4.7: Summary of Key Concepts
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Summary of Key Concepts
Fraction
The idea of breaking up a whole quantity into equal parts gives us the word fraction.
Fraction Bar, Denominator, Numerator
A fraction has three parts:
The fraction bar -
The nonzero whole number below the fraction bar is the denominator.
The whole number above the fraction bar is the numerator.
Proper Fraction
Proper fractions are fractions in which the numerator is strictly less than the denominator.
\(\dfrac{4}{5}\) is a proper fraction
Improper Fraction
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.
\(\dfrac{5}{4}\), \(\dfrac{5}{5}\), and \(\dfrac{5}{1}\) are improper fractions.
Mixed Number
A mixed number is a number that is the sum of a whole number and a proper fraction.
\(1\dfrac{1}{5}\) is a mixed number \((1 \dfrac{1}{5} = 1 + \dfrac{1}{5})\)
Correspondence Between Improper Fractions and Mixed Numbers
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
A method, based on division, converts an improper fraction to an equivalent mixed number.
\(\dfrac{5}{4}\) can be converted to \(1\dfrac{1}{4}\)
Converting a Mixed Number to an Improper Fraction
A method, based on multiplication, converts a mixed number to an equivalent improper fraction.
\(5\dfrac{7}{8}\) can be converted to \(\dfrac{47}{8}\)
Equivalent Fractions
Fractions that represent the same quantity are equivalent fractions.
\(\dfrac{3}{4}\) and \(\dfrac{6}{8}\) are equivalent fractions
Test for Equivalent Fractions
If the cross products of two fractions are equal, then the two fractions are equivalent.
Thus, \(\dfrac{3}{4}\) and \(\dfrac{6}{8}\) are equivalent.
Relatively Prime
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
A fraction is reduced to lowest terms if its numerator and denominator are relatively prime.
The number \(\dfrac{3}{4}\) is reduced to lowest terms, since 3 and 4 are relatively prime.
The number \(\dfrac{6}{8}\) is not reduced to lowest terms since 6 and 8 are not relatively prime.
Reducing Fractions to Lowest Terms
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
A fraction can be raised to higher terms by multiplying both the numerator and denominator by the same nonzero number.
\(\dfrac{3}{4} = \dfrac{3 \cdot 2}{4 \cdot 2} = \dfrac{6}{8}\)
The Word “OF” Means Multiplication
In many mathematical applications, the word "of" means multiplication.
Multiplication of Fractions
To multiply two or more fractions, multiply the numerators together and multiply the denominators together. Reduce if possible.
\(\dfrac{5}{8} \cdot \dfrac{4}{15} = \dfrac{5 \cdot 4}{8 \cdot 15} = \dfrac{20}{120} = \dfrac{1}{6}\)
Multiplying Fractions by Dividing Out Common Factors
Two or more fractions can be multiplied by first dividing out common factors and then using the rule for multiplying fractions.
\(\dfrac{\begin{array} {c} {^1} \\ {\cancel{5}} \end{array}}{\begin{array} {c} {\cancel{8}} \\ {^2} \end{array}} \cdot \dfrac{\begin{array} {c} {^1} \\ {\cancel{4}} \end{array}}{\begin{array} {c} {\cancel{15}} \\ {^3} \end{array}} = \dfrac{1 \cdot 1}{2 \cdot 3} = \dfrac{1}{6}\)
Multiplication of Mixed Numbers
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
Two numbers whose product is 1 are reciprocals.
7 and \(\dfrac{1}{7}\) are reciprocals
Division of Fractions
To divide one fraction by another fraction, multiply the dividend by the reciprocal of the divisor.
\(\dfrac{1}{\dfrac{3}{7}} = \dfrac{7}{3}\)
Multiplication Statements
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:
M = (factor 1) \(\cdot\) (factor 2) Missing product statement.
product = (factor 1) \(\cdot\) M Missing factor statement.
product = M \(\cdot\) (factor 2) Missing factor statement.
Missing products are determined by simply multiplying the known factors. Missing factors are determined by
missing factor = (product) \(\div\) (known factor)