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Summary of Key Concepts

Module by: Wade Ellis, Denny Burzynski. E-mail the authorsEdited By: Math Editors

Summary: This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module reviews the key concepts from the chapter "Introduction to Fractions and Multiplication and Division of Fractions."

Summary of Key Concepts

Fraction ((Reference))

The idea of breaking up a whole quantity into equal parts gives us the word fraction.

Fraction Bar, Denominator, Numerator ((Reference))

A fraction has three parts:

  1. The fraction bar                 size 12{ { {} over {} } } {}
  2. The nonzero whole number below the fraction bar is the denominator.
  3. The whole number above the fraction bar is the numerator.

Four-fifths. Four is the numerator, five is the denominator, and a line between them is the fraction bar.

Proper Fraction ((Reference))

Proper fractions are fractions in which the numerator is strictly less than the denominator.

4545 size 12{ { {4} over {5} } } {} is a proper fraction

Improper Fraction ((Reference))

Improper fractions are fractions in which the numerator is greater than or equal to the denominator. Also, any nonzero number placed over 1 is an improper fraction.

5454 size 12{ { {5} over {4} } } {}, 5555 size 12{ { {5} over {5} } } {}, and 5151 size 12{ { {5} over {1} } } {} are improper fractions

Mixed Number ((Reference))

A mixed number is a number that is the sum of a whole number and a proper fraction.

115115 size 12{1 { {1} over {5} } } {} is a mixed number 115=1+15115=1+15 size 12{ left (1 { {1} over {5} } =1+ { {1} over {5} } right )} {}

Correspondence Between Improper Fractions and Mixed Numbers ((Reference))

Each improper fraction corresponds to a particular mixed number, and each mixed number corresponds to a particular improper fraction.

Converting an Improper Fraction to a Mixed Number ((Reference))

A method, based on division, converts an improper fraction to an equivalent mixed number.

5454 size 12{ { {5} over {4} } } {} can be converted to 114114 size 12{1 { {1} over {4} } } {}

Converting a Mixed Number to an Improper Fraction ((Reference))

A method, based on multiplication, converts a mixed number to an equivalent improper fraction.

578578 size 12{5 { {7} over {8} } } {} can be converted to 478478 size 12{ { {"47"} over {8} } } {}

Equivalent Fractions ((Reference))

Fractions that represent the same quantity are equivalent fractions.

3434 size 12{ { {3} over {4} } } {} and 6868 size 12{ { {6} over {8} } } {} are equivalent fractions

Test for Equivalent Fractions ((Reference))

If the cross products of two fractions are equal, then the two fractions are equivalent.

Three fourths and six eighths, with an arrow from each denominator pointing up at the opposite fraction's numerator. This makes three times eight equals four times six, which is equal to twenty-four on both sides.

Thus, 3434 size 12{ { {3} over {4} } } {} and 6868 size 12{ { {6} over {8} } } {} are equivalent.

Relatively Prime ((Reference))

Two whole numbers are relatively prime when 1 is the only number that divides both of them.

3 and 4 are relatively prime

Reduced to Lowest Terms ((Reference))

A fraction is reduced to lowest terms if its numerator and denominator are relatively prime.

The number 3434 size 12{ { {3} over {4} } } {} is reduced to lowest terms, since 3 and 4 are relatively prime.

The number 6868 size 12{ { {6} over {8} } } {} is not reduced to lowest terms since 6 and 8 are not relatively prime.

Reducing Fractions to Lowest Terms ((Reference))

Two methods, one based on dividing out common primes and one based on dividing out any common factors, are available for reducing a fraction to lowest terms.

Raising Fractions to Higher Terms ((Reference))

A fraction can be raised to higher terms by multiplying both the numerator and denominator by the same nonzero number.

3 4 = 3 2 4 2 = 6 8 3 4 = 3 2 4 2 = 6 8 size 12{ { {3} over {4} } = { {3 cdot 2} over {4 cdot 2} } = { {6} over {8} } } {}

The Word “OF” Means Multiplication ((Reference))

In many mathematical applications, the word "of" means multiplication.

Multiplication of Fractions ((Reference))

To multiply two or more fractions, multiply the numerators together and multiply the denominators together. Reduce if possible.

5 8 4 15 = 5 4 8 15 = 20 120 = 1 6 5 8 4 15 = 5 4 8 15 = 20 120 = 1 6 size 12{ { {5} over {8} } cdot { {4} over {"15"} } = { {5 cdot 4} over {8 cdot "15"} } = { {"20"} over {"120"} } = { {1} over {6} } } {}

Multiplying Fractions by Dividing Out Common Factors ((Reference))

Two or more fractions can be multiplied by first dividing out common factors and then using the rule for multiplying fractions.

5 1 8 2 4 1 15 3 = 1 1 2 3 = 1 6 5 1 8 2 4 1 15 3 = 1 1 2 3 = 1 6 size 12{ { { { { {5}}} cSup { size 8{1} } } over { { { {8}}} cSub { size 8{2} } } } cdot { { { { {4}}} cSup { size 8{1} } } over { { { {1}} { {5}}} cSub { size 8{3} } } } = { {1 cdot 1} over {2 cdot 3} } = { {1} over {6} } } {}

Multiplication of Mixed Numbers ((Reference))

To perform a multiplication in which there are mixed numbers, first convert each mixed number to an improper fraction, then multiply. This idea also applies to division of mixed numbers.

Reciprocals ((Reference))

Two numbers whose product is 1 are reciprocals.

7 and 1717 size 12{ { {1} over {7} } } {} are reciprocals

Division of Fractions ((Reference))

To divide one fraction by another fraction, multiply the dividend by the reciprocal of the divisor.

4 5 ÷ 2 15 = 4 5 15 2 4 5 ÷ 2 15 = 4 5 15 2 size 12{ { {4} over {5} } div { {2} over {"15"} } = { {4} over {5} } cdot { {"15"} over {2} } } {}

Dividing 1 by a Fraction ((Reference))

When dividing 1 by a fraction, the quotient is the reciprocal of the fraction.

1 3 7 = 7 3 1 3 7 = 7 3 size 12{ { {1} over { { {3} over {7} } } } = { {7} over {3} } } {}

Multiplication Statements ((Reference))

A mathematical statement of the form

product = (factor 1) (factor 2)

is a multiplication statement.

By omitting one of the three numbers, one of three following problems result:

  1. M = (factor 1) ⋅ (factor 2) Missing product statement.
  2. product = (factor 1) ⋅ M Missing factor statement.
  3. product = M ⋅ (factor 2) Missing factor statement.

Missing products are determined by simply multiplying the known factors. Missing factors are determined by

missing factor = (product) ÷ (known factor)

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Definition of a lens

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A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

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Who can create a lens?

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