixed numbers with like and unlike denominators

  • What has to be done in order to add or subtract fractions?
  • How do you add and subtract fractions?
  • How do you add and subtract mixed numbers?

The steps that you take to add and/or subtract fractions will vary depending on the type of fractions that you are working with.

When adding or subtracting fractions with a common (or like) denominator, the answer is as simple as adding or subtracting the numerators, while keeping the denominators the same. For example  \frac{1}{5}+\frac{2}{5}=\frac{3}{5} \text
    { or } \frac{7}{9}-\frac{5}{9}=\frac{2}{9} .

In order to add or subtract fractions with different (or unlike) denominators, you must first convert them to fractions with a common denominator. Then add or subtract the numerators, while keeping the denominators the same. For example  \frac{1}{3}+\frac{5}{6}=\frac{2}{6}+\frac{5}{6}=\frac{7}{6} . As you can see, the ⅓ is changed into the equivalent fraction \frac{2}{6}, so that we have a common denominator, then we can add the fractions together.

To review, see:

4h. Solve multiplication and division problems with fractions and mixed numbers

  • How do you multiply fractions?
  • How do you divide fractions?
  • How do you multiply or divide mixed numbers?

When multiplying fractions, finding a common denominator is not necessary –  simply multiply the numerators and the denominators. For example,  \frac{2}{3} \times \frac{5}{7}=\frac{2 \times 5}{3 \times 7}=\frac{10}{21} . You can simplify the answer after multiplying, or by simplifying any common factors between the numerator and denominator before multiplying (see 3c).

When dividing fractions, we can easily change the problem to multiplication. When dividing two fractions, keep the first fraction the same, change the division symbol to multiplication, then flip the second fraction. For example,  \frac{2}{3} \div \frac{5}{7}=\frac{2}{3}
    \times \frac{7}{5}=\frac{14}{15} .

When multiplying or dividing mixed numbers, simply convert them to improper fractions first, complete your multiplication or division, then, if necessary, convert the answer back to a mixed number.

To review, see:

4i. Solve real-world math problems involving fractions

  • How do you solve word problems with fractions?

When solving a word problem involving fractions, it is important to first understand the problem and then transform the problem correctly into an equation that you can solve. Once you have transformed the problem into an equation, you can solve it just as you would solve any other equation involving fractions.

For example: Bill eats \frac{1}{3} of a pizza every day for lunch, how much pizza does he eat in a full 7-day week?

First, we must write this problem as an equation. Since he eats \frac{1}{3} of a pizza each day for 7 days, our equations should be \frac{1}{3}\times 7. Therefore the answer is \frac{7}{3} or 2 \frac{1}{3} pizzas.

To review, see Multiplying and Dividing Fractions Word Problems.

Unit 4 Vocabulary

This vocabulary list includes terms listed above that students need to know to successfully complete the final exam for the course.

  • common denominator
  • denominator
  • equivalent fractions
  • fraction bar
  • improper fraction
  • least common denominator
  • like/unlike denominator
  • lowest terms
  • mixed number
  • numerator
  • proper fraction
  • reducing/simplifying