Multiplying and Dividing Integers

Learn the rules for multiplying and dividing positive and negative numbers.

Intermediate25 minLesson
Printable worksheet

Definition

When multiplying or dividing integers, the sign of the answer depends on the signs of the numbers involved.
The Sign Rules:
OperationResult
(+)×(+)(+) \times (+)Positive
(−)×(−)(-) \times (-)Positive
(+)×(−)(+) \times (-)Negative
(−)×(+)(-) \times (+)Negative
The same rules apply to division!
Simple Rule:
  • Same signs → Positive result
  • Different signs → Negative result
(−3)×(−4)=12(same signs = positive)(-3) \times (-4) = 12 \quad \text{(same signs = positive)}
(−3)×4=−12(different signs = negative)(-3) \times 4 = -12 \quad \text{(different signs = negative)}

Try it now

What is (−4)×(−5)(-4) \times (-5)?

Worked Examples

Calculate: (−6)×(−4)(-6) \times (-4)
1

Identify the signs

Both numbers are negative: (−6)(-6) and (−4)(-4) → Same signs
2

Apply the sign rule

Same signs → positive result → Answer will be positive
3

Multiply the absolute values

6×4=246 \times 4 = 24
4

Apply the sign

Positive result → +24+24

Common Mistakes

Thinking (−3)×(−4)=−12(-3) \times (-4) = -12

Why it's wrong: Students forget that multiplying two negatives gives a positive result.

Correct: Same signs → positive: (−3)×(−4)=+12(-3) \times (-4) = +12

Confusing (−5)×3(-5) \times 3 with 5×(−3)5 \times (-3)

Why it's wrong: Both give the same answer! Different signs → negative in both cases.

Correct: (−5)×3=−15(-5) \times 3 = -15 and 5×(−3)=−155 \times (-3) = -15. Order does not matter.

Forgetting the sign when the answer is positive

Why it's wrong: When two negatives multiply, students may write −12-12 out of habit.

Correct: Always check: same signs = positive, different signs = negative.

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Practice Problems

17 problems
Problem 1 of 17
Easy
What is (−4)×(−5)(-4) \times (-5)?

Why It Matters

Multiplying and dividing integers appears everywhere:
  • Temperature changes: If temperature drops 3 degrees per hour for 5 hours, the total change is (−3)×5=−15(-3) \times 5 = -15 degrees
  • Finance: A monthly debt payment of 200 euros over 6 months means (−200)×6=−1200(-200) \times 6 = -1200 euros total
  • Gaming: Losing 5 points per wrong answer over 4 questions gives (−5)×4=−20(-5) \times 4 = -20 points
  • Elevation: Descending 10 meters per minute for 8 minutes means (−10)×8=−80(-10) \times 8 = -80 meters
These operations help us calculate repeated gains or losses accurately!

Real World Applications

Temperature Changes Over Time

Meteorologists calculate temperature changes using integer multiplication.

Example:

If the temperature drops 2 degrees every hour, after 6 hours the change is (−2)×6=−12(-2) \times 6 = -12 degrees.

1Try It Yourself

A freezer cools food by dropping 5 degrees every 10 minutes. How much does the temperature change in 40 minutes?

What is the total temperature change?

Step 1: Write the mathematical expression

Calculate: (−5)×(-5) \times number of 10-minute periods

Stock Market Changes

Investors track daily gains and losses over multiple days.

Example:

A stock loses 3 euros per day for 5 days: (−3)×5=−15(-3) \times 5 = -15 euros total loss.

2Try It Yourself

An investor reverses a bad trade. They undo a loss of 8 euros that happened 3 times. What is the net effect?

What is the total change when reversing these losses?

Step 1: Write the mathematical expression

Undoing a loss is like: (−1)×(−8)×3(-1) \times (-8) \times 3

Key Takeaways

  • 1Same signs (both positive or both negative) → positive result
  • 2Different signs (one positive, one negative) → negative result
  • 3These rules work for both multiplication AND division
  • 4When multiplying many integers: odd number of negatives → negative, even → positive
  • 5Always calculate with absolute values first, then apply the sign rule

Frequently Asked Questions

Think of it as reversing a reversal. If going backward (−-) is negative, then going backward while facing backward (−×−- \times -) means you actually go forward (positive)!
Think of it as reversing a reversal. If going backward (−-) is negative, then going backward while facing backward (−×−- \times -) means you actually go forward (positive)!
No! Multiplication is commutative. (−3)×5=5×(−3)=−15(-3) \times 5 = 5 \times (-3) = -15. The sign rules give the same answer either way.
Zero times any integer is zero: 0×(−5)=00 \times (-5) = 0. Zero has no sign, so the result is simply 00.

Glossary

Integer
A whole number that can be positive, negative, or zero
Absolute value
The distance from zero, always positive. ∣−7∣=7|-7| = 7
Product
The result of multiplication
Quotient
The result of division
Sign rule
The rule that determines if the result is positive or negative based on the signs of the operands

Formula Card

Positive × Positive

(+)×(+)=+(+) \times (+) = +

Positive times positive equals positive

Negative × Negative

(−)×(−)=+(-) \times (-) = +

Negative times negative equals positive

Positive × Negative

(+)×(−)=−(+) \times (-) = -

Positive times negative equals negative

Negative × Positive

(−)×(+)=−(-) \times (+) = -

Negative times positive equals negative

Today's challengeOne problem, three tries

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