Math - Unit 2

Math 1012 - Unit 2

FRACTIONS

Fractions are numbers used to refer to a part of a whole

• The denominator (the bottom number) represents the number of equal parts that make

  up the whole.

• The numerator (the top number) describes the number of parts that you are describing.

Multiple Choice

Sophia, Daphne, and Charlie are all participating in a relay race to raise money for charity. First, Sophia will run 2 miles. Then, Daphne will run 5 miles. Finally, Charlie will end the race by running 3 miles. What fraction of the race will Daphne run?

• 5/6

• 5/10

• 1/3

Proper and Improper Fractions

Fractions that are greater than 0 but less than 1 are called proper fractions.

In proper fractions, the numerator is less than the denominator.

When a fraction has a numerator that is greater than or equal to the denominator, the fraction is an improper fraction.

An improper fraction is always 1 or greater than 1.

And, finally, a mixed number is a combination of a whole number and a proper fraction.

Multiple Choice

Which fraction below is a PROPER FRACTION?

• 25/20

• 9/9

• 9/13

Multiple Choice

Identify 5/3 as a proper or improper fraction.

• PROPER FRACTION

• IMPROPER FRACTION

Changing Improper Fractions to Mixed Numbers

• An improper fraction can also be written as a mixed number. Mixed numbers contain

  both a whole number and a proper fraction.

• Examples of mixed numbers include 8 1/10, 1 19/20, and 2 1/2.

  
  

Writing Improper Fractions as Mixed Numbers

• Step 1: Divide the denominator into the numerator.

• Step 2: The quotient is the whole number part of the mixed number.

• Step 3: The remainder is the numerator of the fractional part of the mixed number.

• Step 4: The divisor is the denominator of the fractional part of the mixed number.

Writing Mixed Numbers as Improper Fractions

• Step 1. Multiply the denominator of the fraction by the whole number.

  Step 2. Add this product to the numerator of the fraction.

  Step 3. The sum is the numerator of the improper fraction.

  Step 4. The denominator of the improper fraction is the same as the denominator of the fractional part of the mixed number.

Factors and Primes

Factors

• When a natural number is expressed as a product of two other natural numbers, those other numbers are factors of the original number.

• For example, two factors of 12 are 3 and 4, because 3 • 4 = 12.

Factor cont'd

• To find all the factors of a number, you need to find all numbers that can divide into the original number without a remainder

• Suppose you need to find the factors of 30. Since 30 is a number you are familiar with, and small enough, you should know many of the factors without applying any rules. You can start by listing the factors as they come to mind:

  2 • 15, 3 • 10, 5 • 6 and 1 • 30

• The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30.

Multiple Choice

Determine the factors of 165

• 1, 3, 5, 11, 15, 33, 55, 165

• 1, 3, 11, 15, 33, 165

Multiple Choice

When finding the prime factorization of 72, Marie began a tree diagram using the two factors 9 and 8. Which of the following statements are true?

• 1. Marie started the diagram incorrectly and should have started the tree diagram using the factors 2 and 36.

• 2. Marie’s next set of factor pairs could be 3, 3 and 2, 4.

• 3. Marie’s next set of factor pairs could be 3, 3 and 9, 8.

• 4. Marie didn’t have to use a tree diagram.

Multiple Choice

Which of the following fraction pairs are equivalent?

• a. 5/7 and 7/5

• b. 12/30 and 6/10

• c. 4/20 and 1/5

• d. 8/11 and 8/22

Multiple Choice

3/4 and 1/3

Multiply. Simplify the answer.

• 4/7

• 3/3

• 3/12

• 1/4

Multiple Choice

4 ⋅ 3/4

Multiply. Simplify the answer and write it as a mixed number.

• 12/4

• 1/4

• 3

• 3/16

Multiple Choice

1 3/5 ⋅ 3 1/3

Multiply. Simplify the answer and write as a mixed number.

• 5 1/3

• 80/15

Dividing Fractions and Mixed Numbers

If the [product] of two numbers is 1, the two numbers are [reciprocals] of each other.

• Example:

• 3/4 x 4/3 = 12/12 =1

Multiple Choice

What is the reciprocal of 3 2/5?

• 5/17

• 17/5

• 6/5

• 5/6

Dividing a fraction by a whole number is the same as multiplying by the reciprocal, so you can always use multiplication of fractions to solve such division problems.

• Example: Determine 2 2/3 ÷ 4

• 2 2/3 = 8/3 ÷ 4

• 8/3 x 1/4 = 8/12 = 2/3

Multiple Choice

Find 5 1/3 ÷ 2/3. Simplify the answer and write as a mixed number.

• 16/3

• 16/2

• 8

• 4

Adding and Subtracting Fractions and Mixed Numbers with Like Denominators

Adding & Substracting Fractions

• Adding Fractions with Like Denominators

  1. Add the numerators (the number in the top of each fraction).

  2. Keep the denominator (the bottom number) the same.

  3. Simplify to lowest terms.

• Subtracting Fractions with Like Denominators

  If the denominators (bottoms) of the fractions are the same, subtract the numerators

  (tops) and keep the denominator the same. Remember to simplify the resulting fraction,

  if possible.

Substracting Mixed Numbers

• If the fractional part of the mixed number being subtracted is larger than the fractional part of the mixed number from which it is being subtracted, or if a mixed number is being subtracted from a whole number, follow these steps:

• 1. Subtract 1 from the whole number part of the mixed number being subtracted.

• 2. Add that 1 to the fraction part to make an improper fraction.

• 3. Then, subtract as with any other mixed numbers.

• Alternatively, you can change both numbers to improper fractions and then subtract.

Multiple Choice

A stack of pamphlets is placed on top of a book. If the stack of pamphlets is 3 1/4 inches thick and the book is 5 3/4

inches thick, how high is the pile?

• 1 inch

• 9 inches

• 6 inches

• 4 inches

Adding and Subtracting Fractions and Mixed Numbers with

Unlike Denominators

Finding Least Common Multiples

• Sometimes fractions do not have the same denominator. They have unlike denominators.

• You can rewrite one or both of the fractions so that they have the same denominator.

• This is called finding a common denominator. While any common

  denominator will do, it is helpful to find the least common multiple of the two numbers in the denominator because this will save having to simplify at the end.

Finding LCM

• The other method for finding the least common multiple is to use prime factorization.

• Identify the greatest number of times any factor appears in either factorization and multiply those factors to get the least common multiple.

Adding Fractions with Unlike Denominators

• 1. Find a common denominator.

  
  

• 2. Rewrite each fraction using the common denominator.

  
  

• 3. Now that the fractions have a common denominator, you can add the

  numerators.

  
  

• 4. Simplify to lowest terms, expressing improper fractions as mixed numbers.

Multiple Choice

2/3 + 4/5 + 1/12

Add. Simplify the answer and write as a mixed number.

• 93/60

• 1 11/20

• 2 4/5

• 48/60

Summary

• To adding or subtracting fractions with unlike denominators, first find a common denominator. The least common denominator is easiest to use. The least common multiple can be used as the least common denominator.