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Answer:
Hey mate, here is your answer.
Step-by-step explanation:
Turning from addition and subtraction, how do you do multiplication and division with negatives? Actually, we've already covered the hard part: you already know the "sign" rules:
plus times plus is plus
(adding many hot cubes raises the temperature)
minus times plus is minus
(removing many hot cubes reduces the temperature)
plus times minus is minus
(adding many cold cubes reduces the temperature)
minus times minus is plus
(removing many cold cubes raises the temperature
The sign rules work the same way for division; just replace "times" with "divided by". Here is an example of the rules in division:
{20}{4} = 5 4 20
=5
{-20}{4} = -5 4−20
=−5
{20}{-4} = -5 −4 20
=−5
{-20}{-4} = 5 −4−20
=5
(Remember that fractions are just another form of division! "Fractions are division"!)
14 \div 2 = 714÷2=7
14 \div -2 = -714÷−2=−7
-14 \div 2 = -7−14÷2=−7
-14 \div -2 = 7−14÷−2=7
Some people like to think of negative numbers in terms of debts. So, for instance, if you owe $10 to six people, your total debt would be 6 × $10 = $60. In this context, getting a negative answer makes sense. But in what context could dividing a negative by a negative (and getting a positive) make any sense?
Think about having a snack at a café. When you go to pay, the kid has trouble running your debit card. He swipes it six times before finally returning the card to you. When you get home, you check your bank account online. You can tell from the amount that, yes, he actually charged you way more than once. Some portion of that total debit (being a negative on your account) is wrong.
You want to confirm the number of over-charges before you call your bank to correct the situation. How can you figure this out? You can divide the entire amount (let's say, $76.02) by the amount shown on your receipt (say, $12.67), which is the amount of one charge. Each charge is a minus on your account, so the math is:
(- $76.02) ÷ (- $12.67) = 6
So there were indeed six charges in total. The number of charges, 6, being the counting-up of the number of events, should be positive. In this real-world context, dividing a minus by a minus, and getting a plus, makes perfect sense. And now you know to direct customer service to cancel exactly five of the charges.
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You may notice people "canceling off" minus signs. They are taking advantage of the fact that "minus times minus is plus". For instance, suppose you have (–2)(–3)(–4). Any two negatives, when multiplied together, become one positive. So pick any two of the multiplied (or divided) negatives, and "cancel" their signs:
Simplify (–2)(–3)(–4).
I'll start by cancelling off one pair of "minus" signs. Then I'll multiply as usual.
(–2)(–3)(–4)
= (–2)(–3)(–4)
= (+6)(–4)
= –24
If you're given a long multiplication with negatives, just cancel off "minus" signs in pairs:
Simplify (–1)(–2)(–1)(–3)(–4)(–2)(–1).
The first thing I'll do is count up the "minus" signs. One, two, three, four, five, six, seven. So there are three pairs that I can cancel off, with one left over. As a result, my final answer should be negative. If I come up with a positive result, I'll know I've done something wrong.
(–1)(–2)(–1)(–3)(–4)(–2)(–1)
= (–1)(–2)(–1)(–3)(–4)(–2)(–1)
= (+1)(+2)(–1)(–3)(–4)(–2)(–1)
= (1)(2)(–1)(–3)(–4)(–2)(–1)
= (1)(2)(+1)(+3)(–4)(–2)(–1)
= (1)(2)(1)(3)(–4)(–2)(–1)
= (1)(2)(1)(3)(+4)(+2)(–1)
= (1)(2)(1)(3)(4)(2)(–1)
= (2)(3)(4)(2)(–1)
= 48(–1)
= –48
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