How to Classify Fractions
Define the term "fraction., Distinguish the numerator from the denominator., Point out a proper fraction.In any proper fraction, the numerator is less than the denominator., Pick out an improper fraction., Know what a whole fraction is., Identify a...
Step-by-Step Guide
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Step 1: Define the term "fraction.
"A fraction is a specific type of numerical expression in which one quantity is divided by a second quantity.
Another way to think of a fraction is as a value that represents a portion of a whole.
For example, if there are eight apples and you take three of them, you have taken three of the eight, or three eights.
Written in fraction form, the value would be presented as: 3/8 , Simply put, the numerator is the number on the top and the denominator is the number on the bottom.
When treated as a mathematical function rather than an expression, the numerator is divided by the denominator.
For example, with the fraction 5/9:
The number 5 is the numerator.
The number 9 is the denominator. , Example: 2/5 Every proper fraction expresses a value that is less than
1.
A proper fraction represents a partial portion of a whole.
For example, if there are five books and you read two of them, you have read two out of five books, or two-fifths.
Written numerically as a proper fraction, that would be: 2/5 -
Step 2: Distinguish the numerator from the denominator.
For any improper fraction, the numerator is larger than the denominator.
Example: 5/4 Every improper fraction expresses a value that is greater than
1.
An improper fraction represents an amount that is greater than a whole.
For example, if you were only supposed to sleep for eight hours but ended up sleeping for nine hours, you have slept for nine out of eight hours, or nine-eighth hours.
Written numerically as an improper fraction, that would be: 9/8 Also note that an improper fraction can be expressed as a mixed number. , With a whole fraction, the numerator and the denominator are equal.
Example: 3/3 Every improper fraction expresses a value equal to
1.
A whole fraction represents an amount that is exactly equal to a single whole.
For example, if there are three pieces of candy and you eat all three pieces, you have eaten three out of three or three-thirds of the candy.
Written numerically as a whole fraction, that would be: 3/3 In the same example, you could also write that you have eaten the whole amount, or one (1) serving.
Note that a whole fraction differs from a whole number.
A whole number refers to all positive "counting" numbers: 1, 2, 3, 4, 5, and so on.
Whole numbers also include "0" but do not include negative numbers., Example: 1-1/2 Writing a value as a mixed number allows you to avoid the use of improper fractions.
Like an improper fraction, a mixed number represents an amount that is greater than a whole.
Unlike an improper fraction, however, a mixed number expresses the whole separately and identifies the remaining partial whole, as well.
For example, if you are supposed to sleep for eight hours and sleep for nine hours, instead, you have slept for nine out of eight hours.
You could also express this by stating that you have slept a full whole, 1, as well as one hour of an additional eight hour cycle, or one out of eight (one-eighth).
Written numerically as a mixed number, the value would be: 1-1/8 , In a pair or set of fractions, if the fractions all have the same denominator, they are said to be homogeneous.
Example: 1/7, 4/7, 9/7 These fractions represents values of a whole that are divided into the same portions.
For example, if the serving size of one type of candy is seven pieces and the serving size of a second type is also seven pieces, both fractions would be represented in a homogeneous pair of sevenths.
Homogeneous fractions can be added and subtracted as they are written. , Fractions within a pair or set that do not have the same denominator are classified as heterogeneous.
Example: 1/3, 2/5, 7/6 These fractions represent values of a whole when those wholes are divided into different portions.
For example, if the serving size of one type of candy is three pieces and the serving size of a second type of candy is five pieces, those serving sizes will be represented as thirds and fifths, respectively.
Note that heterogeneous fractions cannot be added or subtracted as they are written.
They must first be changed into homogeneous fractions. , Equivalent fractions within a pair or set are those that have the same value when reduced to their simplest forms.
Example: 1/2, 2/4, 3/6, 4/8 If you simplified or reduced each number in the above example set, all four numbers would equal 1/2.
Two is half of four (2/4), three is half of six (3/6), and four is half of eight (4/8), just as one is half of two (1/2).
You can check for equivalent pairs by cross-multiplying the fractions, as well.
When you cross multiply fractions, you multiply the numerator of the first fraction with the denominator of the second, and the numerator of the second fraction with the denominator of the first.
If the two products are equal, then the fractions in the pair are equivalent.
Example: 3/6 and 4/8 3 * 8 = 4 * 6 24 = 24 , Fractions should always be reduced or simplified to their smallest equivalent form.
When a fraction is simplified as much as possible, it is then classified as irreducible.
Example: 4/7, 1/3 To determine if a fraction can be reduced further, divide both the numerator and denominator by the value of the numerator or by the greatest common factor between the numerator and denominator.
For the fraction 3/9 Divide both the numerator and denominator by the number
3. (3/3) / (9/3) = 1 / 3 The irreducible form of the fraction is 1/3.
For the fraction 6/15 Divide both the numerator and denominator by
3. (6/3) / (15/3) = 2 / 5 The irreducible form of the fraction is 2/5. , As with integers, fractions can either be positive or negative.
A negative fraction is represented by a negative sign (-) placed directly to the left of the number.
Example:
-5/9 Other than the negative value, these fractions can be classified using the same sets of rules you would use when classifying individual fractions and fraction sets. , From the following set, identify the proper fraction, improper fraction, whole fraction, and mixed number: 9/5, 3-1/4, 7/9, 3/3 The proper fraction is 7/9.
The improper fraction is 9/5.
The whole fraction is 3/3.
The mixed number is 3-1/4. , From the following set, identify which pair is homogeneous, which is heterogeneous, and which is equivalent: 3/9, 1/3; 2/3, 4/3; 1/6, 2/5 The homogeneous pair is 2/3 and 4/3.
The heterogeneous pair is 1/6 and 2/5.
The equivalent pair is 3/9 and 1/3. , From the following set, identify which fraction is irreducible and which fractions are reducible: 6/12, 3/9, 4/7, 4/2 The fraction 6/12 is not irreducible; it can be simplified to 1/2.
The fraction 3/9 is not irreducible; it can be simplified to 1/3.
The fraction 4/2 is not irreducible; it can be simplified to 2/1 or
2.
The fraction 4/7 is irreducible.
It cannot be simplified any further than it already is. , From the following set, identify which fraction is the negative fraction and which is the positive fraction: 1/2,
-5/7 The number 1/2 is positive.
The number
-5/7 is negative. -
Step 3: Point out a proper fraction.In any proper fraction
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Step 4: the numerator is less than the denominator.
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Step 5: Pick out an improper fraction.
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Step 6: Know what a whole fraction is.
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Step 7: Identify a mixed number.A mixed number contains both a whole number (or integer) and a fraction.
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Step 8: Classify homogeneous fractions.
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Step 9: Identify heterogeneous fractions.
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Step 10: Pick out equivalents.
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Step 11: Identify irreducible fractions.
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Step 12: Note that fractions can be either positive or negative.
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Step 13: Classify each fraction individually.
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Step 14: Classify each fraction pair.
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Step 15: Identify the irreducible fraction.
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Step 16: Identify the negative fraction.
Detailed Guide
"A fraction is a specific type of numerical expression in which one quantity is divided by a second quantity.
Another way to think of a fraction is as a value that represents a portion of a whole.
For example, if there are eight apples and you take three of them, you have taken three of the eight, or three eights.
Written in fraction form, the value would be presented as: 3/8 , Simply put, the numerator is the number on the top and the denominator is the number on the bottom.
When treated as a mathematical function rather than an expression, the numerator is divided by the denominator.
For example, with the fraction 5/9:
The number 5 is the numerator.
The number 9 is the denominator. , Example: 2/5 Every proper fraction expresses a value that is less than
1.
A proper fraction represents a partial portion of a whole.
For example, if there are five books and you read two of them, you have read two out of five books, or two-fifths.
Written numerically as a proper fraction, that would be: 2/5
For any improper fraction, the numerator is larger than the denominator.
Example: 5/4 Every improper fraction expresses a value that is greater than
1.
An improper fraction represents an amount that is greater than a whole.
For example, if you were only supposed to sleep for eight hours but ended up sleeping for nine hours, you have slept for nine out of eight hours, or nine-eighth hours.
Written numerically as an improper fraction, that would be: 9/8 Also note that an improper fraction can be expressed as a mixed number. , With a whole fraction, the numerator and the denominator are equal.
Example: 3/3 Every improper fraction expresses a value equal to
1.
A whole fraction represents an amount that is exactly equal to a single whole.
For example, if there are three pieces of candy and you eat all three pieces, you have eaten three out of three or three-thirds of the candy.
Written numerically as a whole fraction, that would be: 3/3 In the same example, you could also write that you have eaten the whole amount, or one (1) serving.
Note that a whole fraction differs from a whole number.
A whole number refers to all positive "counting" numbers: 1, 2, 3, 4, 5, and so on.
Whole numbers also include "0" but do not include negative numbers., Example: 1-1/2 Writing a value as a mixed number allows you to avoid the use of improper fractions.
Like an improper fraction, a mixed number represents an amount that is greater than a whole.
Unlike an improper fraction, however, a mixed number expresses the whole separately and identifies the remaining partial whole, as well.
For example, if you are supposed to sleep for eight hours and sleep for nine hours, instead, you have slept for nine out of eight hours.
You could also express this by stating that you have slept a full whole, 1, as well as one hour of an additional eight hour cycle, or one out of eight (one-eighth).
Written numerically as a mixed number, the value would be: 1-1/8 , In a pair or set of fractions, if the fractions all have the same denominator, they are said to be homogeneous.
Example: 1/7, 4/7, 9/7 These fractions represents values of a whole that are divided into the same portions.
For example, if the serving size of one type of candy is seven pieces and the serving size of a second type is also seven pieces, both fractions would be represented in a homogeneous pair of sevenths.
Homogeneous fractions can be added and subtracted as they are written. , Fractions within a pair or set that do not have the same denominator are classified as heterogeneous.
Example: 1/3, 2/5, 7/6 These fractions represent values of a whole when those wholes are divided into different portions.
For example, if the serving size of one type of candy is three pieces and the serving size of a second type of candy is five pieces, those serving sizes will be represented as thirds and fifths, respectively.
Note that heterogeneous fractions cannot be added or subtracted as they are written.
They must first be changed into homogeneous fractions. , Equivalent fractions within a pair or set are those that have the same value when reduced to their simplest forms.
Example: 1/2, 2/4, 3/6, 4/8 If you simplified or reduced each number in the above example set, all four numbers would equal 1/2.
Two is half of four (2/4), three is half of six (3/6), and four is half of eight (4/8), just as one is half of two (1/2).
You can check for equivalent pairs by cross-multiplying the fractions, as well.
When you cross multiply fractions, you multiply the numerator of the first fraction with the denominator of the second, and the numerator of the second fraction with the denominator of the first.
If the two products are equal, then the fractions in the pair are equivalent.
Example: 3/6 and 4/8 3 * 8 = 4 * 6 24 = 24 , Fractions should always be reduced or simplified to their smallest equivalent form.
When a fraction is simplified as much as possible, it is then classified as irreducible.
Example: 4/7, 1/3 To determine if a fraction can be reduced further, divide both the numerator and denominator by the value of the numerator or by the greatest common factor between the numerator and denominator.
For the fraction 3/9 Divide both the numerator and denominator by the number
3. (3/3) / (9/3) = 1 / 3 The irreducible form of the fraction is 1/3.
For the fraction 6/15 Divide both the numerator and denominator by
3. (6/3) / (15/3) = 2 / 5 The irreducible form of the fraction is 2/5. , As with integers, fractions can either be positive or negative.
A negative fraction is represented by a negative sign (-) placed directly to the left of the number.
Example:
-5/9 Other than the negative value, these fractions can be classified using the same sets of rules you would use when classifying individual fractions and fraction sets. , From the following set, identify the proper fraction, improper fraction, whole fraction, and mixed number: 9/5, 3-1/4, 7/9, 3/3 The proper fraction is 7/9.
The improper fraction is 9/5.
The whole fraction is 3/3.
The mixed number is 3-1/4. , From the following set, identify which pair is homogeneous, which is heterogeneous, and which is equivalent: 3/9, 1/3; 2/3, 4/3; 1/6, 2/5 The homogeneous pair is 2/3 and 4/3.
The heterogeneous pair is 1/6 and 2/5.
The equivalent pair is 3/9 and 1/3. , From the following set, identify which fraction is irreducible and which fractions are reducible: 6/12, 3/9, 4/7, 4/2 The fraction 6/12 is not irreducible; it can be simplified to 1/2.
The fraction 3/9 is not irreducible; it can be simplified to 1/3.
The fraction 4/2 is not irreducible; it can be simplified to 2/1 or
2.
The fraction 4/7 is irreducible.
It cannot be simplified any further than it already is. , From the following set, identify which fraction is the negative fraction and which is the positive fraction: 1/2,
-5/7 The number 1/2 is positive.
The number
-5/7 is negative.
About the Author
Paul Burns
Experienced content creator specializing in organization guides and tutorials.
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