Just because a pesky negative sign has creeped its way into one of your fraction multiplication problems, it doesn’t mean it’s time to panic. If you know the basic rules for multiplying negative numbers already, multiplying fractions with one (or more) negative signs in the problem is as easy as pi!
It depending on where the negative sign shows up, it’s easier to think about the sign being attached to the numerators and then from there. Let’s look at a few simple examples.
Multiplying with a Negative Numerator
If your problem has a negative sign attached to the numerator (the number on top of the fraction), that’s a good place to start. When you think about a negative quantity, we usually understand that to be an negative offset from zero, for example a distance below where you started, or a debt or some other adjustment.
Negative amounts can still be confusing at times, but the ideas for a negative fraction is the same. Negative one-half just represents something like owing someone one-half of a dollar, or being one-half of a degree below freezing. The key is the negative amount represents the quantity we’re discussing, and for a fraction that quantity is the numerator (the one) measured in terms of the denominator (the two) for the example of one-half.
Thinking about things this way, when you see a fraction with a negative sign, it’s convenient to imagine the negative sign attached to the numerator since that’s the part of the fraction representing what the fraction is counting. So if you have that negative sign on the numerator already, it’s a nice start.
What if the Negative Sign is in Front of the Fraction
If you have a negative sign in front of the fraction, it’s the same thing. For purposes of working out our negative fraction multiplication problem, just consider it as a negative sign on the denominator and go from there.
What you should NOT do is think that a negative sign in front of the fraction means the numerator AND the denominator are both negative. Remember the the denominator is what we’re using to count out what’s represented in the denominator. If you were to take a negative sign in front of a fraction and then move that to both the numerator and the denominator, you’d be creating a second negative sign inappropriately… These would cancel each other out. That would be the same as simply ignoring the negative sign altogether. Don’t do this!
What If the Negative Sign is in the Denominator
To me, if the negative sign shows up for some reason in the denominator, it’s conceptually confusing. What does quantity mean when it’s divided up into a negative number of parts? I’m not sure.
Whenever I see a negative sign show up in the denominator, I simply move it up to the numerator. Whether the sign is in the numerator or the denominator doesn’t matter in terms of the arithmetic used to multiply the fractions, so move it where it makes sense.
What If There is a Negative Numerator and a Negative Denominator
That’s easy. Cancel them out. Now it’s just a positive fraction!
I’ve Moved All These Negative Signs Around… Now What?
Once you’ve moved all those negative signs around and you’ve basically got negative signs all sensibly in the numerators, just go ahead a multiply the fraction just like you would normally. You may get a negative sign in the resulting product’s numerator, and if so just put it in front of the entire fraction to show the answer more consistent with standard presentation (although this would be exactly the same number as-is).
Of course your fractions may have come with other challenges, like mixed numbers, so be sure to check out my other tips for multiplying fractions if negative numbers aren’t the only twist in your fraction problems!
