Comprehensive Study Guide: Operations with Rational Numbers and Fractions

Vocabulary terms, rules, and mathematical procedures covering decimal conversions, rational numbers, and operations with like and unlike fractions.

Repeating Decimal

A decimal in which a sequence of digits repeats endlessly, indicated by placing a bar over the repeating digit or sequence of digits.

Terminating Decimal

A decimal reached by dividing the numerator by the denominator until the remainder is $0$.

Greatest Common Factor (GCF)

The largest factor shared by the numerator and denominator, used to divide both values to express a fraction in simplest form (for example, dividing by $25$ to reduce $\frac{25}{100}$ to $\frac{1}{4}$).

Algebraic Method for Repeating Decimals

A procedure for converting a repeating decimal to a fraction where $N$ is set equal to the decimal, multiplied by a power of $10$ (such as $10N$), and subtracted by $N$ to eliminate the repeating sequence and solve for $N$.

Comparing Rational Numbers

The process of determining whether numbers are related by $<$, $>$, or $=$ by writing fractions as decimals and comparing corresponding place values from left to right (such as showing $0.8 > 0.75$ because $8 > 7$ in the tenths place).

Rule for Subtracting a Negative Fraction

An operational rule stating that subtracting a negative fraction is equivalent to adding the corresponding positive fraction, exemplified by $\frac{3}{4} - \left(-\frac{3}{4}\right) = \frac{3}{4} + \frac{3}{4}$.

Least Common Denominator (LCD)

The least common multiple of the denominators of two or more fractions, used to rename fractions with unlike denominators before adding or subtracting (such as using $24$ for denominators $8$ and $6$, or $36$ for denominators $9$ and $12$).

Adding and Subtracting Like Fractions

The procedure for combining fractions that have the same denominator by adding or subtracting only the numerators while keeping the denominator unchanged.

Renaming a Mixed Number

The process of regrouping $1$ whole from the integer part into fraction units (such as renaming $6\frac{20}{36}$ as $5\frac{56}{36}$) when the fractional part being subtracted from is smaller than the fractional part being subtracted.

Improper Fraction Method for Subtracting Mixed Numbers

A technique for finding the difference of mixed numbers by converting each mixed number into an improper fraction (for example, $5\frac{7}{8} = \frac{47}{8}$ and $8\frac{3}{8} = \frac{67}{8}$), subtracting the numerators, and simplifying the result.

Simplest Form

The form of a fraction or mixed number when the numerator and denominator share no common factors other than $1$, or when an improper fraction is converted to a reduced mixed number (such as simplifying $-\frac{20}{8}$ to $-\frac{5}{2}$ or $-2\frac{1}{2}$).