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Basic Math and Pre-Algebra – Fractions

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    Study Guide Basic Math and Pre - Algebra – Frac Ɵ ons 1. Proper and Improper Frac Ɵ ons 1.1 What Is a Proper Frac Ɵ on? A proper fraction is a fraction where the numerator is smaller than the denominator . Because the top number is smaller, the fraction is less than one . For example: 3/5 Since 3 is less than 5, the value of 3/5 is less than 1. More examples of proper fractions: 4/7, 2/5, 1/9, 10/12 In each case, the numerator is smaller than the denominator. 1.2 What Is an Improper Frac Ɵ on? An improper fraction is a fraction where the numerator is greater than or equal to the denominator . This means the fraction is greater than or equal to one . For example: 12/7 is greater than 1 because 12 is larger than 7. 6/6 is equal to 1 because the numerator and denominator are the same. More examples of improper fractions: 7/4, 3/2, 10/3, 16/15, 12/12 In each of these, the numerator is either larger than or equal to the denominator.

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    Study Guide Key Idea to Remember • If the numerator is smaller than the denominator → proper fraction (less than 1). • If the numerator is greater than or equal to the denominator → improper fraction (greater than or equal to 1). Understanding the difference makes it easier to compare fractions and work with them in math problems. Summary • A proper fraction has a numerator that is smaller than the denominator. • Proper fractions are always less than 1 (example: 3/5). • Examples of proper fractions include 4/7, 2/5, 1/9, and 10/12. • An improper fraction has a numerator that is greater than or equal to the denominator. • Improper fractions are greater than or equal to 1 (examples: 12/7 is greater than 1, 6/6 equals 1). • Examples of improper fractions include 7/4, 3/2, 10/3, 16/15, and 12/12. 2. Mixed Numbers Sometimes a number includes both a whole number and a fraction . When this happens, the number is called a mixed number . For example: 5 ( 1/4 ) 290 ( 3/4 ) Both of these numbers have a whole number part and a fraction part, so they are mixed numbers.

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    Study Guide 2.1 Changing Improper Frac Ɵ ons to Mixed Numbers Remember, an improper fraction has a numerator that is greater than or equal to the denominator. To change an improper fraction into a mixed number: 1. Divide the numerator by the denominator. 2. The quotient (answer) becomes the whole number. 3. The remainder becomes the numerator of the fraction. 4. Keep the same denominator. Example 1 : Solve Change 10/3 to a mixed number. Divide 10 by 3. 3 goes into 10 three times with a remainder of 1. So: 10/3 = 3 1/3 The 3 is the whole number, and 1/3 is the fraction. 2.2 Changing Mixed Numbers to Improper Frac Ɵ ons Now let’s go the other way. To change a mixed number into an improper fraction: 1. Multiply the denominator by the whole number. 2. Add the numerator. 3. Put the total over the original denominator. Example 2 : Solve Change 5 ( 3/4 ) to an improper fraction. Step 1: Multiply 4 × 5 = 20 Step 2: Add 3 → 20 + 3 = 23 Step 3: Place over the original denominator 4 So: 5 3/4 = 23/4

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    Study Guide Summary • A mixed number contains both a whole number and a fraction (example: 5 1/4). • An improper fraction has a numerator greater than or equal to the denominator. • To change an improper fraction to a mixed number, divide the numerator by the denominator. • The quotient becomes the whole number, and the remainder becomes the new numerator (keep the same denominator). • To change a mixed number to an improper fraction, multiply the denominator by the whole number. • Add the numerator to that product and write the result over the original denominator. 3. Renaming Frac Ɵ ons When working with fractions, you often need to rename or simplify them. These skills are especially important when you add, subtract, multiply, or divide fractions. 3.1 Equivalent Frac Ɵ ons Some fractions may look different but actually represent the same value . These are called equivalent fractions . For example: 1/2, 2/4, 3/6, 4/8, and 5/10 All of these name the same number. How to Check if Fractions Are Equivalent A quick way to check is by cross multiplying . Example 1: Simplify Is 2/4 equivalent to 3/6? Cross multiply: 2 × 6 = 12 4 × 3 = 12 Since the cross products are equal, the fractions are equivalent.

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    Study Guide Example 2: Simplify Is 3/4 equivalent to 2/3? Cross multiply: 3 × 3 = 9 4 × 2 = 8 Since 9 is not equal to 8, the fractions are not equivalent. 3.2 Simplifying Frac Ɵ ons When a fraction is your final answer, it should be written in lowest terms . To simplify a fraction: • Divide both the numerator and denominator by the largest number that divides evenly into both. Example 3: Simplify Simplify 15/25. Both 15 and 25 can be divided by 5. 15 ÷ 5 = 3 25 ÷ 5 = 5 So, 15/25 = 3/5 Since 3 and 5 cannot be divided evenly by any number other than 1, 3/5 is in lowest terms. Example 4: Simplify Simplify 8/40. Both numbers can be divided by 8. 8 ÷ 8 = 1 40 ÷ 8 = 5 So, 8/40 = 1/5

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    Study Guide 3.3 Enlarging Denominators Sometimes you need to change a fraction so it has a specific denominator. You can do this by multiplying both the numerator and denominator by the same number. The value of the fraction stays the same. Example 5: Simplify Change 3/4 to eighths. To get 8 as the denominator, multiply by 2. 3 × 2 = 6 4 × 2 = 8 So, 3/4 = 6/8 Example 6: Simplify Express 1/2 as tenths. To get 10 as the denominator, multiply by 5. 1 × 5 = 5 2 × 5 = 10 So, 1/2 = 5/10 Summary • Equivalent fractions have different numbers but represent the same value. • You can check if fractions are equivalent by cross multiplying. • A fraction should be simplified to lowest terms when it is the final answer. • To simplify, divide both the numerator and denominator by the greatest common factor. • You can enlarge a denominator by multiplying both the numerator and denominator by the same number. • Renaming fractions helps you prepare them for operations like addition, subtraction, multiplication, and division.

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    Study Guide 4. Quiz: Proper and Improper Frac Ɵ ons, Mixed Numbers, and Renaming Frac Ɵ ons Question 1. Which of the following is considered a proper fraction? Answer Choices • 3/3 • 6/5 • 9/10 Correct Answer ࿨ ࿩ ࿪ 9/10 Why This Is Correct A proper fraction is a fraction where the numerator (top number) is less than the denominator (bottom number). This means the value of the fraction is less than 1 . • 3/3 = 1 → This equals 1, so it is not a proper fraction. • 6/5 = 1.2 → This is greater than 1 , so it is an improper fraction. • 9/10 → The numerator (9) is less than the denominator (10), so it is less than 1 . Therefore, 9/10 is a proper fraction. 2. Question In order for a fraction to be considered an improper fraction, what must be true? Answer Choices • the numerator is greater than the denominator • the numerator is equal to the denominator • both A and B are correct

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    Study Guide Correct Answer ࿨ ࿩ ࿪ both A and B are correct Why This Is Correct An improper fraction is a fraction where the numerator (top number) is greater than or equal to the denominator (bottom number). • If the numerator is greater than the denominator (like 7/4), the fraction is greater than 1. • If the numerator is equal to the denominator (like 5/5), the fraction equals 1. Both situations make the fraction improper , so the correct answer is both A and B are correct . 3. Question Change 15/4 into a mixed number. Answer Choices • 3 3/4 • 3 3/15 • 3.3 Correct Answer ࿨ ࿩ ࿪ 3 3/4 Why This Is Correct To change an improper fraction into a mixed number, divide the numerator by the denominator. 15 ÷ 4 = 3 remainder 3 This means: • 4 goes into 15 three whole times • There are 3 left over So the mixed number is: 3 3/4 The fraction part keeps the same denominator (4).

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    Study Guide 4. Question Change 7 ( 3/5 ) into a fraction. Answer Choices • 26/3 • 38/5 • 26/5 Correct Answer ࿨ ࿩ ࿪ 38/5 Why This Is Correct To change a mixed number into an improper fraction: 1. Multiply the whole number by the denominator. 2. 7 × 5 = 35 3. Add the numerator. 4. 35 + 3 = 38 5. Keep the same denominator (5). So, 7 3/5 = 38/5 5. Question Which answer has three equivalent fractions?

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    Study Guide Answer Choices • 12/16 ; 15/20 ; 24/30 • 15/18 ; 20/24 ; 35/42 • 12/21 ; 16/28 ; 32/60 Correct Answer ࿨ ࿩ ࿪ 15/18 ; 20/24 ; 35/42 Why This Is Correct Equivalent fractions reduce to the same simplest form. First Choice: 12/16 = 3/4 15/20 = 3/4 24/30 = 4/5 Not all are equal ᤶ ᤷ Second Choice: 15/18 = 5/6 20/24 = 5/6 35/42 = 5/6 All three simplify to 5/6 ࿨ ࿩ ࿪ Third Choice: 12/21 = 4/7 16/28 = 4/7 32/60 = 8/15 Not all are equal ᤶ ᤷ So the correct answer is: 15/18 ; 20/24 ; 35/42

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