How many times does -9 go into 37?

[rose123]

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?

 

[Otis]

How many times does -9 go into 37 … how does it work in case of negative numbers?

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]

I would say that "how many times does a go into b" is simply not a meaningful wording in this case!
37 divided by -9 is $\displaystyle -4\frac{1}{9}$.

 

[Otis]

… simply not a meaningful wording in this case …

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.

$\;$

 

[pka]

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?

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.

 

[HallsofIvy]

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.

$\;$

So your fingers are smarter than your brain?

 

[HallsofIvy]

How many people here remember "Jethro Bodine" of "The Beverly Hillbillys" studying the "gazintas"?