Multiplying and Dividing Integers

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

Intermediate25 min17 problems
Practice Problems

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

Thinking (−5)×3(-5) \times 3 and 5×(−3)5 \times (-3) have different answers

Why it's wrong: Only the signs matter, not which number carries the minus. Both products have different signs → negative.

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

Thinking (−1)×(−2)×(−3)=6(-1) \times (-2) \times (-3) = 6

Why it's wrong: Students remember "negative times negative is positive" and stop there. With a third negative factor the sign flips again.

Correct: Count the negative factors: 3 is odd → negative. (−1)×(−2)×(−3)=−6(-1) \times (-2) \times (-3) = -6

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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 dollars over 6 months means (−200)×6=−1200(-200) \times 6 = -1200 dollars 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 dollars per day for 5 days: (−3)×5=−15(-3) \times 5 = -15 dollars total loss.

2Try It Yourself

A share price falls by 8 dollars every day. You know today's price. Days in the past count as negative days.

How did the price 3 days ago compare with today? Give the difference in dollars (positive = higher than today).

Step 1: Write the mathematical expression

Write the change per day times the number of days (going back = negative):

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 divided by any non-zero integer is zero too: 0÷(−5)=00 \div (-5) = 0. Zero has no sign, so the result is simply 00. But you cannot divide by zero: (−5)÷0(-5) \div 0 has no answer.

Glossary

Integer
A whole number that can be positive, negative, or zero
Absolute value
The distance from zero, never negative. ∣−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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