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Divide
Rational Numbers

Master the rules for dividing integers, fractions, and signed numbers: every quotient of integers (with a non-zero divisor) is a rational number.

7th Grade Math

What Is a Rational Number?

A rational number is any number that can be written as a fraction \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q \neq 0\). This includes positive and negative fractions, whole numbers, and terminating or repeating decimals.

Integers
\(-3,\; 0,\; 7\) are rational because \(7 = \frac{7}{1}\)
Fractions
\(\frac{2}{5},\; -\frac{3}{4}\) are already in \(\frac{p}{q}\) form
Decimals
\(0.75 = \frac{3}{4}\) and \(0.\overline{3} = \frac{1}{3}\)

Why Does Division Matter?

Division appears everywhere in daily life: splitting a bill, calculating speed, converting units, or finding an average. When we divide integers, we always get a rational number (as long as we don't divide by zero).

Sharing Pizza
\(3\) pizzas
 among 4 friends
\( = \frac{3}{4} \text{ each}\)
 Temperature Drop
\(-12^\circ \text{ in } 4 \text{ hours}\)
 rate per hour
\( = -3^\circ/\text{hr}\)
 Splitting Debt
\(-\$24 \text{ owed}\)
 split 3 ways
\( = -\$8 \text{ each}\)

Visualizing Division on a Number Line

When we compute \(-6 \div 4\), we're asking: "Where does \(-6\) land when we partition it into \(4\) equal parts?" The answer is \(-\frac{3}{2} = -1.5\).

-6 -5 -4 -3 -2
\(-1.5\)
 -1 0 1
\(\div 4\)

The Core Rule for Dividing Integers

If \(p\) and \(q\) are integers and \(q \neq 0\), then dividing \(p\) by \(q\) always produces a rational number. The quotient is the fraction \(\frac{p}{q}\).

If \(p\) and \(q\) are integers, \(q \neq 0\)
\(p \div q = \frac{p}{q}\)   (a rational number)

Sign Rules for Division

The sign of a quotient follows the same rules as multiplication: same signs give a positive result, different signs give a negative result.

+ ÷ + = +
\(12 \div 4 = 3\)
- ÷ - = +
\(-12 \div (-4) = 3\)
+ ÷ - = -
\(12 \div (-4) = -3\)
- ÷ + = -
\(-12 \div 4 = -3\)

Where Does the Negative Go?

A negative sign on a fraction can be placed in three equivalent positions: in front of the fraction, on the numerator, or on the denominator. All three forms represent the same value.

\(-\frac{p}{q}\)
 Negative in front
\(=\)
\(\frac{-p}{q}\)
 Negative on top
\(=\)
\(\frac{p}{-q}\)
 Neg. on bottom

Multiplication vs. Division

Division and multiplication are inverse operations. Dividing by a number is the same as multiplying by its reciprocal.

Multiplication Division
Operation \(a \times b\) \(a \div b\)
Equivalent \(a \times b\) \(a \times \frac{1}{b}\)
Sign rule Same signs → +, Different → - Same signs → +, Different → -
Identity Multiply by 1 Divide by 1
Forbidden None Cannot divide by 0

Step-by-Step: How to Divide

1
Determine the sign. Count the negative signs: even count → positive, odd count → negative.
2
Divide the absolute values. Ignore signs and divide the numbers: \(|p| \div |q|\).
3
Attach the sign. Apply the sign you found in Step 1 to the result.
4
Simplify. Reduce the fraction to lowest terms or convert to a decimal if needed.

Example 1: Integer Division

Compute \(-18 \div 3\).

1
Sign: One negative sign (odd) → result is negative.
2
Absolute values: \(18 \div 3 = 6\).
3
Attach sign: \(-18 \div 3 = \boxed{-6}\).

Example 2: Two Negatives

Compute \(-35 \div (-5)\).

1
Sign: Two negative signs (even) → result is positive.
2
Absolute values: \(35 \div 5 = 7\).
3
Attach sign: \(-35 \div (-5) = \boxed{+7}\).

Example 3: Dividing Fractions

Compute \(-\frac{3}{4} \div \frac{2}{5}\).

1
Sign: One negative (odd) → result is negative.
2
Keep-Change-Flip: \(\frac{3}{4} \times \frac{5}{2} = \frac{15}{8}\).
3
Attach sign: \(-\frac{3}{4} \div \frac{2}{5} = \boxed{-\frac{15}{8}}\), or \(-1\frac{7}{8}\).

Real-World Contexts

Quotients of rational numbers show up in rates, averages, and proportional reasoning. Interpreting the sign tells us the direction of change.

Elevation
A submarine descends \(240\) feet in \(8\) minutes: \(\frac{-240}{8} = -30\) ft/min. The negative means going down.
Finance
A company loses \(\$900\) over \(3\) months: \(\frac{-900}{3} = -300\). Each month averages a \(\$300\) loss.

Common Mistakes to Avoid

✕ Wrong sign
Saying \(-12 \div (-3) = -4\). Two negatives make a positive: the answer is \(+4\).
✕ Dividing by zero
Writing \(\frac{5}{0} = 0\). Division by zero is undefined: there is no answer, not even zero.
✕ Flipping wrong fraction
Flipping the first fraction instead of the second in Keep-Change-Flip. Always flip the divisor.

Summary of Key Facts

Rule Example Result
Same signs → positive \(-20 \div (-4)\) +5
Different signs → negative \(20 \div (-4)\) -5
Zero divided by nonzero \(0 \div 7\) 0
Nonzero divided by zero \(7 \div 0\) Undefined
Negative sign placement \(-\frac{p}{q}\) \(\frac{-p}{q} = \frac{p}{-q}\)
Dividing fractions \(\frac{a}{b} \div \frac{c}{d}\) \(\frac{a}{b} \times \frac{d}{c}\)

Question 1 of 5

What is \(-48 \div 6\)?

Question 2 of 5

Which expression equals \(-\frac{7}{9}\)?

Question 3 of 5

A stock lost \(\$72\) in value over 8 days. What was the average daily change?

Question 4 of 5

Compute \(-\frac{2}{3} \div \frac{4}{5}\).

Question 5 of 5

Which statement about division is true?