How many times does -9 go into 37?

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rose123

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How many times does -9 go into 37? I have seen all the answers where they used both positive numbers. But how does it work in case of negative numbers?
 
rose123 said:
How many times does -9 go into 37 … how does it work in case of negative numbers?
Click to expand...
Hi Rose. The basic rule for multiplying or dividing two signed numbers is:

The product or quotient is positive, if the signs of the two numbers are the same.​
​
The product or quotient is negative, if the signs of the two numbers are different.​

The signs of the numbers -9 and 37 are different, so the quotient is negative.

Here's another way to look at it. 37 divided by -9 may be written as a fraction. When there's a negative sign in a fraction, we are free to move it to the numerator, to the denominator, or out in front. That is, we may write the fraction as

\(\displaystyle \frac{-37}{9} \quad \frac{37}{-9} \quad -\frac{37}{9}\)

The last version above shows that dividing 37 by -9 is the same as \(\frac{37}{9}\) multiplied by -1.

Therefore, if you would like to divide 37 by -9 longhand, simply divide 37 by 9 instead and multiply the result by -1.

\(\displaystyle 37 ÷ 9 \; = \;\; \)\(4\frac{1}{9}\) \(\displaystyle \; = \; 4.\overline{1}\)

-9 goes into 37 this many times: \( \; \)-\(4\frac{1}{9} \; \text{or}\) -\(4.\overline{1}\)

?
 
HallsofIvy said:
… simply not a meaningful wording in this case …
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Agree! (Certainly not the first time we've seen silly wording by teachers, parents, or students)

My mind had briefly considered saying something like "-4 times with a remainder of -1", but my brain couldn't force my fingers to type it, ha.

\(\;\)
 
rose123 said:
How many times does -9 go into 37? I have seen all the answers where they used both positive numbers. But how does it work in case of negative numbers?
Click to expand...
Here is something to consider: the modulo-operation.
\(37 \mod 9 = 1\) that is \(37\) divided by \(9\) has remainder of \(1\). See Here.
But \(37 \mod -9 = -8\) that is \(37\) divided by \(-9\) has remainder of \(-8\). See Here.
 
Otis said:
Agree! (Certainly not the first time we've seen silly wording by teachers, parents, or students)

My mind had briefly considered saying something like "-4 times with a remainder of -1", but my brain couldn't force my fingers to type it, ha.

\(\;\)
Click to expand...
So your fingers are smarter than your brain?
 
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