2.2 Operations with Fractions, Decimals & Signed Numbers

Signed Number Arithmetic: Rules of Signs

Mastering signed numbers (positive and negative values) is necessary for every algebraic and arithmetic problem on the HiSET. Operations with signed numbers follow strict structural laws.

Rules for Addition and Subtraction

Every subtraction problem can be rewritten as addition of the opposite (additive inverse):

$a - b = a + (-b)$ $a - (-b) = a + b$

Rules for Multiplication and Division

Multiplication and division of signed quantities depend solely on the count of negative factors:

$\text{Positive} \times \text{Positive} = \text{Positive} \qquad \frac{\text{Positive}}{\text{Positive}} = \text{Positive}$ $\text{Negative} \times \text{Negative} = \text{Positive} \qquad \frac{\text{Negative}}{\text{Negative}} = \text{Positive}$ $\text{Positive} \times \text{Negative} = \text{Negative} \qquad \frac{\text{Positive}}{\text{Negative}} = \text{Negative}$ $\text{Negative} \times \text{Positive} = \text{Negative} \qquad \frac{\text{Negative}}{\text{Positive}} = \text{Negative}$

Parity Rule for Multi-Factor Products: When multiplying or dividing a chain of non-zero terms, count the total number of negative signs:

• Even number of negatives (\implies) Product is Positive (e.g., ((-2)(-3)(-1)(-4) = +24)).
• Odd number of negatives (\implies) Product is Negative (e.g., ((-2)(-3)(-4) = -24)).

Fractions and Mixed Numbers

A fraction (\frac{a}{b}) represents (a) equal parts of a whole divided into (b) equal portions (where (b \neq 0)). Fractions are classified into three core structures:

• Proper Fraction: Numerator is strictly less than denominator (e.g., (\frac{3}{7}), value (< 1)).
• Improper Fraction: Numerator is greater than or equal to denominator (e.g., (\frac{15}{4}), value (\ge 1)).
• Mixed Number: An integer combined with a proper fraction (e.g., (3\frac{3}{4})).

Conversions Between Improper Fractions and Mixed Numbers

• Mixed Number to Improper Fraction: Multiply the whole number by the denominator, add the numerator, and place the result over the original denominator: $w\frac{n}{d} = \frac{w \cdot d + n}{d} \implies 4\frac{5}{8} = \frac{4 \times 8 + 5}{8} = \frac{37}{8}$
• Improper Fraction to Mixed Number: Perform integer division of numerator by denominator. The quotient becomes the whole part, and the remainder becomes the new numerator: $\frac{43}{6} \implies 43 \div 6 = 7 \text{ with a remainder of } 1 \implies 7\frac{1}{6}$

Addition and Subtraction of Fractions

Fractions cannot be directly added or subtracted unless they share a Least Common Denominator (LCD), which is the Least Common Multiple (LCM) of the denominators.

$\text{Compute: } \frac{5}{12} + \frac{7}{18}$ $\text{Step 1: Prime factorize denominators: } 12 = 2^2 \times 3, \quad 18 = 2 \times 3^2$ $\text{Step 2: Find LCD: } 2^2 \times 3^2 = 4 \times 9 = 36$ $\text{Step 3: Convert to equivalent fractions: } \frac{5 \times 3}{12 \times 3} = \frac{15}{36}, \quad \frac{7 \times 2}{18 \times 2} = \frac{14}{36}$ $\text{Step 4: Combine numerators: } \frac{15 + 14}{36} = \frac{29}{36}$

Subtracting Mixed Numbers with Regrouping (Borrowing)

When subtracting mixed numbers where the second fraction is larger than the first, borrow (1) from the whole number:

$\text{Compute: } 8\frac{1}{4} - 3\frac{5}{6}$ $\text{Step 1: Convert to common denominator (LCD = 12): } 8\frac{3}{12} - 3\frac{10}{12}$ $\text{Step 2: Borrow } 1 = \frac{12}{12} \text{ from } 8: 8\frac{3}{12} = 7 + \left(1 + \frac{3}{12}\right) = 7\frac{15}{12}$ $\text{Step 3: Subtract whole numbers and fractions: } (7 - 3) + \left(\frac{15 - 10}{12}\right) = 4\frac{5}{12}$

Multiplication and Division of Fractions

• Multiplication: Multiply numerators straight across and denominators straight across. Always simplify by canceling common factors before multiplying: $\frac{a}{b} \times \frac{c}{d} = \frac{a \cdot c}{b \cdot d}$ $\frac{14}{25} \times \frac{15}{21} = \frac{14 \times 15}{25 \times 21} = \frac{2 \times 3}{5 \times 3} = \frac{2}{5}$
• Division: Multiply by the reciprocal (invert the second fraction): $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$ $2\frac{2}{3} \div 1\frac{1}{9} = \frac{8}{3} \div \frac{10}{9} = \frac{8}{3} \times \frac{9}{10} = \frac{8 \times 9}{3 \times 10} = \frac{4 \times 3}{1 \times 5} = \frac{12}{5} = 2\frac{2}{5}$"

Decimal Arithmetic and Place Value

Decimals represent fractions with denominators that are powers of 10 ((10^1 = 10), (10^2 = 100), (10^3 = 1000), etc.).

 Thousands │ Hundreds │ Tens │ Ones │ . │ Tenths │ Hundredths │ Thousandths │ Ten-Thousandths
   1000   │   100    │  10  │  1   │ . │  1/10  │   1/100    │   1/1000    │    1/10000
     3    │    4     │  7   │  2   │ . │   5    │     8      │      1      │       6

Core Decimal Operations

1. Addition & Subtraction: Align numbers vertically by their decimal points. Fill trailing empty positions with zeros to prevent place-value alignment errors. $\begin{array}{c} 14.800 \\ -\; 6.345 \\ \hline 8.455 \end{array}$
2. Multiplication: Multiply the numbers as if they were integers (ignoring decimal points). Count the total number of decimal places across all factors, then shift the decimal point to the left by that sum in the final product: $3.14 \text{ (2 decimal places)} \times 0.025 \text{ (3 decimal places)}$ $314 \times 25 = 7,850$ $\text{Total decimal places} = 2 + 3 = 5 \implies 0.07850 = 0.0785$
3. Division: If the divisor is a decimal, multiply both the divisor and the dividend by (10^n) (where (n) is the number of decimal digits in the divisor) to make the divisor an integer. Place the decimal point in the quotient directly above the newly positioned decimal point in the dividend: $18.72 \div 0.08 \implies \frac{18.72 \times 100}{0.08 \times 100} = \frac{1872}{8} = 234$

Converting Between Fractions, Decimals & Repeating Decimals

Terminating Decimals to Fractions

Write the decimal digits over the place value of the final digit, then simplify to lowest terms using the Greatest Common Divisor (GCD):

$0.64 = \frac{64}{100} = \frac{64 \div 4}{100 \div 4} = \frac{16}{25}$ $0.0375 = \frac{375}{10000} = \frac{375 \div 125}{10000 \div 125} = \frac{3}{80}$

Fractions to Decimals

Divide the numerator by the denominator using long division until the remainder is zero (terminating) or a remainder repeats (repeating):

$\frac{5}{16} = 5 \div 16 = 0.3125 \quad \text{(terminates)}$ $\frac{5}{6} = 5 \div 6 = 0.8333\dots = 0.8\overline{3} \quad \text{(repeats)}$

Converting Repeating Decimals to Exact Fractions

To convert a repeating decimal into an exact rational fraction, construct an algebraic multiplier that eliminates the repeating decimal tail.

Case 1: Pure Repeating Decimals

$\text{Convert } x = 0.\overline{36} = 0.363636\dots$ $\text{Because the repeating block has 2 digits, multiply by } 10^2 = 100:$ $\begin{array}{rl} 100x &= 36.363636\dots \\ -\quad x &= \phantom{0}0.363636\dots \\ \hline 99x &= 36 \end{array}$ $x = \frac{36}{99} = \frac{36 \div 9}{99 \div 9} = \frac{4}{11}$

Case 2: Mixed Repeating Decimals (Delayed Repeat)

$\text{Convert } x = 0.4\overline{18} = 0.4181818\dots$ $\text{Step 1: Multiply by } 10^1 = 10 \text{ to align before the repeating block: } 10x = 4.181818\dots$ $\text{Step 2: Multiply by } 10^3 = 1000 \text{ to shift one complete repeating period: } 1000x = 418.181818\dots$ $\text{Step 3: Subtract equations to eliminate repeating tails:}$ $1000x - 10x = 418.1818\dots - 4.1818\dots$ $990x = 414 \implies x = \frac{414}{990} = \frac{414 \div 18}{990 \div 18} = \frac{23}{55}$

Benchmark Fraction-Decimal Equivalencies

Step-by-Step Worked Examples

Example 1: Multi-Step Mixed Number Carpentry Problem

Problem: A carpenter begins with a wooden board measuring (14\frac{1}{2}) feet in length. She cuts three pieces from it measuring (3\frac{3}{4}) feet, (4\frac{5}{8}) feet, and (2\frac{1}{2}) feet. Each saw cut wastes (\frac{1}{16}) foot of wood (kerf loss across (3) cuts). What is the length of the remaining board?

Solution:

• Step 1: Calculate total length of pieces cut $\text{Sum of pieces} = 3\frac{3}{4} + 4\frac{5}{8} + 2\frac{1}{2}$ $\text{Convert to LCD of } 8: 3\frac{6}{8} + 4\frac{5}{8} + 2\frac{4}{8} = (3 + 4 + 2) + \frac{6 + 5 + 4}{8} = 9 + \frac{15}{8} = 9 + 1\frac{7}{8} = 10\frac{7}{8}\text{ feet}$

• Step 2: Calculate kerf waste $3 \text{ cuts} \times \frac{1}{16}\text{ ft} = \frac{3}{16}\text{ feet}$

• Step 3: Sum total wood used $\text{Total removed} = 10\frac{7}{8} + \frac{3}{16} = 10\frac{14}{16} + \frac{3}{16} = 10\frac{17}{16} = 11\frac{1}{16}\text{ feet}$

• Step 4: Subtract from original length $\text{Remaining} = 14\frac{1}{2} - 11\frac{1}{16} = 14\frac{8}{16} - 11\frac{1}{16} = 3\frac{7}{16}\text{ feet}$

Final Result: (3\frac{7}{16}\text{ feet})

Example 2: Signed Rational Arithmetic with Complex Fraction

Problem: Evaluate the expression:

$\left( -\frac{3}{4} \right)^2 - \frac{5}{8} \div \left( -\frac{15}{4} \right)$

Solution:

• Step 1: Evaluate exponent $\left( -\frac{3}{4} \right)^2 = \left(-\frac{3}{4}\right) \times \left(-\frac{3}{4}\right) = +\frac{9}{16}$

• Step 2: Evaluate division (multiply by reciprocal) $\frac{5}{8} \div \left( -\frac{15}{4} \right) = \frac{5}{8} \times \left( -\frac{4}{15} \right)$ $\text{Simplify common factors: } \frac{5}{8} \times \left( -\frac{4}{15} \right) = -\frac{5 \times 4}{8 \times 15} = -\frac{1 \times 1}{2 \times 3} = -\frac{1}{6}$

• Step 3: Perform subtraction $\frac{9}{16} - \left( -\frac{1}{6} \right) = \frac{9}{16} + \frac{1}{6}$ $\text{Find LCD of } 16 \text{ and } 6: 16 = 2^4, \; 6 = 2 \times 3 \implies \text{LCD} = 2^4 \times 3 = 48$ $\frac{9 \times 3}{16 \times 3} + \frac{1 \times 8}{6 \times 8} = \frac{27}{48} + \frac{8}{48} = \frac{35}{48}$

Final Result: (\frac{35}{48})"