Signed Multiplication and Division

Signed Multiplication and Division

Signed numbers describe direction as well as size. For multiplication and division, first determine the sign, then calculate the absolute values.

How to solve these problems

Two factors or a dividend and divisor with the same sign give a positive result; different signs give a negative result. A product with an even number of negative factors is positive; an odd number is negative. Zero divided by a nonzero number is zero.

Worked Example 1: 6 × 4

6 × 4

Solution: Both factors are positive: 6 × 4 = 24.

Why it works: Same signs produce a positive product.

Worked Example 2: −6 × 4

−6 × 4

Solution: The signs differ; 6 × 4 = 24. Answer: −24.

Why it works: Four groups of a decrease of 6 total a decrease of 24.

Worked Example 3: 6 × (−4)

6 × (−4)

Solution: The signs differ; the answer is −24.

Why it works: Changing the order of factors does not change the product.

Worked Example 4: −6 × (−4)

−6 × (−4)

Solution: Same signs give a positive result: 24.

Why it works: The negative multiplier reverses the direction of the negative quantity.

Worked Example 5: −35 ÷ 7

−35 ÷ 7

Solution: Different signs give −5. Check: 7 × (−5) = −35.

Why it works: A division answer must reconstruct the dividend when multiplied by the divisor.

Worked Example 6: −35 ÷ (−7)

−35 ÷ (−7)

Solution: Same signs give 5. Check: −7 × 5 = −35.

Why it works: The positive quotient satisfies the inverse multiplication fact.

Worked Example 7: 48 ÷ (−6)

48 ÷ (−6)

Solution: Different signs give −8. Check: −6 × (−8) = 48.

Why it works: Two negative factors are needed to recover the positive dividend.

Worked Example 8: −2 × (−3) × (−4)

−2 × (−3) × (−4)

Solution: First −2 × (−3) = 6; then 6 × (−4) = −24.

Why it works: Three negative factors are an odd count, so the product is negative.

Worked Example 9: −1.5 × (−2)

−1.5 × (−2)

Solution: Same signs give a positive result. 1.5 × 2 = 3.

Why it works: The sign rule also applies to decimal and fractional values.

Worked Example 10: A balance changes by −$8 daily for 5 days. Find the total change.

A balance changes by −$8 daily for 5 days. Find the total change.

Solution: 5 × (−$8) = −$40.

Why it works: Repeated negative changes produce a negative total change; the balance decreases by $40.

Try it again

Cover the solutions and solve the ten examples on your own. Explain which rule you used, then compare your method and answer with the worked solution. Check signs, equivalent forms, and units.