Dividing Integers
Dividing integers follows the same sign rule as multiplying in NSW Year 8 maths: two numbers with the same sign give a positive answer and different signs give a negative one (NESA outcome MA4‑INT‑C‑01). For example, \(-156 \div (-12) = 13\).
You will divide chains from left to right, simplify fractions with products on the top or bottom, cross off zeros in large quotients and work backwards to a missing number. You will substitute values for pronumerals, find a mean or a missing value from a mean, and explain a student's mistake. A netball goal difference, a dam falling over 12 days and a freezer cooling each hour are all integer division. It is part of Year 8 Working with Numbers.
Every question with a fully worked solution.
- Dividing Integers - Video - Dividing Integers | How to Divide Positive and Negative Integers Watch
Theory
Divide the sizes, then use the sign rule: the same signs give a positive answer and different signs give a negative one. \(-20 \div 4 = -5\) and \(-20 \div (-4) = 5\). This Year 8 maths page shows chains of division, the fraction bar and means of integers.
Dividing undoes multiplying, so it uses the same sign rule: \(-91 \div 7 = -13\) because \(-13 \times 7 = -91\).
Equivalent quotients. One minus sign makes the answer negative wherever it sits: \(\dfrac{-56}{7} = \dfrac{56}{-7} = -\dfrac{56}{7} = -8\). The order cannot be swapped: \(7 \div (-56)\) is not \(-8\).
Big numbers. Cross off the same number of zeros from both numbers first: \(-5600 \div 80 = -560 \div 8 = -70\).
The fraction bar means divide, and it groups. Work out the whole top and the whole bottom, then divide.
Chains. \(\times\) and \(\div\) share one priority, so work left to right unless brackets say otherwise. Working backwards: multiplying undoes dividing, so undo the steps in reverse order, then check.
Pronumerals. A letter standing for a number is a pronumeral, and \(-b\) means the opposite of \(b\). With \(a = -72\) and \(b = 8\): \(a \div b = -72 \div 8 = -9\), while \(a \div (-b) = -72 \div (-8) = 9\).
Mean. Mean \(=\) total \(\div\) number of values, so total \(=\) mean \(\times\) number of values. A mean can be negative.
Sign rules for dividing:
| Signs | Example | Answer |
|---|---|---|
| positive ÷ positive | \(12 \div 3\) | \(4\) |
| positive ÷ negative | \(12 \div (-3)\) | \(-4\) |
| negative ÷ positive | \(-12 \div 3\) | \(-4\) |
| negative ÷ negative | \(-12 \div (-3)\) | \(4\) |
How to divide integers
- Divide the sizes (cross off matching zeros first).
- Decide the sign: same signs positive, different signs negative.
- In a chain, work left to right.
How to find a mean
- Add the values (group positives and negatives).
- Divide by how many values there are.
(i) \(20 \div (-4)\)
(ii) \(-20 \div 4\)
(iii) \(-20 \div (-4)\)
| \(20 \div (-4)\) | \(=\) | \(-5\) |
| \(-20 \div 4\) | \(=\) | \(-5\) |
| \(-20 \div (-4)\) | \(=\) | \(5\) |
(i) \(-5\)
(ii) \(-5\)
(iii) \(5\)
(i) \(-168 \div (-14)\)
(ii) \(7200 \div (-90)\)
| \(14 \times 12\) | \(=\) | \(168\) |
| \(7200 \div (-90)\) | \(=\) | \(720 \div (-9) = -80\) |
(i) Same signs: \(12\)
(ii) Different signs: \(-80\)
| \((-5)^2 \times (-4)\) | \(=\) | \(25 \times (-4)\) |
| \(\) | \(=\) | \(-100\) |
| \(-100 \div (-10)\) | \(=\) | \(10\) |
\(10\)
(i) Find the mean.
(ii) What sixth temperature makes the mean of all six \(-4^\circ\)C?
| \(\text{total}\) | \(=\) | \(-28 + 3 = -25\) |
| \(-25 \div 5\) | \(=\) | \(-5\) |
| \(6 \times (-4)\) | \(=\) | \(-24\) |
| \(-24 - (-25)\) | \(=\) | \(1\) |
(i) \(-5^\circ\)C
(ii) \(1^\circ\)C
| \(-90 \div 5 \times (-2)\) | \(=\) | \(-18 \times (-2)\) |
| \(\) | \(=\) | \(36\) |
Yara multiplied first. \(\times\) and \(\div\) share one priority, so work left to right: the answer is \(36\).
(ii) Two integers have a product of \(-50\), and the first divided by the second is \(-2\). Find both possible pairs.
| \(7 \times 2\) | \(=\) | \(14\) |
| \(14 \times (-6)\) | \(=\) | \(-84\) |
(i) \(-84\). Check: \(-84 \div (-6) = 14\), \(14 \div 2 = 7\).
(ii) \(-10\) and \(5\), or \(10\) and \(-5\): \(-10 \times 5 = -50\) and \(-10 \div 5 = -2\).
Common pitfalls
Frequently asked questions
What is a negative divided by a negative?
A positive: \(-24 \div (-6) = 4\).
What is a negative divided by a positive?
A negative: \(-24 \div 6 = -4\).
Does it matter where the minus sign is in a fraction?
No: \(\dfrac{-56}{7} = \dfrac{56}{-7} = -8\).
How do you work out a chain of divisions?
Left to right, one step at a time.
How do you find a missing number in a division chain?
Undo each step in reverse order by multiplying, then check.