Dividing Integers
Dividing Integers uses the same sign rule as multiplying, because division undoes multiplication. It is a core skill in NSW Year 7 maths (NESA outcome MA4‑INT‑C‑01). Divide the sizes, then check the signs: \(-12 \div (-3) = 4\), but \(12 \div (-3) = -4\).
You will divide integers written with \(\div\) and with a fraction bar, work through chains of divisions left to right, find missing divisors, and work with a set of winter temperatures. Sharing a debt equally between friends, or a steady overnight drop in temperature, shows why a negative answer makes sense. It is part of Year 7 Integers.
Every question with a fully worked solution.
- Dividing Integers - Video - Dividing Integers | How to Divide Positive and Negative Integers Watch
Theory
To divide two integers, divide their sizes, then use the sign rule: same signs give a positive answer and different signs give a negative answer. This Year 7 maths page shows why division follows the same rule as multiplication, how to divide from left to right, how to find a missing number and how to work out a change per hour or a mean.
Division undoes multiplication. Because \(-4 \times 3 = -12\), it follows that \(-12 \div 3 = -4\) and \(-12 \div (-4) = 3\). So dividing uses the same sign rule as multiplying.
Same signs \(\Rightarrow\) positive answer: \(-12 \div (-3) = 4\). Different signs \(\Rightarrow\) negative answer: \(12 \div (-3) = -4\).
A fraction bar also means divide: \(\dfrac{-12}{-3} = -12 \div (-3) = 4\).
The mean of a set of values is total \(\div\) number of values. The mean of \(-5\), \(2\) and \(-6\) is \(-9 \div 3 = -3\).
Signs of a quotient (the same as for products):
| \(\div\) | positive | negative |
|---|---|---|
| positive | positive | negative |
| negative | negative | positive |
How to divide integers
- Divide the sizes, ignoring the signs.
- Same signs: positive answer. Different signs: negative answer.
- Several divisions (or a mix of \(\times\) and \(\div\)): work from left to right.
How to find a rate of change
- Find the total change, with its sign.
- Divide by the number of hours, days or steps.
How to find a mean
- Add all the values, keeping the signs.
- Divide the total by how many values there are.
(i) \(-35 \div 7\)
(ii) \(-48 \div (-6)\)
(iii) \(\dfrac{-60}{-5}\)
| \(-35 \div 7\) | \(=\) | \(-5\) |
| \(-48 \div (-6)\) | \(=\) | \(8\) |
| \(\dfrac{-60}{-5}\) | \(=\) | \(12\) |
(i) Different signs: \(-5\)
(ii) Same signs: \(8\)
(iii) Same signs: \(12\)
| \(72 \div (-6) \div (-3)\) | \(=\) | \(-12 \div (-3)\) |
| \(\) | \(=\) | \(4\) |
\(4\)
| \(-4 \times 9 \div (-12)\) | \(=\) | \(-36 \div (-12)\) |
| \(\) | \(=\) | \(3\) |
Left to right: \(3\)
| \(-63 \div (-7)\) | \(=\) | \(9\) |
Check: \(9 \times (-7) = -63\). It dives for \(9\) minutes.
Total change: \(4\) down to \(0\), then \(16\) more, a fall of \(20\). From 9 pm to 2 am is \(5\) hours.
| \(-20 \div 5\) | \(=\) | \(-4\) |
The change was \(-4^\circ\text{C}\) each hour.
| \(-6 + 3 + (-9) + (-4)\) | \(=\) | \(-16\) |
| \(-16 \div 4\) | \(=\) | \(-4\) |
The mean is \(-4^\circ\text{C}\).
(i) \(\square \div (-4) = 8\)
(ii) \(-45 \div \square = 5\)
| \(8 \times (-4)\) | \(=\) | \(-32\) |
| \(-45 \div (-9)\) | \(=\) | \(5\) |
(i) Undo the division: \(-32\)
(ii) Size \(9\); the answer is positive, so the signs match: \(-9\)
Common pitfalls
Frequently asked questions
What is a negative divided by a negative?
A positive. \(-48 \div (-6) = 8\).
What is a positive divided by a negative?
A negative. \(12 \div (-3) = -4\).
Why does division use the same sign rule as multiplication?
Division undoes multiplication. Since \(-4 \times 3 = -12\), \(-12 \div 3\) must be \(-4\).
Does a fraction bar mean divide?
Yes. \(\dfrac{-12}{-3}\) means \(-12 \div (-3)\), which is \(4\).
How do you find the mean of some integers?
Add them, keeping the signs, then divide by how many there are. The mean of \(-5\), \(2\) and \(-6\) is \(-3\).