Comprehensive Review of Integer Rules, Rational Numbers, Decimal Operations, and Word Problems

Unit Review Overview and Vocabulary Framework

• Core Mathematical Topics Covered:

  • Integer operation rules for addition, subtraction, multiplication, and division.

  • Mathematical terminology classification and structural definitions.

  • Rational number operations, including fraction simplification, multiplication, division, addition, and subtraction.

  • Decimal operations, multi-step word problems, and real-life financial calculations.

Section 0.1: Integer Operation Rules and Definitions

• Rules for Addition and Subtraction of Integers:

  • Two Positive Integers: Add their absolute values together; the resulting sum is positive (e.g., $+a + (+b) = +(a + b)$).

  • Two Negative Integers: Add their absolute values together; the resulting sum retains a negative sign (e.g., $-a + (-b) = -(a + b)$).

  • One Positive and One Negative Integer: Subtract the smaller absolute value from the larger absolute value; the sum takes the sign of the integer with the greater absolute value.

  • Double Negative Signs Directly Adjacent: Two consecutive negative signs simplify directly into a positive sign (e.g., $-(-a) = +a$ or $x - (-y) = x + y$).

• Rules for Multiplication and Division of Integers:

  • Two Positive Integers: The product or quotient of two positive numbers is always positive.

  • Two Negative Integers: The product or quotient of two negative numbers is always positive.

  • One Positive and One Negative Integer: The product or quotient of numbers with opposite signs is always negative.

• Mathematical Terminology Definitions and Examples:

  • Integer:

  • Definition: A whole number that can be positive, negative, or zero, containing no fractional or decimal part.

  • Example: $-5$, $0$, $12$

  • Sum:

  • Definition: The result obtained from adding two or more numbers together.

  • Example: The sum of $7$ and $3$ is $10$ ($7 + 3 = 10$).

  • Difference:

  • Definition: The result obtained from subtracting one number from another.

  • Example: The difference between $9$ and $4$ is $5$ ($9 - 4 = 5$).

  • Product:

  • Definition: The result obtained from multiplying two or more numbers together.

  • Example: The product of $6$ and $8$ is $48$ ($6 \times 8 = 48$).

  • Quotient:

  • Definition: The result obtained from dividing one number by another.

  • Example: The quotient of $24$ divided by $6$ is $4$ ($24 \div 6 = 4$).

  • Positive Integer:

  • Definition: Any whole number strictly greater than zero.

  • Example: $1$, $2$, $15$, $100$

  • Negative Integer:

  • Definition: Any whole number strictly less than zero.

  • Example: $-1$, $-4$, $-23$

  • Zero Pair:

  • Definition: A pair of numbers consisting of one positive unit and one negative unit of equal magnitude whose sum equals zero.

  • Example: $+1$ and $-1$ form a zero pair because $1 + (-1) = 0$.

• Categorization of Vocabulary Words by Arithmetic Symbol:

  • Addition Symbol ($+$):

  • add

  • addition

  • altogether

  • both

  • more

  • plus

  • sum

  • total

  • Subtraction Symbol ($-$):

  • difference

  • fewer

  • less

  • left

  • minus

  • subtract

  • subtraction

  • take-away

  • Multiplication Symbol ($\times$):

  • lots of

  • multiply

  • multiplication

  • product

  • times

  • Division Symbol ($\div$):

  • divide

  • division

  • quotient

  • remain

  • remainder

  • share

Section 0.2: Operations of Rational Numbers

• Definition of Rational Numbers:

  • A rational number is any number that can be written in the form $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$. This set includes integers, fractions, terminating decimals, and repeating decimals.

• Part 1: Simplifying Fractions:

  • Procedure: Divide both the numerator and the denominator by their greatest common factor (GCF) until no common factors remain besides $1$.

  • Problem 1: $\frac{3}{6} = \frac{3 \div 3}{6 \div 3} = \frac{1}{2}$

  • Problem 2: $\frac{3}{18} = \frac{3 \div 3}{18 \div 3} = \frac{1}{6}$

  • Problem 3: $\frac{12}{20} = \frac{12 \div 4}{20 \div 4} = \frac{3}{5}$

  • Problem 4: $\frac{15}{35} = \frac{15 \div 5}{35 \div 5} = \frac{3}{7}$

  • Problem 5: $\frac{25}{100} = \frac{25 \div 25}{100 \div 25} = \frac{1}{4}$

  • Problem 6: $\frac{24}{60} = \frac{24 \div 12}{60 \div 12} = \frac{2}{5}$

  • Problem 7: $\frac{28}{63} = \frac{28 \div 7}{63 \div 7} = \frac{4}{9}$

  • Visual Model of Equivalent Fractions:

• As illustrated above: $\frac{1}{2} = \frac{2}{4} = \frac{3}{6} = \frac{4}{8}$.

  • Part 2: Multiplying Fractions:

• Procedural Steps:

  1. Convert whole numbers or mixed numbers into improper fractions if present.

  2. Multiply the numerators together to find the new numerator: $a \times c$.

  3. Multiply the denominators together to find the new denominator: $b \times d$.

  4. Apply sign rules for integer multiplication.

  5. Simplify the resulting fraction to lowest terms.

• Problem 1: $2 \times \frac{3}{5} = \frac{2}{1} \times \frac{3}{5} = \frac{2 \times 3}{1 \times 5} = \frac{6}{5} = 1\frac{1}{5}$

• Problem 2: $-\frac{5}{11} \times \left(-\frac{11}{10}\right) = \frac{(-5) \times (-11)}{11 \times 10} = \frac{55}{110} = \frac{1}{2}$

• Problem 3: $-\frac{6}{7} \times \frac{1}{4} = \frac{-6 \times 1}{7 \times 4} = -\frac{6}{28} = -\frac{3}{14}$

  • Part 3: Dividing Fractions:

• Procedural Steps ("Keep, Change, Flip"):

  1. Keep the first fraction (dividend) unchanged.

  2. Change the division symbol ($\div$) to a multiplication symbol ($\times$).

  3. Flip the second fraction (divisor) to its reciprocal ($\frac{c}{d} \rightarrow \frac{d}{c}$).

  4. Multiply the numerators and denominators, then simplify to lowest terms.

• Problem 1: $\frac{1}{2} \div \frac{3}{5} = \frac{1}{2} \times \frac{5}{3} = \frac{1 \times 5}{2 \times 3} = \frac{5}{6}$

• Problem 2: $\frac{7}{5} \div \frac{2}{5} = \frac{7}{5} \times \frac{5}{2} = \frac{7 \times 5}{5 \times 2} = \frac{35}{10} = \frac{7}{2} = 3\frac{1}{2}$

• Problem 3: $\frac{2}{3} \div (-4) = \frac{2}{3} \div \left(-\frac{4}{1}\right) = \frac{2}{3} \times \left(-\frac{1}{4}\right) = -\frac{2}{12} = -\frac{1}{6}$

  • Part 4: Adding and Subtracting Fractions:

• Procedural Steps:

  1. If denominators are common, add or subtract numerators directly and keep denominator.

  2. If denominators differ, find the Least Common Denominator (LCD).

  3. Convert each fraction to an equivalent fraction using the LCD.

  4. Add or subtract numerators, keep the LCD, and simplify to lowest terms.

• Problem 1: $\frac{2}{7} + \frac{3}{7} = \frac{2 + 3}{7} = \frac{5}{7}$

• Problem 2: $\frac{5}{12} + \frac{1}{12} = \frac{5 + 1}{12} = \frac{6}{12} = \frac{1}{2}$

• Problem 3: $\frac{4}{11} - \frac{7}{11} = \frac{4 - 7}{11} = -\frac{3}{11}$

• Problem 4: $\frac{2}{7} + \frac{3}{7} = \frac{5}{7}$

• Problem 5: $\frac{1}{4} + \frac{2}{3}$

  • Find LCD of $4$ and $3$, which is $12$.

  • Convert fractions: $\frac{1 \times 3}{4 \times 3} = \frac{3}{12}$ and $\frac{2 \times 4}{3 \times 4} = \frac{8}{12}$.

  • Sum: $\frac{3}{12} + \frac{8}{12} = \frac{11}{12}$.

• Problem 6: $-\frac{3}{4} + \frac{3}{8}$

  • Find LCD of $4$ and $8$, which is $8$.

  • Convert fraction: $-\frac{3 \times 2}{4 \times 2} = -\frac{6}{8}$.

  • Sum: $-\frac{6}{8} + \frac{3}{8} = \frac{-6 + 3}{8} = -\frac{3}{8}$.

• Problem 7: $\frac{2}{7} + \frac{3}{7} = \frac{5}{7}$

• Problem 8: $-\frac{5}{6} - \frac{7}{10}$

  • Find LCD of $6$ and $10$, which is $30$.

  • Convert fractions: $-\frac{5 \times 5}{6 \times 5} = -\frac{25}{30}$ and $\frac{7 \times 3}{10 \times 3} = \frac{21}{30}$.

  • Difference: $-\frac{25}{30} - \frac{21}{30} = \frac{-25 - 21}{30} = -\frac{46}{30} = -\frac{23}{15} = -1\frac{8}{15}$.

• Problem 9: $\frac{11}{15} + \frac{1}{20}$

  • Find LCD of $15$ and $20$, which is $60$.

  • Convert fractions: $\frac{11 \times 4}{15 \times 4} = \frac{44}{60}$ and $\frac{1 \times 3}{20 \times 3} = \frac{3}{60}$.

  • Sum: $\frac{44}{60} + \frac{3}{60} = \frac{47}{60}$.

  • Putting It Together (Arranging Expressions from Least to Greatest):

• Expression Evaluation:

  • Expression A: $\frac{1}{3} + \frac{1}{4} = \frac{4}{12} + \frac{3}{12} = \frac{7}{12} \approx 0.5833$

  • Expression B: $10 - 40 = -30$

  • Expression C: $-\frac{2}{5} + \frac{3}{5} = \frac{1}{5} = 0.20$

  • Expression D: $\left(14\frac{1}{3}\right) \div \left(-3\frac{1}{3}\right) = \frac{43}{3} \div \left(-\frac{10}{3}\right) = \frac{43}{3} \times \left(-\frac{3}{10}\right) = -\frac{43}{10} = -4.3$

• Ordering:

  • Comparing numerical values: $-30 < -4.3 < 0.20 < 0.5833$

  • Expressions ordered from least to greatest: B, D, C, A

Section 0.3: Decimal Operations and Applied Word Problems

• Decimal Addition:

  • Rule: Align decimal points vertically, add placeholder zeros where needed, and perform addition.

  • Problem 1: $8.78 + 6.2 = 8.78 + 6.20 = 14.98$

  • Problem 2: $2.89 + 52.5 = 2.89 + 52.50 = 55.39$

• Decimal Subtraction:

  • Rule: Align decimal points vertically, append placeholder zeros, and perform subtraction column-by-column.

  • Problem 3: $46 - 3.63 = 46.00 - 3.63 = 42.37$

  • Problem 4: $21.9 - 4.5 = 17.4$

• Decimal Multiplication:

  • Rule: Multiply values ignoring decimal points. Sum the decimal places of all factors and place the decimal point in the product so it matches this total count.

  • Problem 5: $(7.2)(5) \rightarrow 72 \times 5 = 360$. Total decimal places $= 1$. Result: $36.0 = 36$.

  • Problem 6: $0.13 \times 5.1 \rightarrow 13 \times 51 = 663$. Total decimal places $= 2 + 1 = 3$. Result: $0.663$.

• Decimal Division:

  • Rule: Shift the divisor's decimal point rightward to create a whole number. Shift the dividend's decimal point by the same number of places. Align decimal point in quotient directly above.

  • Problem 7: $7.86 \div 0.2 \rightarrow 78.6 \div 2 = 39.3$

  • Problem 8: $9.5 \div 0.05 \rightarrow 950 \div 5 = 190$

• Real-Life Word Problems:

  • Problem 9 (Cost Allocation): Sam, Jessie, and Emma go out for burgers. Total cost is $\$21.39$. How much will each person pay if they split the cost equally?

  • Calculation: \21.39 \div 3 = \$7.13$

  • Solution: Each person pays $\$7.13$.

  • Problem 10 (Budget Remaining): Nancy has $\$10$. She wants to buy a burger for $\$5.49$, chips for $\$1.29$, and a milkshake for $\$2.79$. How much money will she have left over?

  • Total spent: $\$5.49 + \$1.29 + \$2.79 = \$9.57$

  • Money remaining: $\$10.00 - \$9.57 = \$0.43$

  • Solution: Nancy has $\$0.43$ left over.

Comprehensive Assessment and Worked Practice Problems

• Question 1: Decimal Product:

• Problem Statement: What is the product of $3.5$ and $1.82$?

• Detailed Steps:

  1. Multiply $\mathbf{35 \times 182 = 6370}$.

  2. Count decimal places: $3.5$ has $1$ decimal place; $1.82$ has $2$ decimal places. Total $= 3$ decimal places.

  3. Place decimal point $3$ positions from the right: $6.370 = 6.37$

• Answer Choices Analysis:

  • A) $0.637$ (Incorrect)

  • B) $6.37$ (Correct)

  • C) $63.7$ (Incorrect)

  • D) $637$ (Incorrect)

• Final Answer: B) $6.37$

  • Question 2: Geometric Perimeter Calculation:

• Problem Statement: Determine the perimeter of the rectangle, in centimeters, given length $l = 7.4\,\text{cm}$ and width $w = 3.9\,\text{cm}$.

• Detailed Steps:

  1. Apply perimeter formula for rectangles: $P = 2l + 2w$ or $P = 2(l + w)$.

  2. Substitute dimensions: $P = 2(7.4\,\text{cm}) + 2(3.9\,\text{cm}) = 14.8\,\text{cm} + 7.8\,\text{cm} = 22.6\,\text{cm}$.

• Final Answer: $\text{Perimeter} = 22.6\,\text{cm}$

  • Question 3: Itemized Shopping Total Cost:

• Problem Statement: What is the total cost in dollars, before tax, of buying $4$ pens, $3$ notebooks, and $1$ stapler?

• Price List from Reference Table:

  • Pencil: $\$1.27$

  • Pen: $\$2.38$

  • Eraser: $\$0.42$

  • Notebook: $\$3.49$

  • Stapler: $\$7.82$

• Detailed Steps:

  1. Calculate cost of $4$ pens: $4 \times \$2.38 = \$9.52$

  2. Calculate cost of $3$ notebooks: $3 \times \$3.49 = \$10.47$

  3. Calculate cost of $1$ stapler: $1 \times \$7.82 = \$7.82$

  4. Sum all costs: $\$9.52 + \$10.47 + \$7.82 = \$27.81$

• Final Answer: $\$27.81$

  • Question 4: Equivalent Division Expression:

• Problem Statement: Which expression has the same quotient as $3.5 \div 0.28$?

• Detailed Steps:

  1. Represent division as a fraction: $\frac{3.5}{0.28}$.

  2. Multiply numerator and denominator by $100$ to eliminate decimals in the denominator:        $\frac{3.5 \times 100}{0.28 \times 100} = \frac{350}{28}$

  3. Evaluate quotient: $3.5 \div 0.28 = 12.5$ and $350 \div 28 = 12.5$

• Answer Choices Analysis:

  • A) $35 \div 28 = 1.25$ (Incorrect)

  • B) $35 \div 0.028 = 1250$ (Incorrect)

  • C) $0.35 \div 2.8 = 0.125$ (Incorrect)

  • D) $350 \div 28 = 12.5$ (Correct)

• Final Answer: D) $350 \div 28$