Chapter Review: Rational Numbers and Signed Operations

Vocabulary and operational concepts covering rational numbers, long division for terminating and repeating decimals, and signed arithmetic with fractions and integers.

Negative Product Rule

A product of nonzero numbers is negative if and only if there is an odd number of negative factors, such as $(-) \cdot (-) \cdot (-) = (-)$ or $(-) \cdot (-) \cdot (-) \cdot (+) = (-)$.

Form $\frac{m}{n}$

The standard fractional form of a rational number where $m$ and $n$ are integers and $n \neq 0$.

$\frac{m}{n}$ form of $20.75$

The simplified fraction form $20.75 = 20\frac{3}{4} = \frac{20 \cdot 4 + 3}{4} = \frac{83}{4}$.

$\frac{m}{n}$ form of $-0.48$

The simplified fraction form $-0.48 = -\frac{48}{100} = \frac{-12}{25}$.

$\frac{m}{n}$ form of $4\frac{6}{13}$

The improper fraction form $4\frac{6}{13} = \frac{4 \times 13}{13} + \frac{6}{13} = \frac{58}{13}$.

$\frac{m}{n}$ form of $1.34$

The simplified fraction form $1.34 = 1\frac{34}{100} = \frac{134}{100} = \frac{67}{50}$.

Decimal form of $\frac{7}{56}$

The terminating decimal $0.125$, obtained by dividing $7.000$ by $56$.

Decimal form of $9\frac{13}{20}$

The terminating decimal $9.65$, found by using long division to convert $\frac{13}{20}$ to $0.65$.

Decimal form of $-\frac{5}{12}$

The repeating decimal $-0.41\bar{6}$, where the digit $6$ repeats infinitely.

Decimal form of $-2\frac{9}{55}$

The repeating decimal $-2.1\overline{63}$, where the digits $63$ repeat infinitely.

($52 + (-52)$

Opposite numbers whose sum is $0$.

($-8 - (-50)$)

An operation equivalent to adding the positive opposite: $-8 - (-50) = -8 + 50 = 42$.

$$\frac{1}{4} - \frac{7}{8}$$

The value evaluated by finding the common denominator $8$: $\frac{1(2)}{4(2)} - \frac{7}{8} = \frac{2 - 7}{8} = -\frac{5}{8}$.

$$-6\frac{1}{7} + 3\frac{5}{14}$$

The value evaluated by separating integers and fractions: $3 - 6 + \left(\frac{5}{14} - \frac{2}{14}\right) = -3 + \frac{3}{14} = -2\frac{11}{14}$.

$$\frac{-5}{8} \cdot \frac{8}{15}$$

The simplified product obtained by canceling common factors: $-\frac{1}{1} \cdot \frac{1}{3} = -\frac{1}{3}$.

$$\left(-\frac{3}{4}\right) \cdot \left(-1\frac{2}{3}\right)$$

The product evaluated by converting to improper fractions: $-\frac{3}{4} \cdot -\frac{5}{3} = \frac{1}{4} \cdot \frac{5}{1} = \frac{5}{4} = 1\frac{1}{4}$.

$$5\frac{1}{4} \div \left(-\frac{7}{12}\right)$$

The value evaluated by multiplying by the reciprocal: $\frac{21}{4} \div \frac{-7}{12} = -\frac{21}{4} \cdot \frac{12}{7} = -\frac{3}{1} \cdot \frac{3}{1} = -9$.

$$\frac{\left(-\frac{2}{3}\right)}{\left(-\frac{8}{15}\right)}$$

A fraction whose numerator and denominator are fractions, evaluated as $\frac{-2}{3} \div \frac{-8}{15} = \frac{2}{3} \cdot \frac{15}{8} = \frac{1}{1} \cdot \frac{5}{4} = 1\frac{1}{4}$.

positive or negative

positive

Positive or negative

negative

Positive or negative

negative

Locate -1/2

Locate -3 2/3

Locate 3.6

Locate 2.25

-2.75