Solved Exercises on Review of Integers and Rational Numbers in Grade 8

Master integers and rational numbers: operations, properties, and number line representations through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Integer Operations
Exercise 1
Calculate: (-15) + 8 - (-12) + (-7). Show all steps and explain the rules used.
Definition:

Integer: A whole number that can be positive, negative, or zero.

Opposite: For any integer a, its opposite is -a.

Absolute Value: The distance from zero on the number line, denoted |a|.

Integer operation rules:
  1. Adding integers with same sign: Add absolute values, keep sign
  2. Adding integers with different signs: Subtract smaller absolute value from larger, keep sign of larger
  3. Subtracting integers: Add the opposite (a - b = a + (-b))
  4. Double negatives: -(-a) = a
Step 1: Rewrite subtraction as addition

(-15) + 8 - (-12) + (-7)

= (-15) + 8 + 12 + (-7) [since -(-12) = +12]

Step 2: Group positive and negative numbers

= [8 + 12] + [(-15) + (-7)]

= 20 + (-22)

Step 3: Add numbers with different signs

20 + (-22): Different signs, subtract absolute values

|22| - |20| = 22 - 20 = 2

Sign of larger absolute value is negative

Step 4: Final calculation

20 + (-22) = -2

Final answer:

(-15) + 8 - (-12) + (-7) = -2

Applied rules:

• Subtraction rule: a - b = a + (-b)

• Double negative: -(-a) = a

• Adding different signs: Subtract absolute values, keep sign of larger

-15 -10 -5 0 5 10 15 20 -15 +8 +12 -7 Final: -2 Integer Operations on Number Line
2 Rational Number Operations
Exercise 2
Calculate: (3/4) × (-2/5) ÷ (6/7). Express your answer in simplest form.
Definition:

Rational Number: A number that can be expressed as a fraction a/b where a and b are integers and b ≠ 0.

Reciprocal: For fraction a/b, the reciprocal is b/a.

Simplification: Reducing a fraction to lowest terms by dividing numerator and denominator by GCD.

Step 1: Convert division to multiplication

(3/4) × (-2/5) ÷ (6/7)

= (3/4) × (-2/5) × (7/6) [since a ÷ b = a × (1/b)]

Step 2: Multiply numerators and denominators

= (3 × (-2) × 7) / (4 × 5 × 6)

= (-42) / 120

Step 3: Simplify the fraction

Find GCD of 42 and 120:

42 = 2 × 3 × 7

120 = 2³ × 3 × 5

GCD = 2 × 3 = 6

Step 4: Reduce to lowest terms

(-42) ÷ 6 = -7

120 ÷ 6 = 20

So, (-42)/120 = -7/20

Step 5: Verify simplification

GCD(7, 20) = 1, so -7/20 is in simplest form

Final answer:

(3/4) × (-2/5) ÷ (6/7) = -7/20

Applied rules:

• Division to multiplication: a ÷ b = a × (1/b)

• Multiplication of fractions: (a/b) × (c/d) = (ac)/(bd)

• Simplification: Divide numerator and denominator by GCD

3/4 -2/5 6/7 × ÷ = -7/20 Convert ÷ to × reciprocal: (3/4) × (-2/5) × (7/6) = (-42)/120 = -7/20 Rational Number Operations
3 Ordering Rational Numbers
Exercise 3
Order the following numbers from least to greatest: -3/4, 0.6, -0.8, 2/3, -1/2. Show your work.
Definition:

Ordering Rational Numbers: Arranging numbers in ascending or descending order.

Common Denominator: Converting fractions to have the same denominator for comparison.

Decimal Conversion: Converting fractions to decimals for comparison.

Step 1: Convert all numbers to the same form

Convert to decimals for easier comparison:

-3/4 = -0.75

0.6 = 0.6

-0.8 = -0.8

2/3 ≈ 0.667

-1/2 = -0.5

Step 2: Identify negative and positive numbers

Negative numbers: -0.8, -0.75, -0.5

Positive numbers: 0.6, 0.667

Step 3: Order negative numbers (larger absolute value is smaller)

|-0.8| = 0.8, |-0.75| = 0.75, |-0.5| = 0.5

Since 0.8 > 0.75 > 0.5, we have -0.8 < -0.75 < -0.5

Step 4: Order positive numbers

0.6 < 0.667

Step 5: Combine the orders

From least to greatest: -0.8, -0.75, -0.5, 0.6, 0.667

In original form: -0.8, -3/4, -1/2, 0.6, 2/3

Final answer:

From least to greatest: -0.8, -3/4, -1/2, 0.6, 2/3

Applied rules:

• Comparing negatives: Larger absolute value means smaller number

• Decimal conversion: Makes comparison easier

• Ordering principle: Negative < 0 < Positive

-1 -0.8 -0.5 0 0.5 0.6 0.7 1 -0.8 -3/4 -1/2 0.6 2/3 Ascending Order Ordering Rational Numbers
Number System Concepts, Rules and Methods
a ÷ b = a × (1/b)
Division to Multiplication
Absolute Value
|a| = distance from 0
Always non-negative
Reciprocal
1/a of a/b is b/a
Used for division
GCD
Greatest Common Divisor
For fraction simplification
Key definitions:

Integer: A whole number that can be positive, negative, or zero (..., -2, -1, 0, 1, 2, ...).

Rational Number: Any number that can be expressed as a fraction a/b where a and b are integers and b ≠ 0.

Absolute Value: The distance of a number from zero on the number line, always non-negative.

Opposite: For any number a, its opposite is -a, such that a + (-a) = 0.

Number system methodology:
  1. Classification: Identify if number is integer, rational, or other
  2. Operations: Apply appropriate rules for addition, subtraction, multiplication, division
  3. Comparison: Use number line or decimal conversion to compare
  4. Simplification: Reduce fractions to lowest terms
  5. Verification: Check results using alternative methods
Tip 1: When adding integers with different signs, subtract absolute values and keep the sign of the larger absolute value.
Tip 2: To compare fractions, convert to decimals or find a common denominator.
Tip 3: Remember that dividing by a fraction is the same as multiplying by its reciprocal.
Tip 4: When ordering negative numbers, the one with the larger absolute value is actually smaller.
Key characteristics: Integers are closed under addition, subtraction, and multiplication. Rational numbers are closed under all four operations (except division by zero).
Common applications: Measurement, financial calculations, scientific notation, and mathematical modeling.
Solution: Exercises 4 to 5
4 Complex Expression
Exercise 4
Calculate: [(-2)³ + 4 × (-3)] ÷ [5 - (-1)²]. Show all steps and explain the order of operations.
Definition:

Order of Operations: PEMDAS/BODMAS - Parentheses/Brackets, Exponents/Orders, Multiplication and Division (left to right), Addition and Subtraction (left to right).

Exponentiation: Repeated multiplication (aⁿ = a × a × ... × a, n times).

Step 1: Evaluate expressions inside brackets first

Numerator: (-2)³ + 4 × (-3)

Denominator: 5 - (-1)²

Step 2: Evaluate exponents

Numerator: (-2)³ = (-2) × (-2) × (-2) = -8

Denominator: (-1)² = (-1) × (-1) = 1

Step 3: Perform multiplication in numerator

Numerator: -8 + 4 × (-3) = -8 + (-12) = -20

Step 4: Perform subtraction in denominator

Denominator: 5 - 1 = 4

Step 5: Perform final division

(-20) ÷ 4 = -5

Step 6: Verify the calculation

[(-2)³ + 4 × (-3)] ÷ [5 - (-1)²] = [-8 + (-12)] ÷ [5 - 1] = (-20) ÷ 4 = -5 ✓

Final answer:

[(-2)³ + 4 × (-3)] ÷ [5 - (-1)²] = -5

Applied rules:

• Order of operations: PEMDAS/BODMAS determines calculation sequence

• Exponentiation: Handle powers before multiplication/division

• Sign rules: Negative base raised to odd power yields negative result

[(-2)³ + 4 × (-3)] ÷ [5 - (-1)²] Step 1: Evaluate exponents → [(-8) + 4 × (-3)] ÷ [5 - 1] Step 2: Multiply → [(-8) + (-12)] ÷ [5 - 1] Step 3: Add/Subtract → (-20) ÷ 4 Step 4: Divide → -5 PEMDAS P: () E: ^ M/D: ×, ÷ A/S: +, - Order of Operations
5 Properties and Applications
Exercise 5
A submarine descends 150 meters below sea level, then ascends 85 meters. It then descends another 120 meters. What is its final position relative to sea level? Represent this situation using integers and solve.
Definition:

Real-world Application: Using integers to represent positions, temperatures, debts, and other quantities with direction.

Positive/Negative Convention: Establishing what positive and negative represent in context.

Step 1: Establish convention

Sea level = 0

Below sea level = negative

Above sea level = positive

Step 2: Represent movements

Descend 150m = -150

Ascend 85m = +85

Descend 120m = -120

Step 3: Calculate final position

0 + (-150) + 85 + (-120)

= -150 + 85 + (-120)

= (-150) + (-120) + 85 [rearranging]

= -270 + 85

= -185

Step 4: Interpret result

Final position is -185 meters

This means 185 meters below sea level

Step 5: Verify with number line

Start at 0 → move to -150 → move to -65 → move to -185 ✓

Final answer:

The submarine is 185 meters below sea level.

Applied rules:

• Real-world modeling: Assign positive/negative meanings to directions

• Integer addition: Combine movements sequentially

• Context interpretation: Translate mathematical result back to real-world meaning

Sea Level (0) Below Sea Level Above Sea Level Start Descend 150m Ascend 85m Descend 120m Final: -185m -150m -65m -185m Submarine Depth Problem
Number System Theory: Laws, Methods, Definitions, and Formulas
|a| = \begin{cases} a & \text{if } a \geq 0 \\ -a & \text{if } a < 0 \end{cases}
Absolute Value Definition
Key definitions:

Integer: A whole number that can be positive, negative, or zero (denoted by ℤ).

Rational Number: Any number expressible as a/b where a, b ∈ ℤ and b ≠ 0 (denoted by ℚ).

Absolute Value: The non-negative value of a number without regard to its sign.

Number Line: A visual representation of numbers as points on a line.

Number system methodology:
  1. Classification: Determine the type of number (integer, rational, etc.)
  2. Operation selection: Choose appropriate operation based on problem context
  3. Rule application: Apply relevant mathematical properties and rules
  4. Calculation: Perform operations following order of operations
  5. Verification: Check results using alternative methods or estimation
Tip 1: Remember that all integers are rational numbers (denominator = 1).
Tip 2: When multiplying/dividing, count negative signs: even = positive, odd = negative.
Tip 3: For fraction addition, find a common denominator first.
Tip 4: Always check if your final answer makes sense in the context of the problem.

Key characteristics: Closure properties, commutativity, associativity, distributivity of operations.
Common applications: Measurement, finance, science, engineering, and everyday problem solving.
Essential formulas and rules:

• Addition of integers: (+) + (+) = (+), (-) + (-) = (-), (+) + (-) = sign of larger absolute value

• Multiplication of rationals: (a/b) × (c/d) = (ac)/(bd)

• Division of rationals: (a/b) ÷ (c/d) = (a/b) × (d/c)

• Order of operations: PEMDAS/BODMAS

• Absolute value: |a| ≥ 0 for all real numbers a

Questions & Answers

Question: I'm confused about when to flip the fraction when dividing. Do I flip the first fraction or the second one?

Answer: When dividing fractions, you always flip (take the reciprocal of) the second fraction:

  • Rule: (a/b) ÷ (c/d) = (a/b) × (d/c)
  • Memory trick: "Keep, Change, Flip" - Keep the first fraction, Change division to multiplication, Flip the second fraction
  • Why: Division is the inverse of multiplication, so dividing by a fraction is the same as multiplying by its reciprocal

Example: (3/4) ÷ (2/5) = (3/4) × (5/2) = 15/8

Never flip the first fraction when dividing!

Question: How do I know if a number is rational or irrational? Can you give examples?

Answer: A rational number can be expressed as a fraction a/b where a and b are integers and b ≠ 0. An irrational number cannot be expressed as such a fraction:

  • Rational examples: 3/4, -2, 0.75, 0.333... (repeating), √4 = 2
  • Irrational examples: π, √2, √3, e, 0.1010010001... (non-repeating, non-terminating)

Key characteristics:

Rational numbers have decimal expansions that either terminate or repeat.

Irrational numbers have decimal expansions that neither terminate nor repeat.

All integers are rational (they can be written as integer/1).

Question: Why is the absolute value always positive? Isn't it supposed to be the same number?