Multiplying and Dividing Negative Numbers | KS3 Maths

Start from something you can check for yourself. $3 \times (-5)$ means three lots of negative five — count them up and you get $-5$, then $-10$, then $-15$. Now walk the multiplier down one step at a time and watch the answers:

$3 \times (-5) = -15,\quad 2 \times (-5) = -10,\quad 1 \times (-5) = -5,\quad 0 \times (-5) = 0$

Every time the multiplier drops by one, the answer climbs by five. Keep the pattern going past zero and it has no choice but to carry on climbing:

$(-1) \times (-5) = 5,\quad (-2) \times (-5) = 10,\quad (-3) \times (-5) = 15$

The pattern forces the answers positive. Negative three times negative five is positive fifteen — not because a rule says so, but because anything else would break the staircase.

Down five metres a minute is a rate of $-5$; the finishing depth is $-40$. Time is depth divided by rate:

$-40 \div (-5) = \;?$

Count the five-metre dives down the depth scale — one, two, three… eight of them reach $-40$ m. And the signs agree: a negative divided by a negative is same signs, so positive — which is reassuring, because a dive can't take negative eight minutes!

$-40 \div (-5) = 8 \text{ minutes}$

With a chain of negatives, don't work sign-by-sign — count the minus signs first. Here there are three, and three is odd, so the answer is negative. Then just multiply the sizes:

$4 \times 2 \times 3 = 24 \quad\Rightarrow\quad -4 \times (-2) \times (-3) = -24$

Try it with four minus signs: $-5 \times (-2) \times (-2) \times (-1)$. Four is even, so the answer is positive: $5 \times 2 \times 2 \times 1 = 20$.