Basic Math and Pre-Algebra – Fractions - Quick Guides

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Study GuideBasic Math and Pre-Algebra–FracƟons1. Proper and Improper FracƟons1.1What Is a Proper FracƟon?Aproper fractionis a fraction where thenumerator is smaller than the denominator.Because the top number is smaller, the fraction isless than one.For example:3/5Since 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, 10/12In each case, the numerator is smaller than the denominator.1.2What Is an Improper FracƟon?Animproper fractionis a fraction where thenumerator is greater than or equal to thedenominator.This means the fraction isgreater 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, 12/12In each of these, the numerator is either larger than or equal to the denominator.

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Study GuideKey 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 orequal to 1).Understanding the difference makes it easier to compare fractions and work with them in mathproblems.Summary•Aproper fractionhas a numerator that is smaller than the denominator.•Proper fractions are alwaysless than 1(example: 3/5).•Examples of proper fractions include 4/7, 2/5, 1/9, and 10/12.•Animproper fractionhas a numerator that is greater than or equal to the denominator.•Improper fractions aregreater than or equal to 1(examples: 12/7 is greater than 1, 6/6equals 1).•Examples of improper fractions include 7/4, 3/2, 10/3, 16/15, and 12/12.2. Mixed NumbersSometimes a number includes both awhole numberand afraction. When this happens, the numberis called amixed 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 Guide2.1Changing Improper FracƟons to Mixed NumbersRemember, animproper fractionhas 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:SolveChange 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 1/3The 3 is the whole number, and 1/3 is the fraction.2.2Changing Mixed Numbers to Improper FracƟonsNow 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:SolveChange 5(3/4)to an improper fraction.Step 1: Multiply 4 × 5 = 20Step 2: Add 3→20 + 3 = 23Step 3: Place over the original denominator 4So:5 3/4 = 23/4

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Study GuideSummary•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 wholenumber.•Add the numerator to that product and write the result over the original denominator.3. Renaming FracƟonsWhen working with fractions, you often need torenameorsimplifythem. These skills are especiallyimportant when you add, subtract, multiply, or divide fractions.3.1Equivalent FracƟonsSome fractions may look different but actually represent thesame value. These are calledequivalent fractions.For example:1/2, 2/4, 3/6, 4/8, and 5/10All of these name the same number.How to Check if Fractions Are EquivalentA quick way to check is bycross multiplying.Example 1:SimplifyIs 2/4 equivalent to 3/6?Cross multiply:2 × 6 = 124 × 3 = 12Since the cross products are equal, the fractions are equivalent.

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Study GuideExample 2:SimplifyIs 3/4 equivalent to 2/3?Cross multiply:3 × 3 = 94 × 2 = 8Since 9 is not equal to 8, the fractions are not equivalent.3.2Simplifying FracƟonsWhen a fraction is your final answer, it should be written inlowest terms.To simplify a fraction:•Divide both the numerator and denominator by thelargest numberthat divides evenly intoboth.Example 3:SimplifySimplify 15/25.Both 15 and 25 can be divided by 5.15 ÷ 5 = 325 ÷ 5 = 5So, 15/25 = 3/5Since 3 and 5 cannot be divided evenly by any number other than 1, 3/5 is in lowest terms.Example 4:SimplifySimplify 8/40.Both numbers can be divided by 8.8 ÷ 8 = 140 ÷ 8 = 5So, 8/40 = 1/5

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Study Guide3.3Enlarging DenominatorsSometimes you need to change a fraction so it has a specific denominator.You can do this by multiplyingboththe numerator and denominator by the same number. The valueof the fraction stays the same.Example 5:SimplifyChange 3/4 to eighths.To get 8 as the denominator, multiply by 2.3 × 2 = 64 × 2 = 8So, 3/4 = 6/8Example 6:SimplifyExpress 1/2 as tenths.To get 10 as the denominator, multiply by 5.1 × 5 = 52 × 5 = 10So, 1/2 = 5/10Summary•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 thesame number.•Renaming fractions helps you prepare them for operations like addition, subtraction,multiplication, and division.

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Study Guide4. Quiz: Proper and Improper FracƟons, Mixed Numbers, andRenaming FracƟonsQuestion1. Which of the following is considered a proper fraction?Answer Choices•3/3•6/5•9/10Correct Answer࿨࿩࿪9/10Why This Is CorrectAproper fractionis a fraction where the numerator (top number) isless thanthe denominator(bottom number). This means the value of the fraction isless than 1.•3/3 = 1→This equals 1, so it isnota proper fraction.•6/5 = 1.2→This isgreater than 1, so it is an improper fraction.•9/10→The numerator (9) is less than the denominator (10), so it isless than 1.Therefore,9/10 is a proper fraction.2.QuestionIn 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 GuideCorrect Answer࿨࿩࿪both A and B are correctWhy This Is CorrectAnimproper fractionis a fraction where the numerator (top number) isgreater than or equal tothedenominator (bottom number).•If the numerator isgreater thanthe denominator (like 7/4), the fraction is greater than 1.•If the numerator isequal tothe denominator (like 5/5), the fraction equals 1.Both situations make the fractionimproper, so the correct answer isboth A and B are correct.3.QuestionChange 15/4 into a mixed number.Answer Choices•3 3/4•3 3/15•3.3Correct Answer࿨࿩࿪3 3/4Why This Is CorrectTo change an improper fraction into a mixed number, divide the numerator by the denominator.15 ÷ 4 = 3 remainder 3This means:•4 goes into 15 three whole times•There are 3 left overSo the mixed number is:3 3/4The fraction part keeps the same denominator (4).

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Study Guide4.QuestionChange 7(3/5)into a fraction.Answer Choices•26/3•38/5•26/5Correct Answer࿨࿩࿪38/5Why This Is CorrectTo change a mixed number into an improper fraction:1.Multiply the whole number by the denominator.2.7 × 5 = 353.Add the numerator.4.35 + 3 = 385.Keep the same denominator (5).So,7 3/5 = 38/55.QuestionWhich answer has three equivalent fractions?

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Study GuideAnswer Choices•12/16;15/20;24/30•15/18;20/24;35/42•12/21;16/28;32/60Correct Answer࿨࿩࿪15/18;20/24;35/42Why This Is CorrectEquivalent fractions reduce to the same simplest form.First Choice:12/16 = 3/415/20 = 3/424/30 = 4/5Not all are equalᤶᤷSecond Choice:15/18 = 5/620/24 = 5/635/42 = 5/6All three simplify to5/6࿨࿩࿪Third Choice:12/21 = 4/716/28 = 4/732/60 = 8/15Not all are equalᤶᤷSo the correct answer is:15/18;20/24;35/42

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