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\(×\)
\(÷\)
\(+\)
\(-\)

Multiplying & Dividing
Rational Numbers

Master the sign rules and patterns for multiplying and dividing positive and negative fractions, decimals, and integers.

Pre-Algebra

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 integers, fractions, terminating decimals, and repeating decimals.

Integers
\(-5,\; 0,\; 7\) can be written as \(\frac{-5}{1},\; \frac{0}{1},\; \frac{7}{1}\)
Fractions
\(\frac{3}{4},\; -\frac{2}{5},\; \frac{11}{3}\) are already in \(\frac{a}{b}\) form
Decimals
\(0.75 = \frac{3}{4}\) and \(0.\overline{3} = \frac{1}{3}\)

Rational Numbers on the Number Line

\(-3\)
\(-2\)
\(-1\)
\(0\)
\(1\)
\(2\)
\(-1.5\)
\(0.5\)
\(1.5\)
 Negative Positive

Why Do Signs Matter?

When you multiply or divide two numbers, the signs of those numbers determine whether your answer is positive or negative. Mastering sign rules means you can focus on the arithmetic and never second-guess the sign of your result.

Same Signs
Two positives or two negatives produce a positive result
Different Signs
One positive and one negative produce a negative result

The Sign Rules for \(×\) and \(÷\)

\((+) × (+)\)
\(= +\)
 Same signs
\((−) × (−)\)
\(= +\)
 Same signs
\((+) × (−)\)
\(= −\)
 Different signs
\((−) × (+)\)
\(= −\)
 Different signs

Multiplying Fractions

To multiply two fractions, multiply the numerators together and the denominators together, then apply the sign rule:

Dividing Fractions: Keep, Change, Flip

To divide by a fraction, multiply by its reciprocal. The reciprocal of \(\frac{c}{d}\) is \(\frac{d}{c}\).

1
Keep the first fraction exactly as it is
2
Change the division sign to multiplication
3
Flip the second fraction (take the reciprocal)

Multiplication vs. Division

Multiplication Division
Operation \(\frac{a}{b} \times \frac{c}{d}\) \(\frac{a}{b} \div \frac{c}{d}\)
Method Multiply straight across Keep, Change, Flip
Formula \(\frac{a \times c}{b \times d}\) \(\frac{a \times d}{b \times c}\)
Sign Rule Same signs → + Same signs → +
Different signs → − Different signs → −

Step-by-Step Approach

1
Determine the sign of the answer: count the negatives. Odd count → negative; even count → positive.
2
Convert if needed: turn mixed numbers into improper fractions and decimals into fractions.
3
If dividing: keep the first fraction, change ÷ to ×, and flip the second fraction.
4
Simplify before multiplying: cross-cancel common factors between numerators and denominators.
5
Multiply straight across and attach the correct sign.

Example 1: Multiplying Integers

Compute \((-4) \times 7\).

1
Sign check: one negative, one positive → different signs → negative
2
Multiply the absolute values: \(4 \times 7 = 28\)
3
Attach the sign: \((-4) \times 7 = \) \(-28\)

Example 2: Multiplying Fractions

Compute \(\displaystyle -\frac{3}{5} \times \frac{10}{9}\).

1
Sign check: one negative × one positive → negative
2
Cross-cancel: \(3\) and \(9\) share factor \(3\); \(5\) and \(10\) share factor \(5\).
3
After simplifying: \(\frac{1}{1} \times \frac{2}{3} = \frac{2}{3}\)
4
Attach sign: \(-\frac{3}{5} \times \frac{10}{9} = \) \(-\frac{2}{3}\)

Example 3: Dividing Fractions

Compute \(\displaystyle -\frac{2}{3} \div \left(-\frac{4}{7}\right)\).

1
Sign check: two negatives → same signs → positive
2
Keep, Change, Flip: \(\frac{2}{3} \times \frac{7}{4}\)
3
Cross-cancel: \(2\) and \(4\) share factor \(2\) → \(\frac{1}{3} \times \frac{7}{2}\)
4
Multiply: \(\frac{1 \times 7}{3 \times 2} = \frac{7}{6}\)
5
Result: \(-\frac{2}{3} \div \left(-\frac{4}{7}\right) = \) \(+\frac{7}{6}\) or \(1\frac{1}{6}\)

Example 4: Decimals

Compute \((-1.5) \times (-0.4)\).

1
Sign check: both negative → same signs → positive
2
Multiply absolute values: \(1.5 \times 0.4 = 0.60\)
3
Result: \((-1.5) \times (-0.4) = \) \(+0.6\)

Multiplying Multiple Factors

When multiplying more than two numbers, count the total negative signs to determine the final sign:

Even Negatives
\(0, 2, 4, \ldots\) negative factors → positive result
Odd Negatives
\(1, 3, 5, \ldots\) negative factors → negative result

Special Cases to Remember

Multiply by 0
Any number times \(0\) equals \(0\).
\((-7) \times 0 = 0\)
Multiply by 1
Any number times \(1\) equals itself.
\((-3) \times 1 = -3\)
Divide by 0
Undefined! You can never divide by zero.
\(\frac{5}{0}\) does not exist.
Multiply by −1
Multiplying by \(-1\) flips the sign.
\((-1) \times 4 = -4\)

Common Mistakes

✕ Wrong Sign
Forgetting that \((-) \times (-) = (+)\). Two negatives multiply to a positive, not a negative.
✕ Forgetting to Flip
Dividing fractions by multiplying straight across instead of flipping the second fraction first.
✕ Dividing by Zero
Writing \(\frac{a}{0} = 0\). Division by zero is undefined, not zero.

Question 1 of 5

What is \((-8) \times (-3)\)?

Question 2 of 5

What is \(\displaystyle \frac{2}{5} \div \left(-\frac{3}{4}\right)\)?

Question 3 of 5

What is \((-2) \times (-3) \times (-5)\)?

Question 4 of 5

What is \((-1.2) \times 5\)?

Question 5 of 5

Which statement is true?