Multiply
Rational Numbers
Master the rules for multiplying positive and negative fractions, decimals, and integers: from sign rules to real-world contexts.
7th Grade Math
What Are Rational Numbers?
A rational number is any number that can be written as a fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b \neq 0\). This includes positive numbers, negative numbers, and zero.
Integers
\(-3,\; 0,\; 7\) can be written as \(\frac{-3}{1},\; \frac{0}{1},\; \frac{7}{1}\)
Fractions
\(\frac{2}{5},\; -\frac{3}{4},\; \frac{11}{8}\)
Decimals
\(0.75 = \frac{3}{4}\) and \(-1.2 = -\frac{6}{5}\)
Why Multiply Signed Numbers?
In the real world, negative numbers represent decreases, debts, or opposite directions. Multiplying signed numbers lets us model situations like:
Debt Over Time
Losing \(\$5\) per day for \(3\) days: \((-5)(3) = -15\)
Reversing a Loss
Undoing \(4\) charges of \(-\$6\): \((-6)(-4) = 24\)
Multiplication on the Number Line
The Sign Rules for Multiplication
Why Does \((-1)(-1) = 1\)?
We need multiplication to obey the distributive property. Start with a fact we know:
1
We know \(-1 + 1 = 0\).
2
Multiply both sides by \(-1\): \;(-1)(-1 + 1) = (-1)(0)\).
3
Right side: \((-1)(0) = 0\). Left side by distributive property: \((-1)(-1) + (-1)(1)\).
4
So \((-1)(-1) + (-1) = 0\), which means \((-1)(-1) = 1\). ✓
Adding vs Multiplying Negatives
| Addition | Multiplication | |
|---|---|---|
| Two negatives | \((-3)+(-4)=-7\) More negative |
\((-3)(-4)=12\) Positive! |
| Mixed signs | \((-3)+4 = 1\) Subtract & keep larger sign |
\((-3)(4)=-12\) Negative |
| Key rule | Combine magnitudes by sign | Same signs → +, Different → − |
How to Multiply Step by Step
1
Determine the sign. Count the negative factors. An even count → positive; an odd count → negative.
2
Convert to fractions (if needed). Write decimals and mixed numbers as improper fractions: \(1.5 = \frac{3}{2}\).
3
Multiply numerators together and denominators together: \(\frac{a}{b} \times \frac{c}{d} = \frac{a \cdot c}{b \cdot d}\).
4
Simplify the result. Reduce the fraction and attach the sign from Step 1.
Example 1: Two Integers
Compute \((-8)(5)\).
1
Sign: One negative factor (odd count) → product is negative.
2
Multiply magnitudes: \(8 \times 5 = 40\).
3
Attach sign: \((-8)(5) = \) \(-40\).
Example 2: Two Fractions
Compute \(\displaystyle \left(-\frac{3}{4}\right)\!\left(-\frac{2}{5}\right)\).
1
Sign: Two negative factors (even count) → product is positive.
2
Multiply numerators: \(3 \times 2 = 6\).
3
Multiply denominators: \(4 \times 5 = 20\).
4
Simplify: \(\frac{6}{20} = \frac{3}{10}\). Result: \(+\frac{3}{10}\).
Example 3: Decimal × Fraction
Compute \((-0.6)\!\left(\frac{5}{3}\right)\).
1
Sign: One negative factor → product is negative.
2
Convert decimal: \(0.6 = \frac{6}{10} = \frac{3}{5}\).
3
Multiply: \(\frac{3}{5} \times \frac{5}{3}\). Cross-cancel: the \(3\)s cancel, the \(5\)s cancel → \(\frac{1}{1} = 1\).
4
Attach sign: Result is \(-1\).
Three or More Factors
Compute \((-2)\!\left(-\frac{1}{3}\right)\!(-6)\).
1
Count negatives: Three negative factors (odd) → product is negative.
2
Multiply magnitudes: \(2 \times \frac{1}{3} \times 6 = \frac{2 \times 1 \times 6}{1 \times 3 \times 1} = \frac{12}{3} = 4\).
3
Attach sign: Result is \(-4\).
Properties That Still Apply
Commutative
\(a \times b = b \times a\)
Order doesn't change the product.
Order doesn't change the product.
Associative
\((a \times b) \times c = a \times (b \times c)\)
Grouping doesn't change the product.
Grouping doesn't change the product.
Distributive
\(a(b + c) = ab + ac\)
The foundation of signed-number rules.
The foundation of signed-number rules.
Real-World Contexts
Temperature Drop
Temperature falls \(1.5°\) per hour for \(4\) hours: \((-1.5)(4) = -6°\)
Reversed Charges
A bank removes \(3\) penalties of \(-\$25\) each: \((-25)(-3) = \$75\) returned
Elevation Change
Diving \(\frac{3}{4}\) m per second for \(8\) seconds: \(\left(-\frac{3}{4}\right)(8) = -6\) meters
Scaling a Recipe
Need \(\frac{2}{3}\) of \(\frac{3}{4}\) cup of sugar: \(\frac{2}{3} \times \frac{3}{4} = \frac{1}{2}\) cup
Common Mistakes
✕ Wrong Sign on Two Negatives
Writing \((-3)(-4) = -12\) instead of \(+12\). Two negatives always make a positive.
✕ Confusing Addition Rules
Applying addition sign rules to multiplication. Adding \((-3)+(-4)=-7\) is different from \((-3)(-4)=12\).
✕ Forgetting to Simplify
Leaving \(\frac{6}{20}\) instead of reducing to \(\frac{3}{10}\). Always check for common factors.
Summary of Sign Rules
| Expression Type | Example | Result Sign |
|---|---|---|
| Positive × Positive | \(3 \times 4 = 12\) | + |
| Negative × Negative | \((-3)(-4) = 12\) | + |
| Positive × Negative | \(3 \times (-4) = -12\) | − |
| Negative × Positive | \((-3) \times 4 = -12\) | − |
| Any number × Zero | \((-7) \times 0 = 0\) | 0 |
Question 1 of 5
What is \((-7)(9)\)?
One negative and one positive factor means the product is negative. \(7 \times 9 = 63\), so the answer is \(-63\).
Question 2 of 5
What is \(\displaystyle \left(-\frac{2}{3}\right)\!\left(-\frac{9}{4}\right)\)?
Two negatives give a positive product. \(\frac{2}{3} \times \frac{9}{4} = \frac{18}{12} = \frac{3}{2}\). The simplified, positive answer is \(\frac{3}{2}\).
Question 3 of 5
A stock loses \(\$1.50\) per day for \(4\) days. Which expression models the total change?
The stock loses money, so each day's change is \(-1.50\). It happens for a positive \(4\) days. The product \((-1.50)(4) = -6.00\) represents a \(\$6\) total loss.
Question 4 of 5
What is \((-2)(-3)(-5)\)?
Three negative factors (odd count) → negative product. Magnitudes: \(2 \times 3 \times 5 = 30\). So the answer is \(-30\).
Question 5 of 5
Why does \((-1)(-1) = 1\)?
From \(-1 + 1 = 0\), multiply both sides by \(-1\) and distribute: \((-1)(-1) + (-1)(1) = 0\). Since \((-1)(1) = -1\), we get \((-1)(-1) - 1 = 0\), so \((-1)(-1) = 1\). The distributive property forces this result.