4.7 Summary of key concepts

Summary of key concepts

Fraction ( [link] )

Fraction bar, denominator, numerator ( [link] )

1. The fraction bar $\frac{ }{ }$ $\frac{}{}$
2. The nonzero whole number below the fraction bar is the denominator .
3. The whole number above the fraction bar is the numerator .

Proper fraction ( [link] )

$\frac{4}{5}$ is a proper fraction

Improper fraction ( [link] )

$\frac{5}{4}$ , $\frac{5}{5}$ , and $\frac{5}{1}$ are improper fractions

Mixed number ( [link] )

$1\frac{1}{5}$ is a mixed number $\left(1\frac{1}{5}=1+\frac{1}{5}\right)$

Correspondence between improper fractions and mixed numbers ( [link] )

Converting an improper fraction to a mixed number ( [link] )

$\frac{5}{4}$ can be converted to $1\frac{1}{4}$

Converting a mixed number to an improper fraction ( [link] )

$5\frac{7}{8}$ can be converted to $\frac{\text{47}}{8}$

Equivalent fractions ( [link] )

$\frac{3}{4}$ and $\frac{6}{8}$ are equivalent fractions

Test for equivalent fractions ( [link] )

Thus, $\frac{3}{4}$ and $\frac{6}{8}$ are equivalent.

Relatively prime ( [link] )

3 and 4 are relatively prime

Reduced to lowest terms ( [link] )

The number $\frac{3}{4}$ is reduced to lowest terms, since 3 and 4 are relatively prime.

The number $\frac{6}{8}$ is not reduced to lowest terms since 6 and 8 are not relatively prime.

Reducing fractions to lowest terms ( [link] )

Raising fractions to higher terms ( [link] )

$\frac{3}{4}=\frac{3\cdot 2}{4\cdot 2}=\frac{6}{8}$

The word “of” means multiplication ( [link] )

Multiplication of fractions ( [link] )

$\frac{5}{8}\cdot \frac{4}{\text{15}}=\frac{5\cdot 4}{8\cdot \text{15}}=\frac{\text{20}}{\text{120}}=\frac{1}{6}$

Multiplying fractions by dividing out common factors ( [link] )

$\frac{\stackrel{1}{\cancel{5}}}{\underset{2}{\cancel{8}}}\cdot \frac{\stackrel{1}{\cancel{4}}}{\underset{3}{\cancel{15}}}=\frac{1\cdot 1}{2\cdot 3}=\frac{1}{6}$

Multiplication of mixed numbers ( [link] )

Reciprocals ( [link] )

7 and $\frac{1}{7}$ are reciprocals

Division of fractions ( [link] )

$\frac{4}{5}÷\frac{2}{\text{15}}=\frac{4}{5}\cdot \frac{\text{15}}{2}$

Dividing 1 by a fraction ( [link] )

$\frac{1}{\frac{3}{7}}=\frac{7}{3}$

Multiplication statements ( [link] )

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)