In Grade 6, you learned about integers and how to add and subtract them. Now it's time to master multiplication and division with positive and negative numbers!
Review: What Are Integers?
Integers are whole numbers and their opposites: ..., -3, -2, -1, 0, 1, 2, 3, ...
Positive integers: 1, 2, 3, 4, ... (to the right of zero)
Negative integers: -1, -2, -3, -4, ... (to the left of zero)
Zero is neither positive nor negative
The Rules for Multiplying Integers
The rules for multiplying integers depend on the signs of the numbers.
Rule 1: Positive × Positive = Positive
When you multiply two positive numbers, the answer is positive.
Examples:
5 × 3 = 15
8 × 4 = 32
12 × 6 = 72
This is what you've always known!
Rule 2: Negative × Negative = Positive
When you multiply two negative numbers, the answer is positive.
📚 Practice Problems
1Problem 1easy
❓ Question:
Calculate: (-8) × 5
💡 Show Solution
When multiplying a negative and a positive number, the result is negative.
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Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 5 problems provided, checking solutions as you go. Regular review and active practice are key to retention.
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What course covers Multiplying and Dividing Integers?▾
Multiplying and Dividing Integers is part of the Grade 7 Math course on Study Mondo, specifically in the Rational Numbers section. You can explore the full course for more related topics and practice resources.
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Examples:
(-5) × (-3) = 15
(-8) × (-4) = 32
(-12) × (-6) = 72
Why? Think of it this way: A negative times a negative "reverses the reverse," bringing you back to positive!
Memory trick: "Two negatives make a positive!"
Rule 3: Positive × Negative = Negative
When you multiply a positive and a negative number, the answer is negative.
Examples:
5 × (-3) = -15
(-8) × 4 = -32
12 × (-6) = -72
Order doesn't matter: (-3) × 5 = 5 × (-3) = -15 (commutative property still works!)
Rule 4: Negative × Positive = Negative
Same as Rule 3! One positive and one negative always give a negative product.
Examples:
(-7) × 2 = -14
9 × (-5) = -45
(-10) × 8 = -80
Sign Rules Summary for Multiplication
Same signs → Positive product
(+) × (+) = (+)
(-) × (-) = (+)
Different signs → Negative product
(+) × (-) = (-)
(-) × (+) = (-)
Simple shortcut:
Like signs (same) → Positive
Unlike signs (different) → Negative
Multiplying with More Than Two Integers
When multiplying multiple integers, count the negative signs!
Rule:
Even number of negatives → Positive result
Odd number of negatives → Negative result
Examples:
Example 1: (-2) × (-3) × (-4)
Three negatives (odd number)
Answer will be negative
Calculate: 2 × 3 × 4 = 24, make it negative
Answer: -24
Example 2: (-2) × (-3) × 4
Two negatives (even number)
Answer will be positive
Calculate: 2 × 3 × 4 = 24
Answer: 24
Example 3: (-1) × (-1) × (-1) × (-1)
Four negatives (even number)
Answer will be positive
Answer: 1
The Rules for Dividing Integers
Great news! Division uses the SAME sign rules as multiplication!
Rule 1: Positive ÷ Positive = Positive
Examples:
20 ÷ 4 = 5
36 ÷ 6 = 6
100 ÷ 25 = 4
Rule 2: Negative ÷ Negative = Positive
Examples:
(-20) ÷ (-4) = 5
(-36) ÷ (-6) = 6
(-100) ÷ (-25) = 4
Rule 3: Positive ÷ Negative = Negative
Examples:
20 ÷ (-4) = -5
36 ÷ (-6) = -6
100 ÷ (-25) = -4
Rule 4: Negative ÷ Positive = Negative
Examples:
(-20) ÷ 4 = -5
(-36) ÷ 6 = -6
(-100) ÷ 25 = -4
Sign Rules Summary for Division
Same signs → Positive quotient
(+) ÷ (+) = (+)
(-) ÷ (-) = (+)
Different signs → Negative quotient
(+) ÷ (-) = (-)
(-) ÷ (+) = (-)
Remember: Division and multiplication use the SAME sign rules!
Properties That Still Work with Integers
Commutative Property of Multiplication
Order doesn't matter when multiplying!
Examples:
5 × (-3) = (-3) × 5 = -15
(-4) × (-7) = (-7) × (-4) = 28
Note: Division is NOT commutative!
12 ÷ 3 = 4, but 3 ÷ 12 = 0.25 (different!)
Associative Property of Multiplication
Grouping doesn't matter when multiplying!
Example:
[(-2) × 3] × 4 = (-6) × 4 = -24
(-2) × [3 × 4] = (-2) × 12 = -24
Same answer!
Multiplication by Zero
Any number times zero equals zero!
Examples:
0 × 5 = 0
(-7) × 0 = 0
0 × (-100) = 0
Multiplication by One
Any number times one equals itself!
Examples:
1 × 8 = 8
(-6) × 1 = -6
1 × (-15) = -15
Multiplication by Negative One
Multiplying by -1 gives you the opposite!
Examples:
(-1) × 5 = -5
(-1) × (-8) = 8
(-1) × 0 = 0
Real-World Applications
Temperature Changes
Problem: The temperature drops 3°F per hour for 4 hours. What's the total change?
Solution:
Drop means negative: -3°F per hour
For 4 hours: 4 × (-3) = -12°F
Answer: The temperature dropped 12°F total
Banking (Debt)
Problem: You withdraw $25 from your account 3 times. What's the total change in your balance?
Solution:
Withdrawal is negative: -$25
Three times: 3 × (-25) = -$75
Answer: Your balance decreased by $75
Elevators (Going Down)
Problem: An elevator descends 5 floors per minute for 3 minutes. Where is it relative to the starting floor?
Solution:
Descending is negative: -5 floors
For 3 minutes: 3 × (-5) = -15 floors
Answer: 15 floors below the starting point
Sharing Debt
Problem: Four friends owe $60 total. If they split it equally, what does each person owe?
Then -1 × (-5) is "the opposite of the opposite of 5" = 5
That's why negative × negative = positive!
Pattern thinking:
3 × 2 = 6
3 × 1 = 3
3 × 0 = 0
3 × (-1) = -3 (decreasing by 3 each time)
3 × (-2) = -6 (pattern continues!)
Quick Reference Chart
Operation
Signs
Result
Example
Multiply
Same
+
(-5) × (-3) = 15
Multiply
Different
-
5 × (-3) = -15
Divide
Same
+
(-20) ÷ (-4) = 5
Divide
Different
-
20 ÷ (-4) = -5
Master these rules and you'll be ready for algebra, equations, and advanced math!
When multiplying two negative numbers, the result is positive.
(-6) × (-7) = 42
Answer: 42
3Problem 3medium
❓ Question:
Calculate: 45 ÷ (-9)
💡 Show Solution
When dividing a positive by a negative number, the result is negative.
45 ÷ (-9) = -5
Answer: -5
4Problem 4medium
❓ Question:
Calculate: (-4) × 3 × (-2)
💡 Show Solution
Multiply from left to right:
Step 1: (-4) × 3 = -12
Step 2: (-12) × (-2) = 24
Two negative signs make a positive.
Answer: 24
5Problem 5hard
❓ Question:
A submarine descends 15 meters per minute for 8 minutes. What is the change in depth? (Use negative for descending)
💡 Show Solution
Descending means going down, so we use -15 meters/minute.
Change in depth = (-15) × 8 = -120 meters
The submarine descended 120 meters (or is at -120 meters from starting point).
Answer: -120 meters
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Yes, this page includes 5 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.