Solved Exercises on Sign Rules for Division in Grade 7

Master sign rules for division: positive, negative, zero, absolute value through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Positive ÷ Positive
Exercise 1
Calculate: \( (+12) \div (+3) \)
Definition:

Sign rule: When dividing two positive numbers, the result is positive

Division method:
  1. Divide the absolute values of both numbers
  2. Apply the sign rule: positive ÷ positive = positive
  3. Result is always positive
Original Expression
\((+12) \div (+3)\)
Divide Absolute Values
\(12 \div 3 = 4\)
Apply Sign Rule
Positive
Result
\(4\)
Step 1: Identify the signs

Both numbers are positive: \((+12)\) and \((+3)\)

Step 2: Divide the absolute values

\(|12| \div |3| = 12 \div 3 = 4\)

Step 3: Apply the sign rule

Positive ÷ Positive = Positive

Step 4: Write the final answer

\((+12) \div (+3) = +4\) or simply \(4\)

\( (+12) \div (+3) = 4 \)
Final answer:

\( (+12) \div (+3) = 4 \)

Applied rules:

• Sign rule: Positive ÷ Positive = Positive

• Division: Divide absolute values first

• Positive result: Always results in a positive number

2 Negative ÷ Positive
Exercise 2
Calculate: \( (-15) \div (+5) \)
Definition:

Sign rule: When dividing integers with different signs, the result is negative

Original Expression
\((-15) \div (+5)\)
Divide Absolute Values
\(15 \div 5 = 3\)
Apply Sign Rule
Negative
Result
\(-3\)
Step 1: Identify the signs

One negative \((-15)\) and one positive \((+5)\)

Step 2: Divide the absolute values

\(|-15| \div |+5| = 15 \div 5 = 3\)

Step 3: Apply the sign rule

Negative ÷ Positive = Negative

Step 4: Write the final answer

\((-15) \div (+5) = -3\)

\( (-15) \div (+5) = -3 \)
Final answer:

\( (-15) \div (+5) = -3 \)

Applied rules:

• Sign rule: Negative ÷ Positive = Negative

• Division: Divide absolute values first

• Different signs: Always results in a negative number

3 Positive ÷ Negative
Exercise 3
Calculate: \( (+14) \div (-2) \)
Definition:

Division as multiplication: \(a \div b = a \times \frac{1}{b}\) when \(b \neq 0\)

Original Expression
\((+14) \div (-2)\)
Divide Absolute Values
\(14 \div 2 = 7\)
Apply Sign Rule
Negative
Result
\(-7\)
Step 1: Identify the signs

One positive \((+14)\) and one negative \((-2)\)

Step 2: Divide the absolute values

\(|+14| \div |-2| = 14 \div 2 = 7\)

Step 3: Apply the sign rule

Positive ÷ Negative = Negative

Step 4: Write the final answer

\((+14) \div (-2) = -7\)

\( (+14) \div (-2) = -7 \)
Final answer:

\( (+14) \div (-2) = -7 \)

Applied rules:

• Sign rule: Positive ÷ Negative = Negative

• Division: Divide absolute values first

• Different signs: Always results in a negative number

Rules and methods, laws,...
\( (+a) \div (+b) = +(a \div b) \)
Same Signs Division
Positive ÷ Positive
\( (+a) \div (+b) = +(a \div b) \)
Divide absolute values, keep positive sign
Negative ÷ Negative
\( (-a) \div (-b) = +(a \div b) \)
Divide absolute values, result is positive
Different Signs
\( (+a) \div (-b) = -(a \div b) \)
Divide absolute values, result is negative
Key definitions:

Integer: A whole number including positive, negative, and zero

Absolute value: Distance from zero on number line, always non-negative

Sign rule: Rules for determining the sign of the quotient based on signs of dividend and divisor

Division: The inverse operation of multiplication

Integer division methods:
  1. Divide absolute values: Ignore signs initially
  2. Apply sign rule: Determine sign of result
  3. Same signs: Result is positive
  4. Different signs: Result is negative
Tip 1: Remember: "Same signs give positive, different signs give negative"
Tip 2: Always divide absolute values first, then determine the sign
Tip 3: Negative ÷ Negative = Positive (same as multiplication)
Tip 4: Use multiplication to check your division answer
Common errors: Forgetting sign rules, dividing signs instead of absolute values, mixing up division and multiplication rules.
Exam preparation: Practice all sign combinations, memorize the sign rules, use absolute values first.
Formulas to know by heart:

• Same signs: \( (+a) \div (+b) = +(a \div b) \) and \( (-a) \div (-b) = +(a \div b) \)

• Different signs: \( (+a) \div (-b) = -(a \div b) \) and \( (-a) \div (+b) = -(a \div b) \)

• Identity: \( a \div 1 = a \)

• Zero: \( 0 \div a = 0 \) (where \(a \neq 0\))

Solution: Exercises 4 to 5
4 Negative ÷ Negative
Exercise 4
Calculate: \( (-18) \div (-3) \)
Definition:

Negative ÷ Negative: This always results in a positive number

Original Expression
\((-18) \div (-3)\)
Divide Absolute Values
\(18 \div 3 = 6\)
Apply Sign Rule
Positive
Result
\(6\)
Step 1: Identify the signs

Both numbers are negative: \((-18)\) and \((-3)\)

Step 2: Divide the absolute values

\(|-18| \div |-3| = 18 \div 3 = 6\)

Step 3: Apply the sign rule

Negative ÷ Negative = Positive

Step 4: Write the final answer

\((-18) \div (-3) = +6\) or simply \(6\)

\( (-18) \div (-3) = 6 \)
Final answer:

\( (-18) \div (-3) = 6 \)

Applied rules:

• Sign rule: Negative ÷ Negative = Positive

• Division: Divide absolute values first

• Same signs: Always results in a positive number

5 Dividing Zero
Exercise 5
Calculate: \( 0 \div (-5) \)
Definition:

Division by zero: Zero divided by any non-zero number equals zero

Original Expression
\(0 \div (-5)\)
Apply Zero Property
0 ÷ any number ≠ 0 = 0
Result
\(0\)
Step 1: Identify the dividend and divisor

Dividend is zero \((0)\) and divisor is \((-5)\)

Step 2: Apply the zero property

Zero divided by any non-zero number equals zero

Step 3: Write the final answer

\(0 \div (-5) = 0\)

Step 4: Understand the concept

How many groups of \(-5\) are in \(0\)? Zero groups.

\( 0 \div (-5) = 0 \)
Final answer:

\( 0 \div (-5) = 0 \)

Applied rules:

• Zero property: \(0 \div a = 0\) for any non-zero number \(a\)

• Division: Zero is the multiplicative absorption element

• Independence: Sign of divisor doesn't matter when dividend is zero

Key Concepts: Laws, Methods, Rules, Definitions
\( (-a) \div (-b) = +(a \div b) \)
Negative ÷ Negative Rule
Key definitions:

Integer: A whole number that can be positive, negative, or zero (..., -3, -2, -1, 0, 1, 2, 3, ...)

Absolute value: The distance of a number from zero on the number line, denoted as \(|a|\), always non-negative

Positive integer: A number greater than zero, often written without a sign

Negative integer: A number less than zero, written with a minus sign

Sign rule: Rules for determining the sign of the quotient based on the signs of dividend and divisor

Division: The inverse operation of multiplication - finding how many times one number fits into another

Complete division methodology:
  1. Identify signs: Determine the signs of dividend and divisor
  2. Divide absolute values: Divide the absolute values of both numbers
  3. Apply sign rule: Determine the sign of the quotient based on the signs of dividend and divisor
  4. Same signs: Result is positive
  5. Different signs: Result is negative
  6. Special cases: \(0 \div a = 0\) (where \(a \neq 0\)), division by zero is undefined
  7. Verify: Check with multiplication (quotient × divisor = dividend)
Tip 1: Same signs → Positive result; Different signs → Negative result
Tip 2: Remember: Division follows the same sign rules as multiplication
Tip 3: Always divide absolute values first, then apply the sign rule
Tip 4: Dividing zero by any non-zero number always results in zero
Common errors: Forgetting sign rules, applying addition rules to division, dividing signs instead of absolute values.
Exam preparation: Master the sign rules, practice all four scenarios, memorize special cases.
Formulas to know by heart:

• Same signs: \( (+a) \div (+b) = +(a \div b) \) and \( (-a) \div (-b) = +(a \div b) \)

• Different signs: \( (+a) \div (-b) = -(a \div b) \) and \( (-a) \div (+b) = -(a \div b) \)

• Zero property: \( 0 \div a = 0 \) (where \(a \neq 0\))

• Identity: \( a \div 1 = a \)

• Inverse: \( a \div b = c \) means \( c \times b = a \)

Exercise with Visualization: Sign Rules for Division
Exercise 6: Division Sign Patterns
Observe these division patterns:
\( (+12) \div (+3) = +4 \)
\( (-12) \div (+3) = -4 \)
\( (+12) \div (-3) = -4 \)
\( (-12) \div (-3) = +4 \)

Analysis: The chart shows how signs affect division results.

  • \( (+12) \div (+3) = +4 \): Same positive signs → Positive result
  • \( (-12) \div (+3) = -4 \): Different signs → Negative result
  • \( (+12) \div (-3) = -4 \): Different signs → Negative result
  • \( (-12) \div (-3) = +4 \): Same negative signs → Positive result

Questions & Answers

Question: Why does dividing two negative numbers give a positive result? Is it the same as multiplication?

Answer: Yes, division follows the same sign rules as multiplication! Here's why:

  1. Division as multiplication: \(a \div b = a \times \frac{1}{b}\), so the sign rules must be consistent
  2. Logical reasoning: If \((-12) \div (-3) = ?\), we ask "what number times \(-3\) gives us \(-12\)?"
    The answer is \(+4\) because \((+4) \times (-3) = -12\)
  3. Pattern consistency: Following the same rules keeps our mathematical system logical and predictable

Sign Rules for Division (same as multiplication):

  • Positive ÷ Positive = Positive
  • Negative ÷ Negative = Positive
  • Positive ÷ Negative = Negative
  • Negative ÷ Positive = Negative

The key insight is that division is the inverse operation of multiplication!

Question: What's the easiest way to remember the sign rules for division?

Answer: Since division has the same sign rules as multiplication, here are some memorable ways:

  • Same signs → Positive: "Friends of friends are friends" and "Enemies of enemies are friends"
  • Different signs → Negative: "Friends of enemies are enemies" and "Enemies of friends are enemies"
  • Simple rhyme: "Same signs positive, different signs negative"
  • Remember: Division and multiplication have identical sign rules!

Sign Rules Summary:

  • Positive ÷ Positive = Positive
  • Negative ÷ Negative = Positive
  • Positive ÷ Negative = Negative
  • Negative ÷ Positive = Negative

Always divide the absolute values first, then apply the sign rule!

Question: How do I know if my division of signed numbers is correct?

Answer: Here are several ways to verify your signed number division:

  1. Sign check: Verify you applied the correct sign rule
  2. Estimation: Round numbers to check if answer is reasonable
  3. Multiplication verification: Multiply quotient by divisor to get dividend
  4. Repeated subtraction: Think of division as repeated subtraction

Example verification: For \((-18) \div (-3) = 6\)

  • Sign check: Negative ÷ Negative = Positive ✓
  • Estimation: \(18 \div 3 = 6\), so 6 is reasonable ✓
  • Multiplication: \(6 \times (-3) = -18\) ✓

The most important check is verifying the sign rule and using multiplication to confirm!